mathematical modelling.pdf

ICTMA 11

MATHEMATICAL MODELLING:A WAY OF LIFE

Omnia apud me mathematica fiunt.With me everything turns into mathematics.

Rene Descartes

ABOUT THE EDITORS

Susan J. Lamon, Professor in the Department of Mathematics, Statistics, and ComputerScience at Marquette University, earned her Master of Science and PhD degrees from theUniversity of Wisconsin-Madison. Her research, a combination of mathematics andcognitive science, uses modelling activities to study the development of rational numberconcepts and proportional reasoning. She works on numerous state and national projectsto improve the teaching and assessment of mathematics, and currently serves on theexecutive board of ICTMA.

Willard A. Parker earned his PhD from the University of Oregon, taught in theDepartment of Mathematics at Kansas State University for 28 years, and now teaches atMarquette University. His research interests include abstract harmonic analysis, thehistory of mathematics, and mathematics education. He has worked to improve schoolmathematics at the local, state, and national levels and has served as a governor of theMathematical Association of America.

S. Ken Houston, Professor of Mathematical Studies in the School of Computing andMathematics, University of Ulster, gained his BSc Honours and PhD degrees at QueensUniversity, Belfast. He is a long-standing member and current president of ICTMA andhe serves on national committees in the UK and Ireland. He has a strong interest in allaspects of teaching, learning, and assessing mathematical modelling, and in innovativemethods for preparing students to pursue careers in applied mathematics.

IIleTMA 11

The Eleventh International Conference on theTeaching of Mathematical Modelling and Applications

MATHEMATICAL MODELLING:A WAY OF LIFE

Edited by

Susan J. LamonWillard A. Parker

Ken Houston

Horwood PublishingChichester

2003

First published in 2003 by

HORWOOD PUBLISHING LIMITEDColi House, Westergate, Chichester, West Sussex, P020 6QL England

COPYRIGHT NOTICEAll Rights Reserved, No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical,photocopying, recording, or otherwise, without the permission of Horwood PublishingLimited.

© Horwood Publishing Limited, 2003

British Library Cataloguing in Publication DataA catalogue record of this book is available from the British Library.

ISBN 1-904275-03-6

Printed in Great Britain by Antony Rowe Limited.

Table of Contents

ICTMA Publications

Preface

Section A: Modelling in the Elementary School

Mathematical Modelling With Young LearnersLyn English, Queensland University of Technology, Australia

viii

ix

3

2 Modelling in Elementary School: Helping Young Students to See theWorld MathematicallySusan 1. Lamon, Marquette University, USA

Section B: Modelling with Middle and Secondary Students

19

3 How Mathematizing Reality is Different from Realizing Mathematics 37Richard A. Lesh, Purdue University, USA

4 Environmental Problems and Mathematical Modelling 53Akira Yanagirnoto, Tennoji Jr. & Sf. High School; Osaka KyoikuUniversity, Japan

5 Three Interacting Dimensions in the Development of MathematicalKnowledge 61Guadalupe Carmona, Purdue University, USA

6 Working and Learning in the Real World: A Mathematics EducationProject in Baden-Wuerttemberg 71Hans-Wolfgang Henn, University of Dortmund, Germany

7 Powerful Modelling Tools for High School Algebra Students 81Susan J. Lamon, Marquette University, USA

Section C: Post Secondary Modelling

8 Solving Problems: Perchance to Dream 97Stephen 1. Merrill, Marquette University, US

vi

9 Formal Systems of Logic as Models for Building the ReasoningSkills of Upper Secondary School Teachers 107Paola Forcheri, Istituto di Matematica Applicata e Tecnologie Infonnatichedel CNR, ItalyPaolo Gentilini, Istituto di Matematica Applicata e Tecnologie Infonnatichedel CNR, Italy; Ligurian Regional Institute of Educational Research, Italy

10 Learning Mathematics Using Dynamic Geometry Tools 119Thomas Lingefjard & Mikael Holmquist, Goteborg University, Sweden

II Modelling Search Algorithms 127Albert Fassler, Hochschule fuer Technik und Architektur Biel/Bienne,Switzerland

12 Mathematical Modelling in a Differential Geometry Course 133Adolf Riede, University of Heidelberg, Germany

13 Defending the Faith: Modelling to Increase the Accountability ofOrganisational Leadership 143Peter Galbraith, University of Queensland, Australia

Section D: Research

14 Assessing Modelling SkiIls 155Ken Houston & Neville Neill, University of Ulster, N. Ireland

15 Assessing the Impact of Teaching Mathematical Modelling: SomeImplications 165John Izard, RMIT, AustraliaChris Haines, City University, U.KRos Crouch, University of Hertfordshire, U.KKen Houston, University of Ulster, N. IrelandNeviIle NeiIl, University of Ulster, N. Ireland

16 Towards Constructing a Measure of the Complexity of Application Tasks 179Gloria Stillman, University of Melbourne, AustraliaPeter Galbraith, University of Queensland, Australia

17 Using Workplace Practice to Inform Curriculum Change 189Geoff Wake & Julian Williams, University of Manchester, UK

18 Comparing an Analytical Approach and a Constructive Approachto Modelling 201Toshikazu Ikeda, Yokohama National University, JapanMax Stephens, University of Melbourne, Australia

vii

Section E: Perspectives

19 The Place of Mathematical Modelling in Mathematics Education 215Michael J. Hamson, (Formerly) Glasgow Caledonian University, UK

20 What is Mathematical Modelling? 227Jonei Cerqueira Barbosa, Faculdade Integrada da Bahia e FaculdadesJorge Amado, Brazil

21 Beyond the Real World: How Mathematical Models Produce Reality 235Susana Carreira, Universidade do Algarve; Universidade deLisboa-eIEFUL, Portugal

22 Reconnecting Mind and World: Enacting a (New) Way of Life 245Stephen R. Campbell, Simon Fraser University, Canada; Universityof California, Irvine, U.S.A.

23 ICTMA: The First 20 Years 255Ken Houston, University of Ulster, N. Ireland

VllI

ICTMA PUBLICATIONS

Berry JS, Burghes DN, Huntley 10, James DJG, Moscardini AO (1984) Teaching andApplying Mathematical Modelling Chichester: Ellis Horwood.

Berry JS, Burghes DN, Huntley 10, James DJG, Moscardini AO (1986) MathematicalModelling Methodology, Models. and Micros Chichester: Ellis Horwood.

Berry JS, Burghes DN, Huntley 10, James DJG, Moscardini AO (1987) MathematicalModelling Courses Chichester: Ellis Horwood.

Blum W, Berry JS, Biehler R, Huntley 10, Kaiser-Messmer G, Profke L (1989)Applications and Modelling in Learning and Teaching Mathematics Chichester:Ellis Horwood.

Niss M, Blum W, Huntley 10 (1991) Teaching of Mathematical Modelling andApplications Chichester: Ellis Horwood.

de Lange J, Keitel C, Huntley 10, Niss M (1993) Innovation in Maths Education byModelling and Applications Chichester: Ellis Horwood.

Sloyer C, Blum W, Huntley 10 (1995) Advances and Perspectives in the Teaching ofMathematical Modelling and Applications Yorklyn Delaware: Water StreetMathematics.

Houston SK, Blum W, Huntley 10, Neill NT (1997) Teaching and LearningMathematical Modelling Chichester: Albion Publishing Ltd. (now HorwoodPublishing Ltd.).

Galbraith P, Blum W, Booker G, Huntley 10 (1998) Mathematical Modelling, Teaching.and Assessment in a Technology-Rich World Chichester: Horwood Publishing Ltd.

Matos JF, Blum W, Houston SK, Carreira SP (2001) Modelling and MathematicsEducation ICTMA 9: Applications in Science and Technology Chichester: HorwoodPublishing Ltd.

Ye Q, Blum W, Houston SK, Jiang Q (2003) Mathematical Modelling in Education andCulture: ICTMA 10 Chichester: Horwood Publishing Ltd.

Lamon SJ, Parker WA, Houston SK (2003) Mathematical Modelling: A Way of Life:ICTMA 11 Chichester: Horwood Publishing Ltd.

ix

PREFACE

Mathematical Modelling: A Way of Life

Those who teach mathematical modelling at the university level and those who use it tosolve problems in a wide variety of disciplines, speak of mathematical modelling as a"way of life." This phrase refers to their worldview, their habits of mind, and theirdependence on the power of mathematics to describe, explain, predict, and control realphenomena. The expression suggests that mathematics is indispensable as a way ofknowing about the world in which they live and about the complex phenomena that affectthe quality of their lives. Everything turns into mathematics.

The great difficulty that students face when they study mathematical modelling at theuniversity suggests that it is nearly impossible to adopt this new way of looking at theworld so late in one's education. Without any prior experience in building, interpretingand applying mathematical models, it is difficult to imagine that some students will eversee modelling as a "way of life." It is clearly not enough that students go through themotions of educating themselves by accumulating and remembering a storehouse ofunconnected bits and facts. If students do not develop the spirit of scientificinvestigation---Ionging to know and to understand, questioning all sorts phenomena,conducting logical and systematic investigations, considering premises, and predictingand explaining consequences-they must be helplessly obedient to emotions, pressures,influences, and the authority of other people. At best, they will be reactive and defensivein the face of every problem or crisis that occurs.

Accordingly, one of the chief goals of ICTMA II is to explore the ways in whichteachers at all levels of schooling may provide opportunities for their students to model avariety of real phenomena in ways that are appropriately matched to the students'mathematical backgrounds and interests. Conference participants were invited toexamine from a variety of perspectives what it means to move beyond the efficienttransmission of content in the mathematics classroom, toward creating a classroomatmosphere that conveys critical values, shapes useful processes, and rewards powerfulthinking. This volume contains 23 contributions to ICTMA II, many of which addressthe problems of helping school students to adopt mathematical modeling as a way of life.

ICTMA II has the distinction of marking at least two "firsts." As we write this preface,it is three months before conference convenes in Milwaukee, Wisconsin, USA. Thepresenters/authors and the editors have worked intensively during the year preceding theconference to prepare manuscripts so that conference participants can receive this bookwhen they arrive in Milwaukee. In part, this effort is a response to the ever-lengtheningtime period between the end of a conference and the book's publication-in some cases,almost two years. We suspect that looking ahead after the conference may be moremotivating and productive than looking back. We hope that the extensive review,feedback, and revision process that has already taken place will make for interesting and

x

well-prepared presentations at the conference, and that discussion of these papers inMilwaukee will stimulate ideas and fuel follow-up studies well in advance of the ICTMA12 meeting in London.

Unfortunately, ICTMA II is also the first of our conferences for which participants havehad to make travel plans during wartime. Because of the war in Iraq, the SARSepidemic, and the resulting difficulties with the airline companies, conferenceregistrations are considerably lower than ever before and we are very grateful to ourpublisher, Ellis Horwood of Horwood Publishing, for producing a paperback book. Weappreciate not only his willingness to provide this war-time edition, but the consistentsupport he has shown ICTMA for the last twenty years.

We are grateful to all of the reviewers who freely gave of their time and talents to helpthe authors and the editors prepare manuscripts for publication. We express our gratitudeto Mrs. Pamela Entrikin for her hard work in organizing ICTMA II, to MarquetteUniversity for hosting the conference, and to our friends and corporate sponsors.

Sue Lamon, Bill Parker, and Ken Houston

Section A

Modelling in the Elementary School

��

��

1

Mathematical Modelling With Young Learners

Lyn English Queensland University of Technology, Australia [email protected]

Current research is demonstrating that young children can make significant mathematical and social gains from working authentic modelling problems. This paper argues for the implementation of mathematical modelling activities within the elementary and middle school years. The key features of these activities that make them rich learning experiences for children are explored. Some detailed analyses of how children develop and apply generalizable conceptual systems are then presented. It is argued that analogical and case-based reasoning processes play a powerful role in the construction and application of generalized models.

I wish to thank Helen Doerr for her valuable feedback on an earlier version of this article. The assistance provided by Dr. Kathy Charles and Katrina Lewis during data collection is also gratefully acknowledged. This research is supported by a grant from the Australian Research Council.

��

4 English

INTRODUCTION

Students today are facing a world that is shaped by increasingly complex, dynamic, and powerful systems of information within a knowledge-based economy. As future members of the work force, children need to develop the fundamental components of mathematical modelling-that is, they need to recognise the usefulness of models in today’s world, to develop and use models to interpret and explain structurally complex systems, to develop representational fluency, to reason in mathematically diverse ways, and to use sophisticated equipment and resources (English, 2002a; Lesh & Heger, 2001).

Being able to interpret and work with complex systems involves important mathematical processes that are under-utilised in mathematics curricula, such as constructing, explaining, justifying, predicting, conjecturing and representing, together with quantifying, coordinating, organising, and representing data. Dealing with such systems also requires students to be able to work collaboratively on multi-component projects in which planning, monitoring, assessing, and communicating results are essential to success (Lesh & Heger, 2001). The primary school is the educational environment where all children should begin a meaningful development of these modelling processes (Jones, Langrall, Thornton, & Nisbet, 2002). However, as Jones et al. note, even the major periods of reform and enlightenment in primary mathematics do not seem to have given children access to the deep ideas and key processes they need for dealing with complex systems beyond school.

Modelling activities are the ideal vehicle for developing these ideas and processes- yet elementary and middle school students are being denied important modelling opportunities even though research has shown that young children can engage in complex mathematical and scientific investigations, given appropriate teacher support (Diezmann, English, & Watters, 2002; Doerr & English, 2003). In this paper I show how children in the elementary and lower-middle grades can make significant mathematical and social gains from working authentic modelling problems. 1 first contrast traditional classroom modelling with the mathematical modelling experiences children need for today’s world. I then consider a number of key features that contribute to rich modelling experiences for children. Some detailed analyses of how children develop and apply generalizable conceptual systems are then presented.

TRADITIONAL MATHEMATICAL MODELLING

Traditionally, mathematical modelling in the elementary grades has focused on arithmetic word (story) problems that are represented with concrete materials and then modelled by more abstract operational rules. Solving these word problems entails a mapping between the structure of the problem situation and the structure of a symbolic mathematical expression (e.g., Suzie has saved $12. Lillian has saved 3 times this amount. How mirch has Lillian saved? can be modelled by the expression, 12x3 = 36). Oftentimes, solving these word problems is not a modelling activity for children, rather, it is one that relies on syntactic cues such as key words or phrases in the problem (e.g., times, less, fewer). Furthermore, there is usually only one way of interpreting the problems and hence,

��

Modelling With Young Learners 5

children engage in limited mathematical thinking. While not denying the importance of these types of problems, they do not address adequately the mathematical knowledge, processes, representational fluency, and social skills that our children need for the 2 1st century (English, 2002b).

Numerous studies have shown that children who are fed a diet of stereotyped one- or two-step word problems frequently divorce their real-world knowledge from the solution process, that is, they solve the problems without regard for realistic constraints (Greer, 1997; Verschaffel, De Corte, & Borghart, 1997). In standard word problems, questions are presented to which the answer is already known by the one asking them (usually the teacher). As Verschaffel et al. (1997) commented, questions are not given so children can obtain information about an authentic problem situation, rather, the questions are designed to give the teacher information about the students. Furthermore, both the students and the teachers are aware of this state of affairs and act accordingly.

MATHEMATICAL MODELLING FOR CHILDREN TODAY

Mathematical modelling is frequently viewed as the construction of a link or bridge between mathematics as a way of making sense of our physical and social world, and mathematics as a set of abstract, formal structures (Greer, 1997; Mukhopadhyay & Greer, 200 1). To foster the mathematical modelling abilities children require for today's world, we need to design activities that display the following features: 0 Authentic problem situations;

0 Multiple interpretations and approaches; 0 Opportunities for social development; 0 Multifaceted end products; and

0 Opportunities for model exploration and application;

0 Opportunities for optimal mathematical development.

Authentic Problem Situations For a number of years, mathematics curriculum documents and mathematics educators have been emphasizing the importance of couching children's problem experiences in situations that are motivating, interesting, and relevant to their world, and where there is a genuine need for particular mathematical processes (Boaler, 2002; Kolodner, 1997). Such authentic contexts provide sense-making and experientially real situations for children, rather than simply serve as cover stories for proceduralized and frequently irrelevant tasks.

While the benefits of such experientially real contexts have been well documented (most notably in the Realistic Mathematics Education research, emanating from the Freudenthal Institute; Freudenthal, 1983; Gravemeijer, 1994), there have also been some concerns expressed (Boaler, 2002; Silver, Smith, & Nelson, 1995). A main issue cited is that children are frequently required to both engage with the problem contexts as though they were real and to ignore factors that would pertain to real-life versions of the task (Boaler, 2002). When children situate their reasoning within their own authentic contexts, there are, of course, several correct answers. It has thus been recommended that we need to

��

6 English

not only provide real-world contexts, but also “real-world solutions” (Silver et al., 1995, p. 4 1). Mathematical modelling activities take both aspects into account.

Opportunities for Model Exploration and Application A program of modelling experiences for children is most effective if it comprises sequences of related activates that enable models to be constructed, explored, and applied. The activities should be structurally related, with discussions and explorations that focus on these structural similarities (Lesh, Cramer, Doerr, Post, & Zawojewski, 2002).

One of several sequences of modelling activates that I have used successfully centers on notions of ranking, and selecting and aggregating ranked quantities. In each context, the children analyze and transform entire data sets or meaningful portions thereof, rather than single data points. The sequence begins with an activity that elicits the development of significant mathematical constructs, namely, the Sneaker problem. In this problem, children are asked, “What factors are important to you in buying a pair of sneakers?“ In small groups, children generate a list of factors and then determine which factors are most important. This inevitably results in different group rankings of the factors. The teacher then poses the problem of how to create a single set of factors that represents the view of the whole class.

This activity is followed by a model-exploration activity or activities, where children can consolidate and refine the conceptual systems they have developed as well as construct powerful representation systems for these systems (e.g., the Weather Problem (Appendix A). In essence, at the end of a model-exploration activity, children should have produced a powerful conceptual tool or model that they can apply to other related problems.

The next activity in the sequence is a model-application task (e.g., the Snack Chip Consumer Guide Problem (Appendix B) and the Car Problem (Appendix C), where children deal with a new problem that would have been too difficult without their prior development of a conceptual tool. This new activity requires some adaptation to the tool and involves the children in problem posing as well as problem solving, and information gathering as well as information processing (Lesh et al., 2002).

Multiple Interpretations and Approaches Hatano ( 1997) distinguished “understanding through comprehension” from “understanding by schema application.” Schema application occurs when a known solution procedure is applied to a routine problem that usually involves only one interpretation (p. 385). Because the givens, the goal, and the legal solution steps in word problems are usually specified unambiguously, the interpretation process for the solver has been minimalized or eliminated.

Modelling activities for children involve multiple, simultaneous interpretations. With modelling activities, however, the solver has to not only contemplate which of several approaches could be taken in reaching the goal, but also to interpret the goal itself and the accompanying information. Each of these components might be incomplete, ambiguous, or undefined; there might be too much data, or there might be visual representations that

��


are difficult to interpret. For example, notice how Kate, below, is trying to interpret exactly what the first client desires in the Weather Problem.

Mt. Isa has an extremely high [number of] clear days and that’s what they’re sort of looking for (first client), but I think you might need a bit more rain because they say “We don‘t care if there is a lot of rain.’’ That could mean that they want a bit but they don’t want all that much. But it could mean two things: That they don’t mind and they just don‘t really care; or that they do want some but they just don’t want heaps. It’s a bit tricky to decide whether you want a lot of rain or whether you want not too much rain.

In the above situation, children have to interpret the client’s comment (“We don’t care if there is a lot of rain.”) before they can operationalize any actions on any quantities. Through the interpretation process, the quantification of “how much rain” gains meaning for the children.

When presented with information that is open to more than one interpretation, children might make unwarranted assumptions or might impose inappropriate constraints on the products they are to develop (English & Lesh, 2002). This is where the interactions of group members come to the fore as children interpret and re-interpret the problem information.

Opportunities for Social Development The communication processes inherent in these modelling activities play an important role in children’s social, as well as mathematical, developments (Zawojewski, Lesh, & English, 2002; English, 2002b). Modelling activities are specifically designed for small- group work where children are required to develop explicitly sharable products that are subject to scrutiny by others. This means the children have a shared responsibility to ensure their models meet the desired criteria and that what they produce is informative and user-friendly.

Numerous questions, conjectures, arguments, revisions, and resolutions arise as children develop and assess their models, and communicate their models to a wider audience. Although their discussions during model construction can be off task at times, children nevertheless develop powerful skills of argumentation in which they challenge one another’s assumptions, ask for justification of ideas, and present counter-arguments.

Multifaceted End Products Modelling problems for children call for multifaceted products, in contrast to the solutions required by standard types of problems they meet in class. The modelling problems I have used with younger children present them with a number of criteria that have to be met in producing their final model. In the Weather Problem, for example, the requirements of the two clients serve to guide children’s development of a model for determining which is the best city to locate in. These criteria not only guide model development but also model assessment, both during and following model construction. That is, children can progressively assess their intermediate products, identify any deficiencies, and then revise and refine their models. Or, if several alternative models are

��

8 English

being considered at the same time, then the children are able to assess the strengths and weaknesses of each.

Children's final models are usually expressed using various representational media, including written and oral reports, computer-based representations, and paper-based diagrams or graphs (Lesh & Lehrer, in press). The use of a range of media, in particular the computer-based forms, is especially beneficial to younger children because they engage in purposeful learning. Indeed, representational fluency has been shown to be at the heart of an understanding of many of the key constructs in elementary mathematics and science (Goldin, 2002; Lesh & Heger, 2001) and working flexibly with different representational forms is an increasingly important skill in the workplace.

Opportunities for Optimal Mathematical Development As children work such problems, they engage in important mathematical processes including describing, analyzing, coordinating, explaining, constructing, reasoning critically, and mathematizing objects, relations, patterns, or rules pertaining to the modelled system (Lesh, Hoover, Hole, Kelly, & Post, 2000; Gravemeijer, 1997). Some of these processes are evident in the following excerpt where a group of sixth-graders is deciding on factors to consider in developing their snack chip consumer guide. After the group had discussed a number of factors (with a strong emphasis on flavor), the teacher asked, "But if you were choosing to buy one of those two products [pointing to two different packets of snack chips], what are the things-forget about the individual-like if it's barbeque or not-what other things would you consider? One child responded as follows:

Well, 1 think you should consider .... you should consider how much is in the bag, if you can get that bag ...y ou should also consider ... they might be the same price ... I'm not saying they're the same. Pretend that was 150 grams, and that was 230 grams and that was a bit bigger. That might be 150 grams and $7 and this 230 grams and $4. That might be 130 and might only be $1. And you have to think: What would you rather buy: two of these which is about as much as that one. Because you've got to think "If I'm going to buy that, I pay around the same amount but I don't get as much, but 1 pay only a little bit more to buy two of these and get more than that.

In the above situation the child is comparing the two products in terms of their (estimated) mass and cost, with the aim of determining which item is better value for money. In doing so, she engages in informal proportional reasoning.

GENERALIZABLE CONCEPTUAL SYSTEMS

In this section, I consider one of the key goals of mathematical modelling for children, namely, children's development of generalized conceptual systems. A key criterion in designing modelling problems for children is that the tasks should have the potential to elicit mathematically significant constructs that ultimately become generalizable and re- usable (Doerr & English, 2003). Children are observed to progress through a number of

��


learning cycles on their way towards producing a generalized model (Doerr & English, 2003; Lesh & Lehrer, in press).

When children are presented with modelling problems of the type addressed here, they first of all have to interpret the problem and draw upon their existing contextual and mathematical knowledge in doing so. By contextual, 1 mean knowledge of previously experienced situations in related settings. Sometimes, children will discuss upfront how they should interpret a given problem-through their own perspective or through the perspective of the characters in the problem. For example, in the Car Problem, some children argued that they should interpret the problem only in terms of what Carl and his mother wanted, while others stated, "It helps if you also think about what you would do."

At other times, children's contextual knowledge can be all consuming, taking the children away from the goal of the task. In interpreting the Weather Problem, for example, one group of children spent considerable time discussing their interpretations of dry weather:

Tom: Anne: No, I'm saying dry. Olli: Tom: Anne: Tom:

Clear doesn't mean they're hot.

Clear is basically dry.. . . . . Clear days don't mean dry days or do they? They DO mean dry days. If it's clear it's not going to be wet, is it? Yeah, but it's not going to be dry. That means when you walk on ground it will be dust.

After discussing various alternative interpretations and negotiating meanings of expressions, children normally return to the problem criteria to redirect their efforts (cf., Wyndhamn & Saljo, 1997). Children usually cycle through a number of processes in constructing their models. The processes include sharing, describing, explaining, and justifying their ideas, and rejecting or revising intermediate models depending on how well these meet the problem criteria. As children progress through these learning cycles, they select relevant quantities, create meaningful representations, and define operations that might lead to new quantities (Doerr & English, 2003; Kolodner, 1997).

The reporting-back process, where groups share their models with their class, also provides important opportunities for learning. For example, when Roberta's group was describing their model for the weather problem, there were some inherent difficulties in the model. Roberta took control and modified their system to make it generalizable. She had realized that they needed to apply some operations that would mathematize their actions.

Roberta: Before when Lyndie said nobody knows if Hobart actually is the best .... I put Is', 2nd, 3rd and things for yearly rainfall and then I did it for cities below 15 [degrees], and 1 added those ... and the lowest one would be first and the highest would be last. For example, the yearly rainfall for Hobart came 7Ih and days below 15 came 1'' so I'd add 1 and 7, and that would be pretty low so that would be 8 so that would be pretty low so that would be first.. .. So you only added up two categories? Teacher:

��

10 English

Roberta: Yes, yearly rainfall and days below 15, because they were the ones that were important. Alice Springs got 13 because rain was 9 and days below 15 was 4. Cairns got 9 because the days below 15 were 8 and rainfall was first .... (she continued to describe her procedure).

Teacher: So then how did you decide which was the most suitable city for the first client? Which one did you decide again?

When the teacher asked Roberta to explain how she would use her system to address the first client’s needs, Roberta had in the meantime decided to refine it further. She now incorporated a third factor, namely, days above 30 degrees Celsius. The revised system generated negative numbers, which Roberta handled easily even though the class had not been taught these.

Roberta: Hobart. Oh, I just added this part now. I just realised ... 1 thought a way might be to take away the days above 30.

Teacher: Yes. But Hobart has the same number as Sydney and Canberra, so how would you decide that Hobart was more suitable than Sydney or Canberra?

Roberta: (Referring to Hobart) For the days above, take away 9 because that was what it was, so it would be below zero. Sydney ...... average for that was 8 .... And Melbourne take away 7. And Canberra take away 6.....ln the end Hobart would win because it’s less.

Teacher: So you‘re saying Hobart is the first choice based on three criteria. And Sydney is your second choice?

Roberta: Yes, because it‘s zero. And 3rd would be Melbourne, and 4‘h would be Canberra.

A class member then asked Roberta which client her system would be used for, to which she replied, “It would be for any client who wanted cold weather along with snow.”

Roberta had effectively generated a model that was generalizable. Her class had developed their own term for a generalized system, namely, a universal model. For example, in reference to the Snack Chip Consumer Guide activity, Isaac explained that a universal system “would work for every type of snack chip, not just the ones we looked at.”

Children’s development of generalized systems is, of course, a major goal of mathematics education. The use of sequences of modelling tasks provides opportunities for this development, when accompanied by teacher-initiated class discussions on the structural links between the problems. How children apply their generalized models to new modelling problems is an issue that requires further attention.

APPLYING GENERALIZED CONCEPTUAL SYSTEMS

It is proposed that analogical and case-based reasoning processes facilitate children’s application of generalized models. Case-based reasoning involves reasoning by analogy, where the structural similarities between a known situation and a new situation are identified and utilized (English, 1999). While case-based reasoning has been explored in

��


other domains, such as science education (Kolodner, 1997, Kardos, 2002), it has received little attention in mathematics education.

Case-based reasoning involves reasoning that makes use of previous experiences or cases (Kolodner et al., in press). The cases are fundamentally analogs that represent personally experienced problem situations and include a rich representation of the problem situation, the ways in which the situation was handled, and the outcomes of resolving the situation (Kolodner, 1997).

The models that we have been addressing may be regarded as cases that serve as a basis for reasoning about new problems. For children to make effective use of these cases, they need to reason analogically. That is, children need to identify and match the structural or relational correspondences between their known cases and the new problem. They then need to know how to make any necessary modifications to their existing case in order to accommodate additional features of the new problem (cf. English, 1997; Kolodner, 1997). In working the Car Problem, the final problem in the present sequence, children were observed to make use of the models they had developed in the previous problems as cases for this new problem:

Michelle: We can just use a process of elimination. Group: No, we need to consider all the features. Roberta: We’ll use a rating system. We can use that rating system (the one they had

used on the previous problems); we could rate the features that they consider important and we can do this for Carl and for his Mother. I’ll write down all the features (she drew a table and started to list all the cars down the left-hand side.) We’ll do our old rating system.

Michelle: I don’t know what you mean. Roberta: Well, we got all the features that we considered important for the chips and

this time, we’re looking at the features that Carl and his mother think are important, and then we’ll rate them, like 1 to 10.

Roberta was mapping the key components of the Snack Chip Consumer Guide problem, namely, the selection of product features and the ranking of these features, onto the Car Problem. In drawing a table to assist them, the children had to decide which features they were going to include to define the criteria safe, reliable, andjrn. They used a vote-of- hands to decide this. These features were recorded across the top of their table. The group then completed the respective cells in the table, checking those cars that displayed the particular features. However, the children subsequently returned to the features they had chosen and argued whether all the features should be included; they also questioned their interpretations of some of the features (e.g., whether alloy wheels are a safety feature or a cool feature). As a consequence, they removed some of the columns in their table.

While Roberta continued to stress the need to use their old rating system, other members of her group wanted to use a process of elimination where they would delete those cars that did not meet most of the features desired by Carl and his mother. After much debate, the group accepted that Roberta’s system would be more effective. The system was

��

12 English

applied with some modification, however. The children ranked the quantitative features (mileage and gas consumption) and did a frequency count for the remaining features (antilock brakes, airbags, air-conditioning, sunroof, alloy wheels, and power windows).

The group then applied their system, recording their actions in two additional columns of their table (e.g., in one column they recorded partial results such as C = 2+7 = 9 to indicate that the Nissan Silva had 2 qualitative features that Carl desired and its mileage had a ranking of 7). In the other column, they recorded the final result for the Nissan Silva (13) by adding the previous result (9) to the car’s ranking for gas consumption.

Applying a known model or case to the solution of a new problem is not a simple process for children. It is, in fact, a multifaceted activity that requires children to be able to: (a) Construct models that comprise the necessary structural elements to enable them to

reason analogically with these models; (b) Know to look for related structures in dealing with problems; (c) Know when and how to utilize their existing models in solving new problems. In

this vein, Kolodner, 1997, stressed the importance of students being able to anticipate situations in which a case might be applied; and

(d) Make any necessary modifications to an existing model in applying it to a new problem.

Teacher-guided discussions are important in helping children move beyond just thinking about their models to thinking with them, that is, making their models “explicit objects of thought” (Lesh et al., 2002). These discussions can be included in sessions where children share their models with the class to receive constructive peer feedback, or students might provide written critiques of other student models (after first critiquing their own). In my current research, children submit a critique on our project website (www.ourmathmode1s.com). The website enables classes of Australian children to share their ideas with classes in other countries (at present, USA).

CONCLUDING POINTS

It is imperative that we take children beyond the traditional classroom experiences, where problem solving rarely extends their thinking or mathematical abilities. We need to implement worthwhile modelling experiences in the elementary and middle school years if we are to make mathematical modelling a way of life for our students. As this paper has argued, younger learners, irrespective of their class achievement levels, can successfully complete modelling problems of the type presented here.

Mathematical modelling activities for children should build on their existing understandings and should engage them in thought-provoking, multifaceted problems that involve small group participation. Such activities should be set within authentic contexts that allow for multiple interpretations and approaches. As children work these activities, they engage in important mathematical processes such as describing, analyzing, coordinating, explaining, constructing, and reasoning critically as they mathematize objects, relations, patterns, or rules.

��


Modelling activities not only provide opportunities for optimal mathematical development, but they also facilitate children’s social development. As children collaborate on constructing a model that meets given criteria, they raise numerous questions and conjectures, engage in argumentation, and learn how to resolve issues of disagreement. In doing so, children see different points of view and ways of thinking, which helps them to become more flexible in their own patterns of thinking.

The importance of providing children with numerous opportunities for model exploration and application has been stressed in this paper, with sequences of modelling activities recommended. Completion of these activities facilitates children’s development of generalizable conceptual systems, where they move beyond just thinking about their models to thinking with them.

Analogical and case-based reasoning have been proposed as key processes in children‘s application of generalized models. To effectively apply these reasoning processes, children’s models must comprise the structural elements that enable an existing modelwhich serves as a form of analog or case-to be mapped onto a new, similarly structured problem situation. To facilitate this mapping process, children need to anticipate situations in which their models might be applicable, and know when and how to utilize these models. Finally, children need to be able to make any necessary modifications to their existing models to accommodate the new situation. These processes require specific attention in the classroom through whole-class discussions. As this paper has illustrated, modelling activities for children develop important mathematical ideas and processes that would be left largely untapped in more traditional classroom activities. It is thus imperative that we introduce young children, and their teachers, to the world of mathematical modelling.

REFERENCES

Boaler J (2002) ‘Learning from teaching: Exploring the relationship between reform curriculum and equity’ Journal jor Research in Mathematics Education 33(4),

Diezmann C, English LD, Watters J (2002) ‘Teacher behaviours that influence young children’s reasoning’ in Cockburn AD, Nardi E (Eds), Proceedings of the 261h Annual conference of the International Group for the Psychology of Mathematics Education Vol. 2 Norwich: University of East Anglia, 289-296.

Doerr HM, English LD (2003) ‘A modeling perspective on students’ mathematical reasoning about data’ Journal for Research in Mathematics Education.

English LD (2002a) ‘Promoting learning access to powerful mathematical ideas’ in Edge D, Yeap BH (Eds), Mathematics education for a knowledge-based era. Proceedings of South East Asian Regional Conference on Muthematics Education and Ninth South East Asian Regional Conference on Mathematics Education Vol I Singapore: National Institute of Education.

English LD (2002b) ‘Development of 10-year-olds’ mathematical modelling’ in Cockburn AD, Nardi E (Eds) Proceedings of the 26Ih Annual conference of the

239-258.

��

14 English

International Group for the Psychology of Mathematics Education V01.2 Norwich: University of East Anglia, 100- 107.

English LD (1999) ‘Reasoning by analogy: A fundamental process in children’s mathematical learning’ in Stiff LV, Curcio FR (Eds) Developing mathematical reasoning, K-12 Reston, VA: National Council of Teachers of Mathematics, 22- 36.

English LD, Lesh RA (2002) ‘Ends-in-view problems’ in Lesh RA, Doerr HM (Eds) Beyond constructivism: A models and modelling perspective on teaching, learning, andproblem solving in mathematics education Mahwah, NJ: Lawrence Erlbaum.

Freudenthal H ( 1983) Didactical phenomenology of mathematical structures Boston: Kluwer.

Goldin GA (2002) ‘Representation in mathematical learning and problem solving’ in English LD (Ed), Handbook of international research in mathematics education Hillsdale, NJ: Lawrence Erlbaum, 197-2 18.

Gravemeijer K (1997) ‘Commentary: Solving word problems: A case of modelling?’ Learning and linstruction 7(4), 389-397.

Greer B (1997) ‘Modeling reality in mathematics classrooms: The case of word problems’ Learning and Instruction 7(4), 293-307.

Hatano G (1997) ‘Commentary: Cost and benefit of modelling activity’ Learning and Instruction 7(4), 383-387.

Jones G, Langrall, C, Thornton, C, Nisbet, S (2002) ‘Elementary school children’s access to powerful mathematical ideas’ in English LD (Ed), Handbook of international research in mathematics education Mahwah, NJ: Erlbaum.

Kolodner JL (1997) ‘Educational implications of analogy’ American Psychologist 5( I),

Kolodner JL, Camp PJ, Crismond D, Fasse B, Gray, J, Holbrook J, Puntambekar S, Ryan M (in press) ‘Problem-based learning meets case-based reasoning in the middle school science classroom’ Journal of the Learning Sciences 12(3).

Lesh RA, Hoover M, Hole B, Kelly A, Post T (2000) ‘Principles for developing thought- revealing activities for students and teachers’ in Kelly AE, Lesh RA (Eds) Handbook of Research Design in Mathematics and Science Education Mahwah, NJ: Lawrence Erlbaum.

‘Mathematical abilities that are most needed for success beyond school in a technology based age of information’ The New Zealand Mathematics Magazine 38(2), 1-1 7 .

Lesh RA, Cramer K, Doerr H, Post T, Zawojewski J (2002) Model development sequences in Lesh RA, Doerr HM (Eds) Beyond constructivism: A models and modelling perspective on teaching, learning, and problem solving in mathematics education Mahwah, NJ: Lawrence Erlbaum.

Mukhopadhyay S, Greer B (2001) ‘Modeling with purpose: Mathematics as a critical tool’ in Atweh W, Forgaz H, Nebres B (Eds) Sociocultural research on mathematics education: An international perspective Mahwah, NJ: Lawrence Erlbaum, 295-3 1 1.

Nathan MJ, Kintsch W, Young E (1992) ‘A theory of algebra-word-problem comprehension and its implications for the design of learning environments’ Cognition and Instruction 9 (4), 329-39 1.

57-66.

Lesh RA, Heger M (2001)

��


Silver EA, Smith MS, Nelson BS (1995) ‘The QUASAR project: Equity concerns meet mathematics reforms in the middle school’ in Secada WG, Fennema E, Adajian LB, (Eds) New directions in equity in mathematics education NY: Cambridge University Press, 9-56.

Verschaffel L, De Corte E, Borghart I (1997) ‘Pre-service teachers’ conceptions and beliefs about the role of real-world knowledge in mathematical modelling of school word problems’ Learning and Instnrction 7(4), 339-359.

Wyndhamn J, Saljo R (1997) ‘Word problems and mathematical reasoning-a study of children’s mastery of reference and meaning in textual realities’ Learning and Instruction 7(4), 361-382.

Zawojewski JS, Lesh RA, English LD (2002) ‘A models and modelling perspective on the role of small group learning’ in Lesh RA, Doerr HM (Eds), Beyond constructivism: A models and modelling perspective on teaching, learning, and problem solving in mathematics education Mahwah, New Jersey: Lawrence Erlbaum.

APPENDIX A

The Weather Problem (Doerr & English, 2003) The Global Travel Agency is interested in starting a re-location service to help advise people who are moving to a new area. The travel agency needs your help to develop an advising system for choosing places for their clients to live. The clients are primarily interested in the climate: how much rain, how cold it gets, how hot it gets, and if the days are sunny or cloudy. Each of these factors, however, is not of the same importance to every client.

Two potential clients have sent the following letters to the agency describing their preferences and asking for the agency‘s advice on the best places for them to live. The agency also has gathered some information on the nine cities listed below. 1. Develop a rating system for comparing the climates in different places. Be sure your

system will really help the agency evaluate places, even those not listed below. 2. Write two letters for the travel agency with a recommendation for each of the clients.

You should put the cities into three groups: the best cities, the second best cities, and the worst cities. This way the client will know which cities to consider living in and which cities to avoid. You should explain to the travel agency how your rating system works and why it is a good one.

3.

Dear Global Travel: My wife and I are retiring in several months and would like to relocate in a warm and sunny area. We don’t care if there is a lot of rain and we definitely don’t want to be too cold. What are some cities we should consider living in?

Sincerely, Mr & Mrs Johnson

��

16 English

Dear Global Travel: 1 am looking for some new job opportunities in my field of computer programming. I am quite confident that I will be able to find a job anywhere. I really like all kinds of outdoor sports, especially bushwalking! So I would like to move to a city that has good weather and doesn't get too hot. Where should I consider living?

Sincerely, Donna Smith

Climatic Information

APPENDIX B

The Snack Chip Consumer Guide Problem (English, 2002a) Students are presented with an introductory article on consumer guides, with questions to answer about the article. They are then given the following problem, with various packets of snack chips provided for them.

In this investigation, you will be developing a consumer guide to help people determine which type of snack chip is the best to buy. It is your decision what to focus on in your consumer guide. Your consumer guide must help people in choosing any snack chip, not just the ones you use in this activity.

As a whole class, brainstorm some factors or criteria that you might consider when you are trying to work out which chip is the best to buy. Think about what we could mean by best. Next, in your groups, discuss the following. 1. Describe the nature or type of factors that the whole class brainstormed. What type

of information does each factor give you? 2. How might you categorize the factors? 3. How might you rate the factors to help the consumer determine which packet of

snack chips they should buy?

��


4. Make sure that your guide can be used with any type of snack chip, not just the ones you have on your desk at present. Write clear instructions for the consumer on how to use your guide to compare different kinds of snack chips. Finally, prepare a short report for your class members explaining why the system you developed for your consumer guide is a good one.

5 .

APPENDIX C

The Car Problem (developed by Helen Doerr and Lyn English) Carl and his mother have been out shopping for cars. Carl wants a car that will be fun to drive around in, gets good gas mileage, but doesn't cost too much. But Carl's mother, who is going to help pay for the car, wants him to have a car that is reliable and safe. Your job is to create a list for Carl and a list for his mother showing which cars are the best. Then they will have to decide which one to buy! (Students are given a table of data comprising 9 different cars with their properties listed. These include: year of manufacture, cost, color, mileage, liters per 100 km of city driving, specific features, and body style.

��

2

Modelling in Elementary School: Helping YoungStudents to See the World Mathematically

Susan J. LamonMarquette University, [email protected]

In this chapter, I consider the question "What kind of mathematicalexperiences at the elementary school level prepare students to engage inmathematical modeling?" Implications from cognitive research suggestthat the very activities that facilitate meaningful learning are those thatpromote a mathematical cast of mind, a scientific spirit, and mathematicalcompetence. Using a few general principles based on current knowledgeabout the way people learn, teachers can significantly develop their dayto-day questioning techniques so that young students developmathematical ways of knowing while they are learning essentialmathematical content and processes.

��

20

INTRODUCTION

Lamon

Modelling is a complex activity in which the modeller's course cannot be prescribed inadvance, nor is the total path visible (mentally speaking) from any single vantage point(Resnick, 1987). Although we can recognize this higher-order process when it occursand we can judge the goodness of the achieved model for solving a problem, we cannotprescribe exact methods for socializing young learners into the culture that valuesmathematical modeling as a way of life.

The more general issue concerning how or whether any teaching produces learning isequally elusive. Mathematics teachers have always hoped to teach students to thinkcritically and creatively, to investigate, to analyze, to reason logically, to solve problems,to interpret, to reflect, to refine, and to generalize. Yet, after hundreds of years of goodintentions, all we have learned is that there is a great divide to be bridged betweenteaching and learning, between knowing and valuing something ourselves and helpingothers to know and appreciate it. There is no direct link between theories of learning andtheories of instruction; the first are descriptive and the second is prescriptive (Romberg &Carpenter, 1986).

In the last forty years or so, cognitive research has dramatically improved ourunderstanding of human learning. The implications of cognitive research are socompelling that in recent years, the traditional gap between intentions and results hasbeen diagnosed as an unfortunate case of mistaking a means for an end (Barr & Tagg,1995). Schools and universities have been focused on providing instruction, rather thanfacilitating learning.

It is now possible to outline some guiding principles that bridge the gap betweencognitive research and classroom practice, that is, we can describe the conditions underwhich learning is most likely to occur. Instruction as we have known it for the lasthundred years simply does not support nonalgorithmic, complex, effortful, autonomous,self-regulated reasoning and problem solving in which students must impose meaning,use nuanced judgment, and decide among alternatives (Resnick, 1987). Although thereare no "how-to" manuals for teachers, and no guarantees that every student will learn, atthe very least, mathematics instruction should be based upon conditions most congruentwith the implications of cognitive research.

In this chapter, I will use cognitive research to provide the rationale for some guidingprinciples that I have developed for elementary teachers whose goal it is to facilitate theirstudents' understanding of mathematical content and processes, and to orchestrate a totalenculturation process into mathematics as a way of knowing. As Bishop (1988)observed,

Educating people mathematically consists of more than just teaching them(students) some mathematics. It is much more difficult to do, and theproblems and issues are more challenging. It requires a fundamentalawareness of the values which underlie mathematics and a recognition ofthe complexity of educating children about those values. It is not enough

��

Modelling in Elementary School

merely to teach them mathematics, we need also to teach them aboutmathematics, to educate them through mathematics, and to educate themwith mathematics (p. 3).

21

With conscious, purposeful, sustained effort, teachers can change students' orientation tomathematics through the kinds of questions they ask. By cultivating a rich variety ofquestions, all of which are designed to promote student interaction with mathematics at asubstantially higher level than that expected in their textbooks, the teachers in myprojects have used these guidelines to create change in their classrooms. Afterdeveloping a rationale for the guiding principles, I will provide examples of the kind ofquestions that are easily incorporated into daily classroom activity to help students createdesirable mathematical meanings, attitudes, work habits, and values.

APPROPRIATE GOALS FOR ELEMENTARY SCHOOL STUDENTS

At the elementary level, students encounter their earliest formal experiences, and whilethey are learning basic content and skills, the most compelling case for change applies tothe nature of classroom interaction more than to goals and content. This is primarilybecause the interaction is a vehicle for attitudes and values and a mathematical frame ofmind. The spirit of science, defined in terms of seven underlying values (Woltle, 1966),remains the overarching goal of an education that promotes problem-solving andlearning:

• Longing to know and to understand• Questioning of all things• Searching for data and their meaning• Demand for verification• Respect for logic• Consideration of premises• Consideration of consequences.

Freudenthal (1978) characterized this spirit as something distinct from techniques andscientific instrumentation, and Piaget (1969) referred to it as the spirit ofexperimentation,and the spirit of invention. This orientation to the world is precisely the mindset thatpropels the process of mathematical modelling. But this mathematical cast of mind doesnot result from a haphazard process. The social and psychological aspects of thinkingand acting like a mathematician (Dreyfus, 1990) do not happen by chance and must beintentionally planned and facilitated by the teacher.

In addition to cognitive science research that enhanced our understanding of humanlearning, several other concurrent and interacting forces helped to provide valuableperspectives on the conditions that facilitate mathematical enculturation (Bishop, 1988).These include Piaget's constructivism, Vygotsky's emphasis on social learning, andFreudenthal's process of mathematization.

Piaget: A renewed interest in Piaget's constructivist epistemology (e.g., von Glasersfeld,1987) and its elaboration and application in teaching and learning environments (e.g.,Davis, Maher, & Noddings, 1990) advanced the perspective that we need to "shift the

��

22 Lamon

emphasis from the student's 'correct' replication of what the teacher does, to thestudents' successful organization of his or her own experience (von Glasersfeld, 1983, p.51). It is simply no longer acceptable to have someone else do for the students what wesay we want them to do for themselves. One cannot engage in higher-order thinkingwhen someone else is calling the plays at every step (Resnick, 1987).

Vygotsky: The translation of Vygotsky's work (1962, 1978) had a powerful influence oneducational psychology. Vygotsky related cognitive development to social phenomena,emphasizing the role of language and discourse in mediating learning. When ideas areshared with others and held up to scrutiny and refinement, those ideas are sociallyconstructed. Interpersonal and intrapersonal constructs-neither sufficient unto itselfplaya complementary role in acquiring meaningful concepts, processes, and values. Inexplaining to others, students develop a passion for their ideas. They value and promotethe ideas that they can defend. Pea and Greeno (1990) have argued that learning toparticipate in mathematical discourse is a significant aspect of learning in the discipline,with diagnostic benefits when it is publicly available. Student engagement (or busy-ness)is no substitute for discourse because it does not imply that students are developing thekind of knowledge that will support new learning (Prawaf, Remillard, Putnam,& Heaton,1992). There are important differences between tasks that require hands-on activity andthose that require "minds-on" activity (Greeno, 1991).

Freudenthal: During the 1980s, mathematics educators attempted to move beyondgeneral problem solving toward mathematical modelling in defining appropriatemathematical activity (e.g. Mathematical Association of America, 1981; Cockcroft,1982). This effort was enhanced by the work emanating from the University of Utrecht(later the Freudenthal Institute) that encouraged mathematization as a proper andlegitimate means of doing mathematics across all grade levels (e.g. Freudenthal, 1978,1991; deLange, 1987). Mathematizing is a progressive organizing and structuringactivity in which existing knowledge and skills are used to discover unknownregularities, relations, and structures (deLange, 1987) and it requires an activeconfrontation between the student and the situation that is to be organized. It begins withthe simplest situations, and spirals toward higher and more complex forms oforganization in which lower structures we have imposed become subject to structuringthemselves.

WHAT COGNITIVE RESEARCH TELLS US

A fundamental principle of cognition is that learning requires knowledge (Resnick &Klopfer, 1989). Before knowledge can become generative, that is, before it can be usedto solve problems, provide a useful way of interpreting new situations, and supportthinking and learning, it must be incorporated into existing knowledge structures. Thisknowledge cannot be told and therein lies the challenge: how can we help students tobegin developing a generative knowledge structure so that later they can independentlylearn and solve problems. It seems clear that the only way to deal with this dialecticalrelationship is to begin in the early grades to facilitate formal concepts and procedures,while allowing for mathematization at the same time.

��

Modelling in Elementary School 23

The knowledge that students have when they begin elementary school may be flawed.By the time children come to school, they have already developed sophisticatedpreconceptions about phenomena around them (Wellman, 1990). They do not merelyabsorb information and place it into a depository. Rather, information is filtered,organized and interpreted using models shaped by their past experiences, powerfulengravings from authority figures, memorable experiences, and personal intuitions aboutthe way things work. These student conceptions may consist of powerful, flawedtheories and "buggy" processes (Brown & Van Lehn, 1980; Schoenfeld, 1987) that arestubbornly resistant to change. If so, they may interfere with learning new concepts.

The psychological characteristics of a mathematical task are not reducible to themathematical properties of the task. That is, there may be a gap between the student'sunderstanding and a mature adult's knowledge of a particular concept, or the concept as itis conveyed in a mathematical definition. Rather than tacitly assuming that studentsunderstand ideas in the way that we expect them to, it is important to draw out their preexisting understandings-what they know and how they know it. Students may havesome primitive conceptions that need enhancement, or perhaps they have some clearlyflawed ideas. When this happens, it is useful to devise a problem that will cause thestudents to engage with the troublesome concepts. When student understanding isinappropriate for dealing with the situation, cognitive conflict results, the student has torevise his or her ideas before the problem can be solved. This is Piaget's theory ofequilibration (Flavell, 1963).

Learning is making new understanding or understanding anew, that is, refining ordeepening something previously learned. Understanding, the most essentialcharacteristic of learning, consists of making meaningful connections between ideas,facts, and procedures. When knowledge is well integrated, it is powerful and intelligentin the sense that it becomes part of a system that is then used to support further learning.It increases its own power and effectiveness (Resnick & Klopfer, 1989). In other words,there is a mutually impellent power between existing knowledge and new, meaningfulknowledge.

Research that compared expert and novice performance in various domains emphasizedthat experts are not merely smart people or good thinkers or people who have bettermemories than others. In fact, they have a rich knowledge base that supports everythingthey do. They have the ability to see significant patterns and relationships that are notperceived by novices and to plan multi-step tasks because their knowledge is wellorganized and connected. Through the expert-novice studies, we have come to realizethat an understanding of factual principles is necessary but not sufficient: facts need to beorganized into a meaningful framework before the information can be used to engaged inhigher-order thinking and problem solving. To illustrate meaningful organization,Donovan, Bransford, and Pelligrino, 2000) consider the case oflearning geography. Anystudent may quickly and accurately fill in the names of states, cities, and countries on amap, but when the boundaries are removed, the task is more difficult. A person whounderstands that boundaries often developed where natural phenomena, such asmountains or bodies of water separated people, and that cities arose in locations that were

��

24 Lamon

conducive to trade, along rivers, lakes, and coastal ports, will outperform the novice onthe more difficult identification task.

In mathematics, this structure and overall grasp of significant relationships is the aim ofFreudenthal's mathematization process. By organizing and representing a situation, weare able to see (understand) things we were not able to see before, and this structuringprocess achieves greater and greater depth when appropriate mathematical tools andprocesses are introduced and brought to bear.

Having established this rationale based on cognitive research and influential theories ofthe last several decades, I tum to the central question of this chapter: What kinds ofclassroom activities might help elementary students to build the capacity, value system,and knowledge to enable participation in mathematical modelling? The answer is thatstudents learn to think by thinking and to mathematize by imposing structure; they needto engage in these processes by themselves and with others at the same time that they arelearning the concepts, processes, and tools of mathematics. Because current textbooks donot supply adequate higher-order activities, teachers need other resources. As aresearcher, 1 have found that good questions go a long way toward encouraging childrento think, to reason, and to mathematize, and I encourage teachers to develop theirquestioning techniques around six goals supported by cognitive research. Withconscious, persistent, goal-directed effort, teachers become their own best resource forhelping young learners to see the world mathematically. In the next section, I giveexamples of the principles that 1 use to guide teachers' questions, and examples of thethoughtful activities elementary teachers have devised for their students based on theseprinciples.

PRACTICE THAT SUPPORTS MEANINGFUL LEARNING: THE CHEERSPROGRAM

I ask elementary teachers to incorporate one new question into their mathematics lessonevery day. The questions should rotate among six varieties that are represented by theletters in the acronym CHEERS. Each activity should be carefully designed with regardto mathematical content and goals, and notes should be kept on the effectiveness of theactivity so that it can be improved the next time it is used. This simple procedure causesteachers to think about what really matters-student learning-and about to how tofacilitate it. The following examples came from elementary teachers who were using theCHEERS principles to strengthen their students' mathematical program.

C challenge misconceptionsH head-use it!E explore ideasE extend ideasR reason deductivelyS structure-progressively deepen it

��


Challenge Misconceptions: Grade 5Problem: Squirrels (Figure I)Goal: Clarify the difference between midpoint and any other point that lies between two

othersContent: GeometryNotes: When my students talked about the locations of points on a segment, they usedthe concepts of midpoint and betweenness of points interchangeably, as if they meant thesame thing. Either this is a misconception, or an incomplete notion. This problem isdesigned to help them confront the two ideas side by side and to distinguish them.This puzzle was motivating and the students worked diligently until they could solve it. Idid not announce the reason for the puzzle, yet, when they finished, the students knewthat the Ah-ha! experience that got them over their original hurdle so that they could goon and solve the problem was noting the difference between the midpoint and otherpoints that lie between endpoints.

Squirrels

(

Five squirrels are sitting in a row on a fence. Figure out the order in which they aresitting from the following clues:

l. Click is the same distance from Ajax as she is from Bushy.2. Edgar is seated between Dolly and Ajax.3. Bushy is sitting next to Edgar.4. Edgar is not seated between Bushy and Dolly.

Figure 1. The squirrel problem.

Challenge Misconceptions: Grade 3Problem: The Tractor (Figure 2)Goal: Challenge misconceptions about relationships between wheel size, distance, and

rotationsContent: Quantitative RelationshipsNotes: My students become terribly confused when we used trundle wheels to do somemeasuring in our classroom. We had two wheels that had different diameters and manyof the children insisted that we would get different answers depending on which one weused. I used this problem from Lamon (1999) to help them analyze and discuss multiple

��

26 Lamon

interacting quantities and relationships. We spent an entire class period sorting out all ofthe quantities that students brought into the discussion of this problem. Some interestingmisconceptions were revealed. Some thought that the front wheel would get to the otherend of the field first, so the back wheel has farther to go. Some said that the bigger wheelgoes farther because it turns faster, while some thought that the bigger wheel goes fartherbecause it takes more time to tum completely around. This was very difficult for myclass, but profitable, I think, because they had never thought this way before. As a byproduct of our discussion, the students all know the difference between diameter andcircumference.

The Tractor

When the farmer drives this tractor from one end of the fieldto the other, will both wheels cover the same distance?

Figure 2. The tractor problem.

Using Heads: Grade 5Problem: Twins (Figure 3)Goal: Think about unobservable quantitiesContent: Rate of growthNotes: Until we talked about these quantities in our university class, I had not thoughtspecifically about quantities that are not directly observable and measurable, and I havenever asked my student to think about them either. My students had a very hard timetalking about this problem, but it was because they were dealing with a complexsituation. Specifically, they struggled with the difference between growing more (such asa specific number of inches) and growingfaster, and whether or not these meant the samething.

Using Heads: Grade 5Problem: Dessert Time (Figure 4)Goal: Think without using pencil and paper.Content: Quantitative ReasoningNotes: I used this question to help my students build confidence in the power of theirown heads to do most of the work that they think they need to do by hand. It 'worked'and the students enjoyed it.

��


Twins

The first picture shows Jeb and Sarah Smart when they were younger. The secondfhl"\\lIc th"m as they look now. Who grew faster between then and now?

Jeb and Sarah then Sarah and Jeb now

Figure 3. The twins then and now.

Dessert

Jim family and Nancy's family met at the Sweet Shack for dessert.Here is what Jim's family ordered. Their bill was $9.50 .

Here is what Nancy's family ordered. They paid a total of$6.50.

Figure out the cost of a piece of pie and the cost of a cone.

Figure 4. The dessert problem.

Explore: Grade 5Problem: Traffic Lights (Figure 5)Goal: Explore a situation to learn everything you can about itContent: Reading and interpreting charts and tables.

��

28 Lamon

Notes: Students found it very difficult to interpret this situation. They failed to use theirexperience in cars to help them. They drew exactly opposite conclusions about whichstreets were busiest and which had less traffic.

Traffic Lights

Signal Signal SignalABC

Color

Orchard Ln.

I~"'~.A

~L

]]I

Green 9:01:10 9:01:20 9:01:10Yellow 9:01:34 9:05:09 9:01:39Red 9:01:37 9:05:12 9:01:42Green 9:02: 10 9:05:20 9:02: 10Yellow 9:02:34 9:08:09 9:02:39Red 9:02:37 9:08:12 9:02:42Green 9:03:10 9:08:24 9:03:10Yellow 9:03:34 9:14:09 9:03:39Red 9:03:37 9:14:09 9:03:42Green 9:04:10 9:14:22 9:04:10

r Yellow 9:04:34 9:16:09 9:04:39Pln.St.Red 9:04:37 9:16:12 9:04:42

The table to the right gives information about how the traffic signals A, B, and Careregulated. Draw as many conclusions as you can about the traffic on the streetsshown in the above map.

Figure 5. Traffic Lights

Explore: Grade 4Problem: The Clock (Figure 6)Goal: Explore to discover relationshipsContent: Quantitative ReasoningNotes: Students worked this problem in groups and it was disappointing that none ofthem were able to explain how they knew where the crack in the clock's face occurred.They solved the problem by trial and error and never questioned why their answer madesense or how it could be explained mathematically.

Expand: Grade 5Problem: Climbing Stairs (Figure 7)Goal: Expand student understanding of this problem by encouraging them to make

connections to other contentContent: Number TheoryNotes: Students made hit-and-miss lists at first, but soon developed a way to list thepossibilities systematically. They had to have their work well organized to make theconnections to Pascal and Fibonacci, but about 70% of the class understood theconnections and they were able to explain their solutions using another related but

��


different problem. I'm not sure I would change anything. Organizing is part of theprocess that they need to learn.

The Clock

Our town had a big clock on a tower in the middle of a parkdowntown. One day, a huge storm blew the old clock off thetower and it crashed into the street. The mayor rushed out of hisoffice and picked up the pieces. He noticed something unusualabout the way the clock broke. The face broke exactly in halfand the six numerals on each half had the same sum. Whichnumerals were on each piece and how did you figure it out?

Figure 6. The clock problem.

Climbing Stairs

• There are 8 steps from the family room to the kitchen, and everytime Jack runs up for a snack, he likes to climb the steps a differentway. How many different ways are there to climb 8 stairs, taking Ior 2 at a time?Explore the connection between this problem and Pascal's Triangle.Explore the connections between this problem and the Fibonaccisequence.

Figure 7. The stair-climbing problem.

Expand: Grade 5Problem: Technology (Figure 8)Goal: To develop a deeper understanding of percentagesContent: PercentagesNotes: I used this problem because our textbook exercises develop basic procedures withpercentages but they do not help students to gain sufficient meaning for problem solving.I have to admit, my students did not get very far with this problem. Some realized thatthere was going to be a range for the percentage of people who have both a computer anda cell phone, but no one was able to determine the range for all three pieces oftechnology. Part of the reason is that the students are used to a single correct answer.

��

30 Lamon

Technology

In a certain small business, 70% of the employees have acomputer, 55% have a cell phone, and 40% have a palmpilot. What percent of the company employees have allthree pieces of equipment?

Figure 8. The technology problem.

Expand: Grade 5Problem: The Apple Orchard (Figure 9)Goal: Generalize; go beyond the required answerContent: Number TheoryNotes: My students are always happy when they get the correct answer to a problem. Ithink that they need to be asked to generalize their result and go beyond the immediateconfines of the problem. In this situation, when they were pushed by the final question, itbecame clear that their thinking was actually quite shallow and it did not allow them tosolve a closely related problem. I am sorry to say that none of my students could answerthe second question.

Square's Apple Orchard

Jim Square owns an apple orchard and he sells his apples everyfall. In his first year of business, Jim had only one tree. Eachyear, he planted new trees and his business grew. He alwaysplanted enough trees to keep his orchard square. Here is how itlooked for the first few years.

• o.••

00.00 •• ••

Year I Year 2 Year 3Jim has been in business for a long time, and this year, he planted 31 new trees.How many apple trees are in the Square orchard this year?

Can you explain how to find the sum of the first 20 odd numbers without writinganything down or using a calculator?

Figure 9. The apple orchard problem.

Reason Deductively: Grade 4Problem: The Cube (Figure 10)Goal: Explain all of the steps that led to your conclusion.Content: Logic; deductive reasoning.

��


Notes: I had students set this up in a two-column format: What I know... and How Iknow it... This helped them to explain all of the steps that led to their conclusion. Noteveryone used the same patterns, and so there were different results. During ourdiscussion, we were able to make the point that as long as the results followed from thegiven evidence, they were valid, but the students had never seen a problem that had morethan one path and answer and many were not convinced. I can see we need moreproblems that have multiple solutions!

;::

03l:j:ll':t

9

w I' ...«r-

~>IN't~::l

0::0 IJ.l

'" a:l00::... S <'"l

4

ALMA

2

The Cube

Cut out the figure you see here,fold it on the dark lines, and tapeit together so that it forms a cube.You will notice that one of thefaces of the cube has nothingwritten on it. Determine whatshould be written on the emptyface.

Figure 10. The cube problem.

Structure: Grade 4Problem: Mining (Figure II)Goal: Go beyond surface level observationsContent: Number TheoryNotes: My students have a tendency to "look and tell," that is, they talk about onlyvisible, obvious features of a problem. They have trouble looking below the surface forother characteristics and regularities. Naturally, they all said that the unlike number was19 because it had 9 for one of its digits. The students seemed to get the idea after wediscussed the example, but I have to think up more problems like this so that I can see ifthey really do look below the surface next time.

��

32 Lamon

Which number in the following list is most unlike all of the others?1 3 5 13 11 19 31 33 53

Figure 11. The mining problem.

Structure:Grade 3Problem: Analogies (Figure 12)Goal: Think about relationshipsContent: Cross-disciplinaryNotes: I am having a hard time getting my students to think about relationships, so I triedusing some analogies such as those you might find on a standardized exam, to discuss thedifference between the answer and the relationship that suggests what you might use tofill in the blank.

What is the relation that tells you how to fill in the blank?

Dog is to fur as bird is to _Bee is to honey as snake is to _Hand is to glove as head is to _

Figure 12. Analogies.

REFERENCES

Barr RB, Tagg J (1995, November/December) 'From teaching to learning-A newparadigm for undergraduate education' Change.

Bishop A (1988) Mathematical enculturation: A cultural perspective on mathematicseducation Dordrecht: Kluwer.

Brown JS, Van Lehn K (1980) 'Repair theory: A generative theory of bugs' in CarpenterTP, Moser M, Romberg TA (Eds) Addition and subtraction: A cognitiveperspective Hillsdale, NJ: Erlbaum 117-135.

Cockcroft WH (1982) Mathematics counts. Report of the committee of inquiry into theteaching ofmathematics in schools, London: Her Majesty's Stationery Office.

Davis RB, Maher CA, Noddings N (Eds) Constructivist views on the teaching andlearning of mathematics Reston, VA: National Council of Teachers ofMathematics.

de Lange, J (1987) Mathematics. insight and meaning Utrecht: OW&OC.Donovan MS, Bransford JD, Pelligrino JW (Eds) (2000) How people learn: Bridging

research and practice Washington, DC: National Academy Press.Dreyfus, T (1990) 'Advanced mathematical thinking' in Mathematics and cognition: A

research synthesis by the International Group for the Psychology ofMathematicsEducation Cambridge: Cambridge University Press 113-134.

Flavell JH (1963) The developmental psychology of Jean Piaget New York: VanNostrand Reinhold.

Freudenthal H (1978) Weeding and sowing Dordrecht: d. Reidel.

��


Freudenthal H (1991) Revisiting mathematics education: China lectures Dordrecht:Kluwer Academic Publishers.

Greeno J (1991) 'Number sense as situated knowing in a conceptual domain' in Journalfor Research in Mathematics Education 22(3), 170-218.

Lamon SJ (1999) Teaching fractions and ratios for understanding: Essential contentknowledge and instructional strategies for teachers Mahwah, NJ: LawrenceErlbaum.

Mathematical Association of America's Committee on the Undergraduate Program inMathematics (1981) Recommendations for a general mathematical sciencesprogram Washington DC: Mathematical Association of America, 13.

Pea RD, Greeno J (1990) 'Reflections on the direction of reform in mathematicseducation' Paper presented at the meeting of the American Educational ResearchAssociation, Boston.

Piaget J (1969) Science of education and the psychology of the child New York:Grossman Publishers.

Prawaf RS, Remillard RT, Putnam R, Heaton RM (1992) 'Teaching mathematics forunderstanding: Case study of four fifth-grade teachers' Elementary SchoolJournal 93, 145-152.

Resnick LB, Klopfer, LE (1989) Toward the thinking curriculum: Current cognitiveresearch Washington, DC: Association for Supervision and CurriculumDevelopment.

Resnick LB (1987) Education and learning to think. Washington, DC: National AcademyPress.

Romberg TA, Carpenter TP (1986) 'Research on teaching and learning mathematics:Two disciplines of scientific inquiry' in Wittrock MC (Ed) Handbook ofresearchon teaching New York: Macmillan, 850-873.

Schoenfeld AH (l987)'Cognitive science and mathematics education: An overview' inSchoenfeld AH (Ed) Cognitive science and mathematics education Hillsdale, NJ:Erlbaum, 11-31.

Von Glasersfeld (1987) Learning as a constructive activity in Janvier C (Ed) Problems ofrepresentation in the teaching and learning of mathematics Hillsdale, NJ:Erlbaum, 215-225.

Vygotsky LS (1962) Thought and language Cambridge, MA: MIT Press.Vygotsky LS (1978) Mind in society: The development ofhigher psychological processes

Cambridge, MA: Harvard University Press.Wellman HM (1990) The child's theory ofmind Cambridge, MA: MIT Press.Wolfle D (1966, June 24) 'The spirit of science' Science, 152.

��

Section B

Modelling with Middle and Secondary Students

��

��

3

How Mathematizing Reality is Different from RealizingMathematics

Richard A. LeshPurdue University, [email protected]

The kinds of mathematical processes, understandings, and abilities that areused in modeling are seldom emphasized in traditional mathematicstextbooks, tests, and teaching. In a class of problems that I call mode/eliciting activities, students engage in activities that are recognized andvalued in the world outside the classroom. These activities include higherlevel reasoning, planning, monitoring, and the use of multiplerepresentations and technological tools.

��

38

INTRODUCTION

Lesh

What mathematical understandings and abilities provide the most powerful foundationsfor success beyond school in a technology-based age of information? Why do studentswho score well on traditional standardized tests often perform poorly in more complex,real life situations where mathematical thinking is needed? Why do students who havepoor records of performance in school often perform exceptionally well in relevant, reallife situations?

My experiences in a broad variety of workplaces for more than 25 years have given meinsight into these questions. I have worked inside and outside of departments ofmathematics and in companies ranging from the Educational Testing Service to softwaredevelopment institutes, where various levels and types of mathematical thinking areneeded. Much of my work involved interactions with professors and professionals infields ranging from engineering to management to medicine, where successful peoplerely on mathematics, science, and technology.

When I encounter my former mathematics students, I am sometimes depressed by howlittle is left from what I thought I had taught, but in other cases, I am equally amazed thatsome students whose classroom performances and/or standardized test results wereunimpressive, went on to become very successful people. I have concluded that the kindof mathematical understandings and abilities that are emphasized in textbooks and testsusually represent only a shallow, narrow, and non-central subset of those that are neededfor success in real life situations. Partly because of the preceding kinds of observations,experiences, and interests, a large share of my research during the past twenty-five yearshas focused on the question:

In what ways have powerful computation and communication tools led tofundamental changes in the levels and types of mathematicalunderstandings and abilities that are needed for success in future-orientedfields that are increasingly heavy users of mathematics, science, andtechnology?

A primary goal of this chapter is to explain why a class of problems that I refer to asmodel-eliciting activities emphasizes many levels and types of mathematicalunderstandings and abilities that seldom are emphasized in traditional mathematicstextbooks, tests, and teaching. In particular, I will describe why the knowledge andabilities that are emphasized in model-eliciting activities are even different from thosethat are emphasized when teachers try hard to teach mathematics so as to be useful.When students express, test, and revise their own current ways of thinking aboutimportant mathematical constructs (or conceptual systems), the relevant understandingsand abilities they use tend to be quite different from those that are emphasized whenteachers or textbook authors efficiently guide students toward idealized and highlysimplified portrayals of targeted mathematical constructs.

A corollary to the preceding observations is that, when we recognize the importance of abroader range of mathematical knowledge and abilities, a broader range of students

��

Mathematizing Reality vs. Realizing Mathematics 39

naturally tends to emerge as having extraordinary potential. Furthermore, many of theseextraordinarily capable students come from populations that are highly underrepresentedin mathematics and the sciences. Therefore, at the same time that model-elicitingactivities emphasize new conceptions of what it means to understand mathematics, theyalso provide new ways to deal with issues of diversity and equity.

SUCCESS IN THE 21ST CENTURY

During the last two decades, changes have occurred in the kind of mathematicalunderstandings and abilities that are needed for success in the 2151 century. One aspect ofmy research has entailed visiting graduate schools whose professors are responsible foreducating tomorrow's leaders in future-oriented professions that are heavy users ofmathematics, science, and technology. I have asked these leaders to show me explicitexamples of learning or assessment activities that are simulations of the kind of situationswhere mathematics is used in their fields. I have also asked these experts to describe thekinds of mathematical thinking that they believe is needed for success in thesesimulations, as well as to describe the characteristics of graduating students who are mostsought-after in job interviews following the completion of their programs. The followingcomments are among those I have heard emphasized most often.

Concerning the nature of simulations of real life problem-solving or decision-makingactivities: In fields ranging from medicine to business management, many leadinginstitutions of higher education use case studies for both instruction and assessment.These case studies generally require an hour or more to complete. The context ofteninvolves too little time, too few resources, and conflicting goals, such as those that arerelated to costs and benefits, completeness and simplicity, or quality and timeliness. Theproblem solving is handled by a team of specialists who represent diverse practical andtheoretical perspectives, and who have access to a variety of powerful conceptual tools.The product is designed for a specific decision-maker and for some specific purpose. It isa complex artifact that usually includes a model or conceptual tool for making sense ofsome significant collection of mathematically similar systems. The production processusually involves design cycles in which models are iteratively developed, tested, andrevised. It is often necessary to integrate ideas drawn from a variety of disciplines(Oakes & Rud, 2003).

Concerning the nature of the mathematical ideas that are especially useful in thepreceding kinds of problem-solving/decision-making activities: The mathematicalsystems that are most useful often involve non-linear models, discrete mathematicalmodels, graphic media, iterative functions with feedback loops leading to second-ordereffects that overpower first-order effects, and/or complex, dynamic, and continuallyadapting systems with interacting agents and emergent properties of the systems-as-awhole. However, even when traditional content is required, say that of fractions, ratios,rates, and proportions, the elementary-but-deep understandings are usually those thathave been ignored almost completely in textbooks and tests. Important understandingsoften depend on connections among logically distinct topics. Representational fluency isimportant because the conceptual tools that are most useful involve multi-mediaconstructions, simulations, and transformations (Aliprantis & Carmona, 2003).

��

40 Lesh

Concerning the characteristics of students who are most sought-after in job interviews:The students who consistently emerge as favorites in job interviews are those who areable to make sense of complex systems. They are people who work well andcommunicate meaningfully within diverse teams of specialists, and they are able to adaptrapidly to continually evolving conceptual tools. They are people who are able to mathematize complex systems: by quantifying qualitative information, by dimensionalizing orcoordinatizing space, or by systematizing experience so that powerful conceptual toolscan be used. They are insightful about interpreting practical implications of results thatsophisticated tools produce (Kardos, 2003).

CASE STUDIES FOR KIDS

Another branch of my research has enlisted teachers, professors, and other experts toserve as co-researchers to: (I) design case studies for kids, which are middle schoolversions of the kind of case studies that are used in fields that use mathematics, (2)closely analyze videotapes, detailed transcripts, and live problem-solving sessions inwhich students work on the preceding simulations of real life situations, and (3) developtools that researchers and/or teachers can use to analyze students' work and givefeedback about strengths, weaknesses, and ways to improve. We refer to these sorts ofinvestigations as design experiments for evolving experts because, as interacting groupsof co-researchers go through a series of iterative design cycles in which they test andrevise their case studies, they also clarify their own ways of thinking about the nature ofthe mathematical understandings and abilities that provide foundations for successbeyond school.

The Paper Airplane ProblemThis problem is a middle school version of a case study that we first saw being used inPurdue's graduate program for aeronautical engineering. The original problem involveda wind tunnel and the goal was for graduate students to develop an operational definitionto deal quantitatively with the concept of drag for various shapes of planes and wings.Our experience suggests that finding a way to measure a quantity that cannot be seen andmeasured directly is one of the most common problems that occurs in elementary scienceor in everyday situations where mathematical thinking is needed (Lesh & Doerr, 2003).

To introduce the Paper Airplane Problem, teachers usually ask students to read aspecially created newspaper article (or web site) that describes how to make differenttypes of paper airplanes. Then: (i) students answer warm-up questions about thenewspaper article, (ii] they make their own versions of several of these airplanes; and,(iii) they test the flight characteristics of their planes using the three flight pathssuggested in Figure 1. For each path, the goal is to hit a target from a given startingpoint, in some cases traveling around an obstacle (such as a chair).

Each team is given a data sheet (shown in Table I) with results that another group ofstudents produced for the preceding three flight paths. Notice that for each paper plane,results are shown for each of the three flight paths, and for each flight path, threemeasurements: (l) the total distance flown, (2) the distance from the target, and (3) the

��


Start

Finish

~-..Start

Figure 1. Paths for paper airplanes.

•Start & Finish

Path 1 Path 2 Path 3TEAM AT L D AT L D AT L DTeam 3.1 11 1.8 2.5 7.7 3.2 0.7 1.8 6.8

1 0.1 1.5 8.7 0.9 2.9 8.6 1.2 3.7 6.72.7 7.6 4.5 0.1 1.1 2.4 2.7 8.4 4.4

Team 3.8 10.9 1.7 3.2 9.2 4.6 2.3 8.1 6.12 5.2 13.1 5.4 2.3 9.4 2.9 0.2 1.6 6.9

1.7 3.4 8.1 1.1 2.7 8.8 2.1 6.9 5.2Team 4.2 12.6 4.5 1.7 4 5.9 2.9 8.5 5.5

3 5.1 14.9 6.7 3 10.8 3.1 2.4 7.7 8.73.7 11.3 3.9 2 6 3.2 0.2 1.9 6.7

Team 2.3 7.3 3.25 1.3 4.9 4.4 1.4 4.9 4.94 2.7 9.1 4.9 2.3 7.3 2.4 2.7 7.2 8.1

0.2 1.6 9.1 1.4 3.7 4.1 0.2 1.1 7.3Team 4.9 7.9 2.8 2.7 10.7 1.1 2.5 7.7 5.7

5 2.5 10.8 1.7 3.3 6.3 2.5 2.1 9.8 9.85.1 12.8 5.7 1.3 2.6 7.2 3.2 10.4 5.8

Team 0.2 1.8 8.8 1.8 3.9 4.2 0.1 1.2 8.26 2.4 10.1 4.6 0.2 5.7 4.2 1.3 4.9 4.9

4.7 10.3 5.4 1.6 8.5 3.4 1.8 5.5 2.7

AT = Air time in seconds L = Length of throw in meters.

D = distance from the target in meters.

Length of throw is the straight-line distance between the starting point and the point where theplane landed. The target is the finish point where the plane should land.

Table 1. Data from a Paper Airplane Contest

time in flight. The goal of the problem is for students to write a letter to students inanother class describing how such data can be used to assess various flightcharacteristics, including (1) most accurate, (2) best boomerang, (3) best floater (i.e.,going slowly for a long time), and (4) best overall.

��

42 Lesh

The Volleyball ProblemThe Volleyball Problem is a middle school version of a case study that is used inPurdue's Krannert School of Management. However, it is a problem that is common inmany different fields where it is important to rank things (e.g., products, people, places,or businesses) by aggregating various types of qualitative and quantitative information.For example, such ranking procedures are used in consumer guidebooks that assessautomobiles and other products. They are used in places rated almanacs that rank cities,states, or regions according to their live-ability. They are used when teachers assigngrades to students in their courses by merging information about performances on tests,quizzes, projects, and other assignments. Relevant procedures may involve quantifyingqualitative information, assigning weights to reflect the relative importance of differenttypes of information, and combining these factors using some type of weightedaverage-or some type of iterative process that does not depend on a single formula.

As in the Paper Airplane Problem, students usually are introduced to the VolleyballProblem by reading an article in a math-rich newspaper that describes a summer sportscamp for girls' volleyball. The newspaper article explains that problems arose in the pastbecause it was difficult for the camp councilors to form fair teams that could remaintogether throughout two weeks of camp.

The organizers of the volleyball camp need a way to divide the campers into fair teams.They have decided to do this using information from try-out activities that will be givenon the first day of the camp. The table below shows a sample of the kind of informationthat will be gathered from the try-out activities. Your task is to write a letter to theorganizers where you: (1) describe a procedure for using information like the kind that isgiven below to divide more that 200 players into teams that will be fair, and (2) showhow your procedures works by using it to divide these 18 girls into three fair teams.

As shown in Table 2, the try-outs produce several different types of qualitative andquantitative information. This means that students must develop some way to combinethis information for the purpose of ranking campers from best to worst. This entails notonly developing ways to aggregate information within and across categories, but alsodeveloping mathematical ways to compare scores within categories, campers both withinand across categories, and teams. Solutions may involve procedures that implicitly uselinear combinations (v=ax+by+cz+dw) or vectors (v(a,b,c,d)=i(a)+j(b)+k(c)+l(d)) or aseries of iterative processes that cannot be collapsed to a single function. Note that linearequations differ from vectors in that they involve the same kind of quantities (i.e.,quantities that can be combined).

MODELING AND MODELS

In the workplace, it is rarely the goal to simply produce an answer. Therefore, in modeleliciting activities the most problematic aspect of a task is the purposeful development ofmathematical descriptions (or conceptualizations, or interpretations) of the situation. As

��

Mathematizing Reality vs. Realizing Mathematics

Name Height Vertical 40- Number of Spike Resultsof leap in Meter serves (5 attempts)

player inches Dash in successfully DR dink returnedseconds completed

DU dink unreturnedout of 10

OB out of boundsIN in the net, R returned

Gena 6'1" 20 6.21 8 DR DU Kill IN RBeth 5'2" 25 5.98 7 Kill R OB DR KillJill 5'10" 24 6.44 8 OB R R Kill INArnv 5'10" 27 6.01 9 Kill Kill DU Kill RAna 5'6" 25 6.95 10 OB IN R R DRKate 5'8" 17 7.12 6 Kill DU Kill R KillRhoda 5'3" 21 6.34 5 OB Kill IN IN DRChristi 5'5" 23 7.34 8 IN Kill Kill Kill DUAndrea 5'5" 24 6.32 9 IN OB IN OB RNikki 5'7" 19 8.18 10 DU Kill Kill OB RKim 5'9" 23 6.75 7 DR Kill R OB KillRobin 5'8" 15 5.87 8 Kill Kill Kill DU INEdna 5'4" 21 6.72 8 Kill R OB IN DRLori 5'7" 19 6.88 9 OB IN IN Kill RTina 5'1" 24 6.27 6 DU DR DR Kill OBAngie 5'10" 23 6.54 8 OB Kill OB OB DRRuth 5'3" 26 7.01 9 DU IN Kill Kill KillBecca 5'9" 18 6.78 10 IN OB Kill DR Kill

Volleyball Coach's Comments

Gena: Tall but slow getting to the ball. Robin: The hardest worker we've ever hadBeth: She is very agile on her feet. at the high school. She makes everybodyJill: Height and jumping ability should better she plays with.prove to be an asset for any team. Edna: A girl that others want to be withAmy: Awesome leaper, but she needs to because whatever event she's in, she seems toknow when to use it. always find a way to win.Ana: Not very aggressive partly because Lori: Doesn't always get her serve over theshe's only played on unsuccessful teams. net, but when she does, it's a winner.Kate: Kate has great quickness to get to Tina: One of the most intense players wethe ball after serves. have ever seen. She works hard.Rhoda: Has a tendency to loaf; plays best Angie: Her father coaches at a local school.when the team is playing well. So, she gets the most out of her nativeChristi: She hides her abilities. Her abilities.family life has negatively impacted her Ruth: Her skills are good. She grew upability to play well. watching her sister player at the University ofAndrea: Exceptionally strong for her age. Alabama.Nikki: She does many things well. In Becca: Rebecca is very coachable. She'llparticular, she serves well. improve fast if she gets a chance.Kim: Kim is a great blocker.

Table 2. Data from volleyball tryouts.

43

��

44 Lesh

Meaningful experiences

Symbolically describedsituations

Mathematizing reality:Making symbolic

descriptions of meaningfulsituations.

Rea lizing mathematics :Making meaning out ofsymbolically described

situations.

Figure 2. Realizing mathematics vs. mathematizing reality.

shown in Figure 2, in model-eliciting activities, the relationship between mathematicsand reality is the opposite of that in traditional textbook word problems in which studentsbegin with symbolic statements and try to make sense of them.

When the problem is to create symbolic descriptions of a meaningful situation, theprocesses and abilities that are needed also tend to be nearly the opposite from thecomputation skills emphasized in many textbook word problems (Lesh, Cramer, Doerr,Post & Zawojewski, 2003). Models are purposeful conceptual systems that are expressedusing some (and usually several) representational media, and their purposes generally areto describe or explain the behaviors of other systems so that intelligent predictions,constructions, or manipulations can be made. Consequently, model development tends toemphasize processes such as quantifying, dimensionalizing, coordinatizing, organizing,systematizing, or (in general) mathematizing objects. These processes, tools andrepresentations tend to be ignored when attention is focused on making meaning ofsymbolic descriptions or carrying out symbolic computations or transformations. Asshown in Figure 3, these important processes, tools, and representations are usedrepeatedly as models go through multiple testing and revision cycles.

Ifwe shift our attention beyond modeling processes toward models as knowledge objects,or constructs that are important in their own right, then it becomes clear that the mostimportant goals of mathematics instruction should involve developing powerful modelsfor describing and explaining important classes of mathematical systems. When we askwhat mathematics a student has mastered, answers should go beyond listing the kind ofprocedures that the student can execute flawlessly. Answers also should include thekinds of situations that the student can describe mathematically, that is, the models thatthe student has developed.

��

Mathematizing Reality vs. Realizing Mathematics

Describe, Ex lain , Predi ct

45

Test & Revise

Figure 3: A simplified view of a single model development cycle.

This is why model-eliciting activities shift attention away from mathematics as doing(constructions and computations) toward mathematics as seeing (patterns and regularitieswithin structurally interesting systems). It is also why the models that should be giventhe greatest attention are: (a) those that involve the conceptual systems that underlie thedozen-or-so "big" ideas in a given course, or (b) those that focus on important classes ofsituations that are usually described using those big ideas.

STUDENT THINKING

In solving The Paper Airplane Problem and The Volleyball Problem, students whoemerge as being exceptionally capable at developing powerful and mathematicallysignificant ways of thinking often are not students who have histories of performingexceptionally well on computation exercises. One reason is that the mathematicaldescriptions and interpretations that students develop generally involve very differentkinds of understandings and abilities from those that are involved in the relevantmathematical computations.

For example, when students first attempt the Paper Airplane Problem, difficulties mayoccur when they realize that not all of the given information is equally relevant, and thatsome may be irrelevant, or when they realize the need to merge information from severaldifferent trials , or from several different categories of information. But, the most difficultissues arise when attention shifts toward finding a way to judge which plane is the bestfloater.

Consider a student who begins by thinking of floating ability as speed x time-in-flight.For good floatability, the speed should be low but the time should be high . So, whatwould it mean to multiply speed and time? If speed is calculated using the formula Speed= Distance/Time and floatability is calculated using the formula Floatiness = Speed x

��

46 Lesh

Time, then the floatability formula reduces to a form that isn't sensible (Floatiness =

Distance). In this case, students usually need to go beyond thinking about nakednumbers (which tell "How much?" but not "Of what?"), to thinking about mathematicalrelationships among relevant quantities.

In the Volleyball Problem, students recognize that some information is more importantthan other information, and that: (a) high scores are good for height, but low scores aregood for running, and (b) adding heights (65 inches) and running times (6 seconds) isn'tlikely to make sense. To resolve these dilemmas, students' second-round ways ofthinking often abandon the approach of ranking campers using scores that aggregateinformation across all skill categories. Instead, they often rank campers within each ofthese categories, and then they develop some way to combine these rank scores, perhapsby adding, perhaps by using some type of weighted sums, or perhaps by using some sortof iterative procedure. Table 4 shows one way that the students might rank campers from1 (good) to 18 (bad) according to serving ability.

Name Number of Rank Name Number of RankSuccessful Score Successful Score

Serves Serves

Gena 8 11 Nikki 10 18Beth 7 5 Kim 7 5Jill 8 11 Robin 8 11

Amy 9 15 Edna 8 11Ana 10 18 Lori 9 15Kate 6 3 Tina 6 3

Rhoda 5 I Angie 8 IIChristi 8 11 Ruth 9 15Andrea 9 15 Becca 10 18

Table 4: Campers ranked by serving ability.

Next, such students might produce similar rank scores for running ability, jumpingability, and returning ability, then they might develop some way to produce a total score(i.e., a sort of index of volleyball playing ability) by aggregating information acrossseveral categories. Some might notice that, if teams are formed based on these totalscores, then it is very likely that some teams may have no servers, or no spikers, or nooutstanding defensive players (i.e., returners). Therefore, later solutions often shift awayfrom using only average scores that treat different categories of information as if theywere equivalent. Students' solutions shift toward using some sort of iterative process:

Step I: Deal out the best servers to each team.Step 2: Adjust the preceding list while dealing out the best spikers (i.e. tall campers who

also are good jumpers).Step 3: Adjust the preceding list while dealing out the best defensive players.

��


Step 4: Adjust the preceding list to make sure that servers, spikers, and defensive playersare evenly distributed across all teams.

Students' final solutions to the Volleyball Problem often go beyond the preceding kindsof considerations by dealing with trade-offs between (a) simple solutions that are quickand easy to use, and (b) complex procedures that consider more relevant information butare difficult and time-consuming to use. Final solutions also may deal with issues such asthe fact that, if raw scores are converted to rank scores, then similarities and differencesamong campers tend to be distorted in ways that could impact the equivalence of teamsthat are formed using rank scores. To see why this is true, examine the way the names ofcampers are distributed along the two vertical scales that are shown in Table 5.

18 Ana

l 17 Nikki16 Becca15 Amy14 Andrea

'iJ 13 Lori

'I/; 12 RuthII Gena

Ana, Nikki, Becca 10

~10 Jill

Amy, Andrea, Lori, Ruth 9 9 ChristiGena, Jill, Christi, Robin, Edna, 8 8 Robin

Angie s,Beth, Kim 7 7 EdnaKate, Tina 6 6 Angie

Rhoda 5

~5 Beth

4 4 Kim3 3 Kate2 2 TinaI I Rhoda0 0

Table 5. Vertical scales showing raw serving scores and serving rank.

In follow-up activities to the Volleyball Problem, teachers often use questioningtechniques to help students investigate similarities and differences between scales.Sometimes, they examine ways in which different kinds of scales can be added toproduce weighted sums (or weighted averages) that may result in significant differencesin the ranked lists of campers (Lesh & Harel, 2003).

REPRESENTATIONAL MEDIA

The ubiquitous availability of multi-media conceptual tools (such as graphingspreadsheets and graphing calculators) has led to enormous differences in the daily livesof ordinary people. To see evidence of this fact, look in the pages of a daily newspaper

��

48 Lesh

such as USA Today. From sports, to business, to editorials, to advertisements, many ofthe articles resemble computer displays with a large variety of charts, tables, and graphs.Furthermore, such tools have induced significant changes in the systems that use them.

For example, during the last decades of the 20th century, spreadsheets helped to changethe way grocery stores are run and the way automobiles are bought and sold. Look at theadvertisements for cars in your local newspaper. Thirty years ago, they told how much acar costs if you pay cash. Today, you see information about a confusing collection ofplans for leases, loans, cash returns, and delayed payments. In this technology-based era,the conceptual tools that humans create to make sense of their experiences are used tocreate changes in the worlds in which those experiences occur (Lesh, 200 1).

Today, useful mathematics is multi-media mathematics, and doing mathematics entailsflexibly moving between those media. Partly because of this fact, the diagram in Figure4 has been widely used from kindergarten through college-level mathematics as the basisfor research, curriculum development, assessment, and teacher education (Cramer, 2003;Johnson & Lesh, 2003). This research shows that: (a) meanings tend to be distributedacross a variety of representational media, (b) different media emphasize or de-emphasizeaspects of the constructs and conceptual systems they describe, (c) representationalfluency is part of what it means to understand a given system, and (d) problem solvingoften involves shifting back and forth among a variety of relevant representations.

Equations, tables, and graphs are the "big three" that are emphasized in innovativecurriculum materials such those from the Calculus Consortium at Harvard (Connally,Hughes-Hallett, & Gleason, 2000; Hughes-Hallett, Gleason, Lock, & Flath, 1999). Theother forms of representation have been emphasized in research and developmentactivities aimed at elementary and middle school mathematics (Lesh, Post, & Behr, 1987)and in the K-16 curriculum (Kaput, 1987). Recently, new technology-based layers havebeen developed for each of the media depicted in Figure 4 (Johnson & Lesh, 2003).Because different media emphasize or de-emphasize different aspects of a situation, it isimportant that students develop representational diversity and representational fluency.These are critical abilities for success outside of school.

POWERFUL CONCEPTUAL TOOLS

Some people think that when technical tools are used, the need for mathematical thinkingdecreases. In reality, the opposite tends to be the case. For example, when powerfultechnical tools are used to solve the kind of model-eliciting activities 1 have described,some aspects of tasks do become easier, but they are largely computational in nature.Other aspects of such tasks, those that are more conceptual than procedural, often becomemuch more difficult (Lehrer & Lesh, 2003). These include description, mathematization,communication, and interpretation. For instance, in the Volleyball Problem, somestudents have difficulty trying to figure out how to enter 5' 10" into a calculator, or tryingto make sense of graphs that are produced by their calculators or spreadsheets. Suchdifficulties usually represent far more than simple procedural problems. They ofteninvolve fundamental (mis)understandings about the meanings of basic quantities,relationships, and operations.

��


Figure 4. Meaning distributed across a variety of representational media.

In general, technical tools tend to be good at carrying out automated procedures such asconstruction or computation. Therefore, in the world outside of school, such tasks tend tobe turned over to low-level employees and technicians, enabling higher level employeesto focus on conceptualizing situations in forms so that available tools can be used;communicating and coordinating tasks that are addressed by technicians; and interpretingresults that technicians produce.

Increasingly, it is the case that employees need to produce dynamic tools (for example,spreadsheets and graphs) that may be applied to rapidly changing situations. In otherwords, the process is the product. For example, if someone wants us to develop tools thatthey can use to help them purchase a car, they do not want technicians simply to makedecisions for them about which car to buy and how to buy it. What they want are toolsthat will enable them to make well-informed decisions for themselves, taking intoaccount their own personal values, preferences, resources, and purposes.

In the preceding kinds of problem solving and decision making situations, thedevelopment of tools generally creates second-order effects. As soon as decision-makershave a new tool for making sense of a complex system, they usually start to use that toolto make the system even more complex.

STUDENT UNDERSTANDING

When people work in groups and do not have to do everything themselves, the need formathematical thinking does not decrease. The opposite happens. When realisticallycomplex problems are attacked by a team of diverse specialists, relatively low-levelcomputation is less important and relatively high-level abilities such as planning,monitoring, and communicating are paramount. Furthermore, when a community ofspecialists is making sense of a situation, final solutions may entail sorting outdistinctions that were not apparent from only a single perspective, rejecting ideas that

��

50 Lesh

prove to be less useful, and integrating ideas that are useful. This is true when studentswork in groups, even if they are working on topics that seldom occur in textbooks. Forexample, in transcripts of solutions to the Volleyball Problem, we can see that studentsare able to reach sensible conclusions about the following kinds of questions:(Zawojewski, Lesh & English, 2003):

• What assumptions are made if you try to add unlike quantities (such as heights andtimes)?

• What assumptions are made if you convert raw scores to ranked stores (or otherscales)?

• What is similar and different about sums and averages for comparing individuals orgroups? What different assumptions are made if you use sums versus averages-orother (perhaps iterative) procedures for combining information from differentcategories?

• How can you take into account the fact that you may not want to treat all kinds ofinformation as if they were equally important?

• How can you take into account the fact that a good team probably needs acombination of spikers, setters, and servers, rather than having all players of onetype.

When students' understanding of a mathematical construct increases, it develops indirections different from those that might be developed in mathematics courses ortextbooks. Some meanings depend as much on knowing why a given idea is notappropriate in some situations. Yet, the distinctions, connections, and assumptions whichoccur naturally in situations where students express, test, and revise their own ways ofthinking about big ideas are seldom emphasized when instructors try to guide studentsalong elegant paths to ideas they want to teach.

MONITORING AND ASSESSING

When the products that problem solvers produce involve complex artifacts (simulations,descriptions, explanations), they are usually the result of a series of develop-test-revisecycles. This is similar to what happens when we write descriptions of nearly anycomplex situation. The first draft is seldom adequate. Second, third, and fourth draftsusually are needed. So, the kind of mathematical abilities that tend to be emphasizedgenerally involve planning, monitoring, and assessing (Lesh, Lester & Hjalmarson, 2003;Zawojewski & Lesh, 2003).

Does the ability to plan, monitor, and assess presuppose the ability to produce? To someextend, the answer is yes. For example, a person who lacks any experience inprogramming is not likely to be good at planning, monitoring, and assessing the work ofa group of computer programmers. On the other hand, there are many types ofprogramming experience, and ten years' experience of a single type is generally quitedifferent from ten years of mixed experiences.

Similarly, by observing abilities that are needed for success in model-eliciting activities,it is obvious that problem solvers who are consistently successful go beyond thinking

��


with a model to thinking about their current model. They go beyond doing procedures toplanning, monitoring, and assessing those procedures. They go beyond adopting a singleproblem-solving persona to manipulating their personal roles and styles to suit changingcircumstances that occur from one solution stage to another and from one problemsolving situation to another. For example, sometimes it is wise to be self-critical andattend to details. But, at other times, brainstorming is needed, criticism is postponed, anddetails are largely ignored. The trick to success is to do the right thing at the right time(Middleton, Lesh & Heger, 2003; Zawojewski & Lesh, 2003).

CONCLUSION

How are learning activities in which students mathematize reality different from teachingactivities in which teachers realize mathematics? When teachers use guided questioningtechniques and try to teach mathematics so as to be useful, they seldom try to sort outwhy other plausible ideas are less useful for a given situation and purpose. They seldommake connections among networks of ideas that are drawn from different topic areas.They seldom emphasize the modeling abilities that are needed when complex artifacts areproduced using iterative design-test-revise cycles. Yet, these higher-orderunderstandings and modelling abilities tend to be precisely those that are needed forsuccess beyond school.

REFERENCES

Aliprantis C, Carmona, G (2003) 'Introduction to an economic problem: A models andmodeling perspective' in Lesh RA, Doerr HM (Eds) Beyond constructivism:Models and modeling perspectives on mathematics problem solving, learning, andteaching Mahwah, NJ: Lawrence Erlbaum Associates.

Connally E, Hughes-Hallett D, Gleason, A (2000) Functions modeling change: Apreparation for calculus New York: John Wiley Sons.

Cramer K (2003) 'Using a translation model for curriculum development and classroominstruction perspective' in Lesh RA, Doerr HM (Eds) Beyond constructivism:Models and modeling perspectives on mathematics problem solving, learning, andteaching Mahwah, NJ: Lawrence Erlbaum Associates.

Hughes-Hallett D, Gleason A, Lock P, Flath D (1999) Applied calculus New York: JohnWiley and Sons.

Johnson T, Lesh RA (2003) 'A models and modeling perspective on technology-basedrepresentational media' perspective' in Lesh RA, Doerr HM (Eds) Beyondconstructivism: Models and modeling perspectives on mathematics problemsolving, learning, and teaching Mahwah, NJ: Lawrence Erlbaum Associates.

Kaput J (1987) 'Representation systems and mathematics' in Janvier C (Ed) Problems ofrepresentation in the teaching and learning of mathematics Hillsdale NJ:Lawrence Erlbaum Associates, 19-26.

Kardos G (2003) 'The case for perspective' in Lesh RA, Doerr HM (Eds) Beyondconstructivism: Models and modeling perspectives on mathematics problemsolving, learning. and teaching Mahwah, NJ: Lawrence Erlbaum Associates.

Lehrer R Lesh RA (2003) 'Mathematical learning' in Reynolds W, Miller G (Eds)Comprehensive handbook ofpsychology Vol 7 New York: John Wiley.

��

52 Lesh

Lesh RA (2001) 'Beyond constructivism: A new paradigm for identifying mathematicalabilities that are most needed for success beyond school in a technology based ageof information' in Mitchelmore M (Ed) Technology in mathematics learning andteaching: Cognitive considerations: A special issue ofthe Mathematics EducationResearch Journal Melbourne: Australia Mathematics Education Research Group.

Lesh RA, Cramer K, Doerr HM, Post T, Zawojewski J (2003) 'Model developmentsequences' in Lesh RA, Doerr HM (Eds) Beyond constructivism: Models andmodeling perspectives on mathematics problem solving, learning. and teachingMahwah NJ: Lawrence Erlbaum Associates.

Lesh RA, Doerr HM (2003) Foundations of a models and modeling perspective in LeshRA, Doerr HM (Eds) Beyond constructivism: Models and modeling perspectiveson mathematics problem solving, learning. and teaching Mahwah NJ: LawrenceErlbaum Associates.

Lesh RA, Harel G (2003) . Problem solving modeling and local conceptual development'Monograph for the International Journal for Mathematical Thinking and LearningHillsdale NJ: Lawrence Erlbaum Associates.

Lesh RA, Lester F, Hjalmarson M (2003) 'Metacognitive functioning in mathematicalmodeling activites' in Lesh RA, Doerr HM (Eds) Beyond constructivism: Modelsand modeling perspectives on mathematics problem solving, learning, andteaching Mahwah NJ: Lawrence Erlbaum Associates.

Lesh RA, Post T, Behr M (1987) 'Representations and translations among representationsin mathematics learning and problem solving' in Janvier C (Ed) Problems ofrepresentation in teaching and learning mathematics Hillsdale NJ: LawrenceErlbaum Associates.

Middleton J, Lesh RA, Heger M (2003) 'Interest identity and social functioning: Centralfeatures of modeling activity 'in Lesh RA, Doerr HM (Eds) Beyondconstructivism: Models and modeling perspectives on mathematics problemsolving, learning, and teaching Mahwah NJ: Lawrence Erlbaum Associates.

Oakes W, Rud AG (2003) 'The EPICS model in engineering education: Perspectives onproblem-solving abilities needed for success beyond school' in Lesh RA, DoerrHM (Eds) Beyond constructivism: Models and modeling perspectives onmathematics problem solving, learning, and teaching Mahwah NJ: LawrenceErlbaum Associates.

Zawojewski J Lesh RA (2003) 'A models and modeling perspective on problemsolving' in Lesh RA, Doerr HM (Eds) Beyond constructivism: Models andmodeling perspectives on mathematics problem solving, learning. and teachingMahwah NJ: Lawrence Erlbaum Associates.

Zawojewski J, Lesh RA, English L (2003) 'A models and modeling perspective on therole of small group learning in Lesh RA, Doerr HM (Eds) Beyond constructivism:Models and modeling perspectives on mathematics problem solving, learning, andteaching Mahwah NJ: Lawrence Erlbaum Associates.

��

4

Environmental Problems and Mathematical Modelling

Akira YanagimotoTennoji Jr. & Sr. High School/Osaka Kyoiku University, [email protected]

Mathematical modelling has not been popular in school mathematics inJapan because it is difficult for students to understand and because modellingis not emphasized in the Japanese entrance examination system. However,mathematical modelling is becoming more important in this age ofinformation, and modelling with calculators and computers is necessary forsolving real problems. In addition, there is growing concern aboutenvironmental issues. In this paper I will introduce two examples that applymathematical modelling to environmental problems.

��

54

INTRODUCTION

Yanagimoto

Teaching mathematics in relation to the real world is an international issue. Worldwide,teachers are developing interesting ideas and projects based on real world problems forstudents in elementary, secondary, and tertiary education. In this paper, I discuss someproblems I have developed for junior and senior high school students.

In Japan, a new junior high school curriculum began in April 2002. The total number ofhours for major subjects was decreased. For example, mathematics lessons decreasedfrom 385 to 315 hours a year. This resulted in a five-day school week, an elective lessonsystem, and a new subject, Sougou. Sougou is an interdisciplinary learning class inwhich students might study such topics as the environment, welfare, information systems,international studies, investigation methods, and so on. Although it is difficult toincorporate mathematical modelling and applications into the shortened mathematicscurriculum, Sougou classes are an appropriate alternative. In this chapter, I discuss twomodelling lessons on environmental problems suitable for Sougou.

AVERAGE GLOBAL TEMPERATURE

I first used this unit of study with 9th grade students in Hyogo. My goals were to teachthe process of mathematical modelling and to increase student awareness of globalenvironmental problems.

Teaching ContentWe began with a picture of Bering glacier and we discussed the reason why it is melting.Some students knew about global warming. We broadened the discussion to includeother phenomena such as the rising of the sea level, unusual occurrences in nature,desertification of the world, and the spread of infectious diseases.

Next, we considered how to estimate the average global temperature in 100 years, anddecided to use the records of global temperature averages for the last 140 years. I gavemy students the yearly global temperature averages since 1860, adapted from theJapanese Ministry of Education (1997), as shown in Table I. The students graphed thedata points, as shown in Figure I, and they drew the straight line that best fit the datapoints. When we drew the line through the two points (0, 16.15), (100, 16.75), weobtained the linear function y = 0.006 x +16.15.

Using this formula, we calculated the global temperature averages for the future, and wegot 17.05 degrees for 2050, 17.35 degrees for 2100.

Finally, for comparison, I introduced data from the Intergovernmental Panel on ClimateChange (IPCC, 1995). IPCC estimated that the temperature will rise 3.5 degrees in 100years.

��

Modelling Environmental Problems 55

Years after TemperatureYear 1900 (x) °C(y)

1860 -40 15.97

1865 -35 16.17

1870 -30 16.16

1875 -25 16.39

1880 -20 16.28

1885 -15 16.13

1890 -10 16.09

1895 -5 16.36

1900 0 16.15

1905 5 16.0 I

1910 10 16.09

1915 15 16.00

1920 20 16.18

1925 25 16.11

Year Years afterTemperature

1900 (x ) °C(y)

1930 30 16.33

1935 35 16.35

1940 40 16.49

1945 45 16.69

1950 50 16.38

1955 55 16.33

1960 60 16.61

1965 65 16.27

1970 70 16.42

1975 75 16.31

1980 80 16.59

1985 85 16.53

1990 90 16.66

Table 1. Global temperature averages

, . , ~ • : ' : : ; : : : l.+-:X:/tIf).. .1II ·1O -10 Q 10 10 ID «l lCl jQ 10 III II 1'0

1••1 (.. Ittlll

Figure 1. Student's graph and regression line.

��

56 Yanagimoto

ExtensionWe can deduce another linear function based on the data from 1910-1990. Using thisdata, the temperature rises 0.86 degrees in 100 years. Using Excel, we get the graphs, theregression lines, and the formulae shown in Figure 2.

• V•.. V·.y.-./~~

V V16

y ('t;)

17

16.5

• ••• ----..~• ...,.-... - •

~~~ .•16

y ('t;)

17

16.5

15.5

a 50 10015.5

o 50 100x (years after 1900) x (years af ter 1900)

Regression line from the data 1860-1990y = 0.00375 x +16.2

Regression line from the data 1910-1990Y= 0.00597 x + 16.1

Figure 2. Regression lines using Excel.

According to the third report of the Intergovernmental Panel on Climate Change(Houghton, Meira Filho, Callender, Harris, Kattenberg, & Maskell, 1995), the estimatedrise in average global temperature is between l.0 degree and 3.5 degrees in 100 years(see Figure 3). In the forth IPCC report (Houghton, Ding, Griggs, Noguer, van derLinden, & Xiaosu, 2001), it is between 1.4 degrees and 5.6 degrees in 100 years.

"C5r----------------------,

4

3

2

200Q 2040 2080 2100year

Figure 3. IPCC's estimation in 1995.

��


Student ResponsesThis was the students' first experience in mathematical modelling. They were surprisedthat global temperature average could be predicted by a model as simple as a linearfunction. They were motivated by this lesson because they knew that it had relevance totheir future lives and to the world, and they became interested in other globalenvironmental problems.

BLUEGILLS IN LAKE BIWA

This is another mathematical modelling project based on a local environmental problem.I first taught this lesson to glh & 91h grade students in Tennoji J.H. School attached toOsaka Kyoiku University.

Teaching ContentRecently bluegills have been increasing in Lake Biwa. To decrease the number ofbluegills, the Shiga prefecture planned to catch a certain number of the fish every year.This situation provided another opportunity to teach mathematical modelling, this timeusing graphing calculators. We used the graphic calculator fx-9700GE, with which thestudents were already familiar from some previous work.

The real world problem was stated:According to the report from the fishery section of Shiga prefecture in1999, it is estimated that there are 1500-3000 thousand bluegills in LakeBiwa. Shiga prefecture set a 55 million yen budget to make serious effortsto tackle this problem. They plan to catch 300t every year and to halve thenumber of bluegills in ten years.

First we made explicit our assumptions:1. There are presently 1500t to 3000t bluegills in Lake Biwa.2. The breeding rate of bluegills is constant.3. 300t bluegills are captured every year.4. The breeding rate for bluegills is 15% per year5. Taking the average between 1500t and 3000t, we estimate the present number of

bluegills at 2250t.6. Other conditions are not considered.

Under these assumptions, we derived the recurrence formulaan+J = an + 0.15an- 300.

First, we let n range from 0 to 10 years. The result, shown in Figure 4 suggests that thenumber of bluegills will increase slightly. Next, we let n range from 0 to 100 years. Theresulting graph is shown in Figure 5.

Next we took 5% as the breeding rate of bluegills and input the recurrence formula,an+l=an+0.05an-300. The resulting graph is shown in Figure 6. In this case, the numberof bluegills begins to decrease immediately and the fish die off in eight years. Students

��

58 Yanagimoto

an+1=an+ •

,. ,.+" l~1'I+1

'1 i!&&§B i!'1&q.'1!i i!B'1!i.q

10 3D II. 3

an+1=an+ •~1'I+1

!i'1 1.!i3EB!iB i!.i!IEB!i!i i!.55EB

100 i!.!i3EB

1'1=5 ~=i!50i!.B3!ii!!i&B 1'I=!i0 ~='1i!5'100BI.301

Figure 4. Over 10 years. Figure 5. Over 100 years.

an+1=an+ • an+1=an+ .~1'I+1

l' ..

'+.

·l'1 '1i!3.3'1 '1 15Bi!.B

+ B q5!i.§q B Iqqq.'1!i IBi!.51 s Ii!!ii!. 5

10 -IOB.3 10 I Ii!q.B

...=5 ~=1i!13.!i1l1l11l0& ...=5 ~=IBi!I.'155i!15!i

Figure 6. Breeding rate 5%(for 10 years). Figure 7. Breeding rate 10.23% (for 10years).

tried various breeding rates. And at last they discovered that if the breeding rate were10.23%, the number of bluegills would become half in ten years (see Figure 7).

Students realized that the goals of the Shiga officials may not be met.S I: We need the accurate breeding rate of bluegills and the present number of them. Ourcalculations are not realistic if we don't have these data.S2: It is not clear whether it is possible to catch 300t.S3: I think that the breeding rate of bluegills is not constant. It will depend on someenvironmental conditions, so we must consider the change of breeding rates, living rates,and so on.S4: The breeding rates of living things are not predictable, so it is difficult to determinethe number of captured bluegills.

ExtensionFinally, my students and I considered some extensions. Keeping the breeding rate at10.23 %, we assumed that the present number of bluegills was 1500t and then that it was30001. The resulting graphs are shown in Figures 8 and 9, respectively. We concludedthat the validity of this modelling depends on the accuracy of the breeding rate.Additional explorations of this model might include changing the arbitrary catch numberof 300t per year.

��


an+1=an+ • an+1=an+ •." .. +." ".

l '1 !I!I.'1!1'1 l '1 30&5.!Ia -la!l.!I a 30'1!1.5

"+ !I -50!l.1I !I 30!l1l.&I D -a& I. 5 ID 3111.1

...=5 .a=&01.1!l1!l3511'1 ...=5 .a=301li!.3Iall!l&3

Figure 8. Present 1500t, (for 10 years) Figure 9. Present 3000t, (for 10 years)

Student ReactionsStudents realized that mathematics is useful for dealing with environmental problems andthat a graphic calculator is a powerful tool for doing mathematics. These students hadnot yet learned recursion, because it is taught in high school in Japan. However, theyunderstood the concept when using graphic calculators in this lesson.

SI: It is wonderful to estimate things in future. But it is difficult to implement the planin practice, because living things differ from the calculation.S2: There were some calculations that were made easier by using graphic calculators.Modelling is like a game in a sense. I think it's a good way to get the students interestedin mathematics.S3: As it was an interesting topic for me, I was able to enjoy this lesson. And I becameaccustomed to using graphic calculators. I had the experience of fishing for bluegills inLake Biwa. There were actually so many bluegills.S4: We need mathematics for environmental problems like this, so mathematics is veryimportant. 1 realized it more through this lesson. This lesson was a very good experienceforme.

CONCLUSION

I have discussed two lessons in which students learn about environmental problems andthe process of mathematical modelling, while gaining experience with spreadsheets andgraphing calculators. These lessons demonstrate that it is possible and profitable to teachdata analysis such as linear regression and recurrence formulae using environmentalproblems in Sougou.

Students became deeply interested in these environmental problems. They weresurprised to find out that it was possible to analyze environmental problems scientifically.Needless to say, students enjoyed predicting the future, but they weren't satisfied with themodels and realized that more complicated models were necessary. Traditionally,modelling and application problems have been lacking in the Japanese mathematicscurriculum and they are needed in order to encourage the kind of enthusiasm thesestudents showed, as well as to prepare students for more challenging mathematicalmodelling projects in the upper grades.

��

60

REFERENCES

Yanagimoto

Houghton JT, Ding Y, Griggs DJ, Noguer M, van der Linden PJ, Xiaosu D (Eds) (2001)Climate Change 2001: The Scientific Basis New York: Cambridge UniversityPress.

Houghton JT, Meira Fi1ho LG, Callender BA, Harris N, Kattenberg A, Maskell K (Eds)(1995) Climate change 1995: The science ofclimate change New York: CambridgeUniversity Press.

Ministry of Education (1997) Zudemiru Kankyou Hakusyo (in Japanese).

��

5

Three Interacting Dimensions in the Development of Mathematical Knowledge

Guadalupe Carmona Purdue University, USA [email protected]

I present a framework that relates three critical dimensions of mathematical knowledge building: the individual characteristics of the learner, the social aspects of learning, and the role of context. Next, 1 introduce a modeling activity for middle school students. Finally, I analyze a discussion that occurred while one group of students was testing and revising their model to demonstrate that the interaction of the three dimensions was fundamental to the development of the students’ mathematical knowledge.

��

62 Carmona

INTRODUCTION

In the last few years there have been many advances in our understanding of how mathematical knowledge develops. Using what we know now, we can design activities for the classroom that elicit and document students’ mathematical thinking and the models-internal and external-that students bring to bear on a problematic situation. These are known as model-eliciting activities. In this chapter, I present a framework that draws upon the work of several major authors and links critical dimensions of their mathematical knowledge-building theories. This framework is useful for both developing and analyzing model-eliciting activities. Finally, 1 will give an example of an activity that was designed to elicit middle school students’ models of quadratic relations between variables, and I will present a partial transcript from a group of students working on this task along with an analysis that shows the interaction of individual, social and contextual dimensions of the activity.

ON COGNITION: WHAT DO WE KNOW ABOUT HOW STUDENTS LEARN MATHEMATICS?

Several theoretical perspectives have contributed to what we know about learning mathematics. I will present theory from several major authors and integrate their work into a framework consisting of three interacting dimensions that are critical to the development of mathematical knowledge: the individual characteristics of the learner, the social genesis of learning, and the role of context. Figure 1 shows the three dimensions of knowledge development: individual, social, and contextual. As indicated by the use of dotted lines, the boundaries between the three dimensions are not impermeable.

+-.

--. --.._ _~_.-----_-___~ __.- ,_.-

,_/-

emr-1 DJ.wnsJ#n

--. .-. -.- -_ ._.---. - -... ---- _I-

,,1*1-- Soelat ,.....< DJlMhfShVR

._--- __.___..-. --- --..~. -_ --. -._

Figure 1. Three dimensions that interact in the development of mathematical knowledge.

��

Three Interacting Dimensions 63

Rather, these dimensions simultaneously interact and dynamically change as mathematical knowledge develops. This is a relatively new perspective on knowledge building, since many of the influential learning theories of the past tended to treat the three dimensions separately. For the individual dimension, 1 will discuss Piaget’s genetic epistemology, and his view on the relation between the learner and the object. For the social dimension, I will describe the work of Vygotsky, focusing on the role of language as a mediating tool in learning. Finally, for the contextual dimension, I will highlight the views of Greeno and others who focus on the environment as a complex interacting system, where ideas are connected within a context, and where learning occurs with a specific purpose.

The Individual Dimension in the Development of Mathematical Knowledge Piaget’s genetic epistemology is, among other things, a response to prior theories that tried to explain the acquisition of knowledge as a static relation between two separated entities, subject and object; where the acquisition of knowledge meant the mirroring of reality. For Piaget, knowledge of the world was not a mirroring of reality in the individual’s mind; but rather, an interpretation or a reconstruction.

Sooner or later reality comes to be seen as consisting of a system of transformations beneath the appearance of things. [...I In order to know objects it is necessary to act on them, to break them down and to reconstruct them. (Piaget & Inhelder, 197 1, pp. xiii).

Piaget (Piaget, 1970; Piaget & Inhelder, 197 1; Piaget, 1976) considered the subject-object interaction as an undissociable dialectic unit. Both the subject and the object of knowledge are transformed as a result of their interaction. Every time the subject approaches the object, it will be a new object of knowledge, and a new epistemic subject that will assimilate knowledge through (new) cognitive structures. These cognitive structures are frameworks from which the individual is able to interpret perceptions. They are developed and modified through experience, starting when the individual is born.

Sometimes when the individual approaches an object or a new situation, there may be some features that don’t seem to correspond to the individual’s expectations or predictions, or assumptions. Piaget referred to this mismatch between the individual’s cognitive structure and the new object of knowledge as a state of disequilibrium. It is precisely when disequilibrium occurs that knowledge is constructed, because there is a strong need to modify the current cognitive structure in order to incorporate and understand the new object or the new situation.

The Social Dimension in the Development of Mathematical Knowledge Vygotskian psychology (Vygotsky, 1978, 1986) is currently influential in the field of mathematics education, especially because of its emphasis on the role of language in learning. Vygotsky (1978) was concerned not only with individual capabilities, but the processes by which individuals develop those capabilities. He distinguished two levels of development: current and potential. Current development is indicated by the individual’s

��

64 Carmona

ability to solve a problem alone, and potential development is indicated by the same person’s ability to solve a problem in collaboration with a more capable peer or with an expert. Vygotsky (1 978) defined the zone ofproximal development as

the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers (p.86).

According to Vygotsky, higher mental processes have their origin in social interaction. Symbols, such as spoken language, writing, numeric systems, and other media, reflect social and cultural structures in the environment. He claimed that the interaction between an individual and the world is mediated by these symbols in the development of higher cognitive processes. That is, the symbol is not only a sign, but it holds a meaning that is embedded in a social context. Thus, for Vygotsky and others (Werstch, 1994), language is a shared tool that allows for an inter- and intra-personal mediation in the resolution of problems.

The Contextual Dimension in the Development of Mathematical Knowledge Theorists who ascribe to the notion of situated cognition deny that mathematical thinking can take place apart from the concepts and methods of mathematics and assert that knowledge building is linked to a number of conditions in the environment. That is, the knowledge one brings to bear on a situation is part of an interacting system of elements, that includes individuals, relationships with others, tools, representational media, limiting conditions, goals, and other influences. Greeno (1997) expressed it in these terms: “Situativity focuses primarily at the level of interactive systems that include individuals as participants, interacting with each other and with material and representational systems” (p.7). The situated cognition perspective has implications for learning in schools. From a macro-perspective, it suggests that materials, classroom organization, methods of instruction, and the availability of tools and technology are all part of the context that affects knowledge building. On a smaller scale, the constraints of the given situation, the formulation of the problem, the goal it asks students to achieve, how much information is provided, and a host of other details affect knowledge-building as well. Unlike traditional textbook exercises, when the problem is embedded in a context, students are provided with a goal that is not ostensibly school-related and with a means to make judgments about their progress in reaching that goal.

MODEL DEVELOPMENT SEQUENCES

Model development sequences (Lesh, Cramer, Doerr, Post, & Zawojewski, 2003; Lesh, Hoover, Hole, Kelly, & Post, 2000) are classroom materials that include a series of interconnected activities designed to promote student learning through the use of models and modeling. For the opening activity, students read a newspaper article that helps to familiarize them with the real world context for the problem. The core activity, the model-eliciting activity, has three main goals: (1) that students should develop a model to describe a real-life situation, (2) that students should express, test, and revise their ideas; and (3) that students should use various representational media to explain (and document)

��


their thinking. Students are asked to develop a model for some client who has sought their advice about how to deal with the problem. The final activity asks students to extend, to generalize, or to use in another context the mathematical ideas included in the model-eliciting activity.

Through the design and implementation of model-eliciting activities, the three dimensions: individual, social, and contextual are integrated in practice within a classroom setting. From the social dimension, students are required to solve the activity in teams of 3 4 persons. This interaction among peers creates favorable conditions for students with different skills to help each other develop more powerful mathematical understandings (Zawojewski & Carmona, 2001a, 2001b). In addition, students are required to externalize their thinking using different media in order to communicate with their peers, which, in turn, helps them to select the most appropriate media to clearly present their solution to the client.

Historic Hotels: An Example of a Model-Eliciting Activity An example of a model-eliciting activity intended for middle school students is given in Figure 2 (Aliprantis & Carmona, 2003). The activity called Historic Hotels is based on an economic problem (Aliprantis, 1 999) typically used in undergraduate calculus courses. It deals with economic concepts, such as profit, cost, price, maximization, and equilibrium; and mathematical concepts such as recognition of variables, relation between variables (linear, quadratic relations, direct and inverse variation), product of linear relations, and maximization.

The real-life context of this activity is given in a newspaper article that describes a historic hotel in Indiana. After reading the newspaper article, students are asked to reflect upon the situation by answering some questions.

What do hotels have to accomplish on order to be recommended by the National Trust Historic Hotels of America? How many owners has the French Lick Springs Resort had since it opened? What are the responsibilities of a hotel manager? Why was the Depression a bad time for the French Lick Resort? What are the main features of the French Lick Resort today? What sources of income might a resort have?

.

. . . . . The problem statement explains that Mr. Frank Graham, from Elkhart, Indiana, has just inherited a historic hotel. He would like to keep it, but he has little experience in hotel management. The whole community of Elkhart is willing to help him out because this historic hotel represents a major attraction for visitors, and thus, sources of income for everyone in the town. As part of the community, Elkhart Middle School has been assigned to help determine how much should be charged for each of the 80 rooms in the hotel in order to maximize Mr. Graham’s profit. They know from previous experience that all rooms are occupied when the daily rate is $60 per room. Each occupied room has a $4 cost for service and maintenance per day. They also know that for every dollar increase in the daily $60 rate, there is a vacant room. Mr. Graham would like to know how much he should charge per room in order to maximize his profit, and what his profit would be. Also, he would like to have a tool that would give him this information even if

��

66 Carmona

hotel expenses rise in the future. Students are asked to write a letter to Mr. Graham explaining how he can calculate his profit and how his profit can be maximized. They are asked to be sure that their method will work even if the hotel expenses rise in the future.

Historic Hotels: An Enchanting Vacstlon

into an enchantment. Finding one of these chmming places is P task to which the Nahial f m t Historic Hotels of America is committed.

To k rrcommatdcd by tha Historic Hotels of &nerica, hotels have to prove thal €hey faithfully maintained their historic architchoc and ambience. These hotels hold pride for their simies, myths9 and legends. For atample, the Frm~h Lick Spnitgs Rcwrt m Wan8 is named after an early F d outposa in the MI end nearby sllt lick (a lick is a deposit of exposed MW sait rhat is licked by passing Saimab). This natural lick flourished in the mid-niwieenlh century after Dr. Willim A. B O W ~ B built the nrst French Lick Spingr Hotelq

I Hrlahcr for health reasons, or just for curiosity, visiton were compelled to w i t the rich minerat springs, which w m said to ~ o ~ a c 8 9 curattvc powtrs. Thia omaetion tad Dr.

Waters ail

In 1897 the kotef burned down. It was not untit 1902 that the new French Lick Springs Resort was built on the ruins, under s new owner, Ihomas Taggart. MI. T a m , mayor of Indianapolis. madc Ihe molt gmw in sir and reputation in the early dmxks ofthe 2 0 ~ century. surrounded by lush gmIeas and Imhcnping, the six-story hotel, with its qmwling sitting versnde was a mare than relaxing environment mnny w i W to enjoy. Among the most interesting celebritia buu fnqum)ed the wort wen John Barfymon, Clark Cable, Bing ctosby, The Ttumans, fbc Rcagms, A1 Csponc and Rcsident Franklin Roowelt. In fact, Roowelt even locked up the Demoenlie nomination for president in the hotel's Grand colonnade Ballroom.

Maintenance for a hotel like The French Lick Springs Resort, with all its services. i s not an m y task. In 1929, Mr. T a m died. His only son of six children, Thorn- D. T a m , inheriw it. With the Depression, however, the popular French Lick Springs began to dsclim when the cost of the room bad to be considarbly lower, laving little money for maintenance. E v a che exquisite notaurant lost ?notit of its patrons. World War II brought a monetny mi&, but in 1946 young Tom Taggart Sold out to a New Yo& syndicate.

Today, French Lick springs R c s o ~ i mtt on some 2,600 acres in the breathtaking Hoosier National Forest. Newly q B i d by Sayirin Lodgjng Company, the resort cseffly embnm a "New Bqinning". i t provides 470 rooms, Futl smticc spa, two golf mm, in-housc bowling, alley md arc*, indoor m n i s center rurd ourdoor courta, swimming, crorinct, horseback riding; children's activities; skiing. boating, md Ashlng nearby. Fine and CllSulJ dining is also aMlibbk at a variety of nstatnants. Two mein meeting moms, Grand Colonde Ballroom and Exhibit Center, accommodate large-scde events.

Btsider The French Lick Springs R w r l , the National Trust H W c Hotels of A d c a hns identifd over 140 quality hotels located m 40 states, Cuudr, Uui puato Rico. To find mom information on hiJtaic holds in your date, you only need to go their wtbsite: h _ ~ ~ ~ i u t o r i c b o r c ~ l ~ ~ t . o r ~

Figure 2. A middle school modeling activity.

��


An Analysis of Individual, Social, and Contextual Dimensions Students’ solutions to activities like Historic Hotels make it clear that individual, social and contextual dimensions interact as students construct mathematical understandings. 1 will analyze an excerpt of a transcript of the discussion of a group of students working on this activity to show the interaction of individual, social, and contextual dimensions.

This activity was implemented in a low-income mid-western middle school. The math teacher and 1 worked with 5 mathematics classes of 7Ih grade students, totaling more than 100 students. Students were divided into teams of 3, and were given one class period of 50 minutes to work on the activity. The excerpt given in Table 1 below took just a little over 3 minutes. It occurred at a point when students were going through one of many cycles in which they expressed, tested, and revised their model. M and K are girls and G is a boy.

In this activity, students must struggle with the idea of inverse variation. That is, when the daily rate increases by $1, the number of rooms decreases by one. In this excerpt, one of the students understands inverse variation, but the other two students do not, and the three of them are working to achieve a common understanding.

(To the girls). For every dollar he raises in the daily thing, it goes up 19 dollars, but every time the room number goes down, it goes down 4 dollars.

(The girls listen without making any comments, and the three students work quietly for about 2 minutes, looking down, and working with the calculators. K and M look constantly at each other’s paper, but not at G’s). I have a headache! (G and K look at M, without making any comments, and they continue making more calculations. )

Comments: G, as an individual, has understood the inverse relation between the cost of the room per day, the number of occupied rooms, and the calculation of the profit, including the maintenance fee of $4 per occupied room. To understand where the 19 dollars came from, he first calculated 80 rooms x $60 = $4800. He increased the daily rate to $61, decreased the number of occupied rooms to 79, and multiplied the two numbers to get $4819. Thus the difference in “the thing” is $19. G expresses his thinking to the others in the group (social). The girls listen to what he says, but don’t quite grasp his meaning, and thus, keep making calculations for about 2 minutes. There is no communication between G and K and M (social).

��

68 Carmona

(Looking over at G’s paper) How did you get $48 19?

1 just took what was different in the money, so 1 multiplied the number of rooms by $6 1. So you get $4880.

(While computing with calculator, and addressing G). 60 times 80 is 4800, so if you raise it 1 dollar you get 4880. (Takes the calculator and computes). You did 61 times 80? (K and M nod their heads.)

[For] Every dollar [that] goes up, there is one room less, so you have to multiply by 79.

Oh YEAH!!!!!

M looks over at G’s paper, and sees different calculations than the ones she has, and expresses her doubts to G (social). G explains (social) his procedure (individual) to M and G (social).

K gives her solution (individual), engages in G and M’s conversation (social), and is puzzled by the discrepancies between their calculations. With her statement, she expresses her and M’s shared individual understanding of the problem up to this point, and expresses it to the group (social) to trv to find the discrenancies. M repeats her (and K’s) calculations to try to find the discrepancies in the procedures (social).

G takes M’s explanation, and tries to follow it using the calculator (individual). He questions them to make sure he understands their procedure (social). G understands K and M’s procedure (individual), and is able to give- a better explanation to help them understand the inverse relation statement in the problem (social and contextual). K and M now understand G’s statement, and the three students, individually and as a group, have a common understanding of the inverse relation statement in the problem (individual, social, and contextual).

Table 1. Middle school students revising their models.

For these students, the situation was meaningful. They understood the context surrounding the problem, including the responsibilities of a hotel manager and his need to maximize his profit. From the individual standpoint, though, the students had different interpretations of the problem statement that dealt with the inverse variation between the price per room and the number of occupied rooms. Through the social interactions, the three students were able to achieve a common understanding that would enable them to move toward the problem’s solution.

FINAL REMARKS

In this chapter 1 presented a framework that integrates three dimensions of knowledge development-the individual, the social, and the contextual-that have been identified by

��


different leaning theorists as important components in the development of mathematical ideas. 1 then introduced an example of a modeling activity, Historic Hotels, and I selected a brief excerpt of student discussion to illustrate the interaction of the three dimensions as students test and revise models. I have implemented many model-eliciting activities in different classroom settings for the past four years, and I believe that the integration of the individual, the social, and the contextual dimensions is fundamental in the learning of mathematics. It is important to continue to develop and implement curricular activities, like model-eliciting activities, that integrate the three dimensions in practice.

REFERENCES

Aliprantis C (1999) Games and decision making Cambridge: Oxford University Press. Aliprantis C , Carmona G (2003) ‘Introduction to an economic problem: A ‘models and

modeling perspective’ in Lesh RAA, Doerr H (Eds) Beyond constructivism: Models and modeling perspectives on mathematics teaching, learning and problem solving Mahwah, NJ: Lawrence Erlbaum.

Greeno J (1997) ‘On claims that answer the wrong questions’ Educational Researcher

Lesh RA, Cramer K, Doerr H, Post T, Zawojewski J (2003) ‘Model Development Sequences’ in Lesh RA, Doerr H (Eds) Beyond constructivism: Models and modeling perspectives on mathematics teaching, learning and problem solving Mahwah, NJ: Lawrence Erlbaum.

Lesh RA, Hoover M, Hole B, Kelly A, Post T (2000) ‘Principles for developing thought- revealing activities for students and teachers’ in Kelly AE, Lesh RAA (Eds) Handbook of Research Design in Mathematics and Science Education Mahwah, NJ: Lawrence Erlbaum.

26(1) 5-17.

Piaget J (1970) Genetic Epistemology New York: Columbia University Press. Piaget J (1976) Six Psychological Studies Santa Monica, CA: Random House Inc. Piaget J, Inhelder B (1971) Mental Imagely in the Child: A study of the development of

Vygotsky L ( 1978) Mind in society: The development of higher psychological processes

Vygotsky L (1986) Thought and Language Cambridge: MIT Press. Wertsch, J. (1994). ‘The primacy of mediated action in sociocultural studies’ Mind,

Czrlture, and Activity, 1, 202-208. Zawojewski J, Carmona G (2001a) ‘Social and Developmental Perspectives on Problem

Solving’ in Vale C, Horwood J, Roumeliotis J (Eds) Proceedings of the MAV’s 3gh Annual conference: 2001 A Mathematical O&ssey Mathematics Association of Victoria Australia.

Zawojewski J, Carmona G (2001b) ‘A Developmental and Social Perspective on Problem Solving Strategies’ in Speiser R, Walter C (Eds) Proceedings ofthe Twenty Third Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education Columbus OH: ERIC Clearinghouse for Science Mathematics and Environmental Education.

imaginal representation New York: Basic Books.

Cambridge: Harvard University Press.

��

6

Working and Learning in the Real WorId: AMathematics Education Project in Baden-Wuerttemberg

Hans-Wolfgang HennUniversity of Dortmund, [email protected]

Understanding the relationship between mathematics and the real world isan integral part of general education. The less-than-satisfactory GermanTlMSS results and the new PISA results show that our mathematicseducation does not adequately emphasize this relationship. In this chapter, Ireport some early results of a four-year program in Baden-Wuerttemberg forthe lower secondary level, designed to improve children's understanding ofmathematics in the real world.

��

72

TlMSS AND THE BLK PROJECT

Henn

Germany's disappointing results in the Third International Mathematics and ScienceStudy (TIMSS) (Blum & Neubrand, 1998), and the Program for International StudentAssessment (PISA) (Neubrand, 2001) were the catalyst for a nation-wide debate abouteducational goals and the content of mathematics teaching. In response to this debate, theBund-Laender-Kommission (BLK) initiated a program designed to "further the efficiencyof mathematics and science teaching." This four-year program for the lower secondarylevel (corresponding to grades 5-8 in the US) was launched in the 1998-99 school year.The Federal States of Germany participate in this BLK program, with each state decidingon the nature of its own participation (Blum & Neubrand, 1998; Henn, 1999).

BASIC EXPERIENCES IN MATHEMATICS EDUCATION

The project Promoting Classroom Culture in Mathematics is the contribution of BadenWuerttemberg to the BLK program (Henn, 2002). The Baden-Wuerttemberg project provides experiences related to three fundamental principles that, according to HeinrichWinter (Winter, 1996), can help mathematics education contribute to general education(Borneleit, Danckwerts, Henn, & Weigand, 2001):

• To recognize and to understand phenomena in the world around us. (Thisprinciple recognizes the role of mathematics in acquiring important knowledgeof our world.)

• To learn about and to understand mathematical issues represented in language,symbols, pictures, and formulas as intellectual creations, to recognizemathematics as a deductively ordered world of its own kind. (This principlerecognizes mathematics as a rigorous science).

• To acquire problem-solving (heuristic) skills for tasks that extend beyond thedomain of mathematics. (This principle recognizes mathematics as a school ofthought.)

The three basic experiences are closely related. Deep insight into pure mathematicsresults from application-oriented problems. For example the roots of arithmetic, elementary algebra, and geometry lie in astronomical, surveying, and (regrettably) in militaryissues. Conversely, abstract results and methods of pure mathematics prove to be keysfor understanding our world. Examples include the complex numbers and Riemanngeometry. Some time after their discovery by Gauss, the complex numbers turned out tobe useful for electronic engineering, for example in our TV sets. Riemann geometry became important for the general relativity theory some 60 years after it was discovered.Creative research both in pure and applied mathematics is unthinkable without heuristiccompetence. The necessary competence can only be built through first-hand experience,when students themselves engage in modelling our world and reflect upon those experiences on a meta level.

��

Working and Learning in the Real World

WHAT HAS TO CHANGE?

73

The central question is not "What is to be learned?" but "How should learning takeplace?" "How can mathematical literacy be promoted?" "How can learning processes bemeasured?" Change is accompanied by a willingness to question and to rethink currentteaching, to change one's own perception and to recognize opportunities brought aboutby new practice and teaching techniques.

The issue is what students should actually know by the end of the lower secondary level,when "knowing" is not primarily related to knowledge but rather to basic concepts,abilities, competencies, and attitudes indispensable to functioning in an increasinglycomplex world. We aim at a reasonable transfer of emphasis, balancing betweeninstruction (by the teacher) and construction (by the students themselves), betweenteaching and discovery, between convergent, routine problems and divergent openproblems, reflecting all three basic experiences.

Two guidelines have emerged, which are not necessarily self-evident in teaching at uppersecondary level: to take children and their products seriously on the one hand and toconstruct productive learning environments on the other.

TAKING CHILDREN SERIOUSLY

In the book How Children Compute (Seiter & Spiegel, 1997), the authors contend thatteachers should think and argue with children, listen to them, take their productsseriously, not ignore their mistakes, but rather discuss them productively. We oftenmistakenly assume that children think in the same way that we do. However, Seiter andSpiegel clearly point out that children think in different ways than adults andmathematicians do, in different ways than other children do, and sometimes they eventhink differently than they just did a moment ago within the same problem.

Consequently, it is not enough to question children, but rather to listen carefully to theirquestions, to take them seriously, and to discuss and try to understand them. In this way,teachers are better able to understand their students' cognitive structures.

PRODUCTIVE PRACTICE

Wittmann describes the didactics of mathematics as a design science that simultaneouslydevelops and studies "productive learning environments". In their Handbooks ofProductive Arithmetic Practise, Wittmann and Mueller (1990,1992) provide richproblems for the four elementary school years.

They include a learning section that creates meaningful relations followed by a series ofproblems with different degrees of difficulty. In contrast to traditional step-by-stepinstruction, not all obstacles are removed. Pupils gain experience in using common senseand are challenged to think about the problems, to make their own judgments, and to testwhether their ideas make sense.

��

74

EXAMPLES FROM OUR PROJECT

Henn

The following examples illustrate our approach.

Example I: Illustrate the Sequence of Natural Numbers.

This problem was given to students at the beginning of grade 5 before the introduction ofthe number line. After a rather hesitant start, students were able to develop theillustrations shown in Figure 1. Later, the teacher introduced the number line as a furtherpossib ility.

<' <. -'1 <'; 1 ~ 1L if ... • .• ~ ..

.a /

'ALFigure I. Sequence of the natural numbers.

Example 2: How Many Cars in the Traffic Jam?

Children (grade 6) were shown a photo of a traffic jam on a motorway and were asked,"How many cars are in the traffic jam?" Because there was no further information given,some assumptions were needed. Which motorway could this be? How long was thetraffic jam? Are there two or three lanes? Was it a Sunday, a day without trucks? Whatis the average length of a car? What is the distance between two cars?

Example 3: Did You Know?

The cartoon in Figure 2 came from a daily newspaper and was used to introduce decimalnumbers in a grade 6 class.

Old you know,....thatduringrushhour-name, an averageof only1.2 people are sitting in one car?

Figure 2. Understanding decimals and averages.

��

Working and Learning in the Real World 75

Some questions discussed were:• What does this mean?• There was a large variety of answers, for example, that in a sample of 100 cars

there where 120 passengers.• What was the distribution of passengers if exactly 10 cars were sampled?

Listing all possible answers forced the students to use a systematic approach.Variants with other than 10 cars were also investigated.

• 40 cars were sampled and 39 of them carried only one passenger. Students wereasked to construct logical arguments to prove this situation impossible.

Example 4: Sportsclub Goalgetters

A traditional problem for equations in two unknowns is the following:

The sportsclub Goalgetters has 200 adolescents and 150 adults as members.The monthly fee is $5 for adolescents and $7 for adults. For the followingyear an extra $1600 are needed for renovating the gymnasium. Thereforethe fee is to be increased for the coming year in a way that adults pay $1more than adolescents. Calculate the new monthly fee.

Following a proposal by Arnold Kirsch (Kirsch, 1962), this restricted problem statementthat leads to a unique solution was generalized by substituting the question, "How shouldthe new fees be determined?" There are many possible solutions, and students were todecide upon a new fee structure and to justify their normative modelling. For example,there could be no increase for the adolescents, the increase could be divided equallyamong all members, or the new fees could have the same ratio to each other as the oldones. There are many possibilities. The modified question avoided the misleadingimpression that there is only one correct approach. The students had to talk aboutmathematics, to defend their divergent approaches, and to discuss the merits of eachsolution.

ASSESSMENT: MEASURING GAIN IN COMPETENCE

It is important to differentiate between learning and assessment situations.Understandably so, pupils try to avoid failure in assessment situations. Nevertheless,problems that are open and that require decision-making have to be included in tests. Inour experience, pupils did not see open problems as something new because, in the firstplace, they were accustomed to such problems from their lessons and secondly, thequestions we included in assessments were carefully chosen. In particular, we kept inmind that we cannot force creativity in relatively little time and under stress. Thefollowing assessment problems from our project classes show what is possible.

Example 5. Buying School Supplies (Grade 5)

Armin buys a notebook and two pencils from Beate. The notebook costs$2, and each pencil costs $1. Beate asks for $6 and Armin protests. Beatewrites down an expression and explains her calculation to Armin. Then

��

76 Henn

Armin understands. He tells Beate that the expression is correct but thatshe has violated an important rule in her calculation. Armin pays thecorrect price. Which expression did Beate write down and what rule hasshe used incorrectly?

Of course, she did write the correct expression, $2+$1·2, but violated the rule "domultiplication before addition" $2+$1 = $3 and $3·2 = $6.

Example 6. Problem-Posing (Grade 5)

Write down a problem that includes 2 kg 500 g and 10 days. Then solveyour problem.

Contrary to our expectations, children do not always solve their self-posed problemscorrectly. However, self-posed problems are typically worked with more interest andcommitment. Often teachers can recognize and follow their students thinking. Figure 3shows the problem posed by one of our students and translated into English .

..•..- ~·iS~c:ciJ-~~_.?:.-"1I-l; ~_.~==~~_ ..... -·-.:l";~..·~_~o=-~~A-O.d·~.Jr1.~·~ -.

(After "R" the calculation is given. "A" represents the answer: The ape eats 25 kgbananas in 10 days.)

Figure 3. The monkey in Frankfurt Zoo.

The student's work shows an interesting detail. The student first wrote See/owe (sea lion)and then reconsidered. A sea lion is much too heavy and does not eat bananas, so the sealion became an ape!

Other problems, translated into English, show how highly imaginative the children were:

��


• Laura wants to lose 2 kg 500 g in 10 days. How much weight will she lose inone day?

• A certain monkey in Frankfurt Zoo eats 2 kg 500 g of bananas each day. Howmuch bananas does it eat in 10 days?

• Tim, a lO-month-old baby, weighed 2 kg 500 g 10 days ago. Now he weighs 4kg. How much weight did he gain?

• A baby weighs 2 kg 500 g. What will his weight be in 10 days? (Here, theresult given was 25 kg!)

• Katrin's pony weighs 2 kg 500 g. The pony eats I kg 500 g in one day. Howmuch will it weigh when it eats I kg 500 g for 10 days?

Discussion of the answers is another important preparation for mathematical modelling.The following problem was given in a test on percentages.

Example 7. The Advertising Slogan (Grade 7)

A producer of detergents advertises a product with the following slogan: "Our productwashes 150% whiter." Discuss this statement!

One particularly interesting answer is given in Figure 4 and translated into English.

~r \, - U. ~I..t. .J.. .lQ 1 & l~b.~ '1 Q( ~ r. •t1M ~ 1','1

--..I

t1 II •~ ~ ........ , ~ lo l Id- l- (~

.-: t"1(\" )~~~ r, I~, """I Ill'M-\ hi. f' I ('I I'D 1..L.t: 11"Y'\:l ~ I.' ro 1""-1

~

~)nw " :\\

.U .1 "-It' ,,,. <,l~ .Ill J''h ICJ lPl " oJ,...~ \ro ~ (, .til J\

~ "

(This slogan is strange because colored wash will turn out completely white andsecondly, it is enough if it is 100% white, then it is already totally white.)

Figure 4. A student reaction to the advertising slogan.

Other students observed:• There is no logic behind the slogan because it is 150% from what? But the

customers are impressed because it sounds so much. But actually it does notclean better because 150% of nothing is nothing.

• The slogan is misleading because one does not know how white the washingturned out before or how white other products make the wash.

• The slogan is senseless because the basic value is not known.

��

78 Henn

EXPERIENCES FROM THE PROJECT

There are initially some difficulties when more open problems are posed to the studnets.Pupils want a rule to follow, and without one, they are unsure about how to proceed.They are often afraid of doing something wrong and so they do not start at all. But aftersome time their attitude changes. "During the school year insecurity grew less, pupilssought more and more problem solutions on their own and accepted that there is morethan one way to reach the solution," reported their teachers. Students grew increasinglycomfortable with more complex, more open problem statements. As one of the teachersexplained,

The wide range of solutions was only possible, when I myself as a teacherretreated in the decisive moment - left the problem completely to the classand did not break down the problem into bite-sized pieces by questioninganswering techniques until they were convinced that the problem could besolved only with a linear equation. Is that not what often goes completelywrong in mathematics teaching?

The increase in creative and heuristic abilities is hard to measure. Commonly it wasreported that problems were increasingly handled with perseverance instead ofresignation. All involved were convinced that pupils gained meta-knowledge rather thana collection of easily accessible but quickly forgotten bits of information. The deliberatechange in the teacher role (to refrain from giving hints during the working and solutionphases, to challenge and to accept solutions, to encourage alternatives) was not restrictedto project classes.

SOME PERSISTENT PROBLEMS

In spite of the successes, we realize that ther are some problems. The principle expressedin Matthew's gospel, "to him that has, shall be given" leads to the unavoidable problemthat both able and less able students profit from this teaching, but at different levels, sothat the gap between them is widening. Naturally some parents blamed their children'sfailure on the 'new' style of teaching.

There is a problem with lack of commitment in Germany, of teachers and students, toachieving one's best possible individual results.

Teachers rarely experience in their own university education the kind of teachingmethods we would like them to use in schools.

Only teachers who enjoy and pursue mathematics themselves can succeed in awakeningpupils' interest, in getting students to enjoy puzzling, in encouraging students to achievesomething.

The pressure to achieve high points on the centrally imposed abitur (final examination atthe end of upper secondary level) results in too much rote learning, too little deepthinking. The problem sets on these exams become a hidden curriculum.

��

CONCLUSION


Our preliminary results from the BLK project after less than three years of experiencewith the new teaching methods suggest that the teaching climate and the activeparticipation of most learners have improved. We have reason to believe that students'basic concepts and elementary modelling skills will continue to improve as they solvemore open, and challenging problems.

REFERENCES

Blum W, Neubrand M (Eds) (\ 998) TlMSS und der mathematikunterricht Hannover:Schroede\.

Borneleit P, Danckwerts R, Henn HW, Weigand H G (2001) 'Expertise zummathematikunterricht in der gymnasialen oberstufe' in Tenorth HE (Ed)Kerncurriculum oberstufe Weinheim: Beltz, 26-53.

Henn H W (Ed) (1999) Mathematikunterricht im aufbruch Hannover Schroede\.Henn HW (2002) 'Promoting classroom culture. The BLK-schoolproject in Baden

Wuerttemberg' in Weigand HG et al (Eds) Developments in mathematicseducation in German-speaking countries. Potsdam 2000 Hildesheim: Franzbecker,65-75.

Kirsch A (1969) 'Eine Analyse der sogenannten schlussrechnung' MathematischPhysikalische Semesterberichte 16,41-55.

Neubrand N (200 I) 'PISA: Mathematische grundbildung / Mathematical literacy als kerneiner internationalen und nationalen leistungsstudie' in Kaiser G et al (Eds) Leistungsvergleiche im mathematikunterricht. Ein ueberblick ueber aktuelle nationalestudien Hildesheim: Franzbecker 177-194.

Seiter Ch, Spiegel H (1997) Wie kinder rechnen Leipzig: Klett.Winter H (\996) 'Mathematikunterricht und allgemeinbildung' Mitteilungen der

Gesellschaft fiir Didaktik der Mathematik 61, 37-46.Wittmann ECh, Mueller GN (1990/92) Handbuch produktiver rechenuebungen Stuttgart:

Klett.��

7

Powerful Modelling Tools for High School AlgebraStudents

Susan J. LamonMarquette University, [email protected]

The high school algebra curriculum can be significantly enhanced byincorporating modelling problems, tools, and techniques. In particular,difference equations, matrices, and spreadsheets serve the dual purpose ofenhancing students' content knowledge and socializing students into theworld of mathematical modelling. Especially when used together, thesesimple tools have great power for predicting behavior "in the long run" andfor investigating changes in initial conditions that can affect what happensin the future-a major concern in almost every field of application that usesmathematics today. But merely incorporating new material is not enough;the psychology and previous experience of the learners must be considered.

��

82

RATIONALE

Lamon

Current instruction provides few opportunities for students to decompartmentalize theirknowledge and bring it to bear on a problem, to speculate and innovate, to gain "insiderknowledge" and to experience the "feel" of being a member of the real community ofpractice (Brown & Duguid, 1991). For several reasons this in unfortunate. In real life,no one gets paid to solve problems that follow from the material presented in a precedingchapter, that have a single right answers that can be found at the back of the book.Furthermore, in the language of cognitive science, knowledge is situated, that is, it isstored with markers to the situation in which it was first learned and used, indexed toactivities, culture, practices, and environment in which it was developed (Brown, Collins,& Duguid, 1989, 1999). Fragmented bits of knowledge, unrelated to reality, acquired andpracticed in the sterile, isolated conditions of the classroom lack the markers that allowthe knowledge to be retrieved, interpreted, and transferred to new situations. Withoutlinks to the authentic activities typically organized by a model or a modelling technique,their future usefulness is doubtful. This suggests the importance of linking importantmodels and modelling techniques to mathematics content as it is being taught.

When we speak about enculturating students into mathematical modelling (Bishop, 1988)at least part of what we mean is helping students to develop the ability to think abouttheir problem solving and to ask themselves the right questions to guide themselvesthrough a complex problem-solving activity such as mathematical modelling. Thismanagerial process, known as metacognition (Lester and Garofalo, 1982; Schoenfeld,1985) is an executive function that assesses givens, constraints, and appropriateness ofoperations,and generally monitors and regulates one's problem solving activity. Itincludes the ability to ask and answer for oneself many of the Polya-like questions(Polya, 1957) that are critical to directing and monitoring the process of mathematicalmodelling. Clarifying givens and goals, formulating the problem, seeking additionalinformation, distinguishing parameters, variables, constants, and choosing mathematicaltools and processes, rely heavily on one's being able to ask, to think about, and to answerappropriate questions about the situation at hand. It is important to note that students donot spontaneously develop this skill independently of instruction. Authentic activitiesmust be purposefully built into the mathematics curriculum and the teacher must makethe modelling process explicit, so that students participate in a cognitive apprenticeshipwith the teacher. While participating in authentic activity in the classroom, students learnimportant mathematical content and the language to discuss it, and they experience thespirit of investigation, innovation, and creative thinking that characterizes communities ofpractice (Brown & Duguid, 1991).

Dynamical SystemsIn a multitude of application fields, some of the highest priorities include predicting thefuture state of some system and then manipulating present conditions to change the futureoutcome in some more desirable direction. For example, economists are interested infuture trends so that they can set current policies; the Department of Natural Resources isinterested in the projected numbers of wildlife so that they can set limits on hunting

��

Powerful Modelling Tools 83

licenses this season. In models that help us to predict future developments we are usuallyinterested in a variable X and how it changes in time. If we think of X(t) as a continuousfunction of continuous time t, we would be looking for a differential equation. Clearly,this would be beyond most high school algebra students. On the other hand, if we thinkof X taking values at particular points in time, say every month, or every year, we have adiscrete model and the way that X changes with time is described by a differenceequation. Unlike differential equations that require knowledge of calculus, differenceequations are accessible to high school students with only basic knowledge of algebraicsymbolism.

In this chapter, I share some of the techniques and problem types that I have usedsuccessfully with algebra students who have had no prior introduction to mathematicalmodelling. These include difference equations, matrices, and spreadsheets. Althoughsome brief introduction is needed to make explicit the ideas and goals of mathematicalmodelling, these powerful, basic modelling tools, especially when they are used together,complement the goals of algebra I and can easily be incorporated into the curriculum.These applications, in fact, increase student interest and motivation because they helpstudents to see the uses and benefits of the techniques they are studying in the course. Ihave even used the same problems and methods to initiate my calculus students into theworld of modelling and to help them direct their thinking in useful directions as theyapply their mathematics. I will first describe a technique that quickly and effectivelydraws out onto the table all of the issues that we want students to begin to think aboutduring their first encounter with mathematical modelling.

TAKING STUDENTS FROM THERE TO HERE

Young people have little experience thinking outside of their personal world or sphere ofinfluence. Their chief interest lies in what happens here and now. Predicting futurebehavior and to making changes now that will alter the course of that predicted behaviorare new ways of thinking for most young people. We assume too much when we askmost university freshmen students, for example, to make implicit assumptions explicit orto think about constraints. Their textbook problem solving in high school was socarefully designed to follow the exercises that preceded the problems, and not tointroduce any ambiguities or to require extra thought, that they are, in fact, trained bythose conventional classroom activities not to think, and not to connect their mathematicsto reality. It has been my experience, however, that the apparent absence of a modellingframe of mind is due more to students' prior experiences than it is to innate ability orintelligence. In most cases, no one has ever asked them to think before. Students quicklyadopt useful ways of thinking if we meet them where they are and give them permissionto move in alternate directions.

I have found it helpful to begin with a few small, silly problems to introduce students tothe process of mathematical modelling. Students are immediately motivated toparticipate in the new venture because the problems are clearly not beyond theircapability, but at the same time, because the problems are so simple, it dramaticallyillustrates the shallow perspectives students have about problem solving. By examiningstudent thinking in an uncluttered context, the instructor can better understand the current

��

84 Lamon

nature of student thinking, their misconceptions, and the likely difficulties they will have.Finally, each problem becomes a memorable reference situation to which the teacher canrefer in future discussions to help students remember important lessons learned and waysof thinking. Two such problems are given here.

Boo BugsBoo bugs make a funny little noise as they walk along. They sound like someone quicklysaying "Boo, boo, boo, boo, boo" while they are moving. They also have the specialcharacteristic that the speed at which they crawl depends on the air temperature. Theirspeeds at different temperatures are given in Figure 1. Boo bugs reach their maximumspeed at 34°C and are apparently unaffected by higher temperatures.

44 em.

"

-- ••• _••• '7:05:45

•• • .. ••..·7:05:35

,

.

16222834

temperaturein degrees Centigrade

2345

speedIn cm./sec.

Figure 1. Boo bug data.

One morning as you were eating breakfast, you noticed that a boo bug crawled from thebottom of your kitchen window to the top. Your kitchen clock was nearby and you wereable to note the time at which he entered the bottom pane of glass and the time at whichhe left the top pane. Can you use the bug's trip to predict the outside temperature?

Dean and Della SpaceguardIf our hero and heroine Dean and Della are captured by enemy aliens, they will surely besubjected to the Room of Doom. Before they begin their dangerous mission, their friendsmust be prepared to save them in case they are confined to 'the room.' Here is what theyknow. The room measures 20 meters long and 15 meters wide. When someone is lockedinside, the dimensions of the room begin to change. The length decreases at the rate of 2meters per minute and the width decreases at the rate of 3 meters per minute. How muchtime do their friends have to save them if they should meet this fate?

Students have little difficulty in calculating the bug's speed, 4.4 em/sec, and some caneven interpolate using the values given in the table to find that the outside temperature is30.4°C. But beyond getting the answer, it is clear that many students have few of theways of thinking that will serve them in mathematical modelling. Many students have noidea what they are assuming when they calculate the bug's speed at 4.4 em/sec. It wasnever important for them to ask if the bug moved at a constant speed the entire time hewas within sight, or whether he traveled in a straight line, or whether the bug was moving

��


at its maximum capable speed. Simplifying assumptions are dismissed by most studentsas irrelevant considerations. Even the task of understanding what an assumption is, canbe difficult. Students often have difficulty knowing where to draw the line betweensimplifying assumptions and basic facts. In the bug problem, for example, severalstudents in a calculus class listed 24 hours = 1 day as one of the assumptions.

Most students will not have thought about these issues:• Is t proportional to s or is ~t proportional to ~s, or both?• What is the mathematical model relating temperature and speed of the bug? How do

you get it and why does it make sense?• What are the domain and range for this function?• Is 4.4 em/sec an actual speed or an average speed? What does it mean to say that the

bug's average speed was 4.4.cm/sec, and what does this mean about the accuracywith which your model can predict the outside temperature?

Similarly, in discussing the Dean and Della Spaceguard problem, students' responses are,at first, predictable. Most never considered the minimum width required for a bodybefore being crushed to death, but a discussion of their solutions, the assumptions thatthey made, and the issues that never crossed their minds, had a strong impact on theirthinking in future problem solving.

DIFFERENCE EQUATIONS

A difference equation merely tells how the next term of a sequence can be computedfrom preceding data values. For example, for the Fibonacci sequence, the differenceequation tells us that the difference between the next term in the sequence (An+ 1) and thepresent term (An) is the term immediately preceding the present term (An-I):

An+I-An= An-IEquivalently, An+ 1can be obtained by adding the current term and the one preceding it:

An+I=An+An-t

When difference equations are first introduced, it is useful to analyze some simplesequences to help students distinguish difference equations and functions. For example,consider this sequence: I, 2, 4, 8, 16, ...

Have students describe the manner in which each successive term can be computed usingthe ones before it, then write the relationship symbolically:

An+ I = An + An = 2An

Also have students use pattern recognition to write the pth term of the sequence as afunction of its position, as shown in Table 1.

��

86 Lamon

Position (p) Number(N)I 12 2= 2°3 4= 2<

4 8= 23

5 16= 24

p N=2P- '

Table 1. The functional equation.

Algebra I and Difference Equations

There are many reasons why it is beneficial to incorporate difference equations into thehigh school algebra curriculum. These include useful discussions of many considerationsthat are often tacitly considered during the mathematical modelling process. Some of thepossible advantages and discussion points appropriate for algebra I students areillustrated using the Beanstalk Problem from Edwards and Hamson (1996, p. 61):

A beanstalk grows 3 em on the first day and its growth on each day afterthat is half that on the previous day. If B, is the length of the beanstalk atthe end of day n, write down a difference equation satisfied by Bn.

a) Difference equations use algebraic notations and ways of thinking.b) Translating verbal statements into difference equations requires the active

formulation of models, combating the students' common conception that doingmathematics consists of choosing formulae from a long list of available rules.

Students begin by listing the height of the beanstalk after one interval of time (I day) asBJ, after two time intervals (2 days) as B2, etc.

B1=3

B2=B,+3(.5)B]=B2+3(.5)2B4=B3+3(.5/

c) Elementary difference equations rely on pattern seeking.d) They are useful ways of expressing generality.e) Difference equations are good vehicles for studying recursion.

Students use the patterns visible in the difference equations to write a generalizedformula: Bn+I=Bn+3(.5)n (I)

f) It is sometimes easier to formulate a difference equation than it is to define afunctional relationship. This permits students to examine models that mightotherwise have to wait until they have deepened their mathematical knowledge.

g) Difference equations can be used to approximate continuous change.

For example, as in Table 2, write the height of the beanstalk as a function of time:

��

Powerful Modelling Tools

an_1H =F(n) =3 (-I ),a=.5a-

Day (n) Height of Beanstalk (H) (cm)I 32 3+3(.5)3 3+3(.5)+3(.5)L

4 3+3(.5)+3(.5t+3(.5)3...n 3(1+.5+(.5)L+(.5)J+(.5t+ ... )

Table 2: Height as a function of time.

87

To produce this functional equation is beyond most algebra I students, and an examplesimilar to this one is usually not taught until algebra II (in the US, this is year 3 of thehigh school mathematics curriculum) because it entails the sum of a geometric series.Unlike the difference equation that relies on recursion, the chief advantage of the functionis that it enables us to immediately find any data point without first listing all of theprevious points. But algebra I students can gain insights into the behavior ofa continuousvariable by studying change between discrete data points.

For example, students may rearrange (I) and consider what the symbols say about thebehavior of the system:Bn+I-Bn= 3(.5)"

Is plant growth a continuous or discrete process?What does the difference equation suggest about future growth?

h) They provide opportunities to discuss the differences between constants,variables, and parameters and to discuss what might be sensible values for acertain parameter.

i) They help students to look for form and structure, rather than attending tosurface-level, problem-specific characteristics.

If the growth each day were less than half the previous day's growth, say, Y4 of theprevious day's growth, what would be the difference equation?If the initial day's growth is 2 em rather than 3 em, what would the difference equationbe?If we allow some of the numbers to change, what other problems that we have studiedmight this difference equation fit?

SPREADSHEETS

Spreadsheets can considerably amplify students' ability to analyze a situation, and today,because they are so ubiquitous, most students have already had some experience with aspreadsheet before high school. One of the best problems for convincing students of the

��

88 Lamon

power of the spreadsheet is based on some information that de Lange (1987) obtainedfrom the book Rats by the Dutch biologist, Maarten 't Hart.

Counting Rats

Let's estimate the number of offspring produced by one pair of rats underideal conditions. Although relatively little is known about rats in nature,their reproductive habits have been observed in the laboratory. The averagenumber of young in a litter is 6. The period of gestation is 21 days;lactation also lasts 21 days. However, a female may conceive again duringlactation; she may even conceive again on the very day she drops heryoung. To simplify matters, let's assume that the number of days betweenone litter and the next is 40. If the female drops 6 young on the first day ofJanuary, she will be able to produce another litter 40 days later. Thefemales from the first litter will be able to produce offspring themselvesafter 120 days. Assuming that all females who are able to give birth do soevery 40 days, that a litter will always consist of 3 females and 3 males, andthat none of the rats die, what will be the total number of rats by nextJanuary? (Adapted from de Lange, 1987, p. 46)

Solving the problem using brute force, that is, by listing pairs and counting, you willquickly see the need for carefully schematizing the rat offspring. At surface level, itappears to be a simple counting problem, but one quickly realizes that the challenge liesin adopting a representation that facilitates the counting process when multiplegenerations of rats are reproducing simultaneously. Table 3 shows one usefulschematization. But have we gotten them all? Students can verify their results usingdifference equations and/or difference equations and spreadsheets. Table 4 shows the useof difference equations in Excel. We note without demonstration, that this problem issolvable using a Leslie matrix on a spreadsheet as discussed in the next section.

SYSTEMS THINKING

After a suitable introduction to the modelling process and the ways of thinking thatstudents need to develop, it is important to move away from simple situations and toencourage students to think in terms of larger systems, where there are many interactingelements. Environmental issues provide an important and accessible field of application.

The term environmental mathematics (Fusaro & Sward, 1990) was coined just over adecade ago to reflect the growing importance of mathematical modelling in analyzing andunderstanding environmental problems. Before then, modelling environmental issueswas taught in some university mathematics courses (Fusaro, 1992), but usually only bypersons who considered themselves "environmentalists." Today, the environment is oneof the most important problems facing the human race (Rutherford & Ahlgren, 1990;Hadlock, 1998) and young people are genuinely concerned about environmental issues(Broussard 2001). Unfortunately, news media reports of some emotional partisanposition on pollution, recycling, timber management, water quality, or energy, are stillthe main source of information for most Americans. The issues are unavoidably value-

��


Time New TotalPeriod Mice Mice

01M 3M

6 8IF 3F

13M

6 143F

••.................•. ·····..·i

23M i

6 203F i

33M 9M ......;......... ........ ...........

24 443F 9F i

43M 9M 9M i 42 863F 9F 9F I

53M 9M 9M 9M ........•........ ......... ........... 60 1463F 9F 9F 9F

·········1;

63M 9M 9M 9M 36M I 132 2783F 9F 9F 9F 36F I

73M 9M 9M 9M 36M 63M • 258 5363F 9F 9F 9F 36F 63F

83M 9M 9M 9M 36M 63M 90M

"438 974

3F 9F 9F 9F 36F 63F 90F

93M 9M 9M 9M 36M 63M 90M 198 M

834 18083F 9F 9F 9F 36F 63F 90F 198 F

Table 3. One schematization for counting rats.

Time TotalPeriod Rats0 J 81 4}.... 8+62 ( ~"'- 14 + 63 (4~"'- "- 20+J.2.4 86 <,<, 44~5 146 <, 86 -t\6Ql.6 278 146 -t132>

An+J An+ 3An_2

Time Total RatsPeriod

-I 20 8I 142 203 444 865 1466 2787 5368 9749 1808

Table 4. Solving Rats using difference equations and Excel.

��

90 Lamon

laden, but most people have never studied environmental issues in a principled, scientificway, assessing trade-offs between alternatives (economics vs. wildlife, jobs vs. spottedowls, erosion and logging vs. salmon runs). This involves modelling systems andcomplex relationships, rather than making decisions about isolated phenomena based onfeelings or opinions. It also entails problem-based learning in which a problematicsituation precedes and motivates the understanding and the resolution of the problem(Barrows & Tamblin, 1980). It is then the student's job to identify the crux of the issue(problem identification or problem finding), to ask the critical questions that must beanswered (problem formulation), to seek needed information, and to use suitable tools toinvestigate the situation.

In other disciplines, there is nothing new about the use of problem solving as a means oflearning. Business, medical, and engineering schools, for example, have used casestudies for years. Case studies present a stark contrast to the narrow, obsolete, untypicalconcepts and operations taught in separate classes as unrelated bodies of facts. In casestudies, students spend time investigating real situations and building mental models withthe greatest power and utility for understanding, interpreting, explaining, answeringquestions, or making predictions about the real situations (Lesh and Lamon, 1992).

Case Studies and Systems 'In the Long Run'

Student who have had little experience with true problem solving are often overwhelmedwhen they encounter a truly complex situation. Having already prepared them with a fewsimple tools, I like to introduce them to environmental problems that require someanalysis of what might happen "in the long run."

In the real world, models are likely to involve more than one variable. Linear differenceequations involving more than one variable can be expressed as vectors and matrices.For example, if a population of animals consists of babies, adults, and seniors, we mightwish to consider a model for predicting population growth that keeps track of the numberof individuals in each one of those age groups.

In that case, in the difference equation Xn+I=TXn, X, and Xn+1 are vectors and matrix T,called a transition matrix, defines the transitions of the population subgroups during eachtime period to the next subgroup in terms of percentages that advance to the next stage.This special matrix T that applies to population dynamics is called a Leslie matrix(Leslie, 1945).

Spreadsheet capabilities make difference equations expressed as matrices even morepowerful and convenient than matrices on hand-held calculators. In the case that youwish to obtain data points for large values of n, the matrix multiplications can becometedious on the calculator, while the spreadsheets' ability to reference other cells and tocopy and extend operations and apply them recursively, enables students to minimizetime spend on calculations and to better spend their time analyzing the long-termbehavior of a system. We consider the following problem:

��

Powerful Modelling Tools

Save the Wallitups

A group of protestors surrounded the Marpleton courthouse and shoutedobscenities at a group who marched on the other side of the street. TheSave-the-Wallitups group had recently learned that a nearby wooded areahad the only surviving population of Wallitups in the world: 2 babies, 3adults, and 4 seniors. A land developer wanted to buy the woods in whichthe animals lived to build a strip mall. The position of the "Wallies" wasthat the state should buy the property to let the animals exist peacefully andto help them flourish. The opposing citizens argued that it was too far todrive from Marpleton to the next town for shopping and that it was abouttime that the town had some stores. The Wallies argued that the townwould become famous, tourism would flourish, and that as the animalsmultiplied, they could be sold to zoos around the world. The would-beshoppers argued that the local economy would become more robust bybuilding the strip mall because the residents of neighboring towns wouldcome to Marpleton to do their shopping. How does one begin to deal withthis emotionally charged issue?

91

Wallitups are capable of reproducing at the age of 1 year, and reproduction rates drop intheir third year. This suggests that we might represent the population at time step n interms of the number of animals in each of 3 categories:

Yn= number of young animals up to I year oldAn = number of young adults up to 2 years oldM, = number of mature adults of age 2 years or older

Additional information about annual birth and death rates for the 3 groups is given inTable 5.

Group Birth Rate Death Rate

Y 0 .1A .3 .2M .1 .3

Table 5. Birth and death rates for population subgroups.

Using the given data, we can form a transition matrix, T, that shows the percentage ofeach age group that will proceed to the next time period:

YAM

: l·~ .~ .~jM 0 .8 .7

It is now a simple task to enter the matrix into a spreadsheet and use it to predict whatwill happen to this population of animals in the long run. To determine what happens in

��

92 Lamon

each new time period, we multiply the previous year's population by T, the transitionmatrix. This difference equation is

r:1= r·~ .~ '~1 r:1M n+1 0.8.7 M n

The matrix multiplications are easily accomplished using the cell referencing and clickand-drag capabilities of the spreadsheet. The extended behavior of the system is shownin the Excel spreadsheet in Table 6.

yearsmatrix T start I 2 3 4 5 60 0.3 0.1 2 1.30 1.06 0.86 0.74 0.62 0.530.9 0 0 3 1.80 1.17 0.95 0.77 0.66 0.560 0.8 0.7 4 5.20 5.08 4.49 3.91 3.35 2.88

Table 6. Predicted Wallitup population for the next 6 years.

It can be seen that the Wallitups will soon become extinct. It is predicted that after 3years, there will be only 4 seniors (and seniors typically have a low birth rate). Thisknowledge suggests some strategies for compromise between the two groups ofprotestors. Because the animals will soon become extinct, perhaps the shopping mall canbe postponed for 3 years. At that time, it might be useful to investigate the gender of theremaining animals to determine "next steps."

Another powerful aspect that aids system analysis is the ability to change initial values toobserve the effect in the long run. The following kinds of questions help student toinvestigate precisely the kinds of issues that are essential in the mathematical modellingand most real applications:• How does the system behave in the long run?• Does it grow (or decrease) without bound?• Does it oscillate?• Does the system have any equilibrium values?• Are the equilibrium values relative stable, or do they react to changes in initial

values?• Do small changes or large changes in initial values have dramatic effects?

Discussing questions like these helps students to know where to focus their attention asthey begin to investigate dynamical systems. It is this kind of thinking that fuels andregulates the mathematical modelling process and helps students to see modelling as athinking process, rather than as mechanical substitutions into formulae.

��

CONCLUSION


High school students (and university students as well) often see no value in themathematics they are learning. They go through the motions required by their courses,taking it on faith that the material they are learning will be valuable someday. Instead ofusing mathematics to control their world, the mathematics controls them! By workingwith the students we have (not the ones we wish we had), by not assuming too muchabout their preparation, and by taking the time to guide their entry into spirit ofmathematical modelling, dramatic progress can be achieved in changing these attitudes ina relatively short period of time.

Freudenthal (1978) characterized this spirit as something distinct from techniques andscientific instrumentation, and Piaget (1969) referred to it as the spirit ofexperimentation,and the spirit of invention. This orientation to the world is precisely that which propelsthe process of mathematical modelling. To facilitate its growth, we need to "shift theemphasis from the student's 'correct' replication of what the teacher does, to thestudents' successful organization of his or her own experience (von Glasersfeld, 1983, p.51). It is simply no longer acceptable to have someone else do for the students what wesay we want them to do for themselves. In this chapter, I have suggested that it is bysimultaneously emphasizing tools and their applications in authentic problems, whilecarefully socializing students into the modelling process, that we can help them toexperience for themselves the power of mathematics in organizing their world.Environmental problems are particularly motivating, and difference equations, matricesand spreadsheets are powerful and accessible starting tools for novices.

REFERENCES

Arthur MA, Thompson JA (1999) 'Problem-based learning in a natural resourcesconservation and management curriculum: A capstone course' Journal ofNaturalResources and Life Sciences Education 28, 97-103.

Barrows HS, Tamblyn R (1980) Problem-based learning New York: Springer.Bishop A (1988) Mathematical enculturation Dordrecht: Kluwer Academic Publishers.Broussard SR, Jones SB, Nielsen LA, Flanagan CA (2001) 'Forest stewardship education

with urban youth' Journal ofForestry 99(1), 37-42.Brown JS, Collins A, Duguid P (1988), Situated cognition and the culture of learning

(Research report no IRL88-0008). Palo Alto, CA: Xerox Palo Alto ResearchCenter, Institute for Research on Learning.

Brown JS, Collins A, Duguid P (1989) Situated cognition and the culture of learningEducation Researcher, 18( I), 32-42.

Brown JS, Duguid P. (1991) Organisational learning and communities of practiceOrganisation Science Vol. 2 No. I ppAO-57.

Dahlgren MA, Oberg G (200 I) 'Questioning to learn and learning to question: Structureand function of problem-based learning scenarios in environmental scienceeducation' Higher Education 41, 263-282.

de Lange J (1987) Mathematics insight and meaning Utrecht: OW& OC.

��

94 Lamon

Dooley KE, Neill WH (1999) 'Systems modelling by interdisciplinary teams: Innovativeapproaches to distance education' Journal ofNatural Resources and Life SciencesEducation 28, 3-8.

Edwards D, Hamson M (2001) Guide to mathematical modelling 2nd Edition London:Palgrave.

Freudenthal H (1978) Weeding and sowing Dordrecht: D. Reidel.Fusaro B (1992) 'Environmental mathematics' in Steen L (Ed.) Heeding the call for

change: Suggestions for curricular action. MAA Notes #22 Washington, DC:Mathematical Association of America, 83-92.

Fusaro B, Sward M (1990) 'Solving environmental problems: Where are themathematicians' Focus, April.

Hadlock C (1998) Mathematical modelling in the environment Washington, DC:Mathematical Association of America.

Lesh R, Lamon SJ(Eds) (1992) Assessment of authentic performance in schoolmathematics Washington, DC: American Association for the Advancement ofScience.

Leslie, PH (1945) 'On the uses of matrices in certain population mathematics' BiometricaVolume XXXIII, 183-212.

Lester F, Garofalo J (1982) Mathematical problem solving: Issues in researchPhiladelphia: Franklin Institute Press.

Piaget J (1969) Science of education and the psychology of the child New York:Grossman Publishers.

Polya G (1957) How to solve it New York: Doubleday.Rutherford FJ, Ahlgren A (1990) Science for all Americans New York: Oxford

University Press.Schoenfeld AH (1985) 'Metacognitive and epistemological issues in mathematics

understanding' in Silver EA (Ed) Teaching and learning mathematical problemsolving: Multiple research perspectives Hillsdale, NJ: Lawrence ErlbaumAssociates, 361-380.

Tan SC, Turgeon AJ, Jonassen DH (2001) 'Developing critical thinking in group problemsolving through computer-supported collaborative augmentation: A case study'Journal ofNatural Resources and Life Sciences Education 30, 97-103.

Von Glasersfeld E (1983) 'Learning as a constructive activity' in Bergeron JC,Herscovics N (Eds) Proceedings ofthe fifth annual meeting ofthe North Americanchapter of the International Group for the Psychology ofMathematics EducationMontreal, Canada, 41-69.

��

��

9

Formal Systems of Logic as Models for Building theReasoning Skills of Upper Secondary School Teachers

Paola ForcheriIstituto di Matematica Applicata e TecnoIogie Informatiche del CNR, [email protected]

Paolo GentiliniIstituto di Matematica Applicata e Tecnologie Informatiche del CNR, ItalyLigurian Regional Institute of Educational Research, [email protected]

We describe the main features of an in-service teacher-training proposalaimed at helping the teachers (and their students) to use logic to developvaluable models of thinking. We take into account the evolution ofcommon sense reasoning and the reasoning capabilities required by thecomplexity of global rationality. Referring to the situation in Italianschools, we first discuss the levels of competence to be achieved byteachers in relation to the formal reasoning objectives of education. Wepropose innovations in teacher training in accordance with the current viewof rationality. Finally, we report some of our results when we used ourmodel with teachers.

��

108

INTRODUCTION

Forcheri & Gentilini

Mathematical logic constitutes the basis for modelling all reasoning, includingmathematical and scientific reasoning, thus offering a powerful tool to help studentsdevelop valuable models of thinking. To fully exploit its pedagogical value, it should becompared to common sense reasoning. Moreover, the teaching of mathematical logicshould be connected, at least to some extent, to the complexity of global rationality, inwhich common sense sometimes clashes with the results offormallogic.

However, at least in Italian schools, the teaching of mathematical logic has nottraditionally followed this approach. In the lower school, procedures, rules, andbehaviours are taught, with little attention to reasoning; in the upper school, limitedattention is paid to logic, as it usually is not included in the final examination. Onlyrecently, the growing need to renew instruction on reasoning has led the Italian school tofocus attention on the design of curricula aimed at helping students to become aware oflogical tools which are part of the present culture and technology, and to acquire theexpressive and linguistic capabilities related to them. To create effective changes inschools, however, in-service teachers need to construct knowledge about the variety offormal logics at their disposal and to gain awareness of their possible influence oncommon sense reasoning.

In this chapter, we will discuss a plan for re-designing in-service teacher training inlogical modelling, including a contemporary view of rationality. The limited attention tologic education is, in our opinion, a major cause of the decreasing capability of the Italianupper secondary school to provide a valuable education in reasoning. This situation isconfirmed by questionnaires sent to teachers of all Ligurian upper secondary schools bythe Ligurian Regional Institute of Educational Research (lRRE) (Gentilini & Manildo,200 I). The work described in this chapter is partly based on the project Education toRationality of IRRE Liguria, a project aimed to improve education in reasoning in uppersecondary schools (Gentilini, 200 I).

OUR TEACHER TRAINING PROPOSAL

FrameworkActive involvement in current socio-cultural and technological processes requires thatpeople be able to reason by choosing among different rationalities that apparentlycontradict each other, the appropriate one for the situation at hand. Global rationality is,in fact, complex. It includes two major types of reasoning, common sense reasoning andscientific reasoning, with the latter category including classical, intuitionistic, andparaconsistent systems of logic.

Common sense reasoning: This includes the reasoning capabilities that help to selectbehaviours or arguments considered obviously reasonable by the majority of people(including students), thus determining obvious deductions. It is difficult to define'obviously reasonable' (Fagin, Halpern, Moses, & Vardi, 1995), as it depends on the timeand on the cultural context.

��

Systems of Logic 109

Scientific reasoning: This is the type of reasoning canonically established by thescientific world. It can be divided into two sub-categories: classical and modernrationality. Classical reasoning is based on the theory of inference and on the theory oftruth established by classical thought, essentially Aristotelian, without the use of anyartificial languages. Modern rationality, dating from the end of the nineteenth century tothe present, is dominated by mathematical logic. It is built upon symbolic-formallanguages, on the mathematical approach to classical logic, and on non-standard logics.Examples of conflicts between different kinds of rationalities are shown in Figure I.

Example tBoth the Euclidean straight line and plane include an infinite number ofpoints. Are there more points on the plane or on the line?

Common sense reasoning would answer that the plane includes more points than the line orthat nothing can be said.Cantor's set theory proves that the two sets of point can be put in a bijectivecorrespondence. Thus, the plane and the line are sets with the same infinite cardinality.

Example 2Common sense reasoning would say: high and low exist as absolute orientationsin space.Physics states: absolute high and absolute low do not exist.

We note that the common sense notions of high and low are employed in everyday life.

Example 3As shown by experimental psychology (Evans, Newstead, & Byrne, 1993), common sensereasoning often leads people to make inferences of the following types:

PremisesIf there is a circle on the left, then there is a square on the right.There is not a circle on the left.Conclusion: There is not a square on the right

PremisesIf there is a circle on the left, then there is a square on the right.There is a square on the rightConclusion: There is a circle on the left.

In both cases, the inferences applied are not valid in classical logic.

Figure 1. Three examples of conflicts between common sense and scientific reasoning.

Thus, the goals of teaching reasoning should be to create awareness of the various typesof rationality and to develop strategies for handling the conflicts between them. To reachthese objectives, new models for in-service teacher training on logic should be devisedwhich are suitable for helping teachers to: I) acquire knowledge of the logic underlyingthe different types of rationality that are part of our present culture; 2) use this knowledgeto help students develop awareness of the types of rationality that are functioning in our

��

110 Forcheri & Gentilini

present culture. These are the goals of the project Education to Rationality of IRRELiguria. The project trains in-service upper secondary school teachers (mathematics,science, humanities and philosophy teachers) in mathematical logic.

OverviewOur teacher education program is organised in two parts.• The first phase includes an introduction to modem classical logic tools, including

logical consequence, and the analysis of heuristic methods suitable for teaching thisnotion.The second phase introduces teachers to formal rationality produced in the lastcentury, and it addresses the modelling power of the complex global rationality ofmathematical logic via a comparison between classical, intuitionistic, andparaconsistent logics.

Each phase includes two kinds of activities: theoretical lessons and activities designed tohelp the teachers tum theory into meaningful classroom activity for learning. We brieflyoutline the content of both phases.

Logical ConsequenceTheoretical Lessons: We propose the following topics that could be adapted for studentsin mathematics and philosophy (last year of upper secondary school).• Introduction to the idea of artificial language, via the predicate logic LK. A rigorous

approach, appropriate for teachers of different backgrounds, might be inspired, forexample, by Barwise and Etchemendy, (1992).

• A presentation of the deductive apparatus of LK, possibly via Gentzen's naturaldeduction and an examination of the distinction between syntax and semantics for atheory T axiomatized in the language of LK and basic definitions of the Tarskisemantics for the language of LK.

• Presentation of the notion oflogical consequence.• Analysis of the links between syntax and semantics for the first-order logic LK, via

the result of completeness for LK.

In teaching heuristics, we observe that a number of pedagogical difficulties have to beovercome to introduce to the students the notion of logical consequence, in particular theconcept of semantics of a symbolic language. A (rigorous) heuristic approach, and itsadaptation to arguments expressed by using the natural language, can help to overcomethe problem. Thus, teachers can introduce the idea of logical consequence to students,even if students do not know the formal language of LK.

Accordingly, in our training proposal we introduce the concept of possible world,corresponding to a Tarski formal interpretation:

A possible world Q interpreting a statement B is a state of affairs alternativeto the real world of references such that: i) Q is endowed with an intrinsicfactual coherence; ii) and it includes objects and situations to which termsand predicates occurring in B can be referred.

For classroom practice, the main difficulty is the introduction ofthe following definition:

��

Systems of Logic

Proposition C is a logical consequence of hypotheses BI, ...,Bn if in everypossible world (or circumstance) where Bl,...Bn hold, C also holds.

This definition has to be discussed by using arguments expressed in natural language.

III

Applications: We report here some examples that help teachers to better understand thetheoretical part of our lessons. In Figure 2, we give i) the content or topic to be addressedand ii) an example of an activity in which the content is applied.

Classical, Intuitionistic and Paraconsistent LogicsClassicallogic: We begin with a review of classical logic in comparison to other logics.In particular classical logic can be presented as a logic oriented towards ontology andmight be used to describe mathematical entities within a Platonist conception ofmathematics. This presentation, moreover, helps to point out that the lack of awarenessof classical deduction rules and diffused tautologies (for example, the excluded middleprinciple, the double negation law, the non contradiction principle) preventsunderstanding of non-classical logics.

Intuitionistic logic (Troe1stra & Van Dalen, 1988): This logic is introduced to showteachers a formal logic different from the classical one, and to help them to experimentwith the possibility of applying different logics to the same problem. More specifically,the content includes:• presentation of historical and philosophical roots of intuitionism, a logic centred on

constructive proofs as objects produced by the human mind.• a discussion of the lack of validity of the excluded middle principle and of the double

negation law.• a presentation of Hetyting's axiomatization of the formal system for the intuitionistic

predicate calculus LJ, limited to pure predicate logic, without function symbols.• an essential presentation of Kripke models for LJ, with particular emphasis on the

examples of falsification of classical tautologies (Van Dalen, 1986). This trainingstep is particularly important as it proposes a different notion of the truth, withrespect to that of Tarski for the classical logic.

• an example of mathematical reasoning that employs intuitionistic logic at the theorylevel and classical logic at the meta-theory level. (See Appendix A.)

Paraconsistent Logic: This topic includes:• the introduction of a paraconsistent logical system, for example the system C I of Da

Costa (1974) and the role that logic can play in modelling the construction ofrelevant conjectures in scientific discovery.

• an example showing the 'break' of Lagrangian mechanics with Newtonianmechanics. (See Appendix B.)

• Paraconsistent predicate logic systems can be extended by means of contradictionsF 1\ ---,F without trivializing. Thus, in the semantics of paraconsistent logic, acontradiction is not necessarily false. This characteristic makes paraconsistentrationality a powerful tool to model conjectures.

��

112

TopicTranslation of natural languagesentences into the language ofpredicate logic and the analysis ofthe representational power ofartificial languages.

The idea of a possible world.

The notion oflogicalconsequence.

Deductive apparatus of LK

Notion of logical consequence.


ActivityTranslate the statements then show that other sentencesexpressed in natural language having similar structures,can be represented by means ofthe translationsobtained.

If a teacher teaches logic to students then he has tobe brave.

Each minute a person is robbed in Genoa.

Discuss the fact that the following statement admitspossible worlds where it is false according to theheuristic definition of possible world.

Napoleon Bonaparte died in 1821.By using the heuristic notion of possible world, discussthe fact that, in the following arguments expressed innatural language, the conclusion is a logicalconsequence of the premises.

I. All mammals have lungs. All whales aremammal. Thus, all whales have lungs.2. All individual with 10 paws have wings. Allspiders have 10 paws. Thus, all spiders have wings.

Discuss then the following facts: in argument I) allpropositions are true with respect to the 'standard'world; in argument 2) all propositions are false withrespect to the standard world.Why do these facts not influence the existing relationoflogicalconseauence?Given the following syllogism, expressed in naturallanguage:

Every budgie is an animal with wings.All animals with wings are white.:. Every budgie is white.

Verify that a translation into the language of thePredicate logic is:

Vx(C(x) ~Af'l»Vy(A(y) ~B(y»Vz(C(z) ~I,l(:«:»

On the basis of the inference rules of the naturaldeduction for LK, show that: I) a proof of theconclusion from the hypotheses exists; 2) theconclusion, in the context of the classical logic, is alogical consequence of the hypotheses.Show that the following argument, expressed innatural language does not establish a relation oflogicalconsequence between the premises and the conclusion.

If! had all the gold ofFort Knox I would berich. I do not have all the gold of Fort Knox.Thus, I am not rich.

Figure 2. Examples using the theoretical perspectives of our lessons.

��

Systems of Logic 113

We observe that the most interesting and innovative conjectures with respect to a wellestablished theory often present contradictions. Let us consider for example classicalmechanics and general relativity. They are classically mutually inconsistent. If weinclude in classical mechanics Galileo's principle of addition of velocities, we can derivefrom classical mechanics that "a light signal exists having a velocity greater than c" whilefrom general relativity we obviously derive that "light signals having speed greater than cdo not exist." Therefore, the simultaneous application of the two theories produces acontradiction. However, we cannot simply say that the relativity postulates trivializeclassical mechanics.

A TEACHING EXPERIMENT

Some Examples for Comparing the Three Systems of LogicIn Figure 3, we provide some activities to help teachers better understand classical,intuitionistic, and paraconsistent logics and to provide examples of the types of activitiesthat make them meaningful for students. The activities focus on the criticism ofarguments expressed in natural language, from the points of view of modem logic.

The course outlined above was used in a training experiment, in the scholastic year 2000200 I, organised by lRRE Liguria. Twenty upper secondary school in-service teachers(mathematics, science, philosophy, humanities teachers) took part in the course. Thecourse included 24 hours of lessons and about 20 hours of discussion of arguments.

The teachers were first given a questionnaire to determine their opinions about both thereasoning capabilities of their students and the changes they thought they needed to makein their teaching to improve student reasoning. Thus, teachers were motivated to takepart in the course, as a way to address perceived problems. The reaction of the teacherswas quite positive. They showed interest in the theoretical notions introduced,notwithstanding the difficulties they faced. There were, however, two difficulties.

It was difficult for the participants to understand that the syntactic and the semanticanalysis of a reasoning process are independent of each other. We observed that themajority of our teachers often confused the truth and the syntactic correctness. A typicalmisunderstanding of this type was the belief that a deduction is not sound if all sentencesin the deduction are false with respect to the standard world. Teachers were alsoperplexed by the existence of various logics, all of which were valuable, and about thepossibility of choosing the one to adopt according to the situation at hand.

The examples we discussed were valuable in helping teachers to overcome thesedifficulties, and, at the same time, they formed a basis for devising instructional activitiesfor students. In fact, as a result of the course, teachers developed some modules that theyintended to use with their students. An example is given in the following section.

��

114

Analyze this argument.


If individuals are good, laws are not necessary to prevent them fromacting badly; on the contrary, if individuals are bad, laws will not beable to prevent them from acting badly. Thus the laws are notnecessary, or useless.

The argument is implicitly based on the equivalence between not-good and bad. Apartial syntactic reduction transforms the initial argument into:

Hypotheses: individuals are good -7 laws are not necessaryindividuals are not-good -7 laws are useless

Logical axioms: (individuals are good) or (individuals are not-good)(laws are not necessary) or (laws are useless)

Syntactic analysis: The argument is correct from the point of view of classical logic.In fact, from the hypotheses we deduce:

(individuals are good) or (individuals are not- good) -7 (laws are not necessary) or(laws are useless).

Then, by applying the logical rule of modus ponens with the axiom of the excludedmiddle we have the thesis.

Semantic analysis: However, we can criticise the argument from the semantic pointof view. In fact, it can be argued that the hypothesis

individuals are not-good -7 laws are uselessis false in all possible worlds, or that there are not possible worlds in which bothhypotheses are simultaneously true. Thus, the hypotheses constitute a contradiction.Therefore, the argument is correct but useless, as we can prove everything fromcontradictory hypotheses.

Comparison with other logics: It can be observed that, according to theintuitionistic view, the syntactic analysis illustrated above does not work, as theaxiom of the excluded-middle cannot be accepted. Moreover, the argument refers tothe potentially infinite set of all possible individuals. From the paraconsistent pointof view, finally, the semantic criticism carried out above is not possible, since inparaconsistent logic, in general, from a contradiction it is not possible to deriveevery sentence.

Figure 3. An analysis of an argument from multiple perspectives.

��

Systems of Logic

RELATING THE PROPOSAL TO CLASSROOM PRACTICE

115

One module was designed to help students in the first two years of upper secondaryschool (14-16 years old) to develop competence with proofs. Typically, there is littledevelopment of reasoning abilities at this age. The module used an interdisciplinaryapproach to basic linguistics and mathematics. Students were encouraged to reflect onthe knowledge underlying the procedures introduced, by 1) giving written (linguistic)explanations of their meaning; 2) acquiring awareness of the variety of uses of someterms; 3) inventing and writing elementary mathematical proofs. Explanations werebased upon definitions, theorems (in scientific arguments natural language is used), andaxioms. Terms sometimes had a twofold meaning: they could be viewed as notions of aspecific discipline or as words of the natural language (e.g.. energy, straight line,product). Thus, the students' mathematical proofs were to be analysed by the humanitiesteachers for syntactic correctness as well as for the validity of their arguments. Examplesof the activities teachers developed for students are shown in Figure 4. These arecurrently being used in two Ligurian schools.

Activity IWhat does it mean to compare two numbers? Write two fractions, ofyour choice. Consider various methods to compare them. Write anexplanation of these methods and apply them to the fractions at hand.

Activity 2The word set refers to a group of objects that exhibit a commonproperty. In mathematics, a set is defined if this property can bedefined avoiding ambiguity and contradiction. Taking into account thisfact, indicate which expressions among the following define a set in themathematical sense, and explain why.1) the most important Italian towns2) the Italian towns that are provincial capitals3) the small numbers4) the numbers that can be divided by 35) the active students in a classroom.

Activity 3Consider the definition:

If an integer number m is the product of an integer number nandanother integer number (different from 0) then we say that m ismultiple ofn.

Write a proof of the following propositions:I)45 is multiple of 152)45 is not multiple of 6.

Figure 4. Activities to help students develop basic mathematical declarative knowledge.

��

116

REFERENCES


Barwise J, Etchemendy J (1992) The language offirst order logic Stanford, CA: CSLlPublications.

Da Costa NCA (1974) 'On the theory of inconsistent formal systems' in Notre DameJournal ofFormal Logic 15(4),497-510.

Evans J B, Newstead SE, Byrne R M (1993) Human reasoning: The psychology ofdeduction Hillsdale NJ: Erlbaum.

Fagin R, Halpern J Y, Moses Y, Vardi M Y (1995) Reasoning about knowledgeCambridge MA: MIT Press.

Gentilini P (200 I) Critique and innovation in science education 1I. Mathematics in uppersecondary school: Methods, curricula, interdisciplinary interactions Genova: DPSEditions (in Italian).

Gentilini P, Manildo G (Eds) (2001) Protocol about education to reasoning in uppersecondary school Genova: DPS Editions (in Italian).

Goldstein H (1965) Classical mechanics Reading, MA: Addison-Wesley.Troelstra A S, Van Dalen D (1988) Constructivism in mathematics, Vol. 1 Amsterdam:

North-Holland.Van Dalen D (1986) 'Intuitionistic logic' in Gabbay D, Guenthner F (Eds) Handbook of

philosophical logic Dordreeht: Kluwer Academic Publishers, 225-339.

APPENDIX A: Using Classical Logic to Prove Theorems about Intuitionistic Logic.

Prove that ---,---,'v'x(Q(x) v ---,Q(x» (a double negation of an instance of the excluded middle

principle) can be falsified in Kripke (Van Dalen I986).semantics for intuitionistic logic.

Proof. Let us consider the model K: { 11'0, wI, 11'2, 11'3"" }, with an infinite number of nodes

11'0:5: WI :5: 11'2:5: ... . For every i, i>= 0, the domain D(w) is the set {O, ... ,i}. The formula Q(i)

can be asserted in every w;.j>i: at the node 11'0 no atomic statement that can be asserted.

We have the following heuristic scheme:11'3 {0,1,2,3} Q(O), Q(I),Q(2)

I11'2 {O,I,2} Q(O), Q(l)

IWI {O,I} Q(O)

I11'0 {O}

Let us suppose: w oll----,---,'v'x(Q(x) v ---,Q(x».

Then, ---,'v'x(Q(x) v ---,Q(x» cannot be asserted in any node ofthe model.

Thus, for every node wp, a node 11'" W, ~ wp , exists, such that'v'x(Q(x) v ---,Q(x» can be asserted

in Wt. Let us suppose that 'v'x(Q(x) v ---,Q(x» can be asserted in wt.

In particular, we have: w,ll-Q(t) v ---,Q(t).

By construction Q(t) cannot be asserted in wI"

��

Systems of Logic

Moreover, w,II--,Q{t) does not hold, as, by construction: w .+.II-Q(t).

Thus, there does not exist W, ~ wp such that V'x(Q(x) v -,Q(x» can be asserted in Wt.

Thus, -,-,'17'x(Q(x) v -.Q(x» cannot be asserted in woo

II?

From a cognitive point of view, the main difficulty here concerns the possibility of using classicallogic (reduction to absurdity and the principle of the excluded middle) to prove results aboutintuitionistic logic, without provoking lack of coherence in the overall reasoning.

APPENDIX B: Modelling the Break of Lagrangian Mechanics with NewtonianMechanics

D'Alembert principle includes a contradiction, but classical mechanics does not trivialize

Let us consider a physical system of particles m.; i=1,2,....n.The virtual works principle states that:

LFj lir j = 0where: <'i'j: virtual shift of the particle m, that is a shift "without time"; F; the active forces.That is: in the equilibrium state the virtual work ofthe active forces must be zero.

In a dynamic state, (with impulses pj <m, Vi ) the principle above becomes the D 'Alembertprinciple:

Lj(Fj Pi) Sr,> 0

where: F, real forces; Pi opposite forces; (by the notion of virtual shift dynamics is reduced to

statics).The D 'Alembert principle allows us to define the Lagrange equations of the system:

!!-( oL J- oL = 0dt oqj oqj

where: L = T-V, T: kinetic energy; V: potential energy(see (Goldstein 1965)).

We remark that the virtual works principle is a very productive principle but it is a strongconjecture W.r.t. Newton Mechanics since the central notion of virtual shift implies a contradiction.In fact, by assuming that in Newton Mechanics only one absolute time t is admitted, we have:

orj virtual requires: !!- (Sr, ) =0dt

however, or, is a shift requires: !!- (8rj ) >0dt

thus, the principle must be formulated:

(Lj F; Orj = 0 ) /\ ( !!- (8r;) =0 )/\ ( !!- (8r;) >0 )dt dt

which implies a contradiction, since !!- (8r; ) >0 implies -, !!- (8r; ) = O.dt dt

Note. It could be observed that the common-sense use of the results of theoretical classicalmechanics must have some paraconsistent reasoning resources.

��

10

Learning Mathematics Using Dynamic Geometry Tools

Thomas LingefjardGoteborg University, [email protected]

Mikael HolmquistGoteborg University, [email protected]

We describe our work with prospective teachers who are learning to solvecomplex problems using dynamic geometry tools. In particular, we relatestudent experiences and discoveries as they investigate a problem based onMorgan's theorem. Dynamic tools facilitated the rapid construction ofmany examples that otherwise would have been too time-consuming, thusallowing students to concentrate on modeling and making conjectures.��

120

INTRODUCTION

Lingefjiird & Holmquist

Courses in mathematics are normally well structured and well articulated. The sequenceof the content is prescribed, and recommended textbook exercises follow a given methodor theorem. When creating a course in mathematics that is more open in terms of itsproblems, examination, and teaching and learning structure, it is important to justify itsbenefit for students and teachers. In this chapter, we examine the benefits of usingdynamic geometry programs to investigate complex problems. As teachers ofa course inmathematical modeling for the last 7 years, we have encouraged students to start solvingopen and complex problems early in the course and not merely in a final exam at the end.Our hope is that they wiIllearn more mathematics, learn how to write about mathematics,and learn how to defend solution strategies. Pollak (1970) argued that we seldomchallenge students to study a situation and try to make a model of it for analyzing thesituation.

A carefully organized course in mathematics is sometimes too much like ahiking trip in the mountains that never leaves the well-worn trails. The tourmanages to visit a steady sequence of the "high spots" of the naturalscenery. It carefully avoids all false starts, dead-ends, and impossiblebarriers, and arrives by five o'clock every afternoon at a well-stocked cabin.The order of difficulty is carefully controlled, and it is obviously a mostpleasant way to proceed. However, the hiker misses the excitement ofrisking an enforced camping out, of helping locate a trail, and of making hisway cross-country with only intuition and a compass as a guide. "Crosscountry" mathematics is a necessary ingredient of a good education (p.329).

Most students in elementary or secondary schools and in undergraduate levels at theuniversity experience mathematics as a subject where the task is to learn a specificmethod, to follow a given path, or to memorize some facts or procedures. Little time isdevoted to guessing, seeking patterns, or making and testing conjectures.

There is something odd about the way we teach mathematics. We teach itas if assuming our students will themselves never have occasions to makenew mathematics. We do not teach language that way... the nature ofmathematics instruction is such that when a teacher assigns a theorem toprove, the student ordinarily assumes that the theorem is true and that aproof can be found. This constitutes a kind of satire on the nature ofmathematical thinking and the way new mathematics is made. The centralactivity in the making of new mathematics lies in making and testingconjectures. (Garry, 1997, p. 55)

Modern visualization tools are powerful aides in helping students to make newmathematics. The availability of new tools for studying, learning and teaching geometryhas resulted in a virtual rebirth of geometry and its visualization not only at theuniversities, but also in elementary and secondary schooling. Some of the better-knowntools are Geometer's Sketchpad and Cabri-geometre, which are used around the world in

��

Dynamic Geometry Tools 121

both compulsory and tertiary mathematics education. These are complemented withpowerful curve fitting tools to help facilitate the modeling process, both as software forcomputers and in any graphing calculators.

With the emergence of dynamic geometry software, Euclidean geometry ingeneral, and later theorems in particular, have aroused renewed interest. Inthese 'rnicroworlds,' geometrical theorems can become much more thanpropositions waiting to be proven, they can become projects forinvestigation, which rely on the ease with which many instances of aproposition can be obtained, analyzed, measured, and compared. All theseat the service of conjecturing and testing (empirical 'proof). (Bruckheimer& Arcavi, 2001).

It is our belief that the assessment procedure in a course affects the way students study,learn and appreciate the content of that course. Assessment is probably the mostpowerful tool a teacher has for influencing the way students respond to a course andbehave as learners. With this in mind, we decided that for students to adequately explorea problem in an exam situation, they must be given an extended time period. We allowedour students several weeks.

MATHEMATICAL MODELING

The possibility of visualizing mathematical concepts and experimenting with numericaltools allow us to go beyond traditional deductive reasoning methods and to use modelingfor solving geometrical problems. deLange (1996) describes the modeling process as ahorizontal and vertical organizing and structuring activity called mathematization. Weconsider several of these components to be of great importance when students areworking at solving geometrical problems in the presence of technical tools. In order tostrengthen the visual rendering, in Figure I we have positioned the elements of deLange's scheme (1996, p, 69) along horizontal and vertical axes.

At Goteborg University, students preparing to become teachers of mathematics andnatural science for Grades 4 to 9 or for the gymnasium (Grades 10 to 12) takemathematics courses that are offered either by the department of mathematics or by thedepartment of mathematics education. One such 5-credit course called Geometry andMathematical Modeling entails 5 weeks of full-time study and contains a strong emphasison the use of graphing calculators and a variety of software. The mathematicalbackgrounds of the students may vary, but at the present time, most have had courses innumber theory, Euclidian geometry, linear algebra, and real analysis, which correspondto approximately 30 weeks of full-time studies. Before entering the teacher program, theystudied mathematics for 12 years, including 9 years in the compulsory school and 3 yearsin the gymnasium. Although they may have experienced reasoning and problem solvingin geometry in several different courses, they have usually not taken any specificgeometry courses. Taking into account their background in mathematics, the students inthe class of fall 2001 were given an examination problem known as Walter's Theorem.

��

122 Lingefjard & Holmquist

Activities ('rmtnining vertical component«

GencmlizingFonnulnting a new m:"hcm.1tic,,1conceptCombining and ill1 cgral inll modd~Usin g diffcrelll modelsRelining and adjust ing modd~

Pro,'ing regularitiesReprescntmg a retation in :1 I"nnnw

Tra nsferring a real world problem 10 a known nunhemalical modelTra ns ferring a real world problem 10 :1 nUlthCntllical problem

Re~"()gnizin!t isomorphic ;l!;JX'C1S in diffcre nt problemsDiscovering rcgularuics

Disco\cnng relationsFormulating a l'd \'isllalizinjl ' I problem in different \,,~ S

Schc nl;llizingIdel\l if~ i ng lhe spec mc m,"hcmati cs ill a gi,en co ntext

Activities containing IlOriUlnfttl components

Figure I: The mathematization process .

WALTER'S THEOREM

Walter's theorem (after Marion Walter) was first presented in the November 1993 issueof the Mathematics Teacher in the section called Reader Reflections (Cuoco, Goldenberg,and Mark, 1993). It is presented in Figure 2.

Area (Triangle ABC )= 13,87 square em

Area(Hexagon) = 1,39 square em

(Area (Triangle ABClYArea(Hexa9on) = 10.00

A

B"""'::::-- - - L

If the trisection points of the sides of anytriangle are connected to the opposite vertices,the resulting hexagon has area one-tenth the areaof the original triangle .

c

Figure 2: Walter's theorem.

��


Marion Walter's theorem was mentioned again in the May 1996 issue of the MathematicsTeacher in an article about Morgan's theorem (Watanabe, Hanson, and Nowosielski,1996). The article tells the story of young Ryan Morgan, a ninth grader with a goodmathematical sense and a desire to explore a problem to its limits. When his mathematicsteacher (Frank Nowosielski) presented Walter's theorem to his class in the fall of 1993and asked them to see if it would hold for various types of triangles, Ryan Morgan wasnot satisfied with just verifying Walter's theorem. Ryan was interested in finding outwhat would happened if the sides of the triangles were partitioned into more than threecongruent segments. According to the article, Ryan and his teacher called the process ofdividing a side of a triangle into n congruent segments n-secting, and using Geometer'sSketchpad, Ryan experimented with different n-sections (Watanabe et al., p. 420). Thearticle further explains Ryan's methodology and where it led him. Not surprisingly,Ryan was invited to present his conjecture at a special mathematics colloquium atTowson State University in 1994.

We became so fascinated by this ninth grader and his lust for exploring and seekingpatterns, that we decided to give the very same problem to the students in our modelingclass. As one out of three problems in a take-home exam, we presented the followingproblem in December 2001 to 37 pre-service mathematics teachers.

EXAM PROBLEM: TRIANGLES

Work on your constructions and investigations using geometry tools. Your report shouldbe detailed and accurate. Use solid arguments for the conclusions you present in items athrough e.

a) Given an arbitrary triangle L'1ABC. Trisect (in congruent segments) each side in thetriangle. Chose for each side the first trisected point (clockwise relative to thevertices A, Band C) and connect each such point with the opposite vertex.Investigate the relation between the area of the central triangle that results from thisconstruction, and the area of the triangle L'1ABC.

b) Given an arbitrary triangle L'1ABC. Trisect (in congruent segments) each side in thetriangle. Connect the trisection points to the opposite vertices. Investigate therelation between the central figure's area and the area of the original triangle L'1ABC.

c) Given an arbitrary triangle L'1ABC, five-sect (in congruent segments) each side in thetriangle. On each side of the triangle connect the two central points of sectioning tothe opposite vertex. Investigate the relation between the resulting central figure'sarea and the area of the original triangle L'1ABC.

d) Using the same construction as in c, divide each side in the triangle into n partswhere n is and odd number larger than 5. Investigate the relation between the centralfigure's area and the area of the original triangle MBC for each such selected valueofn.

��

124 Lingefjard & Holmquist

e) Repeat the procedures in c and d but connect instead the two outermost points ofsection on each side to the opposite vertex.

Results and ConclusionsWe wanted our students to use their mathematical knowledge together with availablesoftware (such as the Geometer's Sketchpad and CurveExpert) and with graphingcalculators to explore and investigate this problem. They were expected to use bothvisual and algebraic methods, try out ideas, seek mathematical connections, look forpatterns, make conjectures for verification, and find good arguments to support theirconclusions. We found that this way of working with a geometrical modeling problemput new demands on the students' thinking ability. They needed to think like a modelerand to discuss patterns and structures.

Out of the 37 students who were given the problem, 12 did not pass, 22 passed, and 3students scored a high pass. The 3 students who earned a high pass were the only oneswho managed to come close to the solution of Ryan Morgan. It is worth noting that thesestudents became unrestrainedly fascinated with the problem and with the connectionsthey discovered. One of them remarked:

My finding says that the relation between the two areas when one dividesevery side in the triangle in n sections is (9n2

- I)/8, a truly amazing findingwhich I unfortunately haven't been able to support by a strict mathematicalproof.

Among the students who earned a high pass, a common characteristic was their desire fora proof of some sort. They seemed to want to find some evidence beyond the capacity ofthe software. It is an interesting dilemma that the formal concept of proof is not possiblein dynamic geometry software.

While dynamic geometry software cannot actually produce proofs, theexperimental evidence it provides users with produces strong conviction,which can motivate the desire for proof (King & Schattschneider, 1997, p.xiii),

Analysis of student solutions also revealed that they used elements of both vertical andhorizontal mathematization. Some of these components seemed more important thanothers. Important horizontal components were formulating and visualizing a problem indifferent ways, discovering relations, and discovering regularities. The prominentvertical components were representing a relation in a formula, proving regularities,refining and adjusting models, and generalizing.

It is difficult to assess and grade open-ended problems, especially when the students havea take-home exam. Besides valuing the result a student presents as a solution, the teachermust also appraise the quality of the report, the way the student communicates themathematical content in written form. Consequently, it was not enough to just presentthe relation (9n2

- I )/8. An equally important criterion was the way the studentsdescribed their solution process. We explained to our students that open-ended problems

��


are used in the compulsory school where the scoring criteria are those used on theSwedish national tests in mathematics. Our students were also encouraged to do peerassessment and to develop their own criteria for assessment. It is curious to note thatthey ended up with criteria similar to the national standards.

Some questions considered in assessing student work on Walter's theorem were:• Did students' attempt to search for a pattern?• What were students' ideas about the n-secting the side of a triangle?• What was the quality of their mathematical language?

We expected that students, who had worked with the same problem over a long time,would be able to engage in meaningful discourse. However, this was not the case.Students who were not successful in finding a pattern or structure when working with thedifferent triangles, obviously failed in some aspect of both horizontal and verticalcomponents of the mathematical modeling process. Although they arrived at the samemeasurements, they could not make reasonable conjectures. For example, one of thestudents who did not pass the investigation wrote:

I have investigated a variety of triangles, but I don't see any relation hiddenin here. When n=7, the larger triangle is about 55 times as big. Whenn=II, the triangle is about 136 times as big, and when n=IS, the triangle isabout 253 times as big. Well, so what? What am I supposed to see?

Nevertheless, we saw good opportunities for students to learn more mathematics anddeepen the understanding of mathematics they have already learned by engaging in thiskind of problem-solving activity. It seems to be of major importance that the studentsconstantly train themselves to think in a modeling way and thereby enhance thepossibility of using their mathematical knowledge to solve new problems. However, theinvestigation of an advanced mathematical problem such as the triangle problem we havepresented, would not be feasible without the aid of modem technical tools. Because thestudents were able to generate examples so quickly and easily with the dynamic geometrytools, they were able to concentrate on the tasks of seeking patterns and makingconjectures, rather than on tedious process of producing examples using straight edge andcompass constructions.

REFERENCES

Bruckheimer M, Arcavi A (200 I) 'A Herrick among mathematicians or dynamicgeometry as an aid to proof International Journal ofComputers for MathematicalLearning (6) Dordrecht: Kluwer, 113-126.

Cuoco A, Goldenberg P, Mark J (1993) 'Reader reflections: Marion's theorem'Mathematics Teacher 86(8),619.

de Lange J (1996) 'Using and applying mathematics in education' in Bishop AJ,Clements K, Keitel C, Kilpatrick J, Laborde C (Eds.) International handbook ofmathematics education Dordrecht: Kluwer, 49-97.

��

126 Lingefjllrd & Holmquist

Garry T (1997) 'Geometer's sketchpad in the classroom' in King J, Schattschneider D(Eds.) Geometry turned on: Dynamical software in learning teaching andresearch Washington DC: Mathematical Association of America, 55-62.

King J, Schattschneider D (1997)' Preface: Making geometry dynamic' in King J,Schattschneider D (Eds.) Geometry turned on: Dynamical software in learningteaching and research Washington DC: Mathematical Association of America, ixxiv.

Pollak HO (1970) 'Applications of mathematics' in Begle E (Ed.) The sixty-ninthyearbook of the National Society for the Study ofEducation Chicago: Universityof Chicago Press, 311-334.

Watanabe T, Hanson R, Nowosielski F (1996) 'Morgan's theorem' Mathematics Teacher89(5),420-423.

��

11

Modelling Search Algorithms

Albert FasslerHochschule fuer Technik und Architektur BiellBienne, [email protected]

In this paper I begin with a combinatorial search problem that presents asuitable challenge for students. Introducing abstract notation andconsidering limiting conditions, I show how this problem can lead to aclass of problems, conditions for their solutions to exist, and optimalalgorithms for their solutions.

��

128

A SEARCH PROBLEM

Fassler

There are n coins, among which at most one is forged. The weight of the coins may becompared by using a beam balance, shown in Figure I.

//

(

Figure 1. A balance beam and some coins .

How do you proceed in order to decide in a minimum number of weighing s, in a worstcase scenario, whether or not there is a forged coin?If there is a forged coin, which one is it? And is it lighter or heavier than the others?

Modelling with Stepwise AbstractionLet w be the unknown number ofweighings required to find the solution.We number the n coins : 1,2,3 ... • n.The set of all possible (2n+ I) solutions can be described as follows:{I ',1 \2',2+, ... ,n',n+,OI2' means: coin number 2 is the solution and it is light, 0 means that all coins are"normal" .What are the outcomes ofw measurements?They can be represented abstractly with vectors or lists oflength w of three elements e, d,u. Here is an example: (e, e, u, e, d, .. ., u). This means, that the first and second measureare e (= equilibrium). the third u (= up), the fifth d (= down), etc., as indicated in Figure2.

e u d

Figure 2. Balance beams showing conditions e, u, and d, respectively .

Now we reach a peak of abstraction : An algorithm is a mapping from the set M of allpossible outcomes (vectors) of the measurements onto the set of all possible solutions S,as shown in Figure 3.

��

Modelling Search Algorithms 129

3"vectorsAlgorithm

Figure 3. Defining an algorithm.

Necessary Conditions: InequalitiesAs a consequence, the cardinality of the set M has to be at least as large as the cardinalityof the set S:

3w::::: 2n+1 (A)

This condition is necessary. It gives a minimal lower boundary for the number ofweighings w in case of n coins. It is not known at this stage, whether or not it is alsosufficient. In other words: nothing is said about the existence of a possible solution. Thisquestion will be treated in the next sections.

Let us also consider two modified problems:If it is known, that there exists exactly one forged coin among the n coins, we have thecondition

~:::::~ ~If it is known, that there exists exactly one forged coin among the n coins, which is toolight, we have the condition

3w ::::: n (C)Such inequalities are essential to answer questions about the complexity of an algorithm.

The Example with 12 Coins

The inequality 3w ::::: 2*12+1 = 25 implies w ::::: 3. It turns out that the problem can besolved with w = 3. Because 33 is quite close to 25, the problem is not too easy.Here are the essential hints for the solution:First of all, we introduce the following notations:If a coin is a candidate for being light, we use the symbol L.If a coin is a candidate for being heavy, we use the symbol H.If a coin has standard weight, we use the symbol S.Ifwe have no information about a coin, we use the symbol X.

First measurement: Take four coins on each side of the balance. Ifnot in equilibrum, wehave four candidates for light, four candidates for heavy and four for normal coins: 4L,4H,4S.Second measurement: Take LLH to the left side and HLS to the right side of the balance.

Have fun finishing this case with an appropriate third measurement and work out thecomplete solution analyzing all combinations!

��

130 Fassler

How About 13 Coins?Is there a solution for w = 3 with n = 13 coins? It turns out, that in a way the answer isyes and no.First of all, we realize, that our necessary condition is just fulfilled at the limit:

33=27~2*13+1 =27This is a limiting case. If we would start with four coins on each side, we could not solvethe problem:In the case of equilibrium, we would have 5 coins left. As a consequence, we shouldhave to fullfill the condition 32

~ 2*5+1 for the remaining two weighings, which isimpossible.If we started with five coins on each side, again, a solution is impossible. In theunbalanced case, we would be left with 5L and 5H. But with 10 possible solutions, thecondition 32 ~ 10 is not fullfilled.

However if we have an additional 14th reference coin S (which is known to be standard),a solution for w = 3 exists! Here is the hint for the first measurement:Take five coins on each side, among them the reference coin. Again, have fun solvingthis harder problem for all possible combinations. We know it is right at the limit ofsolvability.

The Limiting Cases of ProblemsBy limiting cases we mean problems in which the inequality (A), (B) or (C)corresponding to the full problem or to at least one of the partial problems turns into anequality.

For better reading, we introduce the following suggestive notations that are inspired bythe three inequalities:A(n,w) denotes the weighing problem for n coins, all identical except for at most one andfor which at most w weighings are needed to decide whether or not there is a forged coin;and if there is one, to determine which coin it is and whether it is heavier or lighter thanthe others.B(n,w) denotes the weighing problem A(n,w) modified as follows: It is known thatamong the n coins there is exactly one that is forged.C(n,w) denotes the weighing problem B(n,w) modified as follows: It is known that theforged coin is heavier than the others.

If in a given problem an additional (n+ l )" standard coin is at disposal, we use a thirdparameter IS, such as in B(n,w,IS).

Examples of Limiting Cases

Problem B(4,2) obviously is not solvable, if the first weighing with one coin on each sideis balanced. On the other hand, B(4,2,IS) is solvable, with XX vs. XS in the firstweighing.Problem B(l3,3) is not solvable, because:If the first weighing 4X vs. 4X is balanced then the remaining partial problem B(5,2)must satisfy 32 ~ 2*5.

��

Modelling Search Algorithms 131

If the first weighing 5X vs. 5X is not balanced then the remaining partial problemC(1O,2)would have to satisfy 32 2:2*5.

Problem A(13,3,IS) has been solved earlier.

Problem A(40,4, IS) is solvable:Let the first weighing be 14X vs. 13X+ISIn the case of an equilibrium, the remaining problem is A(13,3,IS), which has beensolved earlier.If the left side is heavier, then 14H+l3L remain. This is Problem B(27,3), a limitingcase, because 33=27.

The second weighing is 4H+5L vs. 5L+4H.In the case of an equilibrium, 3L+6H remain. The third weighing 3H vs. 3H finishes theprocess.Ifin the second weighing the left side is heavier, then 4H+5L remain.The third weighing LLH vs. HHL finishes the process. Similarly, if the right side isheavier.If in the second weighing the right side is heavier, then 13H+14L remain. Proceed aspreviously.

It may surprise a non-expert that A(4,2) is not solvable while A(40,4,IS) is.

Algorithms for Large nHere are the solutions of the problems A(n,w) and B(n,w):If we take approximately n/3 coins on each side, we reduce the number of solutions fromapproximately 2n to 2n/3 with the first measurement, because(a) In the case of equilibrium, we reduce the problem to the remaining n/3 coins, but wehave no information about light or heavy. Therefore, there remain approximately 2n/3possible solutions.(b) If the first weighing is unbalanced, approximately 2n/3 coins are left for a possiblesolution. But we know which coins are a candidates for light and heavy. Thus again, wehave approximately 2n/3 possible solutions.

By proceeding recursively in this way, we get that each measurement reduces the numberof possible solutions by 113. Therefore the algorithm is asymptotically optimal.

The problems C(n,w) are left to the reader as an exercise.

Another way to look at the problem is from the viewpoint of information theory. Themeasurements must be done in such a way, that as much information as possible isgained.

DIDACTICAL EXPERIENCES

In view of the ubiquity of this problem in books and websites, it is not surprising thatadults, adolescents and children alike find pleasure in tackling combinatorial challenges

��

132 Fassler

like these. This holds for pupils and students of any field, including mathematics. Infact, it holds for everybody.

Here are two extreme cases:In my math course for highly gifted students aged II to 13, two out of fourteen studentsindependently solved Problem A(13,3,1S).

One day a mathematician phoned me, because he needed a hint to solve one of theweighing problems I had posted in our school bulletin.

The simple inequalities that serve as necessary conditions for the solvability may be newto the reader. They are a good means to discuss with students of all levels the simple factthat solutions to a problem may not exist. In the history of mathematics, it is only inmore recent centuries, that Evariste Galois brought to our attention the fact that problemsmay not have solutions. He stated, for example that, in general, it is impossible to solve apolynomial equation of degree greater than 4 by a formula containing radicals. If onlythe people who spent years trying to square the circle had known about the non-existenceof solutions! Furthermore, describing the solution deductively requires skilful andabstract coding of various steps. Other than that, my observations have confirmed thatemotions and an element of fun play a vital role not only in solving riddles, but also indoing mathematics in general.

PERSONAL REMARKS

I was confronted with the topic treated here for the first time when I was an adolescent.A classmate posed to me Problem A(l2,3), offering me 10 Swiss francs for solving it. Atruly remarkable prize in those days! When 1 first tried in vain to solve it, 1 asked myclassmate where he got the problem. From a professor, whom he had met by coincidenceat the coffee shop, was his answer. The crucial idea for solving the problem suddenlyoccurred to me when I was riding my bike home, after having failed for hours to solve theproblem by pencil and paper. Only later, I was told that my classmate had been offered20 francs by the professor for solving the problem. My classmate became a businessman.

Riddles like the one described intrigued me and I studied mathematical tools so that Icould solve them. They served as a crucial motivation for me to take up studies inmathematics.

Why was the bike ride such a key experience to me? In my professional work as amathematician, I have often had the experience that it was only after intense work withpaper and pencil, or on the computer, that the crucial ideas occurred to me at momentswhen I was left on my own, deprived from all tools.

��

12

Mathematical Modelling in a Differential GeometryCourse

Adolf J I RiedeUniversity of Heidelberg, [email protected]

This paper describes how a critical study of traditional curves of pursuitleads to new curves that reflect a pursuing process in a more realistic way.Also it contains an investigation of the design a railway route in a realmountain as a curve of constant slope. As a side result, the model yields amethod for constructing contours of a topographic map. The second topicshows that good mathematical comprehension is necessary in order to usethe computer in a reasonable way.

��

134

INTRODUCTION

Riede

Students took part in two projects while taking a course on differential geometry. Thefirst one dealt with curves of pursuit. Modem methodology of mathematical modellingprovided two new models of curves of pursuit. This methodology requires that we dealwith realistic situations. Therefore the second project investigated the design of a railwayroute in a real mountain. Also in this paper a mathematical method is described fordesigning topographic maps. This example demonstrates that notions and theorems ofpure mathematics have not only academic value, but are essential in practice too.Conversely, practice promotes motivation for pure mathematics. Namely, calculatingcontour lines for a topographic map applies a sophisticated theorem in a student's firstyear analysis course, the implicit function theorem.

Laymen and especially students at the beginning of their mathematics study often askwhat a professional mathematician is working on. These models present some examplesand may stimulate interest in mathematics and its applications. The paper also takes alook at the didactical value ofa CAS, namely, Mathematica.

CURVES OF PURSUIT

The ProblemAs a project in this course, students studied curves of pursuit. This section shows howthe methodology of mathematical modelling and the use of a computer algebra systemcan lead to an exiting engagement in mathematics. As background, students readSchierschers article (1997) on the historical scientific and didactical aspects of the topic.As Schierscher reports, the problem was formulated for the first time in 1732 by Bouguerin the following form: Two ships cruise at constant speed on the high seas; one of themsteadily sets course towards the other which moves in a straight line. What curve doesthe pursuing ship trace?

One hundred years later the same problem was investigated, but instead of a shippursuing a second ship a dog was running towards his master. The pursuit curve becamethe dog's curve. With today's weapons that set their course automatically, the curves ofpursuit become cruel reality.

Mathematical ModelsThe mathematical situation: Figure 1 shows a mathematical overview of the situation andthe notations used. The orbit of the pursuer is denoted by h(t) = (x(t),y(t», which isplotted as a thick line, with the time t as parameter. The absolute value of the velocity h'is v = WI where the apostrophe means differentiation with respect to t. The vector wpoints from the pursuer to the pursued. At time t = 0, the pursuer is positioned on the y-

axis at the point a > 0. The orbit of the pursued on the x-axis is Z(t) = (Xz (t),O) with

Xz = u t and constant speed u.

The classical model: In the derivation of the classical model it was assumed that thevelocity h' was directed from the pursuer towards the pursued and had constant absolute

��

Modelling in Differential Geometry

Figure 1. The notations.

135

value v. Then the classical differential equation of the dog's curve could be set up andlead to a formula for its solution.

Discussion of the assumptions: Usually the next point in a lecture on differentialgeometry is to calculate the curvature of the curve of pursuit. The curve of pursuit that isnot a part of a circle has non-constant curvature. This makes one doubt the assumption ofconstant velocity. For, if the curvature of the orbit is varying it will take a varying forceon the part of the pursuer to get the bend and he will have more or less force left to speedup. Therefore the speed must depend on the curvature. So differential geometricalreasoning is needed at the very first beginning of the modelIing procedure. In order to beconvinced of this fact let us regard the motion on a circular orbit: h(t) = (r cos(at), r

sin(at», r,a>Oconstant. Let K=c, T+c z N bethesplittingoftheforceKofthe

dog in a tangential and a normal component, where T = h'I I h'l is the tangential unitvector pointing in the same direction as h' and N =J(1) is the normal vector; here J isthe rotation in the mathematically positive sense by 90 0. Let c = IKI. The frictional forceR is modelled by R = -b h' with the frictional coefficient b > 0, According to Newtonianmechanics we get the equation: R + K = m h"; where m is the mass of the dog, orrespectively, the mass of the pursuing ship. It folIows: c

2=mra 2

, and <bra-v c, =0.

Inserting a = vir and r = Ilk (k = curvature) we get C2 = mkvi, and -bal k +c j = O.

The solution of these equations for the variables v, Cjand c2 shows how v depends on k.

This is visualized in Figure 2 where the speed v is plotted against curvature k for circularmotions.

D.! 0.2 OJ1

0.4

Figure 2. Speed versus curvature.

��

136 Riede

At this stage the students were asked to discuss the procedure thus far. They did not onlyquestion the assumption v = constant, but they said that the assumption that the velocityh ' was directed towards the pursued could be accepted in the dog problem but not for theship. They found that it was not a favourable strategy for a ship.

Finally we agreed to replace the assumption v = constant by the assumption that C =constant, since the dog runs with his maximal force towards his master (or biker orjogger) and does not get tired within such a short distance. In a similar way one ship willpursue the other one with full and hence constant power. In addition to the classicalassumption concerning the direction, a new one has presented itself: (a) h' is directedtowards the pursued or (b) K is directed towards the pursued.

~ = x - ut ,with time t as parameter.y' y

with time t as parameter; (2) dx = x - ut, with y as parameter;dy y

The model using assumption (a): Taking the vector product of both sides of theNewtonian equation

(*) -bvT + c1 T + c1N = mh"

with h' on the left yields the equation C2 h ' x N = m h' x h". Here Ih' x NI =Ih'IINI = v, because the ordered pair of vectors (h ',N) is positively oriented, Ih'l = v

and INI = 1. So we find.c, = (m/v) I h ' x h"l and C1 =~Cl -c/. Ifwe

inserted these formulas for C 1 and C 2 into equation (*) we would get a horrible implicit

differential equation because c2

contains h". A teacher at this point needs to proceed

pedagogically in order to encourage the students to continue studying the problem. It isvery important for students studying Mathematics to learn not to give up when problemsarise, even when the problems seem to be unsolvable. Frequently this indicates that awrong strategy has been chosen. A different choice may have prevented from suchhorrible difficulties. Pointing out such non-mathematical skills stimulates people to keepon doing mathematical modelling. In our case we must realise that we have not yet usedthe assumption (a). There are four possibilities for modelling this assumptionmathematically:

(I) T = (tu,O) - h(t) ,l(tu,O) - h(t)1

(3) dy =_y_, with x as parameter; (4)dx x-ut

At this stage the search for alternative mathematical formulations justifies itself as animportant modelling skill. It is only in the fourth formulation that we easily get anexpression containing h' x h", namely by differentiating (4) with respect to t. Thus we

can eliminate h" in C2. We get Ih"xh'l y(x'-u) - y'(x - ut) and insert this in·2 2Y y

and the formulaThis formula forC · (m)y'(x-ut)-y(X'-U),22' C2 = - Y .

V y2

('1 = ~""C-2-_-(-'1-1 inserted in (*) is the differential equation that models the assumptions.

��

Modelling in Differential Geometry 137

A student argued that it was the wrong way to insert the first formula for c2

into (*)

because the formula was derived from equation (*). Indeed he had detected that the firstreasoning was heuristic. Nevertheless, in combination with equation (4) the heuristic firsttry led to a good model.

The following figure shows a solution. It looks qualitatively like the classical curve ofpursuit. A comparison with the classical curve, which has the same starting point and thesame point where it meets the pursuit object, shows that the classical one has the largercurvature. By using the new differential equation we have thus detected a means to findless curved and shorter curves of pursuit. A posteriori the investigation has provided auseful purpose. This fact is demonstrated in Figure 3 where the thick line is a new curveof pursuit.

Figure 3. Under assumption (a).

The model under assumption (b): Setting K = c (tu,O) - h(t) and inserting this expressionl(tu,O) - h(t)1

into (*) provides the second new differential equation describing curves of pursuit. InFigure 4, the curve of pursuit is plotted. Since an animation is not representable in abook the straight lines from the pursuer to the pursued are also drawn for some discretevalues of time. The dots indicate the position of the pursued when the pursuer passes theother endpoint of the line.

Figure 4. Under assumption (b).

In the case of a dog pursuing a biker and directing his force towards the biker, Figure 5shows that the dog misses the biker and again can approach the biker but this time fromthe opposite side. You may have noticed that sometimes a dog pursuing a biker changessides. Certainly this cannot be seen as a clever manoeuvre of the dog. He only acts

��

138 Riede

according to his hunting drive. The conjecture is that this model shows that the dogmisses the biker because of the inertial or centrifugal force, which pushes him out of thebend.

Figure 5. The dog changing the sides and overtaking the biker.

Realistic data: The students' task was to find realistic parameters for the dog problem onthe web. Depending on the breed of dog, it can be found that a dog needs about 4.5 secto cover 50 m.

Flying sport: If a pilot (as a beginner) sets his course steadily towards a certaindestination while wind drives him sideward, the airplane follows a curved line. Thenflying experts joke: "He flies a dog's curve." In fact, this notion of a dog's curve is thesame as the one we studied under assumption (b) if we use a Cartesian coordinate systemin the air that moves with the wind. Take the destination point as the origin and the yaxis orthogonal to the speed of the wind. Denote the absolute value of the speed of thewind by u; assume that the speed of the wind is constant. Force K directed towards thedestination is modeled by K = -C h(t)~h(t)I. For the frictional force R the speed of theairplane relative to the air has to be taken into account: R = -b(h'(t) - (u,O». We get the

differential equation:(*) -b(h '(t)-(II,O» - c h(t)~h(t)1 = m h "(z)

Let (x,y) denote the coordinate system that moves with the wind. Then we have the

coordinate transformation (x,y) = (x,y) - (II,O)t. Let h(t) denote the orbit with respect

to the coordinate system (x,y), we get: h(t) =h(t)-(ut,O) , h'(t) =h(t)-(u,O) ,

h"(t) =h"(t). Insertion of these equations into (*) provides equation (I) for the (x,y)system:

- bh'(/) +C«lI/,O) - h (1» /I(11/,0) - h (I) 1= mh"(t) (I)

CURVES OF CONSTANT SLOPE

Review of Theoretical BackgroundWe denote a space curve by h: I ~ RJ with I an interval and arc length s as a parameter.h(s)=(x(s),y(s),z(s». According to the main theorem on space curves the curvature k andthe torsion r determine h up to proper motions of Euclidean three-space. A basic class ofcurves consists of those curves whose quotient t / k (k *-0) is constant. This conditionmeans that the angle between the tangent vector h'(t) and a suitable fixed vector v is

��


constant. If we orient the curve so that v. is vertical, it is a curve of constant slope . Suchcurves have to be determined for the design of a road or a railway route in the mountains.Locally a surface in Euclidean three-space can be approximated by a paraboloid witharbitrary precision. Thus a mountain pass can be approximated by a hyperbolicparabolo id. Thus, in order to learn how to design curves of constant slope for a realmountain, it seems a good exercise to compute such curves on a hyperbolic paraboloid.

Curves of Constant Slope on a Hyperbolic ParaboloidWe used slope a = 0.4. The first differential equation is easily found: z ' = a. The saddlesurface is represented by the equation

(*) z =cx 2 + dy' , c > 0, d < O.

The second equation emerges from the fact that the desired curve should stay on thesurface . Insert ing x = x(s), y = y(s), z = z(s) in (*) and differentiating with respect to syields the differential equation: z' = 2 c x x ' + 2 d y y '. Since s is arc length, the tangentvector has modulus I and we obtain the third equation: X·2+y .2+Z·2 = 1

These implicit differential equations can be transformed into explicit equations using aprecise by hand procedure and using the CAS for only some of the steps. We used c = I,d = -1. The result is shown in Figure 6. Changes of the signs occur for example on thegradient curve that starts at the saddle point and on the ridge curve marked in Figure 6with little lines.

Figure 6. Curves of constant slope on a hyperbolic paraboloid.

Experiences with the CASThe first sense of delight on seeing a curve plotted on the surface was lessened becausethe curve ended after a short distance. The first explanation for this was that the curvehad reached such a flat part that there did no longer exist a curve of slope a. Proceedingfurther we observed a curve that left the surface. The students settled this problem asfollows: The transformation into explicit differential equations was not unique . Therearose a choice between two signs and it had to be decided which was the right one. Sign I= + <::> y'<ady /(2c 2x 2 + d ' y2) > 0, sign2 = + <::> x ' > O. This might also be the

reason why the CAS does not solve implicit differential equations by transforming theminto explicit ones. The computer cannot know which case the user wants to calculate.On the one hand the user influences the signs by choosing the way to the right or to the

��

140 Riede

left downwards at slope a. On the other hand the signs are influenced by the fact that thetangent vector to the curve must also be tangent to the surface. In addition it turned outthat the CAS cannot convert equations (I), (2), (3) in the variables x ', y' and z' to getexplicit differential equations. In this example the use of a CAS only leads to correctresults if the user has a good understanding of the mathematics of the situation.

Realistic SituationAt first sight it seems very doubtful that one can construct a railway route of constantslope in a real mountain by the same method. Fortunately, Weidig (1994) showed byanother method how to get a piecewise linear approximation to such a curve. This isshown in Figure 7. This encouraged us to attempt a solution. We used the samemountain so we could compare our results with Weidig's result as a test of our procedure.

Figure 7. Topographic map with piecewise linear route.

Two-dimensional interpolation of point data: A point at height 324.0 m in thetopographic map (Figure 7) I:25000 of Germany, sheet 3925 (Westlicher Auslaufer desHarz), was taken as point (I, I) of a Cartesian coordinate system. Instead of the moredifficult approximation by paraboloids the surface was locally approximated by tangentplanes. As approximate tangent plane to the lattice point (a,b) E R2 we used the planethrough (a,b), (a+l,b) and (a,b+l) for a = 1,... ,7 and b = 1,... ,9. These approximatetangent planes were described by functions 7;,.h(X,y) and glued together with bell

functions u a.b : Put u(t):= (t + 1)2(1 _1)2 for ItI< I and u(t):= 0 for ItI> I and set7 9

ua.h(x,y):= u(!x-a 1,1 y-al) and K".t>(x,y):= u".h(x,y)/L LU;,/x,y)' then the K a.h;=1 j=1

form a partition of unity. Then the graph of the function7 9

G(x,y):= L LK".t>(x,y)T".t>(x,y) represents the mountain. The use of bell functionsu=1 h=1

and the partition of unity exhibits the height of the mountain at(xo,Yo)as a weighted

��


average of the height of all tangent planes Tu,b at (xo,Yo) in such a way that the height of

T,',b has higher weight if (a,b) is closer to (xo,Yo)'

Construction ofa railway route: As Weidig did, we constructed a curve ofconstant slope4% from an assumed city called Bergheim, and from the point where the landscapebecame too flat we extended the curve linearly as shown in Figure 8. We transformed to

explicit differential equations as follows. Let (*) be the equation x'= ±~l- a2_ y'2 .

Figure 8. Design of a railway route in a real mountain.

Calculate z=G(x,y), z'=G,x'+G,.y', z'-G,,y'=G,x', (a-G,.y')2=G;(l-a2_y'2).

The CAS solves the last equation with respect to the variable y'. We get a first explicitdifferential equation of the form y' = j(x,y). Inserting this in (*) provides the seconddifferential equation of the form x ' = g(x,y). Herefand g are certain functions calculatedby the CAS. The third differential equation is z' = a. Signs occur: signl = +~ y' -aGIo /(G; + G;) > 0, sign2 = + ~ x'> 0 .

In this procedure the number of calculations is very high. For a square region it growsquadratically with the length of the edge of the square.

The railway route in the map: With the aid of a CAS command it is possible to drawcontour lines G(x,y) = c = constant. This is very closely related to the theorem of theimplicit function. Thus elementary differential geometry provides motivation for thisimportant theorem. This method gives an idea how topographic maps can be designed.For this purpose geographers use a software called GRASS, which is described by Netelerand Mitasova (2002). Although the method used and the lattice of points were verycoarse the procedure provides a topographic map that is in principle correct and is fairlysimilar to the official one shown in Figure 9. Also the railway route as a differentialcurve of constant slope is similar to Weidig's piecewise linear curve.

��

142

CONCLUSION

Riede

Figure 9. Contour plot with curve of constant slope.

There are other problems in which mathematical models of surfaces as graphs of afunction or as contour plots are of interest. For example representing the surface of thecornea in order to create a means for better-adapted individual contact lenses. Stammlerand Buchsteiner-KieBling (1996) compute continuous contours and mention manyapplications especially to metal surfaces.

REFERENCES

Neteler M, Mitasova H (2002) Open source GIS: A GRASS GIS approach Dordrecht:Kluwer Academic Publishers.

Schierscher G (1997) 'Verfolgungsprobleme Mathematik und Unterricht 3, 49-78.Stammler L, Buchsteiner-Kiellling E (1996) Globale optimierung von niveaulinien

Heidelberg, Leizpig: Johann Ambrosius Barth.Weidig I (I994) 'Gebirgsbahnen - Ein Anwendungsfeld fllr den Mathematikunterricht in

Blum W, Henn W, Klika M, MaaB J (Eds) Schriftenreihe der ISTRON-Grllppe:Materialien fur einen realitdtsbezogenen mathematikunterricht, Band 1,Hildesheim: Franzbecker, 136-143.

��

13

Defending the Faith: Modelling to Increase the Accountability of Organisational Leadership

Peter Galbraith University of Queensland, Australia [email protected]

A situation has emerged in which individuals who do not claim particular mathematical expertise take elements of mathematical theories e.g. ‘Chaos Theory’ and apply them uncritically to support preferred policies or actions. The challenge this poses includes building models to test the nature of generalised claims, and hence to challenge recommendations and policies that can vary between the dubious and the dangerous. This paper tests claims arising within the field of educational administration through the design of a non-linear simulation model, illustrating a different role for mathematical modelling in an educational environment.

��

144 Galbraith

INTRODUCTION

Traditionally the emphasis in modelling and applications studies has been related to the teaching of associated skills to students within mathematics programs at school and undergraduate levels. This paper addresses a different challenge that has emerged as a consequence of the popularisation of mathematical developments such as ‘chaos theory.’ The audience for this endeavour continues to include those interested in using modelling to address issues arising in the real world. However, it also includes a new component - professionals working in the area of organisational leadership and management, and postgraduate students enrolled in courses with cognate interests, who are called upon to evaluate critically literature and policies placed before them. Thus the nature of the ‘real world’ motivation for the modelling activities is different, as is the type of educational outcome sought.

There have long been severe reservations about the ‘scientific approach’ to leadership and management in organisations, an approach, which sought to identify generalised skills that would provide the key to successful management practice. Such approaches typically adopted a positivistic stance within which the nature of an enterprise or the characteristics of individuals within it were deemed of small importance. So-called ‘paradigm wars’ emerged as the basis of such assumptions were challenged.

The publication of the Fifth Discipline (Senge 1990) provided an impetus for a reconceptualisation of organisational leadership. From our viewpoint the most interesting feature is that the foundational discipline of ‘systems thinking’ underpinning the approach is based upon an understanding and application of simulation models developed within the System Dynamics tradition. During the same decade we find that the popularisation of Chaos Theory in the wider literature and in the public consciousness has resulted in a number of writers attempting to apply its principles and insights to the field of organisational management and leadership in education and other enterprises. It appears that most of these writers have no particular expertise in mathematics, and some of the claims they made vary from the heroic to the outrageous. The existence of such claims provides the ‘real world’ motivation for the approach taken in this paper, which adopts the following structure.

Firstly the context will be set through reference to content from four articles from a leading international journal in the field of educational leadership (Journal of Educational Administration). This will be achieved through a consideration of selected texts, which will provide basic ‘data’ from which to motivate the modelling that follows. Secondly, specific assertions found in the articles will be used as motivation for designing a model through which the assertions may be tested. Finally implications of the model output will be used to review claims made with respect to leadership and management in organisations, and hence to reflect on the educational contribution of such modelling approaches.

��

Defending the Faith

Source: Reilly, D.H. (1999) Non-linear systems and educational development in Europe, Journal of Educational Administration, 37(5), pp. 424-440.

Table I Linear System Non-linear System Initial conditions Not important Very important Equilibrium Stability Chaos Prediction Deterministic Chance Feedback Negative Positive

A fourth and very critical difference ... is that feedback is negative in linear systems. In non-linear systems, feedback tends to be positive. Positive feedback in a non- linear system is the mechanism that serves on a continuing basis to actuate the difference between an initial condition of a system and a resulting one (p. 429).

Essentially a non-linear system demonstrates an irregular but oscillatory pattern of behaviour.. .There are four stages that range on a continuum from linearity and predictability through two stages of mixed linearity and non-linearity to a final stage of chaos where the behaviour is characterized by non-repeating periodicity @. 430).

Because the set of behavioural interactions cannot be predicted, it is not possible to accurately forecast future directions of the system, its behaviours, or their outcomes. Each of these characteristics and stages is related to current conditions of educational systems in both Eastern and Western European nations (p. 431).

,-

145

SETTING THE CONTEXT (LITERATURE SOURCES)

Source: Gunter, Helen. (1995) Jurassic management: Chaos and management development in educational institutions, Journal of Educational Administration, 33(4), pp. 5-30.

Chaos Theory allows us to see that education managers have a third choice to either stability or disintegration and that is to operate within “bounded instability.” A successful school or college would therefore operate away from equilibrium between stability and disintegration.. .The future is created by the sensitive response to fluctuations in the environment or the ‘Butterfly Effect’ - the flap of a butterfly’s wings could cause a thunderstorm in another part of the world (p. 14).

The butterfly effect allows us to recognise that one person can have an impact and therefore schools and colleges must tap into and encourage the whole skills of colleagues.. . When events or crises hit individuals and groups there is a spontaneous capacity to organise and respond (p. 15).

��

146 Galbraith

Source: Sungaila, Helen. ( 1 990) The New Science of Chaos: Making a New Science of Leadership, Journal of Educational Administration, 28(2), pp. 4-23.

The new science of chaos has alerted us to the butterfly effect; to the very considerable impact tiny fluctuations in a non-equilibrium system can have on its output. It is already generally recognised in the literature that it is the function of leadership to bring about qualitative change in the system. The new science of chaos suggests that the creative input of a single individual who is prepared to stand his or her ground can be enormously effective (p. 12).

If the leader is to succeed in reinforcing the fluctuations from within, to the point where the system is driven over the threshold into a qualitatively new regime, then the leader must also deal in culture; destroying old myths, stories, legends.. .and where appropriate creating new ones (p. 17).

Source: Sullivan, T.J. (1999) Leading people in a chaotic world, Journal oj Educational Administration, 37(5), pp. 408-423.

When some organisational groups accepted new influences, they also took on new expectations and acted different. In so doing they changed the expectations that other groups had of them. This process was repeated many times along the communication and action networks until radical deviations to the evolutionary paths of some groups soon emerged as a chaotic threshold. At some point near this chaotic threshold, the open dynamic system was dominated by a chaotic attractor, which literally attracted additional self-referential communicative influences around it. ... The chaotic attractor, on which so much self-referential communication was focused, was the system of policy ... The policy acted as a chaotic attractor by becoming the focus of attention for people in the organisation. ... Soon the various groups were unaligned and in a state of chaotic order. The system remained in this state until the dynamics finally stabilised into a new transformed order. The chaotic attractor (in this case the system of policy) was the power drive of change in the school organisation (p. 415).

Summarising the main features of these selections we note: Statements concerning the mathematical structure and behaviour of non-linear systems (e.g. Reilly). Description of the ‘butterfly effect’ and suggested implications for educational leadership (e.g. Sungaila and Gunter). Claims that chaotic modes both describe and offer opportunities relating to the general operation of educational systems - as for example in eastern, and western, Europe (e.g. Reilly). Identification of specific structures and behaviours alleged to represent manifestations of deterministic chaos within a particular organisation (e.g. Sullivan).

��

Defending the Faith 147

DESIGNING A MODEL

In order to examine issues such as those raised in the literature excerpts, we note that claims vary from the specific to the general, from detailed assertions about precise structures and behaviours, to arguments constructed at the level of metaphor. This then has implications for the kind of model needed to illuminate hrther discussion. A useful model should be capable of adding insight at both these levels. That is, the model should be of a non-linear system capable of testing specific dynamic assertions, but also able to generate behaviours useful to inform debates in which chaos and non-linear behaviours are invoked in a generic sense. A simple model to serve these purposes is described below. I t has been designed to incorporate aspects of the problem of matching teacher supply and demand.

Model Structure

delay (coume length)

new graduates seeking work (NGSW)

population (PUP)

Figure 1: Feedback Loop Structure

Figure I contains a representation of a simple model whose principal components are described below. The behaviour of a non-linear model such as the above is ultimately determined by the cumulative effect of interacting feedback loops. This simple model contains three negative (balancing) loops and a positive (reinforcing loop). The signs on the arcs indicate the nature of causal relationships. A + (-) sign indicates that the change or tendency to change in the variable at the head of an arrow is in the same (opposite) direction as the change in the variable at the foot of the arrow that is impacting upon it. In a balancing loop an initial change in a variable works its way around the closed circuit to eventually cause a change in the same variable in the opposite direction. In a reinforcing loop the final impact is in the same direction as the initial change. Loop B1: T+ SR+ ATT+UI-+NGSW+T An increase in teacher numbers reduces the shortage ratio, resulting in less attractive job prospects, leading to a reduction in university intake to teacher education courses,

��

148 Galbraith

resulting (after a delay) in fewer graduates seeking employment as teachers, leading to fewer new teachers being employed, and hence leading eventually to a decrease in teacher numbers. Loop B2: T+SR+VAC+NTE+T An increase in teacher numbers reduces the shortage ratio, resulting in fewer vacancies, hence to fewer new graduates being employed, leading to a decrease in teacher numbers. Loop B3: T+RES+T An increase in the teacher population leads to an increase in resignations, which leads in

turn to a decrease in the teacher population. Loop R1: T-+RES+VAC-+NTE-+T An increase in the teacher population leads to an increase in resignations, hence to an increase in vacancies, thence to an increase in new teachers employed, and thus to an increase in the teacher population.

The structure summarised in the loops is written precisely as sets of non-linear integral equations using specialised software such as Powersim. The model equations are solved iteratively to simulate behaviour over time. For present purposes interest is in using the model to test claims made about the structure and behaviour of non-linear systems, and resulting inferences concerning organisational leadership. To achieve this it is necessary to examine the model output in relation to the conditions under which it is simulated. For this purpose the time-histories of selected variables are plotted as shown in Figure 2(a), (b) below. The plotted variables are selected on two grounds. Firstly to enable the sense’of the model to be appreciated, and secondly to enable some of the claims made in the literature to be precisely addressed. It is important to recognise what the model does not claim to do. It does not set out to predict precise quantities in a supply-demand situation. Rather it enables (at the level of behaviour mode) some understanding of the behavioural consequences of changes in the operating environment.

The model is started in steady-state with a constant resignation rate of five percent of the teaching workforce, and a graduate supply tuned to replace them. The dynamics are activated by an increase in this resignation rate of two percent for a period of three years from time=3, after which it returns to the former value. Initial values of the variables are chosen arbitrarily, as this is a policy analysis model, not for purposes of point prediction. The graphs in Figure 2(a) then provide insight into the way the real world of supply and demand is reflected in the model. The change in the resignation rate triggers a shortage followed closely by an increase in advertised vacancies as shown in graphs 1 and 2. The enhanced teaching opportunities signaled result in an increased enrolment in teacher education programs with the subsequent response in new graduates seeking work (graph 3) delayed by the length of the training program (4 years). As the resignation ‘shock’ passes the system returns again to steady state. By way of linking with the real situation we note:

Surpluses and shortages tend to be cyclic. Short-term cycles are associated with the economic cycles, and longer-term cycles and trends experienced in Australia are associated with the attractiveness of teaching compared with alternative activities (Preston, 1997).

��


EXAMINING THE LITERATURE CLAIMS

We may divide the literature comment into two categories: that making general statements about the structure and behaviour of non-linear systems, and that making specific reference to chaos. Addressing the former we note in particular the various statements in the Reilly paper. Noting the summary in Table 1 we observe that linear systems in fact contain no feedback and hereafter confine our interest to non-linear systems. The system model in Figure 1 is strongly non-linear and contains both negative and positive feedback as illustrated in the description above. The behaviour modes of

-1- 1.061

-3- 350) 0 5 10 15 20 25 30

ycars

shortage-ratio

vacancies

-3- new-grad-seek-wrk

’2’3- 1 1.23-1.23- 1

I I I 0 5 10 15 20 25 30

ycars

I

5 10 15 20 25 30

years

Figure 2. (a), (b) Model output.

this model shown in Figure 2 (a) and (b) return to a state of stable dynamic equilibrium after the initial resignation shock has been dissipated. The graphs in Figure 2 (b) represent teacher numbers for three different simulations. Graph 2 is an output of the ‘standard run’ that also produced the results shown in Figure 2(a) for the other variables. Graph 3 in Figure 2 (b) was produced by changing the initial value of ‘new graduates seeking work’ from 375 to 350. The graph is virtually identical and this is a consequence of the policy that adjusts vacancies so as to eliminate a shortage or surplus in terms of the ratio of pupils to teachers. Subsequent university enrolments responding to demand eliminate the discrepancy between the initial conditions in the two runs. Consequently we note that sensitivity to initial conditions is not necessarily a property of non-linear system behaviour, and the behaviour is not necessarily chaotic. Run 1 in Figure 2 (b) has

��

150 Galbraith

been generated by stepping up the average yearly progression rate of university students through their degree, from 75 to 85 percent. The effect is to increase the responsiveness of the system by providing more graduates more quickly. This illustrates that sometimes a significant change in behaviour can be activated by a parameter change. Put simply, almost every generalised statement in the literature excerpt about the structure and behaviour of non-linear systems is wrong.

LOOKING FOR CHAOS

The general robustness of behaviour modes in non-linear complex systems has been illustrated above. However chaotic modes sometimes emerge, as shown in Figure 3 below. To generate this behaviour a particular ‘policy’ was enacted in the system represented by the model. This involved changing the time-scale for action depending on whether a teacher shortage or surplus was current. Under conditions of surplus the vacancy level was adjusted over a two year period and new teachers employed over the same time frame. Under conditions of shortage the period for action was set at three months. Thus the supply and demand system was subjected to a continuing series of sudden jerks that caused teacher numbers to fluctuate in a chaotic mode. (In order for the graphs to show adequate resolution only a five-year period has been selected for plotting.) Here we do note the sensitivity to initial conditions, as graph 1 (in Figure 3) was generated with new graduates seeking work set initially at 375, and the second graph with the corresponding initial value at 374. Notice the subsequent variations are random and unrelated, in contrast with the behaviour observed in Figure 2 (b). What then can we reasonably infer from the behaviours noted?

6,Ol

6,OO

2 0 6,OO

J 5.99

5,99

Q) 2=

25 I 26 27 28 years

29 30

Figure 3. Chaotic modes.

In fact we have an illustration of a general property of non-linear systems, namely, that chaotic modes emerge only within a restricted range of parameter space. As Anderson (1988) reminds us following extensive experimentation, “even in those systems that do contain chaotic modes the chaotic mode appears only elusively.” Furthermore, in this case the chaotic mode was generated by parameter values that had numerical rather than practical significance. Nothing approaching chaos emerged when policy actions remained in a normal operating range.

��


REFLECTIONS

It has been known for many years that “Complex systems differ from simple ones in being counter intuitive, i.e. not behaving as one might expect them to” (Forrester in Miller 1972). It seems likely that this property is mistakenly linked to the existence of chaos, which is then evoked indiscriminately to support a wide range of interpretations and claims at the level of metaphor for management and leadership in organisations. There is no basis at this for ascribing properties of education systems and school operations to manifestation of chaos in action, or to suggest that “a successful school or college would therefore operate away from equilibrium between stability and disintegration.” Nor to ascribe disagreements within a school to the operation of a ‘chaotic attractor’ or to suggest that the science of chaos indicates “that signs of disorder might well be signs that the system of education is healthy and on its way to a much improved new order.” One might rather prefer to look for inept management! More dangerously, arguments based on the ‘butterfly effect’ stand to encourage megalomaniacs to introduce bizarre policies on the grounds that a flap of their wings will create an organisational thunderstorm to change the face of the future. This is, in fact, in direct opposition to the learning organisation concept that seeks alignment and collegiality in leadership and management. So the purpose of building models as illustrated in this paper is to call to account claims based on metaphors involving chaos, and supposed properties of non-linear systems. The model developed here is not in any of the specific areas addressed in the selected literature. The point here is that arguments based upon appeals to generalised properties of non-linear systems can be tested by demonstrating that such systems do not possess the properties claimed. The model represents a counterexample at the level of generic or metaphorical argument. Without such an approach it is difficult to see how claims of the type discussed can be critiqued, and this implies the need to build simulation models in coursework designed for students in management and leadership programs.

REFERENCES

Andersen D (1988) ‘Chaos in system dynamic models’ System Dynamics Review 4,3-13. Forrester J quoted in Miller J (1972) ‘Living systems: the organization’ Behavioral

Science 17, SO. Gunter H (1995) ‘Jurassic management: Chaos and management development in

educational institutions’ Journal of Educational Administration 33(4), 5-30. Preston B (1 997) Teacher supply and demand to 2003: Projections, implications. and

issues Canberra: Australian Council of Deans of Education. Reilly D ( 1 999) “on-linear systems and educational development in Europe’ Journal of

Educational Administration 37( 5) , 424-440. Senge P (1 990) The fifrh discipline: The art and practice of the learning organization

New York: Doubleday. Sullivan T (1999) ‘Leading people in a chaotic world’ Journal of Educational

Administration 37(5), 408-423. Sungaila H ( 1 990) ‘The new science of chaos: Making a new science of leadership’

Journal of Educational Administration 28(2), 4-23.

��

Section D

Research

��

��

14

Assessing Modelling Skills

Ken HoustonUniversity of Ulster, N. [email protected],

Neville NeillUniversity of Ulster, N. [email protected]

This chapter reports on the development of test instruments for theassessment of modelling skills. It summarises the progress made in the lastfew years and gives a rationale for the latest test used with students at theUniversity of Ulster at the beginning of the academic year 2002/03.Results obtained from testing undergraduates in the honours course inMathematics, Statistics and Computing are analysed and inferences aredrawn about how students understand the various stages in the modellingcycle.

��

156

INTRODUCTION

Houston & Neill

At the University of Ulster, mathematical modelling plays a central role in the teaching ofmathematics. Bachelor of Science students in the four-year honours course inMathematics, Statistics and Computing study modelling modules in both years one andtwo. Experienced tutors initiate these undergraduates into the classical modelling cycleby means of group-based case studies. Using examples from texts such as Edwards andHamson (1996) and Berry and Houston (1995), the students study how to take thestatement of a real world problem and

1. formulate an appropriate model,2. produce a corresponding mathematical solution,3. interpret possible outcomes,4. evaluate an (optimal) solution,5. report this solution and its implications,6. revisit and refine the original model in the light of the analysis.

The students spend their third year in an industrial placement and return to university fortheir final year.

Most undergraduate modelling courses concentrate on steps 1, 2, 4 and 5 above andhence, we might expect to see students' skiIl levels increase in these areas. Currentresearch is designed to test whether or not this is the case, and this paper reports on thedevelopment of an instrument, namely a multiple-choice modelling questionnaire, whichcan test students' awareness of the steps in the modelling cycle.

DEVELOPMENT OF THE TEST

At ICTMA 9, Haines, Crouch and Davis (2001) reported on the production andimplementation of two test papers each consisting of six multiple-choice questions. Eachof the questions addressed a particular modelling skill. The tests were designed to analysestudents' understanding of the stages involved in moving between the real world and themathematical world. An example of each question type is given here. Further details maybe found in Haines, Crouch, and Davis (2000).

Type t: Making Simplifying AssumptionsA tram stop position has to be placed along a new tram route. A covered shelter will beprovided. Where should the stop be placed so that the greatest number of people will beencouraged to use the service? The transport company wants people to use the service butof course cannot add trams on demand.

Which one of the following assumptions do you consider the least important informulating a simple mathematical model?

1) Assume that tram passengers will not walk great distances to catch a tram.2) Assume that the trams run to a twenty-minute timetable.3) Assume that the tramline is single track.4) Assume that the tram driver can drive the tram from either end of the tram.5) Assume that the tram stop could be placed at any position.

��


Type 2: Clarifying the GoalWhat is the best size for pushchair wheels?

157

Which one of the following clarifying questions best addresses the smoothness of the rideas felt by the child?

I) Does the pushchair have three or four wheels?2) What is the distance between the front and the back wheels?3) Is the seat padded?4) How old is the child?5) Is the pavement tarmac or paving slabs?

Type 3: Formulating the ProblemA large supermarket has a great many sales checkouts, which, at busy times, lead tofrustratingly long delays especially for customers with few items. Should expresscheckouts be introduced for customers who have purchased fewer than a certain numberof items?

In the following unfinished problem statement which one of the five options should beused to complete the statement?Given that there are five checkouts and given that customers arrive at the checkouts atregular intervals with a random number of items (less than 30), find by simulationmethods the average waiting time for each customer at 5 checkouts operating normallyand compare it with

I) the average waiting time for each customer at 1 checkout operating normallywhilst the other 4 checkouts are reserved for customers with 8 items or less.

2) the average waiting time for each customer at 4 checkouts operating normallywhilst the other checkout is reserved for customers with fewer items.

3) the average waiting time for each customer at I checkout operating normallywhilst the other 4 checkouts are reserved for customers with fewer items.

4) the average waiting time for each customer at some checkouts operating normallywhilst other checkouts are reserved for customers with 8 items or less.

5) the average waiting time for each customer at 4 checkouts operating normallywhilst the other checkout is reserved for customers with 8 items or less.

Type 4: Assigning Variables, Parameters, and ConstantsThe time required to evacuate an aircraft after an emergency landing at an airport needsto be known by the emergency and safety services. There are conflicting needs of aircraftconstruction, safety, access, and ease of exit.

Consider an aircraft fuselage wide enough for two seats either side of a central aisle, withpassengers exiting singly at the front and the rear of the aircraft. Which one of thefollowing options contains parameters, variables or constants, each of which should beincluded in the model?

1) Time elapsed after the emergency landing: Number of people evacuated at time t:Time of day at which the landing occurred

2) Speed of people leaving their seats: Initial delay in unbuckling seatbelts beforethe first person can leave: Amount of personal items carried out

��

158 Houston & Neill

3) Number of people evacuated at time t: Time of day at which the landing occurred:Width of the emergency exits

4) Total time to evacuate everyone: Space between passengers leaving: Width of theemergency exits

5) Number of people in the aircraft: Time elapsed after the emergency landing:Number ofpeople evacuated at time t.

Type 5: Formulating Mathematical StatementsThere are two queues at a supermarket checkout. In the first queue there are m, customersall with n, items in their baskets, while in the second queue there are m, customers allwith n2 items in their baskets. It takes t seconds to process each item and p seconds foreach person to pay and customers wish to know which queue to join.

Which one of these options gives the condition for the first queue to be the better queueto join?

1) ml(p+n,t) = m2(p+n2t)2) m.tp-rn.t) <mip+n2t)3) m2(p+nZt):S; m.Iprn.t)4) m2(p+n2t) < ml(P+nlt)5) ml(p+n1t) :s; m2(P+n2t)

Type 6: Selecting a ModelWhich one of the following options most closely models the distance fallen by an objectfrom a tall building (in terms of time t)?

I) eSt_I 2) (1- 5t)2 3) 5t 4) 5t2 5) 1/(1 +eSI).

As can be seen, each of the questions had five potential solutions, some of which weredistractors. Haines et al. (200 I) proposed a partial-credit scoring method that allowedmarks to be awarded for non-optimal solutions, hence better reflecting studentachievement. They gave the tests to a small group of II students during the second yearof their mathematics course and an analysis of their responses confirmed the importanceof testing the instruments with a larger population. A second study, this time by Hainesand Crouch (200 I), tested 42 students using the same two instruments. They concludedthat

• the student answers provided a continuum in terms of how demanding thequestions had been; and

• all but one pair of corresponding questions elicited comparable responses.

Houston and Neill applied these questionnaires to larger cohorts at the University ofUlster. All students were tested at the beginning (October) and end (May) of each of theacademic years 1999/2000 and 2000/2001. A third questionnaire, similar in structure andof comparable standard to those of Haines et al. (2000), was developed by Houston andNeill (2003). This third questionnaire allowed more flexibility when selecting the test tobe given to students. Approximately 200 students were tested and the results have helpedin the preparation of a new modelling module at Ulster, which attempts to address theareas of weakness identified over the two-year period. In particular, it was found that

��

Assessing Modelling Skills 159

many students lacked the ability to clarify what is to be accomplished by a model (Type2) and had difficulty in relating a physical situation to a mathematical formulation (Type6). The new module will encourage students to reflect more carefully about theassumptions they make, and will give them more practice at this task. Furthermore,students will be encouraged to think more deeply about describing the behaviour of avariable using both graphs and functions.

Although considerable use had now been made of the modelling tests described above,analyses suggested that they were too short to be effective measures of progress.Therefore, the three tests were combined and four additional questions, which had beentested by Haines, Crouch, and Fitzharris (2003), were included to produce an instrumentcomposed of22 questions. Examples of the two new question types are given here.

Type 7: Using Graphical RepresentationsAn aircraft is waiting to land at a busy airport. It has been stacked at a constant heightflying on an approximately circular path at a fixed speed. At a particular moment theaircraft is instructed to land and to taxi some distance to the airport terminal.

Which one of the following graphs best represents the variation in the speed of theaircraft as the distance covered increases, from the stacking situation to the arrival at theterminal?

speed

speed

o

speed

D

distance

speedE

distance

o distance distance

Type 8: Relating Back to the Real SituationTwo pylons on either side of a busy two-way road support a high voltage electric cable.This places restrictions on the height and size of objects/vehicles that can pass safelybeneath it.

In this diagram, which is symmetric, the carriageway is shown to be 20m wide in eachdirection and the height of the cable is modelled by the function 10F(x), which takes thevalues (metres) given in the table below:

��

160 Houston & Neill

f('1

10

Which one of the following objects can pass beneath the cable on a low loader, assumingthat the base of the object is one metre above road level?

A B c n

On D" ~1 0"6 4 4 5

The composite test was given to 64 students enrolled in years 1-4 of the BSc (Honours)course in Mathematics, Statistics and Computing. Students were allowed 45 minutes forthe test, which was taken in the first week of the first semester under the supervision of amember of academic staff. Students in year three, who were off-campus for the year,returned their test papers by post. Analyses (Izard, Haines, Crouch, Houston and Neill,this volume) checked internal consistency of the 22 questions and calibrated them on acommon scale.

RESULTS BY COHORT

The correct answer for each question gained two marks and a partially correct answergained one mark. Thus, the maximum score possible was 44. The results are summarisedin Table 1.

YearofBSc Number of Mean score Standard Rangecourse students Deviation

I 20 24.75 5.88 15 to 332 14 26.93 4.18 18 to 343 10 30.20 4.10 23 to 364 20 29.85 2.37 25 to 36

Overall 64 27.67 4.65 15 to 36

Table 1. Analysis of test results by cohort.

��

Assessing Modelling Skills 161

CommentaryThe first year students, having just arrived at university, had no formal trammg inmathematical modelling. Year 2 students had taken the mathematical modelling modulein year I. The final year students had taken both modelling modules and had spent a yearin industry; thus, we expected clear evidence of their mathematical maturity. While theexperienced student modellers achieved higher average marks than their novicecounterparts, approximately one third of the novice group scored 30 or more. It seemedthat modelling was an intuitive skill in some students. Clearly, modelling ability may beenhanced by appropriate educational methods, but the skill level of novice modellersembarking on their first taught module varies enormously. Only 10 out of24 students ofthe year 3 cohort returned their test papers. These achieved the highest average, althoughthey did not take the test under the same controlled conditions as their peers at theuniversity.

RESULTS BY QUESTION GROUPS

The 22 questions in the composite test fell into 8 groups, according to the modellingskills they tested. Since 64 students took the tests, each set of three questions had 192responses and each set of two questions had 128. Table 2 summarizes the results.

NumberNumber

Modelling Skills Questionsagreeing with

partiallyNumber Not

the experts' incorrect answeredsolutions

correct

Type I1,2,3 90 48 54simplifying assumptions

Type 24,5,6 48 68 76

clarifying the goalType 3

7,8,9 123 27 42formulating the problemType 4assigning variables, 10,11,12 150 23 19parameters, constantsType 5formulating mathematical 13,14,15 148 25 19statementsType 6

16,17,18 53 53 73 13selecting a modelType 7

19,20 59 37 25 7graphical representationsType 8relating back to the real 21,22 54 36 21 17world

Table 2. Analysis of test results by skill group.

��

162 Houston & Neill

CommentaryQuestions 7-15 posed few problems. Students, even novice modellers answering solelyon prior personal experience, could

• formulate precise problem statements,• assign parameters within a model, and• describe problems in mathematical language.

Questions on simplifying assumptions, were generally answered successfully, with overtwo-thirds of the students obtaining correct or partially correct answers. Questions 4-6confirmed the results of earlier studies. Students, even those who had studiedmathematics for two years, had taken two mathematical modelling modules, and hadspent a year in industry, had difficulty in clarifying the goal in a seemingly simplesituation. Not one student got all three questions correct and 28 scored 2 or less out of apossible 6. Clearly this is an important skill area that needs to be given much moreattention when teaching the subject.

When selecting the mathematical equation that best models a problem, there wereinconsistent results across the three questions. When modelling the rate of growth of asunflower, 42 out of the 61 respondents selected either the correct or the partially correctoption. Modelling the speed of a car starting from rest resulted in 37 correct or partiallycorrect responses from 60 respondents. Modelling the distance fallen by an object from atall building produced a fairly even spread of answers. Out of 58 respondents, 16 choseA, 10 B, II C, II 0 and 10 E. Students' difficulty in relating basic equations of motionto simple physical situations may be explained by the decline in the teaching of appliedmathematics in the United Kingdom.

In question 19, although only 9 of the 61 respondents chose the optimum solution, 32chose the partially correct option perhaps due to the fact that the two graphical answerswere quite similar in appearance. 50 out of 60 respondents selected the correct option forquestion 20. Comparing these results with those of earlier studies (Houston and Neill,2003) shows that students have difficulty in drawing graphs ofphysical situations but canchoose the correct option if given a range of possible solutions.

The higher number of unanswered questions toward the end of the test is almost certainlydue to time pressure.

STATISTICAL ANALYSIS

Table 3 shows the results of an elementary data analysis carried out on the skill groups tosee if there were differences between their mean scores. The table indicates that themean scores of groups 2 & 6. are lower than those of the other four groups. The results ofan ANaYA (Table 4) show that there are indeed significant differences between some ofthe six means and a subsequent Tukey's test (Table 5) identifies where these differenceslie.

��


Skill Group Mean N Std. Deviation

1 3.6094 64 1.35242 2.5469 64 1.25903 4.2031 64 1.63474 5.0625 64 1.43515 2.5625 64 1.24076 2.5625 64 1.5000

Total 3.8333 384 1.7398

Table 3. Summary data by skill group.

Sun ofdf

MeanF Sig.

Squares SquareBetween

407.396 5 81.479 40.960 .000GroupsWithin

751.938 378 1.989Groups

Total 1159.333 383

Table 4. ANOVA analysis of skill groups.

Subset for a = .05Skill

N 1 2 3Group

2 64 2.54696 64 2.5625I 64 3.60943 64 4.20315 64 5.01564 64 5.0625

Siz, 1.000 .163 1.000Means for groups in homogeneous subsets are displayed.

Table 5. Tukey analysis data.

163

Table 5 shows that there is no significant difference between the mean scores in skillgroups 2 and 6, between groups land 3 and between groups 4 and 5. It also confirms thatthe mean scores of groups 4 and 5 are significantly higher than those of groups 1 and 3,which, in turn, are significantly higher than the means of groups 2 & 6.

Another area of interest was whether there was any correlation between the scores ofeach individual across groups, Analysis showed no significant correlation however, andeven those who scored well in group 4, did not necessarily score well in group 5. This

��

164 Houston & Neill

leads to the conclusion that the question groups are certainly testing different skills andthat none of the groups are redundant when ranking students.

CONCLUSIONS

Two of the key questions that have to be addressed by teachers of modelling are: Howcan a student's modelling ability be characterised? and How can it be developed overtime? This paper reports on the development of a composite test instrument, the results ofwhich help to answer these questions. The various aspects of the modelling cycle havebeen identified and questions have been produced to test student understanding of someof these areas. Preliminary results show one or two specific topics that cause concern forthe complete spectrum of undergraduates, notably, the ability to clarify what is to beaccomplished by a given model. If the results of the international study, of which thiswork is a part, produce similar conclusions, it should lead to a greater appreciation ofhow the teaching of mathematical modelling may be adapted to overcome suchweaknesses.

REFERENCES

Berry J, Houston SK (1995) Mathematical modelling London: Arnold.Edwards D, Hamson MJ (1996) Mathematical modelling skills London: McMillan.Haines C, Crouch R (2001) 'Recognizing constructs within mathematical modelling'

Teaching mathematics and its applications 20(3),129-138.Haines CR, Crouch RM, Davis J (2000) 'Mathematical modelling skills: A research

instrument.' University of Hertfordshire, Department of Mathematics TechnicalReport No. 55, Hatfield: University of Hertfordshire.

Haines CR, Crouch RM, Davis J (2001) 'Recognising students' modelling skills' inMatos JF, Blum W, Houston SK, Carreira SP (Eds) Modelling and mathematicseducation [eTMA 9: Applications in science and technology Chichester: HorwoodPublishing, 366-380.

Haines CR, Crouch RM, Fitzharris A (2003) 'Deconstructing mathematical modelling:Approaches to problem solving' in Ye Q-X, Blum W, Houston K, Jiang Q-Y (Eds)Mathematical modelling in education and culture, Chichester: HorwoodPublications, 41-53.

Houston K, Neill N (2003) 'Investigating students' modelling skills' in Ye Q-X, BlumW, Houston K, Jiang Q-Y (Eds) Mathematical modelling in education and cultureChichester: Horwood Publications, 54-66.

��

15

Assessing the Impact of Teaching MathematicalModelling: Some Implications

John IzardRMIT University, [email protected]

Ros CrouchUniversity of Hertfordshire, [email protected]

Chris HainesCity University, [email protected]

Ken HoustonUniversity of Ulster, N. [email protected]

Neville NeillUniversity of Ulster, N. [email protected]

Curriculum statements describe intentions. Without valid studentassessment practices the actual achievements are never compared in alegitimate way with the intentions. Assessment strategies have beendevised to gather evidence of growth of competence in mathematicalmodeJIing and applications. Problems with teacher-made assessmentstrategies are explored and data collected to overcome some of theseproblems. Item response modelling was used to develop scaled scoreequivalents for raw scores on three tests and examples are provided on theiruse.

��

166 Assessing the Impact of Teaching Modelling

THE NATURE AND PURPOSE OF THE STUDY

Curriculum statements about the teaching of modelling and applications describeintentions. Without valid student assessment practices, the actual achievements are nevercompared in a legitimate way with the intentions. A group of researchers, concernedabout recognising student achievement, in Australia and the UK have devised assessmentstrategies including tests for modelling and applications. These tests attempt to gatherevidence of growth in competence in mathematical modelling. Reports of research aboutthese assessment approaches have been presented at ICTMA conferences since 199I.This paper presents the latest developments in the continuing quest for knowledge aboutachievement of competence in mathematical modelling.

Valid student assessments provide quality assurance for certification of achievement orprofessional recognition, for informing management, and for evaluation of innovationsand development intervention. Student assessment is a powerful influence on citizenperceptions of what is important. Successful educational achievement gives access togreater income, better work conditions, and further education, and allows greaterparticipation in national, regional and local decision-making. Consequently, universityand school examinations are usually high-stakes assessments. Those who havesucceeded consider it important to protect the quality of the education system and keepthe assessment requirements just as demanding for those who follow.

Some Problems with Our Assessment StrategiesIfwe asked you to measure the changes in the height of some plants, you would probablyreach for a measuring tape, metre-rule or some other measuring device. You wouldrecord the initial height, record the height at later times, and look at the differences toshow how much the plant has grown. This strategy is well known, and is based onconsiderable experience using measuring scales for length (whether the units are ininches, feet, yards and miles or millimetres, centimetres, metres, and kilometres). Thereare agreed standards defining the size of the units and community expectations about theaccuracy of the measures.

When teacher-made tests are used to assess students in a classroom, there are no definedunits for the measuring tapes or rulers. One tape may have large units while another mayhave small units and the relationship between these units is unknown. The tapes willvary in length and the units will probably not be in equal intervals along the tape.Measuring progress is fraught with difficulty when tests are used on two or moreoccasions.

If a test is easier, then scores will tend to be high. If a test is more difficult, then scoreswill tend to be low. If two tests are given at the same time to the same students, then itwill be possible to see which items are easy and which are difficult. If two tests are givenat different times (without being given together) any changes in the score cannot beinterpreted. One does not know whether the difference in scores was because the testsdiffered in difficulty, whether learning occurred over the time interval, or whether somecombination of these events occurred. (If you wish to demonstrate progress, give themore difficult test first, then give the easier test. Scores after teaching will rise whether

��

Izard, Haines, Crouch, Houston, & Neill 167

the teaching was effective or not. The reverse applies if the easier test is given first.)Many teachers do not know the relative difficulties of the tests they give, so interpretingthe results of their tests is impossible.

Teachers assume that the tests measure what they are intended to measure and seemunaware of measurement errors. Short true/false tests are popular, even though highscores can be achieved without seeing the items. Many teacher-made tests are conceivedand written without concern for adequate sampling of topics. Sub-tests often have toofew items to provide meaningful information. If we use the analogy of a ruler torepresent a test, teacher-made tests are rulers of different lengths with different markingson each ruler. Since rulers are never placed together, a reading on one ruler has nomeaning in relation to a reading on another ruler.

Test items constructed by teachers without technical support do not always distinguishbetween students with relevant knowledge and students lacking such knowledge. Testsdiffer in number and format of items. Items vary in difficulty but are treated asequivalent in difficulty. Two tests of the same topic are assumed to cover the same workwithout empirical validation. Difficulty is not controlled from one occasion to the next.So measuring progress becomes problematic: one does not know whether a scoredifference is due to changes in test difficulty, student progress, or the tests beingunrelated to each other. For example, some teachers use tests of differing lengths. Therationale is that less of the work has been covered at the beginning so the earlier testsshould be shorter. If you ask 20 questions at the start, 50 questions at the end, and thequestions are of comparable difficulty, then the average score will rise since the later testhas more opportunities for students to be successful.

The task for universities is more complex where courses extend over several years.There is no explicit documentation of the extent to which tests in year I overlap with thetests of year 2, and so on. To extend the ruler analogy used earlier, there are severaldistinct rulers but we do not know how the marks at one year-level relate to marks onanother year-level. We assume that the year 2 tasks are somehow more difficult than theyear I tasks. Further, when test items are prepared to suit minimum competencerequirements, many students miss out on the opportunity to demonstrate their skills andknowledge.

How is a teacher to indicate progress? What tasks can a student now do that could not bedone before? If we know what the student can now do, what learning is a good "bet" forthe immediate future? Before we can indicate the progress that has been achieved in ateaching program, we have to know the initial achievement status of each student and thesubsequent assessments have to include tasks representative of the skills we intendedteaching.

Where a certification process for student assessment becomes well known, the certifyingauthority serves to reassure the public that meaningful learning is occurring and thatassessment of that learning is systematic, valid, and fair. In some constituencies, studentsare blamed if they do not learn. In others, teachers are blamed for not teaching well.Others criticize the provision of teaching materials or equipment. Assessment

��


instruments and procedures are criticized too, if students cannot achieve (as judged bypass rates on the assessments). In fact, a sound education system requires all of thesecomponents:

• Well-trained teachers who can help students learn,• Well-designed curriculum that is coherent and builds on past success,• Teaching and support materials that suit the curriculum and are accessible to all

students,• Students who are motivated to learn and who persevere in their pursuit of

knowledge, and• Assessment strategies that give due credit to the quality of achievement and

generalisable skills.

In accreditation systems, factors like these are checked to see whether a course or ateaching institution is worthy of being trusted to operate for a limited period (often 3years) without more frequent review. The assessment strategies are only part of the storybut they (or at least the public notices about the results) are often more visible to thegeneral public. When the general public or educational institutions place little trust in theassessment practices of schools and universities, alternative external schemes are set upto exercise additional quality control of the certification.

Many teachers and administrators accept and support analysis of test data to improveteaching and learning, but practical implementation has been found wanting (Izard, 1998;Black and Wiliam, 1998a, 1998b). The constant threat of litigation makes manyexamination boards cautious about publishing results before they check (through itemanalysis) that every item (as scored) distinguishes between able candidates and less ablecandidates in the right direction (able candidates scoring higher on the item than less ablecandidates). (But see Ludlow, 200I [at http://epaa.asu.edu/epaa/v9n6.html] for anexample in teacher licensure testing in the United States of America where this did notapply.) Technical limitations of current assessments include limited knowledge about thetechnical properties of the tests and associated assessment strategies, and tests that arelimited with respect to demonstrating student progress. This study has collected testitems and data from several universities in order to analyse the data, identify the technicallimitations and implement a plan to overcome these limitations.

THE THEORETICAL FRAMEWORK

Validity refers to the extent to which meaningful, appropriate, and useful inferences canbe made about scores from a test as used for particular purposes. Evidence of validity fora purpose can be construct-related, content-related, and criterion-related (both concurrentand predictive evidence). The key validity issue is the extent to which meaningful.appropriate. and useful inferences can be made about scores.

But the validity of the assessments is only part of the issue. The actions that are taken toimprove learning and teaching as a consequence of the assessment evidence are also

��


important. Black and Wiliam (\ 998a, 1998b) have reported that studies of formativeassessment show effect sizes (Cohen, 1969, 1988) between 0.4 and 0.7 on standardizedtests, larger than most known educational interventions. However, the correspondingeffect sizes calculated using raw scores for research reported at ICTMA conferences areerratic (and in some cases negative, implying that student achievement is goingbackwards). Raising standards in mathematical modelling implies improving learning aswell as assessment so a study of assessment should address the learning implications.

THE METHODOLOGY USED

Assessment information should inform the teacher (and the student) of what tasks can beattempted successfully, what skills and knowledge are being established currently, andwhat skills and knowledge are not yet within reach. Assessment strategies that showprogress in these curriculum terms are available now. This approach has been shown towork from the first years of schooling through to university entrance level, inundergraduate studies, and at graduate level (for example, see Haines & Izard, 1993;Izard, 1994). The methods of analysis are well established (for example, see Wright &Stone, 1979; Wright & Masters, 1982; Wilson, 1992). The approach can handleperformance tasks as well as traditional pencil-and-paper tests (for example, see Izard,1997; Haines, Izard, and Berry, 1993; Izard & Haines, 1993), and has been used in suchinternational studies as the Third International Mathematics and Science Study (TIMSS)(for example, see Lokan, Ford & Greenwood, 1996) and the current OECD PISA study(see www.pisa.oecd.org).This "new" approach is more than 40 years old (Rasch, 1960).It makes use of the same data used in traditional approaches (and can therefore provideall the traditional statistics as well, if required). The approach has been referred to asItem Response Modelling (or Item Response Theory) and is widely used by examiningboards throughout the world (Izard, 1992). It is appropriate to use Item ResponseModelling to analyse assessments of students in modelling courses. Constraints in givingcandidates due credit for their work will be addressed as part of the strategies for qualitycontrol in assessment (Izard, 2002a).

Traditional analyses focus on the items correct for each candidate. The "new" approach(more than 40 years old) also investigates the candidates correct for each item. Theextent to which each item distinguishes between "more knowledgeable" candidates and"less knowledgeable" candidates (as judged by their scores on the test itself) is stillinvestigated, but the candidates and the items are placed on a common linear continuum(known as a variable map). QUEST, a test analysis software package (Adams and Khoo,1993) can be used to produce a variable map.

Item response modelling (or item response theory) was the only way for tackling theanalyses for the research described in this paper. But there is more to be done here, too.For example, the effects of expanding the number of items at a given year level need tobe compared with the effects of expanding items across year-levels. A further example isthe validation of teaching strategies to make sure that they do improve learning.

��


THE NATURE OF THE DATA COLLECTED

Existing partial credit multiple-choice items developed by Haines, Crouch, Davis andFitzharris at City University and the University of Hertfordshire have been used in anumber of studies. Houston and Neill (Ulster University) have expanded the number ofitems by creating further examples. Commentaries on the constructs and the items appearin university technical reports and in papers presented at ICTMA 9 (Lisbon), ICME 9(Tokyo) and ICTMA 10 (Beijing). Existing data include pre- and post-test results(referred to as Tests A and B) with items collated as a person-item matrix. Some items(referred to as Test C) lacked linking data until this research was carried out so their usefor pre- and post-testing was problematic. For example, Houston and Neill (ICTMA 10,Beijing) used Test A as a pre-test and Test B as a post-test to evaluate progress over anacademic year. The following academic year they used Test C as a pre-test and Test B asa post-test.

It was decided to administer all 22 multiple-choice items from the tests to further pre-testgroups in a single application late in 2002. The intention was to use the analyses of thesedata in further analyses of earlier data. Items were also translated into Japanese andadministered in a Japanese university. This would enable an exploration of the extent towhich the results generalise across languages.

THE RESULTS

Analyses checked internal consistency, and calibrated all items on to a common scale sothat existing data could be re-interpreted on a sounder technical basis. The results fromthe administration of the 22 items are presented as variable map diagrams (see Figures Iand 2) that allow different items to be placed in order of difficulty alongside studentsplaced in order of achievement. This type of presentation was chosen to enhance the useof the assessment information for formative purposes.

In Figures I and 2, each student attempting the 22-item test is shown (as an X) adjacentto a vertical scale representing a continuum from the least proficient at mathematicalmodelling (at the bottom of the diagram) to most proficient (at the top). Note thatstudents who scored the maximum of 12 cannot be shown on the graph. We expect thatthey are more proficient than those who scored 11 but we do not know how much betterbecause we have no evidence of what they cannot do. No students scored O. If therewere students with zero scores, we could not show them on the diagram either. Weexpect that they would be less proficient than students with some correct items but cannotestimate their level of achievement without having evidence of what they can do.

Responses to items in the 22-item test were given a score of 2 if completely correct, 1 ifpartially correct, and 0 if incorrect. Up to two thresholds are shown for each item in thisinvestigation. The position of a threshold on the vertical scale shows the relativedifficulty of achieving that score on that item. Easier items are lower on the diagram andmore difficult items are higher. The lower threshold (represented by n.l, for example1.1) shows the zone of the variable map where a score of 0 and a score of I on this item

��

Izard, Haines, Crouch, Houston, & Neill

3.0II 4.2III

III2.0IIIII 5.2XI11.21.0 X IxxxIXXXX IXXXXXXXXXX I 6.2XXXXXXXXXXXXXXXXXXX I 2.2I 3.2 6.\ 8.29.2XXX I 8.\0.0 XXXX 2.\ 5.1XII 9.1XXX I 10.2XX I 3.14.1XIXX 11.1I 7.2 10.\I 11.2 12.2I-1.0

III 11.\ 12.1IIIII-2.0

III 7.\IIIII-3.0

Figure 1. Variable map showing item estimates (thresholds) for Items 1-12 of the 22item test (Ulster data, 2002).

171

��


3.0 IIIIIII 22.2

II

2.0 II 19.2

IIII

x I 18.2II 16.2

1.0 X Ixxx I

IXXXX I 17.2

XXXXXXXXXX IXXXXXXXXXXXX 18.1 22.1

XXXXXXX II

xxx I0.0 XXXX I

XI 17.1 19.1 21.2I 13.2 15.216.1

XXX Ixx IX I 15.1 20.2XX I

I 20.121.1

I \4.2

I-1.0 I 13.1

III 14.1

IIIII

-2.0 I

Figure 2. Variable map showing item estimates (thresholds) for Items 13-22 of the 22item test (Ulster data, 2002.)

are equally probable. A student below this zone is expected to score 0 on this item and astudent above this zone is expected to score at least 1 on this item. The further thestudent is below this threshold, the greater the probability that a score of zero will beobtained on this item. Conversely, the further a student is above this threshold, thegreater the probability that a score of at least I will be obtained. The higher threshold for

��


the same item (represented by n.2, for example 1.2) shows the zone of the variable mapwhere a score of 1 and a score of 2 are equally probable. A student below this zone isexpected to score 1 or less on this item and a student above this zone is expected to score2. The further the student is below this threshold, the greater the probability that a scoreless than 2 will be obtained on this item. Conversely, the further a student is above thisthreshold, the greater the probability that a score of 2 will be obtained. Items are shownto vary in difficulty in two ways, the position of the thresholds relative to other itemthresholds, and the separation of the thresholds within an item. For example, gaining ascore of 2 on item I represents a greater increase in difficulty over gaining a score of 1than the corresponding interval for item 2.

It is useful to compare the range of item difficulty of these tests with the range of studentachievement. These variable maps show that the range of student achievement isbounded by the difficulty of the items for most students tested. Only those obtainingperfect scores are unable to receive due credit for their achievements. Longer tests withitems reflecting higher levels of skill would be necessary to distinguish between theachievements of these more proficient students.

The raw scores on each of tests A, Band C are not comparable because the items in eachtest vary in difficulty. Using the results from the administration of the 22-item test inUlster in 2002 it was possible to obtain comparable information about tests A, Band Csince all of the 18 items in these 3 tests were included in the 22-item test. Thisinformation relates the raw scores on each test of 6 items (each with a maximum score of12) to the continuum determined by the 22-item test as administered to the Ulster groupin 2002. These scale score equivalents and associated errors are provided for each of theTests (see Table 1).

Note that, on the basis of the Ulster 2002 data, test A covers a wider range ofachievement on the logit scale than the other tests (-2.56 to 2.56, in Table 1). In tum, testB covers a wider range of achievement than Test C. Further, a particular raw scorerepresents different levels of achievement on each test. For example, a score of 6 on TestA has a lower estimate (-0.22) than the same score on Test B (-0.11) and Test C (-0.02).The associated errors (as shown in Table I) are relatively large for these tests, beingapproximately equivalent to 2 raw score units near the middle of the score range.

Using the information from Table I further analyses of earlier data are possible for itemsin test A, Band C that were the same as items in the 22-item test. The effects of variationin the tests can be illustrated using the Houston and Neill (ICTMA 10, Beijing) data fortest A as a pre-test and test B as a post-test in the first academic year and test C as a pretest and test B as a post-test in the second academic year for a number of courses. Onlystudents with complete data over the two academic years were used for this analysis.Students obtaining perfect scores on any test were excluded, since perfect scores cannotbe scaled in this way. The results presented in Table 2 show the scale score means,standard deviations, and magnitude of the changes (effect sizes) for the sampled studentsby course over two academic years. The descriptors for effect sizes have been assignedaccording to the magnitude of the effect size [after Cohen (1969, 1988)]. An effect size

��


Test A (Haines & Crouch)(N = 54, L = 6 Probabilitx Level=0.50)

Score Estimate Error(max= 12) (logits)

11 2.63 1.1910 1.61 0.879 0.97 0.738 0.51 0.657 0.12 0.606 -0.22 0.585 -0.55 0.574 -0.89 0.603 -1.27 0.652 -1.77 0.771 -2.56 1.05

Test 8 (Haines & Crouch)(N = 54, L = 6 Probabilit Level=0.50)

Score Estimate Error(max= 12) (loaits)

11 1.82 0.9710 1.17 0.699 0.77 0.598 0.45 0.547 0.17 0.526 -0.11 0.525 -0.39 0.544 -0.69 0.563 -1.03 0.602 -1.44 0.691 -2.05 0.91

Test C (Houston & Neill)(N = 54, L = 6 Probability Level=0.50)

Score Estimate Error(max= 12) (logits)

11 1.83 0.9310 1.22 0.689 0.83 0.588 0.52 0.547 0.24 0.516 -0.02 0.515 -0.28 0.514 -0.55 0.543 -0.86 0.582 -1.25 0.671 -1.85 0.91

Table 1. Score equivalence tables for tests A, 8 and C (Ulster, 2002 data).

of less than 0.1 is negligible. (The two group means are almost the same or the same;there is almost total overlap between the two score distributions for the groups.) An effectsize of 0.1 to 0.3 is small. (The two group means differ; the overlap between the twodistributions is considerable.) An effect size greater than 0.3 but not exceeding 0.5 ismedium. (The two group means differ; the overlap between the two distributions ismoderate.) An effect size greater than 0.5 but not exceeding 0.7 is large. (The two groupmeans differ; the overlap between the two distributions is small.) An effect size greaterthan 0.7 is very large. (The two group means differ; the overlap between the twodistributions is negligible.)

��


Means (Standard deviationsNumber of Year I Year 2

Course Students in Oct 1999 May 2000 Oct 2000 May 2001Sample TestA TestB Test C Test B

HNDIIHND2 8 0.14(1.00) 0.16(0.50) -0.24(0.51 ) 0.33(0.49)HND2/BSc2 5 0.03(0.72) 0.87(0.31 ) 0.3 \(0.37) 0.47(0.30)BScl/BSc2 10 0.16(0.78) 0.37(0.52) 0.62(0.44) 0.47(0.31)BSc2/BSc3 II 0.52(0.74) 0.79(0.57) 0.61(0.57) 0.79(0.56)BSc3/BSc4 10 0.50(1.06) 0.46(0.69) 0.07(0.50) 0.46(0.46)

Differences (Effect sizes)Course Number of Oct 1999- May 2000- Oct 2000- May 2000-

Students May 2000 Oct 2000 May 2001 Mav 2001HNDI/HND2 8 0.03<0.03) -0.40(-0.75) 0.56(1.00) 0.17 0.34)HND2/BSc2 5 0.84(1.22) -0.56( -1.28) 0.16(0.48) -0.40 -1.13)BScl/BSc2 10 0.21<0.32) 0.24(0.50) -0.15( -0.40) 0.09 0.22)BSc2/BSc3 II 0.26(0.40) -0.18(-0.32) 0.18<0.33) 0.00 0.01)BSc3/BSc4 10 -0.04(-0.05) -0.39(-.62) 0.33(0.66) -0.06 -0.10)

Table 2. Re-analysis of Houston-Neill ICTMA 10 data.

The patterns in these results raise many issues worthy of investigation. For example, inthe case of the HNDI/HND2 group when scaled scores are used, the change betweenOctober 1999 and May 2000 is negligible. When raw scores are used the difference was0.25 with an effect size of 0.12. This is a case of an instrumentation effect - thedifference in score was due to the difference in the tests and not due to the learning by thestudents. The change between October 1999 and May 2000 for the BScl/BSc2 groupwas 0.21 with an effect size of 0.32 (medium), and further progress was made from May2000 to October 2000 (0.24 with an effect size of 0.50 medium). Sadly, from October2000 to May 200 I the changes are negative (-0.15 with an effect size of -0.40 medium),implying that the students have not retained all of their skills.

The changes between May 2000 (end of first academic year) and May 2001 (end ofsecond academic year) also need exploration. In these cases the results are the same forboth raw scores and scaled scores (because the same test was used on both occasions).The decline in scores from the end of the first academic year to the end of the secondacademic year cannot be explained by an instrumentation effect because the test was thesame on each occasion.

IMPLICATIONS FOR LEARNING AND TEACHING MODELLING

Where assessments provide information for high-stakes decisions there is considerablepressure to ensure that the results are comparable across teaching institutions, thatstudents have a fair access to comparable facilities in preparing for their assessments, andthat students are assessed fairly on their own work.

��


The results from the analyses confirmed some of the findings in the Houston and NeillICTMA 10 paper. They concluded that:

There was not, as had been anticipated, an increase in scores as the studentsprogressed through the programmes. Despite having taken severalmodelling modules and worked in industry for a year, the final year degreestudents did not perform significantly better than their more junior peers,indeed their attempts at some of the questions were very disappointing.

The re-analyses reported in this paper differed in that they controlled for variation in thedifficulty of the test items in each test. The progress of students is reported on a commonscale regardless of fluctuation in test difficulty. The task of interpreting the results interms of how teaching might improve to assist students to become more competent inmathematical modelling is the essential next step.

This approach provides evidence of what is already known by that student, as a basis forfurther learning by that student. It also provides a strategy for gauging the progress madeover time (expressed in curriculum terms) as a consequence of the intervention. Thedirect information about what is known and what is yet to be learned is far betterinformation than the old "normative" approach that compares student with studentwithout mentioning what each can do and what each has yet to learn. Describing studentachievement in teacher-friendly ways has implications for formative and summativeassessment (Izard, 2002b). The approach explored in this research is a better practicalmodel for describing progress, and taking action to improve learning.

REFERENCES

Adams RJ, Khoo ST (1993) Quest: The interactive test analysis system. (Computersoftware & manual) Melbourne, Victoria: Australian Council for EducationalResearch.

Black P, Wiliam D (1998a) 'Assessment and classroom learning' Assessment inEducation, Vol. 5,7-74.

Black P, Wiliam D (1998b) 'Inside the black box: Raising standards through classroomassessment' Phi Delta Kappan, 139.( www.pdkintl.org/kappan/kbla98l9.htm)

Cohen J (1988) Statistical power analysis for the behavioral sciences. (2nd Ed) Hillsdale,NJ: Lawrence Erlbaum Associates.

Cohen J (1969) Statistical power analysis for the behavioural sciences New York:Academic Press.

Haines CR, Izard JF (1993) 'Authentic assessment of complex mathematical tasks' inHouston SK (Ed.) Developments in curriculum and assessment in mathematicsColeraine, Northern Ireland: University of Ulster, 39-55.

Haines CR, Izard JF, Berry JS (1993) 'Rewarding student achievement in mathematicsprojects' (Research Memorandum 1/93) London: City University.

Izard JF (2002a) Constraints in giving candidates due credit/or their work: Strategies/orquality control in assessment Valetta, Malta: Ministry of Education, Malta and the

��


University of Malta for the Association of Commonwealth Examinations andAccreditation Bodies.

Izard JF (2002b) Describing student achievement in teacher-friendly ways: Implicationsfor formative and summative assessment. Valetta, Malta: Ministry of Education,Malta and the University of Malta for the Association of CommonwealthExaminations and Accreditation Bodies.

Izard JF (1998) 'Validating teacher-friendly (and student-friendly) assessmentapproaches' in Greaves D, Jeffery P (Eds.) Strategies for intervention with specialneeds students Melbourne, Vic.: Australian Resource Educators' Association Inc,101-115.

Izard JF (1997) 'Assessment of complex behaviour as expected in mathematical projectsand investigations' in Houston SK, Blum W, Huntley !D, Neill NT (Eds.)Advances and perspectives in the teaching and learning mathematical modelling:Innovation, investigation and applications Chichester, UK: Albion Publishing,109-124.

Izard JF (1994) 'Strategies for assessing projects and investigations: Experience inAustralia and United Kingdom' in Mauritius Examinations Syndicate (Eds.)School-based and external assessments Mauritius: Mauritius ExaminationsSyndicate, 214-222.

Izard JF (1992) 'Assessment of learning in the classroom' (Educational studies anddocuments, 60) Paris: United Nations Educational, Scientific and CulturalOrganisation.

Izard JF, Haines CR (1993) 'Assessing oral communications about mathematics projectsand investigations' in Stephens M, Waywood A, Clarke D, Izard JF (Eds.)Communicating Mathematics: Perspectives from classroom practice and currentresearch Hawthorn, Victoria: Australian Council for Educational Research, 237251.

Lokan J, Ford P, Greenwood L (1996) Maths & science on the line: Australian juniorsecondary students' performance in the Third International Mathematics andScience Study Melbourne, Victoria: Australian Council for Educational Research.

Ludlow LH (February, 2001) 'Teacher Test Accountability: From Alabama toMassachusetts' Education Policy Analysis Archives, Vol 9 No 6.[http://epaa.asu.edu/epaa/v9n6.html]

Rasch G (1960) Probabilistic models for some intelligence and attainment testsCopenhagen: Danmarks Paedagogiske Institut.

Wilson M (1992) 'Measurement models for new forms of assessment.' in Stephens M,Izard J (Eds.) Reshaping assessment practices: Assessment in the mathematicalsciences under challenge Melbourne, Victoria: Australian Council for EducationalResearch, 77-98.

Wright BD, Masters, GN (1982) Rating scale analysis Chicago, IL: MESA Press.Wright BD, Stone MH (1979) Best test design Chicago, IL: MESA Press.

��

16

Towards Constructing a Measure of the Complexity of Application Tasks

Gloria Stillman University of Melbourne, Australia [email protected]

Peter Galbraith University of Queensland, Australia [email protected]

A key characteristic of application and modelling tasks involves the embedding of mathematics in problem contexts, and its extraction and interpretation within the solution process. A central construct in addressing factors involving the successful solution of application and modelling problems is the notion of complexity. In this paper we discuss complexity as it is construed by a group of senior secondary school students and their teachers. A profile detailing sub-categories of complexity is described and illustrated.

��

180 Stillman & Galbraith

INTRODUCTION

A key characteristic of applications and modelling tasks is the embedding of mathematics in a problem context. The extent to which the d$fictrIry of a modelling task is influenced by its complexity remains a matter of central interest. In this paper we explore aspects of complexity, and begin by considering alternative contextualisations of a common piece of mathematics. The examples chosen have been set within the senior (pre-university) preparatory mathematics course within the state of Queensland, Australia. The inclusion of applications and modelling is mandated within this two-year (grades 1 1 & 12) program. We then refer to research involving an intensive analysis of student performance on a range of application tasks, augmented by interview and video data in which students provided judgements of the complexity of tasks together with assessments of their approaches and expectations. Teacher interview data provided parallel judgements of complexity from the teachers’ viewpoint. The outcome is a profile for estimating the complexity of an application or modelling task as viewed by students and teachers. The research took place in classrooms of two secondary schools across the two years of the senior mathematics program, and comprised part of the PhD studies of the first author while enrolled at the University of Queensland. We begin by introducing a pair of problems set during the first phase (year 11) of the program.

THE INFLUENCE OF CONTEXT

Recalling that our interest in exploring complexity includes the influence of context, the procedure involved firstly a pre-test of mathematical skills during class time, in which a range of exercises was presented for solution by the students. Some of the mathematical techniques and skills required for the solution of these exercises were later embedded in applications contexts and presented to selected students for solution in video-taped sessions. Since the mathematical analysis in the two situations was the same, we could consider the extent to which the mathematics which had been successfully employed in the standard exercises was later recognised as relevant within the contextualised problem. Figure 1 contains one such standard exercise (Arc problem) that can be solved for example, by a straightforward application of Pythagoras’ Theorem supplemented by rudimentary knowledge about symmetry properties of chords of a circle.

ACB is an arc of a circle with centre 0. D is the midpoint of chord AB which is 20 cm in length. CD = 5 cm. Find the length of OD.

0

Figure 1. Arc problem.

��

Complexity of Application Tasks 181

The mathematics embodied in this exercise subsequently appeared in the following contextualised Road Accident problem. Note, of course, that this problem is solvable by students who fail to solve the Arc problem, or do not recognise its relevance. Such students treat it as an independent problem in its own right.

Road Accident (adapted from Smith & Hurst, 1990) A car screeched round a bend and ended up in the ditch by the side of the road. The police were called and when they arrived they made detailed measurements of the skid marks left on the road by the car. These measurements were used to draw a plan of the scene of the accident as shown in Figure 2.

f X

Figure 2. Road accident plan.

A reference line was used to measure the skid marks (also shown in Figure 2). The distance x was measured along the reference line and the distance y perpendicular to it. For the outer skid mark the values obtained were:

x 1 0 1 3 1 6 1 9 1 1 2 1 1 5 1 1 8 1 2 1 1 2 4 1 2 7 1 3 0 ] 3 3 y I 0 I 1.19 12.15 12.82 13.28 I 3.53 13.54 13.31 12.89 I 2.22 I 1.29 I 0

All distances were measured in metres. The police also measured the incline of the road and found that this particular stretch was flat. When a car moves round a curved path with its wheels rotating but slipping sideways as in the accident above, a simple model to obtain its speed is to use the equation s’ = dr: where s is the speed, d is the drag factor for the road surface and r is the radius of the curve. Test skids conducted at the scene found

��


the drag factor to be 6.64 mls is . Using the data above the police determined the driver's speed allowing a 10% error in the final calculation in the driver's favour (to compensate for any errors in measurement). Did the police calculations show that the driver was telling the truth about the speed of the car?

Student Work We will discuss the approach of two students (Andrea and Bruce) who were successful on the Arc problem. Andrea was the only student in the group not to draw a graph using the table of values, but calculated a value halfway between the y values for x = 15 and x = 18 to work out the y value at x = 16.5. She then drew Figure 3, but despite its similarities to Figure 1, she could not recall the method. That is, Andrea was unable to proceed from a stage (generated by her as an intermediate step) that was essentially the same as the initial condition from which she had succeeded in the standard task.

Figure 3. Andrea's diagram for the Road Accident problem.

Bruce was successful even though he did not recognise the task as a contextualised example of the Arc problem which he had solved using Pythagoras' Theorem. At first, he thought the curve was a parabola but then "figured it had to be a circle because it had a radius." His first approach was algebraic. However, he rejected this approach, as there were too many unknowns in his equations, and then opted for what he considered was a much less reliable method.

Bruce: 1 figured I would go for the unreliable method of attack - the unreliable way of measuring it from the graph. The way I did it I figured was unusual because I never actually measured off a graph before to get an answer.

In trying to fit the entire circle on the page so he could read the length of the radius easily, he made the same error common to all the students (except one) who drew a graph. He did not use the same scale for both axes and did not realise at the time of doing the task how this would transform his figure. He had difficulty trying to scribe a circle through all the points but thought this was due to possible errors in the real measurements. The intersection point of the bisectors of two chords from two pairs of adjacent points was used to find the centre of the circle and then the radius was read off. The completion of the task then entailed a simple substitution into the given equation, the conversion of units and an allowance for 10% error. His physics knowledge helped him to convert units successfully. The problem was not similar to anything he recognised but, in spite of this, he showed both a willingness to draw on as many techniques as were necessary to complete the task and to develop his own set of procedures for his solution.

��


From the above we infer that the introduction of complexity (through contextualisation) has impacted on the task demand. While Andrea and Bruce solved the standard task directly, Andrea was unable to transfer her method to an isomorphic situation in the Road Accident problem. Bruce did not even recognise the similarity and proceeded independently, and with some flair, to construct a solution. In both cases the difficulty of the contextualised problem clearly exceeded that of the Arc problem, which they solved quickly and directly.

There have been various attempts to separate task difficulty and task complexity. The issue is whether task complexity is inherent within a task itself, or is determined by the requirements of the solution to the task. Williams and Clarke (1997) distinguished between task complexity and task difficulty on the basis that task complexity is "an attribute of the task alone," whereas task difficulty is "a consequence of the interaction of student with task" (p. 451). In this view, task complexity is fixed, whereas task difficulty can change from student to student. Williams and Clarke suggest task difficulty for a particular student on a specific task can be represented "as a co-ordinate point on a two- dimensional axis system," where the vertical axis shows task complexity, and the horizontal axis, student skill level. This is problematic, however, as it appears to imply that student skill level is constant for a given problem. In fact, much learning can occur through the very activity of problem solving, with the effect of increasing the skill level in the course of achieving a solution. This appeared to happen with Bruce in solving the Road Accident problem. It can be argued that the iterative process common in solving modelling problems has a boot-strapping effect in enhancing the skill of a student within the same task, as well as in general.

CONSTRUING COMPLEXITY

The above illustrates quite directly the impact of contextualisation, but contextualisation is far from the only factor contributing to task complexity. In an earlier paper (Galbraith & Stillman, 2001) we illustrated an approach to defining the complexity of an application or modelling task by an analysis of the mathematical and interpretative elements necessary for its solution. This approach has an objective aspect in that tasks are coded in terms of defined criteria relating to mathematical operations, cues, and assumptions that are invariant across tasks. The coding system is, of course, theoretically determined and hence contains a subjective component. In this paper we discuss another approach to estimating the complexity of application and modelling problems. This approach represents a construction based on the responses of students and teachers when they are asked to comment on the perceived complexity of a range of such problems.

Pursuing this approach to the goal of profiling complexity we use as a starting point the position elaborated by Williams and Clarke (1997, p. 452), who proposed the following categories of complexity for a mathematical task:

Contextual Complexity: Perception of the relationship between the situation

Conceptual Complexity: The types and combinations of concepts utilised in

Linguistic Complexity: Vocabulary and lor sentence structure

described and the required mathematical procedure

developing the task

��


Intellectual Complexity: Bloom's Taxonomy: knowledge; comprehension; application; analysis; evaluation; synthesis Mathematical Compkxity: The types and combinations of operations required to perform the task. (This category has been broadened from its original designation as numerical complexity.) Representational Complexity: The symbols, diagrams, graphs and other representational forms, which need to be used and interpreted to understand and develop the problem

Williams and Clarke sought to validate their scheme by surveying the opinions of six experts (two academics in tertiary mathematics, three academics in mathematics teacher education and a mathematics teacher) about the relative complexity of two senior secondary mathematics tasks. While the overall framework was accepted, expert opinion with respect to the relativities of the respective tasks differed within all the above categories. The greatest consensus was reached with respect to intellectual complexity for which all but one expert agreed. While the authors were encouraged by the experts' acceptance of their overall profile and its potential utility as an analytical framework for the consideration of mathematical task complexity, of concern was the diversity of opinion among the sample of experts with respect to the two tasks. Given the diversity of opinion noted within such a small sample, it is reasonable to infer that a theoretical consensus on the relative complexity of mathematical tasks is unlikely to be agreed within the informed community. That is, while general notions of complexity properties held within individual heads may be in agreement globally, they will differ substantially at the level of specifics. Accordingly, we seek another approach to assessing complexity.

EMPIRICAL CHARACTERISATION OF TASK COMPLEXITY

The approach we describe here involves the use of empirical data, in which an initial framework of categories is tested and refined as the definitions of the categories are modified-that is, as an emerging theory is grounded in data. In this study the Williams and Clarke (1997) structure provided the provisional initial categories, which were refined and augmented using a grounded theory approach based on Strauss and Corbin (1990). Rather than comparing tasks, students and teachers were asked to judge what it was that made particular tasks more or less complex. Their responses were informed on the one hand by the construction of the tasks (teachers), and on the other by attempting to solve them (students). In this way categories were refined and saturated, with relationships among the data identified and validated using analyses supported by NUD.IST software. The data sources comprised 41 volunteer mathematics students in years 11 and 12 at two provincial secondary schools, together with their eight teachers. The final definitions stabilised using this process are presented in Figure 4. While a variety of data types were generated, those most relevant here involved the conduct of sixty-four videotaped student task solving sessions, immediately followed by clinical interviews as a stimulus to recall, together with teacher interviews. Approximately twenty contextualised problems were used as tasks, the Road Accident problem being a typical example.

��


'ASK COMPLEXITY ;enera1 Properties

EVEL OF COMPLEXITY 'ONCEPTUAL COMPLEXITY

Complexity of concepts involved Number of topic areas involved Pedagogical development

IATHEMATICAL COMPLEXITY Number of techniques used Degree of rehearsal of techniques Obscurity of choice of techniques Complexity of each technique Complexity of combination of techniques Type(s) of combination of techniques Perceived visibility of links between techniques Number of steps involved Length of solution Familiarity of problem type Type of problem

Type of application Amount of mathematical information given

Number of topic areas involved JNGUISTIC COMPLEXITY

Amount of guidance given Vocabulary Sentence structure Amount of information given in written form Familiarity of wording Amount of reading involved Orientation of wording Format

Relevance of information VTELLECTUAL COMPLEXITY

The solution requires analysis The solution requires synthesis Decision making Amount of thinking required Level of challenge of task Number of steps to integrate involving mental coordination

Number of visual representations Type of visual representation(s) Task can be represented in a diagradgraph Degree of difficulty to draw diagradgraph

$PRESENTATIONAL COMPLEXITY

Dimensional Ranges

simple ... complex

basic ... abstract one ... many early.. .complete

one ... many cursorily treated ... well rehearsed fairly obvious ...q uite obscure quite simple ...q uite complex all quite simple ... most quite complex conjunction, composition, inverse fairly apparent ...q uite obscure one.. . many sho rt... long familiar ... unfamiliar direct taught, reverse taught, direct novel, reverse novel procedural ... true application sufficient ... information must be imported one ... many

none ... high simple.. .complex simple ... complex a little ... a lot familiar ... unfamiliar a little ... a lot mathematical, everyday, technical point form, 1 paragraph, several paragraphs all relevant, extra information

no, yes no, yes none ... a lot little ... a lot straightforward ...p erplexing

few ... many

one ... many none, diagram, graph yes, no easy ... hard

��


CONTEXTUAL COMPLEXITY Familiarity of task cont Obscurity of mathematical formulation Type of task Level of contextualisation Amount of contextual info to process & integrate Assumptions for model formulation Reality of context

familiar ... totally unfamiliar obvious.. .quite obscure application ... modelling task border. .. tapestry a little ... a lot given ... all made by student contrived ... real life

Figure 4. Attributes and dimensions of task complexity.

The attributes of a task that are perceived to contribute to its complexity are numerous. They contribute to the overall complexity construct as part of the mathematical, linguistic, intellectual, representational, conceptual or contextual complexities of the task. For a particular task, both students and teachers tend to focus on only a few attributes in assessing complexity, as would be expected from the known limits on working memory. This is why a grounded theory needs to include data from many individuals responding to a range of tasks-so that a cumulative and stable picture might emerge.

Included below are sample comments that indicate the way in which student and teacher opinion addresses the question of complexity. As noted above, students typically focused on just a few attributes in assessing complexity. In the examples given, the focus is on intellectual complexity (IC) and mathematical complexity (MC) identified from responses to questions about the Road Accident problem.

I: So what is complex about this particular task? Alan: That, the difference was, normally with this type of work, the [pause] we are given

I : Uh huh. Alan: ... and here we are not, and that is what we have to find in order to find the radius

I : So you didn't think you were told the length of the chord? Alan: No, we were given different variables like. I : So, you would see this as certainly towards the complex end of the spectrum, but not

because of the maths that's in it? Alan: Yes. It is the actual thinking to get to find out what to do. (IC) Amy: Umm, because it takes a lot of thinking. And probably a lot of work to do it, it

would take time. Because it is something I have never seen before. I mean, it probably relates to something that we have done, but I just couldn't get to it, to relate it. (IC)

the actual [pause] uhmm [pause] the length of the chord.

and that was hard.

Bruce: Because you used more than the one formula. (MC)

With respect to teacher judgments the head of department at one school for example, distinguished between complex and simple tasks on the basis of intellectual complexity.

Teacher 1 : The characteristic, I think, of an unfamiliar applications task that makes it simple is that when a student looks at it a student is able to quickly

��


get to the nub of the problem. To do it may be a different matter but they are very quickly able to identify what to do, whereas a complex unfamiliar [application], I think, needs a fair bit of thought about where you are going to go next. (IC)

The Road Accident problem highlighted this focus on intellectual complexity although both linguistic complexity (LC) and representational complexity (RC) were also key attributes of the complexity of the task from teachers' perspectives.

Teacher 1 : This one has got a lot of words and reading involved and then you would have to synthesise a lot of information. This one actually illustrates what I was trying to say before, how a student has to do a lot of thought before they have to write down their answer. The answer may be relatively straightforward.. .But to me there is a lot of analysis and synthesis. (IC & LC)

Teacher 2: I'd probably say at this stage it's verging on complex for them because just the amount of data presented to them and the various ways in which the data is presented to them and the amount of reading in it, that they would have to go through in order to do it. (LC & RC)

Other teachers considered the task to be simpler based on the multiple representations contained in the task statement. Their judgement was therefore based on aspects of its representational complexity.

Teacher 3: I'd probably say it is a simple but unfamiliar application. Equations are given to them; data is presented to them. There's a graph drawn for them, so there's a lot of visual information to go along with the written stuff. (RC)

While both teachers and students appear to use only a subset of the total attributes of a task as indicative cues on which to base their judgments of complexity, it can be seen in the above responses, that this subset is not necessarily the same for everyone.

REFLECTIONS

The outcomes reported here direct attention to two important avenues for further work. The detail evident in the complexity profile (Figure 4) and the fact that individuals are able to attend to only so many cues at a time, draw attention to the impact of complexity on such factors as cognitive load (Sweller, 1988) and its impact on problem solving ability. The consequent link between task complexity and task difficulty is at the heart of the challenge to improve the ability of students in application and modelling, a challenge that has been addressed in Stillman (2002). Secondly, what is presented here is essentially theory creation. While the grounded theory approach has tested the emerging theory in terms of the extensiveness and robustness of the categories comprising the complexity profile, another form of testing awaits. This involves the use of the profile in constructing application and modelling tasks of pre-determined complexity, and

��


subsequent evaluation of insights into how teaching and learning effectiveness may be enhanced thereby.

REFERENCES

Galbraith P, Stillman G (2001) ‘Assumptions and context: Pursuing their role in modelling activity’ in Matos J, Houston S, Blum W, & Carreira S (Eds) Modelling and mathematics education: Applications in science and technology Chichester: Horwood Publishing, 3 17-327.

Smith R, Hurst J (1990). ‘Accident investigations’ in Huntley ID, James GJG (Eds.) Mathematical modelling: A source book of case studies Oxford: Oxford University Press, 67-80.

Stillman G (2002) Assessing higher order mathematical thinking through applications (Ph.D. thesis) Brisbane: The University of Queensland.

Strauss A, Corbin J (1990) Basics of qualitative research: Grounded theory procedures and techniques Newbury Park, CA: Sage.

Sweller J (1988) ‘Cognitive load during problem solving: Effects on learning’ Cognitive Science 12,257-285.

Williams G, Clarke D (1997) ‘The complexity of mathematics tasks’ in Scott N, Hollingsworth H (Eds.) Mathematics: Creating the future. Proceedings of the 1 6‘h Biennial Conference of The Australian Association of Mathematics Teachers Adelaide: AAMT. 45 1-457.

��

17

Using Workplace Practice to Inform Curriculum Change

Geoff Wake University of Manchester [email protected]

Julian Williams University of Manchester [email protected]

We report here implications for curriculum and pedagogy drawn from the research project Using College Mathematics in Understanding Workplace Practice. This project investigated how the practices of a range of workers might be understood by ‘outsiders’ using mathematics. In this paper we therefore look to the world of work and ask how this might inform mathematics curricula so that future workers may perhaps be better equipped to use mathematical thinking in the workplace. Many of the suggested changes may be considered mathematical modelling and problem solving skills and attitudes. We use our case study evidence to support and illustrate our call for these to be included in mathematics curricula.

��

190 Wake & Williams

INTRODUCTION

The focus of attention in our research project ‘Using College Mathematics in Understanding Workplace Practice’ (supported by a grant from the Leverhulme Trust) was mathematics at the ‘boundary‘ of college and workplace (Wake and Williams, 2001). By joining with students and workers in examining and discussing the mathematical practices of workplaces, we aimed to break out of previous research lacunae in which the two systems (college and work) were separately investigated and the differences between them emphasised. Instead we sought to build bridges for them in the discourses about workplace practices.

THEORETICAL PERSPECTIVES

Our theoretical perspectives were informed by the debate about transfer of knowledge in the education literature between two contrasting positions. The first we characterise as the classical psychological information processing position, that learning involves individuals in constructing cognitive structures undschemus that are effective in a variety of situations. These classical cognitivist ideas (Anderson, Reder, & Simon, 1996, 1997),

general mathematical competence and resources (skill) (Williams, Wake,& Jervis,1999; Wake & Williams, 2000), mathematical modelling, problem solving and inquiry strategies, and mathematical dispositions, and confidence

were useful in describing the demands placed on students in the case studies we developed. We were able to use these to identify skills which we researchers had available, and which the students and workers often did not have.

On the other hand we found the situated cognition perspectives (e.g., Lave 1988,1996), particularly cultural historical activity theory (Engestrom & Cole, 1997), helpful in analysing differences in workplace and college practices that cause dissonance. In particular, the notions of conflicting activity systems at work and in college have been helpful in pointing to the different structures of workplace and college activity and discourse. We find it useful to bear in mind the complexity of each workplace viewed as an activity system that has historically and socioculturally ‘moulded’ the practices that we observed. In college for example, the ‘object’ of activity is the education of individuals, with outcomes of qualification and certification. This contrasts with the workplace where the object of activity centres around production (of products or services) with the outcome of profit. Because of this primary function all the elements of the activity interact to ensure success. So, for example, workplace instruments (such as graphs, spreadsheets and so on) and the division of labour is organised to ensure mathematical mistakes are never, or rarely made. This contrasts starkly with the college where assessment, by its very nature, ensures that we test individuals to the limits of their capability and consequently they expect to make mathematical errors. Generally speaking, learning and communicating are implicit at work. Indeed, the mathematics itself is often hidden in systems and instruments designed to avoid mathematical demands of the workers.

��

Using Workplace Practice to Inform Curriculum Change 191

In this chapter we will focus in the main on the former of these two perspectives as we highlight the skills and competences useful in understanding practices in workplaces. But we bear in mind the situated structure of workplace mathematics in analysing the demands it makes on outsiders’ understandings as they attempt to bridge the gap.

METHODOLOGY

The project researchers acted as teacher-researchers and in a few cases were actually the students’ teacher. Each case study involved a researcher in: (i) observing and joining in discussions with workers about their working practices and the place of mathematics in them, (ii) discussing with teachers and students the mathematics in college, (iii) joining students with workers in the workplace, to discuss the mathematical work identified there previously, and (iv) revisiting the student and workplace to check understandings of the mathematical practices involved. The vital generative ingredient in each case study was the bringing together of researcher, worker and student(s) to explore together the mathematical activity in the workplace. These exchanges and other interviews were audio recorded and transcribed.

The nature of the dozen or so workplaces and the mathematical practices was varied as much as possible: a student with very little mathematical knowledge placed one day a week in a gardening department, (working with plans, making cement, planting out gardens and marking out pitches) on the one hand, and on the other, an industrial chemistry laboratory, in which an advanced mathematics student was on a short term work placement, (carrying out controlled experimental explosions with sophisticated technology). This variety of cases is designed to test theoretical propositions to empirical ‘destruction,’ but it has the added spin-off that it provides us with a variety of illustrations that we draw on here.

MAKING SENSE OF MATHEMATICAL PRACTICES IN WORKPLACES

In this section, we draw on our case studies to make some observations about the nature of mathematics in workplaces and the demands placed on the workers and college who were involved.

Workplaces develop mathematics in forms that do not conform or align with practice in schools and colleges. Mathematical conventions and methods are either adapted or indeed developed idiosyncratically. On some occasions the adaptation of conventions and methods from those that are familiar to us as mathematicians or maths educators is for a specific reason that is clearly rooted in workplace requirements. As an example of this, consider the change of variable plotted on the axis of a graph from one that is familiar to another that is an important measure in the workplace.

i’he controlled explosion and its unconventional graph: Figure 1 shows a graph, encountered by a highly able student on placement in an industrial chemistry laboratory. Although familiar with graphical work from her studies in both mathematics and chemistry, this graph from the workplace had features for which the student was unprepared. In discussion we identified these:

��

192 Wake & Williams

(i) one of the variables plotted on the horizontal axis has its lowest value at the right- hand end of the axis (and this being the more familiar variable, temperature);

(ii) more than one variable plotted on each of the horizontal and vertical axes; (iii) the trace is not an exact straight line or that of an exact function; (iv) the use of a logarithmic scale on the vertical axis.

Ln CVapour Press-peia)

6

5

4

3

2

1

0

- 1

TRAINING EXERCISE TOLUENE

Plot Of Log P r ~ - s u r s CPSI4) Against. 1/ C T ~ m p + 2 7 3 > -868875

Figure 1. Industrial chemistry laboratory graph.

One difficulty was due to the fact that, in effect, the graph in Figure 1 should be 'read' from right to left, i.e. the starting point of the experiment is at the right-hand end of the trace. The industrial chemist creates this difficulty by redefining the variable that is of interest from that of temperature to a coefficient that has importance in analysis of the

This new variable increases from left to right along the 10000 temperature + 273 ' situation, i.e.

horizontal axis, but this leads to confusion as the new variable is low when temperature is high and is high when temperature is low. This has the effect of the more familiar quantity, temperature, having high values at the left of the horizontal axis and having low values at the right of the horizontal axis. This is the reverse of the student's common experience of graphs in both mathematics and chemistry-so much so that her reading of the graph was from left to right and to be consistent with this she was willing to suggest that during the experiment the materials cooled down although she knew that during the experiment the sample was heated at 2 Wmin and hence throughout the temperature was

��


rising. For a more comprehensive description of this particular case study refer to Williams, Wake, & Boreham (2001).

An idiosyncratic spreadsheet formula to estimate gas consumption: On other occasions idiosyncratic mathematics occurs because that is the way the mathematics was constructed or developed. For example, consider this formuldmodel in a single cell of a spreadsheet developed by an engineer in a power plant. He uses this to predict the gas required by the plant over a twenty-four hour period:

{ { { { {r‘2nd INTEGRATING READING”-“0600 INTEGRATING READING”}+

READING”)}/T2}*TIME4)}/100000}/3.6*CALCV* 1000000/29.307 1 } { { {6r2nd INTEGRATING READING"}-{“^ St INTEGRATING

Without recounting here the lengthy explorations and discussions of worker, researcher and students, during which they attempted to make sense of this particularly complex formula, it is worth noting the features of the idiosyncratic algebra that the engineer has constructed. Cells in the spreadsheet have been assigned names based on terminology familiar in the workplace thus making it easier for the engineer to recall the reasoning encapsulated by his formula. The use of so many brackets lengthens the formula and makes it look very complex but perhaps this is because it has been built in stages. The brackets may well be useful in emphasising this and helping him to ‘unpack’ the logic in the construction of the formula when necessary.

Programming a punch press: As a final example of adaptation of conventions and methods, consider the activity of a worker who uses technical drawings supplied by customers to produce a programme to drive a punch press that stamps components from sheets of metal. The programs he develops are particularly idiosyncratic; they result in command lines such as

“ X 2 5 . Y 172.5 TI2 (390.”

Here “T12” designates the tool to be used whilst X and Y axes are unconventionally developed from an origin in the top left of a metal sheet with X being positive in the direction left to right, and with Y being positive in the direction from top to bottom. The worker, although he has no formal understanding of vectors, flips with ease (using instructions G90 and G9 1) between absolute and relative referencing in this coordinate system.

On occasions, during workplace conversations, the researcher and/or worker tried several strategies to assist those present to develop a mathematical understanding. These strategies can be considered as basic modelling or problem solving skills, or alternatively, as pedagogic approaches to mathematical explanation. It is clear from our analysis of transcripts that, in general, the students did not have available to them such a range of strategies to assist them in making sense of the mathematics with which they were unfamiliar. We summarise and illustrate these here.

A formula to calculate percentage error: To exemplify this we return to the engineer in the power plant who as part of his daily routine predicts how much gas the plant will

��

1 94 Wake & Williams

require from its supplier, over each twenty-four hour period. In one of his spreadsheets this engineer calculates how efficient his predictions were, using the formula:

{ "BG THERMS"/"MONTHLY THERMS'') * 100- 100.

Here "BG THERMS" represents "British gas therms," the amount of gas actually used by the plant; "MONTHLY THERMS" represents the engineer's estimate of the use of gas. Following the visit to the power plant, the researcher attempted to make sense of this with a student. They first explored the idea of error. In doing so, the researcher appealed to an everyday context:

R(esearcher)

S(tudent) Right. R

S Oh right, twenty.

If 1 said, 'I'm going to town to buy a television set', and I think it'll cost E300, yes? [anchors in the familiar case of shop purchase]

Now, I go down, go to [well-known store], find one 1 like, and it's let's say E320. What was my error? How much was I out by?

The researcher and student proceeded to consider the idea of percentage error, and again make progress by considering the everyday context of shopping with the researcher asking, "So if we go back to this where I was out by E20, now I want to know, what percentage was that of the actual value?" The researcher goes on to build bridges from this example to the actual workplace problem, "Now, if you look at his way of doing it, what he would have done.. ."

I t becomes apparent as the researcher and student consider the workplace transcript that, in engineering terms, the fraction required is that of error / estimate. This contrasts with the usual practice in mathematics of using error / accurate; we may well have included this illustration in the section above that looks at mathematical conventions and methods.

Linear interpolation-a spreadsheet formula and graphical representation: One of our case studies centered on an engineer's modelling of potential damage due to the fracturing of steam pipes. At one point, this resulted in researcher, worker and student investigating the use of linear interpolation. Figure 2 shows the formula used by the worker and the sketch the worker drew to attempt to explain her method to the student.

Note that the worker translates spreadsheet notation to the axes of her sketch graph. This graph is perhaps more familiar as 'college mathematics' and helps the outsider to make sense of it. The student struggled to understand the method used in the linear interpolation; the worker had worked from the later time (B 16) rather than using the more familiar college practice of working from the earlier time (B10). However, the college practice of using a graphical representation assisted everyone in eventually coming to terms with the mathematics involved.

In a number of case studies, we found formulae being used by workers to calculate a

��


z((B16- B12)/((B16- BIO) c:p/ CI ".'.

~

*(ClO -CI 6)) f C16

f

BIO BI B,

Figure 2. Spreadsheet formula and the worker's graphical interpretation.

range of measures of performance. Some of these (both formulae and measures) are relatively straight-forward. However, outsiders, such as researchers and students often have difficulties in understanding what exactly is going on. One way of attempting to make sense of them is to consider special, or extreme, cases.

A formula to indicate pay back time: As an example, we turn to a case study where we investigated finance office workers analysing performance data in a medium-sized retail company. Here researcher and workers struggled to make sense of an indicator "debtor days." This particular indicator is used to give an indication of how long customers are taking to clear their debts and is found using the formula:

debtor days = (outstanding debts / annual turnover)*365.

The worker who calculated this measure each month had a sense of what the measure conveys, but was not able to relate the measure related to the data involved. The researcher and office manager, however, were able to gain an understanding by substituting the simplified values, "annual turnover = 2 million [pounds]" and "outstanding debts = 1 million [pounds]" giving "debtor days = 182.5" or half a year. This gives an indication of how long customers are taking to clear debts. Perhaps the confusion was compounded because the worker actually knew how long each customer takes to clear debts associated with their purchases.

Calculatingprofit: In the same case study, the office manager tried to clarify for students the ideas of gross margin percentage and uplifj on cost by again referring to simplified values:

W(orker)

S(tudent) Is it loo?

What's the gross margin of something? If you bought something for a El00 and sold it for E200, what's the gross margin percentage?

��

196 Wake & Williams

W No, that’s uplift on cost. Gross margin is the profit divided by what you sold it for. So in that case it would be 50%. Uplift on cost is 100% because you’ve got a cost of 100 and you sold it for 200, so they are two different things.

Calculatingpercentage error: As another illustration of such a strategy, we return to the researcher working with a student who was struggling to make sense of percentage error in calculations associated with use of gas in the power plant.

Now, the way it works, e m , you can actually see that if I make my numbers a bit easier. Let’s say instead of 200 over 300 which isn’t very nice, lets say my estimate is 20 pounds out, and the actual price is just El00 then, if you worked that out it would be, erm 20%.

On a few occasions the validity of the mathematics used came under scrutiny as illustrated in the following example,

Calculating the mean: This case study concerned a police inspector calculating and interpreting performance measures. The inspector found the average of a particular measure for his whole division by averaging the averages for each of the five towns in the division. The researcher questioned the validity of doing this because each town had a different number of teams of police officers. This did not concern the police officers and students until the researcher suggested, “if you imagine ... one of your divisions has only got one [team] in it, and the other one’s got a 100 [teams] in it ...”. All involved in the conversation could then identify the potential problem. The police officer asked:

Do you think it’s unfair because you‘ve got more [teams] in one area than another ... So somebody’s carrying a bigger burden..? ... I know what you’re getting at, yes. We have to take the weighting out of it.

Although both students and police had neither the correct mathematical language to talk about the problem nor the technical competence to surmount it, they did have, in this context, at least an intuitive understanding of its nature based on the notion offhirness. Perhaps questioning the validity of mathematics presented to them is not something we should expect of students. After all, much of what they experience at college is concerned with the presentation of a correct method leading to a correct answer.

As a final case study, consider the following example that demonstrates a mix of some of the activities considered so far.

Modelling an accident in a workplace: The weighting of data became significant when an engineer was explaining how she had modelled how much space was filled by equipment across thirteen separate areas that she and her colleagues called nodes. This activity was part of a larger problem in which the engineer was trying to determine the effect that a break in pipes carrying steam would have. The engineer had calculated thirteen individual percentage values and went on to explain to researcher and student:

So, what we could have done, is just added up these 13 numbers and divided it by 13 and found an average of the 13. But, when you look at it, I

��


mean, the percentage.. . If node nine is 25% full of equipment, then that’s a lot of equipment compared to, say, node one having 25% of equipment. (A scale diagram of the situation showed that area 9 is much larger than area 1 .) So we couldn’t really just add them all up and divide by 13.. .

The student was unsure about why it wasn‘t correct to simply find an average of the values for each area. Using the sketch in Figure 3, the researcher attempted to explain.

If you imagine you’ve just got two nodes, and you’ve got one huge node, and it’s 50% full of equipment. And then you’ve got a tiny little node next door, and it’s only 10% full of equipment. If you just average the 10% and the SO%, you’d get, what? 30%. But, in fact, that big node is much more important than the little one.

50% filled 10% filled

Figure 3. Researcher’s sketch of extreme cases.

The engineer suggested considering the more extreme case where one node is completely full of equipment and the other completely empty and then went on to offer an explanation in a context with which the student may be more familiar:

I suppose it’s a bit like an election, in a way, in certain things, if you said, ‘Well, we’ll give every house one vote’, and you could end up with a university block with 200-300 people going in, then there’s 200 people there who are important, as opposed maybe to the one person living on his own next door. You would say, well, half of the votes went for this and there was one vote either way, whereas there’s really 200 people against one.

DISCUSSION: IMPLICATIONS AND SUGGESTIONS FOR CURRICULUM AND PEDAGOGY

Our work to date leads us to suggest that we need to develop appropriate mathematics curricula that would better prepare students to make sense of workplace practices and that the approach adopted in our research is one way of informing such development. Previous research used to inform curriculum design too often involved secondary data, and relied on accounts by industrialists of the mathematics they think their workers need. We doubt the validity of such an approach. Other research in the workplace goes further and overcomes this weakness to some extent: workplace ethnography establishes

��

198 Wake & Williams

mathematical practice somewhat ‘as it is lived’ by the insiders. However, there is always an ‘outsider’ effect, and the problem of subject and object in ethnography is a critical concern in anthropology (for example, see Wagner, 2001). Rather, then, we believe the study of ‘appropriate outsiders’ is needed: in this case teachers and students, to generate valid accounts of mathematics in workplaces from the perspective of the college learner and teacher.

Our investigations suggest that curricula and pedagogy change to reflect our experiences in the following respects:

workplaces involve ‘complex and messy’ problems but often only use relatively simple mathematics; workplaces present a diversity of conventions and idiosyncrasies which make the mathematics seem strange to outsiders; workplaces present outsiders with a need to ‘decode’ the mathematics of others; and making sense of workplace mathematics in complex situations makes demands on outsiders’ problem solving strategies, e.g., considering simple cases, and demands a critical, inquiring disposition which many students seem to lack.

We found that the workplaces we visited hide many interesting practices that challenge outsiders’ understanding of mathematics. The workplace activity masks mathematics with its special conventions and tools, its division of labour, and its ruledways of communication. Mathematics tends to become invisible over time (some say transparent) to the workers involved and fused with their daily actions, jargon and technology. Because these are generally unfamiliar to the outsider however, they cause substantial demands. To overcome these difficulties students and other outsiders need to develop flexible attitudes to the way mathematics is used (and often specially constructed) in the workplace.

We found that students were not always well prepared for this challenge. For instance students:

had little awareness that the college mathematics they had learnt was ‘conventional’, and that conventions might be varied; had little awareness of the extent and limit of their own mathematical knowledge; were not always well prepared with the inquiry skills needed to uncover the mysteries of the mathematics embedded in the workplace situation.

We suggest that to better prepare students to be able to make sense of mathematics they may meet in their future employment, their curricula (and associated assessments) should:

challenge students to explore a diversity of mathematical conventions and methods; encourage students to develop mathematical thinking in contexts that reflect realistic workplace situations; encourage students to develop a range of mathematical modelling and problem solving strategies; and require students to investigate, make sense of, and explain the mathematical activity of others.

��

Using Workplace Practice to lnform Curriculum Change 199

REFERENCES

Anderson JR, Reder LM, Simon HA (1996) “Situated learning and education” Educational Researcher 25(4), 5-1 1.

Anderson JR, Reder LM, Simon HA (1 997) ‘Situative versus cognitive perspectives: Form versus substance’ Educational Researcher 26( 1 ), 18-2 1.

Engestrom Y, Cole M (1997) ‘Situated cognition in search of an agenda’ in Kirschner D, Whitson JA (Eds) Situated cognition: social semiotic and psychological perspectives NJ: Lawrence Erlbaum, 30 1-309.

Lave J (1996) ‘Teaching as learning’ Practice Mind Culture Activity 3(3), 149-164. Lave J (1988) Cognition in practice: Mind mathematics and culture in evevday life

Wagner R (2001) An anthropology ofthe subject London: University o f California Press. Wake G D, Williams J S (2000) Mathematics in pre-vocational courses in Bessot A.,

Ridgeway J (Eds) Education for Mathematics in the workplace Dordrecht: Kluwer.

Wake GD, Williams JS (2001) Using College Mathematics in Understanding Workplace Practice Manchester: University o f Manchester.

Williams JS, Wake GD, Jervis A (1999) ‘General mathematical competence in vocational education’ in Hoyles C, Morgan C, Woodhouse G (Eds) Mathematics Education for the 2Ist Century London: Falmer Press.

Williams JS, Wake GD, Boreham NC (2001) ‘College mathematics and workplace practice: an activity theory perspective‘ Annual Journal ofthe British Society for Research into Learning Mathematics.

Cambridge UK: Cambridge University Press.

��

18

Comparing an Analytical Approach and a ConstructiveApproach to Modelling

Toshikazu IkedaYokohama National University, [email protected]

Max StephensUniversity of Melbourne, Australiam. [email protected]

This study investigates the effects when an analytical approach is used inmodelling as compared to a constructive approach. In the first approach,students were given a simple mathematical representation of the situationbeing modeled, while in the second, students were given some keyquestions to guide them in creating a suitable representation. Although theresults were not totally in favor of a constructive approach, it does appearthat students using an analytical approach tended to focus too much on thegiven mathematical representation without paying sufficient attention to theassumptions and limiting conditions implicit in the situation.

��

202 Ikeda & Stephens

BACKGROUND TO THE STUDY

Mathematical model1ing and applications were gradual1y introduced into the teaching ofmathematics in junior and senior high school in Japan in the 1990s. Japanese documentsof 1990s emphasized a variety of goals in mathematical model1ing and applications(Ikeda, 2001); namely, appreciation of mathematical model1ing, fostering mathematicalthinking to promote mathematical model1ing, and acquiring particular mathematicsknowledge and skills through mathematical model1ing.

Teachers want to foster mathematical thinking and to promote mathematical model1ing,but they need to know how. In this study, we assume that group discussion promotesmathematical model1ing (Ikeda and Stephens, 2000). However, after presenting amodel1ing task and letting students discuss it in groups, how does a teacher best supportstudents' progress?

We investigate two different approaches to teaching mathematical model1ing. The firstapproach is to have students analyze and interpret a simple mathematical model and adiagram provided by someone else. We cal1 this an analytical approach. The secondapproach is to ask students key questions to elicit ideas that promote mathematicalmodel1ing. Students then construct a mathematical model by themselves. We call this aconstructive approach.

What are the likely differences in students' performance between the two approaches?The fundamental purpose of this study is to compare the two approaches to the teachingof model1ing. After students obtain the mathematical solution, is there any differencebetween the two approaches in students' ability to interpret, validate and modify amathematical model? How do these different approaches help students to take intoaccount the original assumptions and conditions, and to change a specific problemsolution into a more general solution?

Modelling Task: Bushwalking with Kim

The task for this study is adapted from a problem developed for Year 12 mathematicsstudents in the Victorian Certificate of Education (VCE) in Victoria, Australia (Board ofStudies, 1997). In the task Bushwalking with Kim, students must minimize the time ittakes Kim to travel from one point to another by selecting the best route, while takinginto consideration the location of a clearing in which a bushwalker can walk faster than ispossible in the bush.

Bushwalkers travel through different types of country. The denseness ofthe bush and the ruggedness of the terrain influence the speed of travel. Byplanning a route to take such factors into consideration, the total time takento travel from one point to another can be reduced. In calculating estimatesof the time for a particular route, a walker uses his/her average speed foreach different type of country. Kim is planning to walk from Ardale (A) toBrushwood (B). The direct route, a distance of 14km, wil1 take her entirelythrough rugged bush country. However, as shown here, there is a large

��

Analytical vs. Constructive Approaches

square clearing between the two towns, withA 8 side length estimated by Kim to be 7 km.

Kim assumed that this clearing has onediagonal along the perpendicular bisector ofthe direct route from A to 8 and one comer,C, at the midpoint of the direct route.Further, Kim estimates that she travels at anaverage speed of Ikm/h in the bush and

5km/h through the clearing. Find and describe the route for which hertraveling time will be least.

DESIGN OF THE STUDY

203

Procedures and SettingWe used two classes consisting of 48 and 50 students, respectively. In each class,

students formed working groups of four or five members. All students were given ageneral diagram illustrating the global process of mathematical modelling. This studywas carried out in four stages:(I) Testing (60 minutes) to establish baseline data regarding students' modelling ability

using test items developed by Haines, Crouch and Davis (200 I).(2) Presenting students the problem Bushwalking with Kim and an additional description

according to the type of group, and having them discuss it in their working groups for90 minutes.

(3) Having students write a short reflection about their discussion and a full report oftheir own solutions, using both in-class and out-of-c1ass time over the course of oneweek. In the reflective report, they were asked to describe the issues they saw asimportant in developing a mathematical solution to the original modelling task and togeneralize their solution.

(4) Taking a test one week later (90 minutes) that focused on mathematical modellingspecific to the task Bushwalking with Kim.

To analyze the differences among the two classes' achievement on the modelling task, weused a partial-credit scoring rubric, similar to that used by Ikeda and Stephens (200 I).The partial-credit data was complemented by a second source of data, namely, the studentreport. If several groups got high (or low) scores in the partial-credit analysis, we couldlook for reasons for this by analyzing each student's report on the discussion in his/hergroup. The student reports were also used as qualitative tools to investigate thefundamental question of the study: to what extent did the teacher's approach facilitate orrestrict students' thinking about the modeling task? Finally, we compared students'achievement across the two classes to determine the effects of the two treatments, and tointerpret the magnitudes of these effects by comparison with the baseline data.

Five Questions for Each ClassBoth classes were given five questions to promote mathematical modelling. Ananalytical approach was used in class A and a constructive approach was used in class B.(QI) and (Q2) are intended to help students to clarify the problems. Both questions are

��


the same for each class except that Figure 1 was provided to class A and not to class B.

(Ql) The simplest case is for Kim to take the direct route from A to B.Compare the time taken by Kim using the direct route and the time taken ifKim takes the shortest path from A to the clearing, traveling in a straightline across the clearing, and, after leaving the clearing, heading directly forB.(Q2) If Kim decides not to take the shortest route to the square clearing, thatis, she heads toward another point on the side of the square, can youestablish a specific time for this route?

N

1

Figure 1. Diagram provided to class A.

The next question marks a crucial difference between class A and class B. In class A,using the analytical approach, a mathematical formula and a simple mathematicalsolution with a diagram were presented to students.

(Q3-A) When we assign a variable x as shown Figure 2, the total timefunction t(x) is t(x)= 2x/5+2-V{(7-x)2+X2}. Explain how this formula isobtained.Further, we can show that the minimum time is 11.2 hours when x=3.Check this result by using a graphing calculator or by some other method.Please explain what your results mean.

c 7q,----""lr"':~----1'8

A

Figure 2. Solution presented to class A.

��

Analytical vs. Constructive Approaches 205

In the solution of (Q3-A), students who are presented with this simple model need only tointerpret the formula and validate whether or not the answer satisfies the assumptions,and then go on to consider and evaluate other possibilities.

In class B, using the constructive approach, question 3 was designed to help them thinkabout how to construct a mathematical formula by assigning x as some appropriatedistance.

(Q3-B) Is it possible to establish a general formula involving x (suitablydefined) giving the total time taken by Kim to travel from A to B utilizingsome part of the square clearing? Then can you use this formula to find itsminimum value? Please relate your finding to a point on the squareclearing to which Kim must head from A. Compare the minimum timewith what you found in (QI) and (Q2) above.

(Q4) and (Q5) are the same in both classes. (Q4) is intended to analyze a student's abilityto take into account the limitations of the original assumptions and conditions. (Q5) isintended to analyze their ability to consider a real problem by changing a specificproblem situation into more general problem situation.

(Q4) What information is useful for Kim in planning her route from A toB? For example, what if Kim does not know the side length of the squareclearing?(Q5) Is it possible to establish a general formula giving the minimum timetaken by someone else (not KIM) to travel from A to B? Can you then usethis formula to find some minimum times taken by this other person?Please explain what your results mean.

The students were given 90 minutes to discuss the problem and the related informationprovided. At the end of this discussion period, each student was asked to write a shortreflective report about their group's discussion and to write a full report about their ownsolution. Both classes received general guidelines for writing up a report. We decided toanalyze the differences between two classes by focusing on (Q3), (Q4) and (Q5)specifically.

The Post Test and Partial-Credit CriteriaThree test questions were developed to provide an indication of how well students hadarrived at the key ideas that are essential to developing a successful model.(Question I) Formulate the total time function and find the minimum time.(Question 2) If Kim knows that the side length of the square is exactly 4 KM what should

her strategy be?(Question 3) Consider someone else whose relative speeds are not the same as Kim's. As

the person's relative speed through the clearing increases (or decreases),how should they plan their journey?

The scoring criteria for three test problems were as follows.Question 1• I point Formulating the total time function f(x).

��


• I point Paying attention for the range of x.• I point Getting the correct answer by using graphic calculator or by some other

method.Question 2• I point Noticing that 4km is shorter than the route used in question I.• I point Noticing that the total time function is monotonically decreasing for x<3,

but not explaining clearly or• 2 points Explaining clearly that the total time function is monotonically decreasing

for x<3 by checking the shape of graph, or by systematically substituting severalvalues.

• I point Finding that the minimum time occurs when x=2'./ 2.• I point Explaining that Kim should go toward the vertex of the square.Question 3• I point Noticing that the route will change by changing the relative speed.• I point Finding the general formula.• I point Checking the fluctuation by substituting one or two values or

2 points Checking the fluctuation systematically by substituting several values.• I point As the relative speed increases, the walking distance through the square

clearing is getting longer.• I point As the relative speed gets smaller, the route is closer to the direct route from

A to B.• I point As the relative speed is getting larger, the route is closer to the shortest route

to the square clearing (that is, at an angle of45° to the direct route and perpendicularto the side of the clearing.)

ANALYSIS OF THE STUDY

Pretest (Baseline Data)In the first stage of the study, we used modelling questions developed by Haines, Crouchand Davis (200 I) in order to check whether or not there were any substantive differencesin students' modelling abilities between classes. Two problems were selected for each of6 key components of mathematical modelling ability (Haines, 2001, pp.368-369): (I)making simplifying assumptions; (2) understanding of the nature of the problem; (3)formulating a precise problem statement; (4) assigning variables, parameters andconstants; (5) formulating equations or mathematical statements; (6) choosing the correctmathematical model.

There were 12 problems. A correct response received 2 points, and a partially correctresponse received I point. The results for each working group in class A are reported inTable I, and the results for the working groups in class B, in Table 2. There was nosignificant difference in performance on the pretest between class A and class B. (TTest: p=.13153 >.05, one side). This result suggests that the two randomly assignedclasses, A and B, are roughly similar in modelling ability.

Results and Analysis of Post TestTable 3 show the average and (unbiased) standard deviation for classes A and B on thepost test. There is a significant difference in the average scores on question 2 between the

��


GroupI 2 3 4 5 6 7 8 9 10 II Tot.number

#ofstudents 5 5 5 5 5 4 4 4 4 4 3 48in zroun

Total72 78 79 85 85 69 63 65 61 73 54 784score

Average 14. 15.6 15.8 17 17 17.3 15.8 16.3 15.3 18.3 18 16.3score 4

Table 1. Results of pretest in class A.

GroupI 2 3 4 5 6 7 8 9 10 II Tot.

number#of

students 4 4 4 5 4 5 5 5 5 5 4 50in group

Total 61 60 63 93 66 77 94 89 85 89 72 849scoreAverage 15.3 15 12.6 18.6 16.5 15.4 18.8 17.8 17 17.8 18 17.0

score

Table 2. Results of pretest in class B.

two classes (T-Test, p=.042<.05, one side). There is no significant difference on theother questions (T-Test, Question I: p=.409>.05, Question 3: p=.327>.05, total:p=.l08>.05, one side). These results suggest that students using the constructiveapproach may have some advantages over those using the analytical approach inconsidering the limitations imposed by the original assumptions and conditions.However, there appears to be no difference between the classes in changing a specificsolution into more general solution. There is no significant difference in the variance onquestions 1,2 or 3 between class A and class B (F-test, Question I: p=.391>.05, Question2: p=.197>.05, Question 3: p=.241>.05, total: p=.273>.05, dfA=47, dfB=49).

The average scores of each working group in class A for questions 1, 2, and 3 arereported in Table 4. Similarly, the average scores for the groups in class B are given inTable 5.

Ouest ion 1 Question 2 Ouestion 3 TotalAve. S.D. Ave. S.D. Ave. S.D. Ave. S.D.

Class A 2.21 0.62 3.08 1.78 3.60 1.63 8.90 2.76Class B 2.18 0.60 3.68 1.60 3.76 1.77 9.62 2.96T-T,

0.409 0.391 0.042 0.197 0.327 0.241 0.108 0.273F-T

Table 3. Analysis of the post test.

��


Gl G2 G3 G4 G5 G6 G7 G8 G9 GIO GIl Ave. S.D.QI 2.40 2.20 2.20 2.40 2.40 2.25 1.50 2.25 2.25 2.00 2.33 2.20 0.25

Q2 3.60 2.60 2.80 3.80 2.40 1.00 3.25 3.50 4.75 3.00 3.33 3.09 0.90

Q3 3.00 4.60 2.80 2.60 4.00 2.25 2.50 4.25 5.25 3.75 5.33 3.67 1.06

Tot. 9.00 9.40 7.80 8.80 8.80 5.50 7.25 10.0 12.2 8.75 11.0 8.96 1.72

Table 4. Results of the three questions in each small group in class A.

GI G2 G3 G4 G5 G6 G7 G8 G9 GIO GIl Ave SOQI 2.50 1.25 2.25 2.40 2.00 2.00 2.00 2.20 2.00 3.00 2.25 2.17 0.41

Q2 5.00 2.50 2.75 3.80 4.50 2.80 4.20 3.00 4.60 4.20 3.00 3.67 0.84

Q3 5.00 2.25 3.00 3.20 4.50 1.80 4.00 4.60 5.60 4.20 3.00 3.74 1.13

Tot 12.5 6.00 8.00 9.40 11.0 6.60 10.2 9.80 12.2 11.4 8.25 9.58 2.07

Table 5. Results of the three questions in each small group in class B.

The results of further analysis are given in Table 6. In this table, B>A means that thevariance in class B is greater than that in class A. There is no significant difference invariance of the working groups between class A and class B.

Value ofF ProbabilityQuestion I 2.71 B>A 0.066Question 2 1.15 A>B 0.414Question 3 1.14 B>A 0.421Total 1.45 B>A 0.285

Table 6. Results of statistical analysis of variance.

Analysis of students' reportsWe analyzed students' reports by focusing on questions 3 and 4. Under the constructiveapproach, we observed two methods of defining x in order to formulate the total timefunction. These are shown in Figure 3. They were different from the method provided inthe analytical approach.

Figure 3. Two other methods of defining x.

��


Further, among those in class 8 who could fonnulate the total time function, there werequite a few who could not analyze the total time function suitably. For example, it wasobserved that one group found the minimum value of the formula (7_X)2+X2 under thesquare root by mistake, after formulating the total time function y=2x/5+2"'«7-x)2+X2).There was one more interesting feature of students' approach to question 3. Under theanalytical approach, we found that most students substituted x=3 into t(x) at first toconfirm whether or not the value is correct. Then, we observed that some students, whogot a low score on question 2, used the total time function given in the task and obtainedthe minimum value by differentiating the total time function, without trying to interpretthe meaning oft(x). One student's work is shown in Figure 4.

..-;l '"\ ~

3 "~ :Jif",,)

..- 0 t ,

i~,I

{be) \t ;:r ,

: :::1..••

.";i(ij~~f"~1 . ~t';-l~ffl ~ r!7rI.:fl.'i- ¥ ~ 0 ,~.~.W..., .... ~f:J'j't, ..t;'b~):Qr:.tn )';1)~l1t f~~JI::I'f(f) JoI) ...

rRrr-rJ?'+ x." 1'10;(· H '" 0 ... ~

<D~~r----... m ••• )~~::Z}·-t.::t:' ~ -:.(I.O~.~.J.S? ':~ .. (j>/ ..

:: : T' (j) t~'" .. ",..... --. .. . . . .. _.. -,.. -1l; . ... ..- llo:.< - is] /'0 <::; I' t < To'" @

, .,. ,

Q).f)~iJt Lit..7. ']:2 _ 14-~.~'tt= ItJ.O Xl ~ ..~0OJ:~ . .'.~1S. ....

,g:t?- 'f:( t 12) =-0

qf('i-3](I-'f)~O 1f~~L:::O .-r.11 tit It C@~1TD~(;r

-j. = 3

Figure 4. A student in class A gives no explanation for the total time function.

��


In another small group from class A who got a low score on question 2, we found somestudents who substituted x=3 into the total time function t(x), and got the value of t(3),namely t(3)=11.2. Then they concluded that t(x) has a minimum value when x=3,offering no explanation. In addition, we found many students using the analyticalapproach who claimed that they found the minimum value for t(x) by using a graphingcalculator. The explanation "by using a graphing calculator" was suggested by thewording of question 3. It is not clear whether or not these students really took intoaccount of the range of x by actually using the calculator. Of course, these students couldnot adequately answer question 4, namely, they could not refer to the case when x<3.

On the other hand, we observed that many students using the constructive approach paidattention to the range of x. For example, one student, whose work is shown in Figure 5,said that when x is more than 6 km, it takes more time for Kim to walk than when thevalue of x is 6 km. If x is less than 6 km, it takes less time for Kim to walk on thediagonal of the square. (In this case, x is defined as type A in Figure 4.)

Figure 5. A student under the constructive approach explores the range of x.

From these results, it might be said that more students under the constructive approachappear to consider the problem by taking account the original assumptions andconditions. The reason might be that they have been required to build up themathematical model. However, students working under the analytical approach, wherethe mathematical model and the solution process were given, appear to focus almostexclusively on the meaning of the total time function, and how to get its minimum value.As a result, there may be no impetus for students to check whether or not their solutionsatisfies the assumptions and conditions. We conjecture that this may be a major reasonwhy there is a significant difference on question 2 between the two approaches.

RECOMMENDAnONS FOR TEACHING MATHEMATICAL MODELLING

In a constructive approach, there are several key issues for students to discuss in order toget a mathematical solution that are not required in an analytical approach. Therefore,the teacher needs to support students' activity, without specifically telling them what todo. Students using a constructive approach also tend to pay more attention to thelimitations imposed by initial assumptions and conditions as compared to students usingan analytical approach. Therefore, it appears all the more important when using ananalytical approach for the teacher to set up the situation so that students must take

��


account of the limitations imposed by assumptions and conditions. Asking probingquestion is essential in the teaching of mathematical modelling. In the modelling task weused in this study, the question, "If Kim knows that the side length of the square isexactly 4 km what should her strategy be?" is important for making students discuss thelimitation of assumptions and conditions. There was no clear difference in students'performances between two approaches in students' ability to change a specific problemsituation into more general problem situation.

While these recommendations are based on one specific modelling task, they tend tosuggest that students can have a tendency to look only at the mathematics if they arepresented with a simplified mathematical formulation of the problem. Teachers shouldavoid using any approach that may lead students to believe that getting a mathematicalformulation is the essence of modelling. Clearly, teachers are not faced with a simplechoice of selecting a constructive method over an analytical method. Our study did notshow one method to be obviously superior to the other, and in fact, both methods may beused in the classroom for different purposes. Whatever method is used, teachers need tofoster working arrangements that promote deep questioning and reflection by the studentsduring the modelling process.

REFERENCES

Board of Studies (1997) Mathematical methods common assessment task (CAT):Investigative project Student Booklet. Carlton, Australia: Author.

Haines C, Crouch R, Davis J (2001) 'Understanding students' modelling skills' in MatosJF, Blum W, Houston SK, Carreira SP (Eds) Modelling and mathematicseducation Chichester: Horwood Publishing, 366-380.

Toshikazu I, Stephens M (2001) 'The effects of students' discussion in mathematicalmodelling' in Matos JF, Blum W, Houston SK, Carreira SP (Eds) Modelling andmathematics education Chichester: Horwood Publishing, 381-390.

Toshikazu I (2002) 'A study of the teaching goals of mathematical modelling andapplications in Japanese documents of the 1990s' Journal ofthe Japanese SocietyofMathematical Education Vol LXXXIV (5), 2-12 (in Japanese).

��

Section E

Perspectives

��

��

19

The Place of Mathematical Modelling in MathematicsEducation

Michael J. HamsonFormerly, Glasgow Caledonian University, [email protected]

This paper will first consider the current content of mathematics curriculain schools and universities. Although the author's views are influenced byfirst hand knowledge of the UK educational system, the problems inmathematics education discussed here are universal. Current deficienciesin the content of secondary school mathematics and the difficulties inrecruiting mathematics undergraduates are discussed. The place ofmathematical modelling within mathematics curricula at school anduniversity levels is then outlined. A strong case is made for includingmodelling and problem solving in these curricula.

��

216

INTRODUCTION

Hamson

There is currently a crisis in mathematics education. There has been a downturn instudent applications to read mathematics at universities and colleges, markedly so in theUK. This has already caused and wi1l continue to cause the closure of many universitymathematics departments. Reasons for the decline in student interest in math are notdifficult to locate: unattractive curricula over-full with techniques of little use or interest,teachers who have a poor grasp of the use of mathematics in real life, students who areunaware of the careers open to mathematics graduates, and the fact that virtually notraining is provided at any stage on how to apply the subject. I shall review mathematicseducation in the following section and then consider how mathematical modelling canplaya part in alleviating the crisis.

MATHEMATICS EDUCATION

From age 6 to 16 pupils study computation and quantification, communication andlanguage, and other key skills needed in life. Thus a student's initial mathematicseducation occurs largely within this IO-year period. It is hoped that the mathematicscurriculum for these years consists of much more than mere content. Equally importantis how the mathematics is delivered and why particular topics are included. There is alsothe underlying message about what mathematics is what it means to use mathematics inadult life. If these features are well presented then many will want to extend theirmathematics beyond the IO-yearperiod.

What is Mathematics?Mathematics itself is an abstraction. Was it invented by its practitioners, or was itinherently present on earth waiting to be discovered? There was a radio programmebroadcast recently in the UK by the SSC on this matter and the (famous) participantscould not easily agree on the truth here! Surely we need to count, buy and sell, measureand calculate, and even gamble successfully in order to live in the modern world. At aminimum we must handle household mathematics. So who needs algebra, geometry andtrigonometry? This question is worth considering and we should all be able to answer itbecause there are plenty of folk around (even some educators) who like to boast of theirsuccess in life starting with the claim "I was hopeless at school math and dropped thesubject as soon as possible - look where I am today!" The answer is in the need toprovide a firm foundation of mathematical techniques at school level but at the same timeto help students see the inherent usefulness of mathematics, as well as its beauty, patternand mystery. If algebra, geometry and trigonometry are only taught in a mechanical waywithout discussing applications, there wi1l be few converts. It is essential during the 6-16period that teachers emphasize the usefulness and application of the content as it is beinglearned.

Mathematics Curriculum 11-16Let us now briefly outline the math topics that we want to see in the 11-16 curriculumand note that there is good practice already in operation (see Edexcel GCSE MathematicsA). We should be clear that the seeds of mathematical interest are sown during thisschool period and acknowledge that if the teaching therein is poorly directed from a

��

Modelling in Mathematics Education 217

boring curriculum, potential higher math study will be irrevocably rejected by manystudents.

From a post-secondary perspective, the following topics appear to be essential:• Numbers and calculations, indices.• Using calculators.• Formulas, algebraic manipulation, equations; linearity and quadratics.• Sequences, functions, graphs.• Geometry: shape and space, measurements, coordinates, constructions.• Trigonometry: sin, cos, and tan for simple two & three-dimensional figures.• Data collection and handling, statistical averages.• Probability: enumeration and simple ideas of adding or multiplying

probabilities.• Familiarity with IT as a support for math, including graphs and data handling.• Problem solving: simple contextual situations.

The importance of statistics cannot be overestimated; everyone needs to use data andunderstand statistical measures. Also we include simple probability. We hope all thetopics listed above are supported at every opportunity by use of a computer and of coursea calculator. The list given here is not intended to be comprehensive, but merely toprovide an idea ofwhat should be taught. It is vital to make the subject alive and relevantto the pupils, whether they show aptitude or not! It must be made clear why each topic isbeing taught as it comes along. There is a need for contextual examples throughout sothat our theme is maintained. Of course we must not deny the place of math rigour,logical thinking and proof where appropriate. Problem solving should be incorporatedseamlessly throughout the mathematics curriculum and assessment should include smallindividual projects, again using the computer. There are of course many contexts thatprovide motivation and relevance. Here are some examples. Of course, there are manyother possibilities.

• 'Think of a number, double it. .. ' (algebra)• TV game shows: e.g. make 796 from 50,6,5,4,3,2,1. (Can this always be done

with the 4 rules?)• Timetable reading• Gas bills• Car prices and depreciation• Home decorating• Sports tables/ cup draw/fixtures• Cooking• Borrowing money• Gardening problems• Bicycle gears

This is also the time for the involvement of the professional mathematician! It helps ifmathematics forums are arranged where pupils aged about 14 can learn whatmathematicians do and what career you might pursue after studying math at university or

��

218 Hamson

college. The role of mathematical modelling in industry and business must beemphasized by the forum speakers. If we fail to arrange these kinds of activities, all thebest students will take up such subjects as accounting, economics, psychology, mediastudies and medicine and health care merely because it is more obvious what people do inthese jobs.

Mathematics 16-18: The CrisisThere has been a great deal of concern over recent figures in the UK for the mathenrollment post 16. At face value, the curriculum 11-16 seems to be attractive followingrecent revisions in 2000 (Edexcel GCSE, Mathematics A). However to accommodate thedifferences in aptitude over the entire pupil cohort, mathematics is examined at threeseparate levels in England for 16-year-olds (Foundation, Intermediate and Higher). Nowas more and more students are encouraged to seek university entrance at 18, the schoolmath courses offered to the 16-18 year-olds need to be accessible and relevant to a widerstudent group than hitherto. However instead of taking this into account, the revisedcurriculum offered (known as 'Advanced Subsidiary' (AS) for 16-17 and'Advanced'(A2) for 17-18) remains overfull with a broad range of traditional math topics. The resultis that 17-year-olds are presented with an Everest of material to be learned in a relativelyshort period of time. Many are quickly disenchanted by this diet, particularly if they holdonly an Intermediate GCSE pass at 16, and so tum away from math study probably neverto return.

It is therefore necessary to considerably prune the content of AS and A2 mathematics orrisk further decline in the number of students continuing math up to A2 level. If thisimplies dilution of mathematical knowledge for intending university entrants in math(and science and engineering) then this price must be paid. There will of course alwaysbe some school students who can be fast-tracked in mathematics to acquire greater mathknowledge at 18 and so perhaps gain entrance to study math at 'Ivy League' or 'RussellGroup' universities. There is obviously nothing wrong with this, but the main marketwhere the majority of the students are situated must be re-captured. Clearly we should beencouraging all or nearly all students remaining in school for the 16-18 period to continuewith some mathematics. At any rate the cohort who want to continue will include somewhose knowledge of key underpinning topics is less than other students.

The current offering in the UK is depressingly out of date! As already mentioned above,we see a curriculum stuffed with techniques (said to be pure mathematics), as in theEdexcel GCE Advanced specification that were the content of study 40 years ago:standard derivatives and integrals, coordinate geometry of conics, harder trigonometricformula and even hyperbolic functions. All these items are readily available in anycomputer algebra package such as DERIVE. In addition, we find that appliedmathematics still includes a large diet of classical mechanics! Some of the examinationproblems set in 2002 are no different from those set 50 years ago. We are competing inan ever-growing market of choice on what to study post 16 years, and the failure topresent modem and meaningful applications of mathematics is a costly mistake. In thepost-war days, there was a choice between the Arts and the Sciences. Boys frequentlytook up science, and pure and applied mathematics were championed as essentialsubjects, unless the student was preparing for medical studies. In 2002, we have from

��


newspaper reports (Independent Newspaper), the alarming data that in the UK there were53.9 thousand candidate-entries in mathematics for the A2 examination (at age of 18) outof a grand total of 701.4 in all subjects-a mere 7.7%. As many students pickedpsychology and sociology as picked mathematics, while 20 thousand chose mediastudies!

The solution to this crisis must be a radical revision of the 16-18 mathematics curriculum.Ifwe drive potential students away with old-fashioned courses, the eventual outcome willbe closure of university mathematics departments and, more seriously, a diminishingnumber of potential mathematics teachers.

Part of the problem in the UK has been an attempt to broaden the curriculum at 16-18 byexpecting students to undertake five different subjects in their first year (referred to as ASlevel), as opposed to the previous requirement of three. In the second year students thendrop two of these subjects and continue with the other three (referred to as A2 subjects).The implications for the study of mathematics are considerable. The time available forstudy is reduced by about 13% in the first year. Since these students are then assessed byformal exams it would be reasonable to reduce the amount of material to be covered byabout 20% at AS level.

However, as already explained, there has been no reduction of material in mathematics atAS level with the disastrous result that there has been a very large failure rate and muchdisillusionment for students. The outcome is that mathematics has become the subjectthey drop because it is too concentrated, too difficult, and too time-consuming. Anothersubject, applied mathematics, made up of one or two of mechanics, statistics and decisionmathematics may be studied at AS level. While the inclusion of statistics and decisionmathematics is very welcome and both these courses naturally contain contextualsettings, there is no evidence within this applied mathematics programme that studentslearn how to apply mathematics through problem solving and modelling, although boththe terms modelling and problem solving are paid lip service in the syllabi.

There is now a new course offered at AS level called Use of Mathematics (Sangwin,200 I). Is this the alternative we have been seeking to expose students to usefulmodelling and applications? Not really. This course is "not intended for the beststudents, only for the those NOT continuing with mathematics after the first year." Thesyllabus includes "use of the modelling cycle and simulation methods." The specimenexamples of simulation methods are good and what we all want to see, but closeexamination of the examination papers is disappointing. Essentially, students arerequired to use pure math techniques. For example, in a sub-unit within Use ofMathematics entitled 'Modelling with Calculus' we read that the cross-section of a river

is given by D = x4

-5x3 + 71x' -105x' where D is depth (em) and x the distance in meters4 2

across the river. The student is asked only to 'do the calculus' to find where the river isdeepest, and to find the total volume of water. Despite the use of modelling rhetoric thereal goal is to use standard math methods without any thoughtful analysis. This kind oftask conveys the wrong message about the modelling enterprise. Not only is noknowledge of real river flow needed here, but the model equation given is palpably

��

220 Hamson

nonsensical. In the opening chapter of the textbook Mathematical Modelling Skills(Edwards and Hamson, 1996) there is another example illustrating the difference betweena mathematical model and its imposters.

It is the author's view, as indicated above, that the use and application of mathematics toreal world situations should permeate mathematics curricula at all levels from II to 18years in school. However because the term modelling is open to misuse andmisunderstanding it is necessary to sort out what should actually be going on in themathematics classroom. The essential aim must be to provide good math training to theyoung, but also to awaken in them the interesting possibilities in the subject before it istoo late. The issue here is one of critical timing: when to introduce what without killingoff the delicate seedling and keeping in mind that time available is limited by othersubjects.

MATHEMATICAL MODELLING

The Modelling ProcessDefinitions of mathematical modelling appear in all the books written on the subject overthe past 20 years. Edwards & Hamson (2001), for example, define modelling as follows:

Mathematical Modelling is the activity of translating a real problem into amathematical form. The mathematical form (or model) is solved and theninterpreted back to help explain the behaviour of the real problem.

The procedure is succinctly illustrated in the book by Burghes, Galbraith, Price &Sherlock (1996) by the simplified flow diagram shown in Figure I.

/" formulation

Coal ProblC::=S<,

interpretation

MathematicalProblem

Figure 1. Flow diagram of the mathematical modelling process.

A model is formed from the real problem by making various assumptions. Themodelling process is one of iteration in that following the analysis and interpretation ofthe first simple model, it is usually necessary to modify assumptions to create a bettersecond model, and so on.

The Origins of ModellingWe should consider next why there is now such an uproar over modelling. The wordappears everywhere! Before about 1975 the terms mathematical model and modelling

��

ModelIing in Mathematics Education 221

were rarely used. Studying what was simply calIed 'Applied Mathematics' (first atschool then at university) we learned alI about applications via Newton's Laws of Motionand how to solve those endless exam questions concerning light inextensible strings andsmooth planes. Most of these exam questions amounted to pure math tests in which g=32factored into the calculations! The questions were not always easy: smooth chainssliding off smooth tables and compound pendula that never stopped oscillating. Weobeyed the rules and some of us did welI in this cosy world. Graduates returned to schoolteaching, while the best did research in relativity or electro-magnetic theory. There wasusualIy a diet of 19th century fluid dynamics in university mathematics courses whichended with the Navier-Stokes partial differential equations.

This scene was disturbed by advances in technology. Computers had a tremendousimpact on applied mathematics (Andrews & MeLone, 1976). It became possible toprogram algorithms that solved problems we previously found impossible or thatinvolved lengthy calculations. Numerical Analysis moved into centre spot with thedevelopment of a library of reliable computer programs that were readily available to alIworkers. Industrial mathematics expanded and there was also growth of applications ofmath in business through statistics and operational research work. At the same time, acommon complaint among employers was that mathematics recruits to industry were tooinhibited to tackle raw problems. The training they had received in applied mathematicsdealt exclusively with using classic equations presented as fixed truths of a situation. Theformulation that preceded the equations was often omitted or accepted as gospel. Noble(1967) discusses the impact of mathematics on engineering and the need formathematical modelIing based upon the liberating effect of computer power on industrialmaths. We can now solve the models. The drive for mathematical modelling skills istherefore firmly set in industrial applications where we mean both engineering andbusiness. The professionals require sharp skills and performance to a schedule. (Don'tforget that time = money.) In addition, they require good presentation skills tocommunicate the results of their problem solving. A typical industrial flow chart isshown in Figure 2.

IndustrialProblem

Form a mathematicalmodel and obtain itssolution via computerpower.

Figure 2. Industrial mathematics flow chart.

Report to highermanagement(who may not bemathematicians)and implementthe results.

Industrial problems needing a mathematical treatment come in alI sorts of shapes andsizes. Sometimes the math input is minimal. Many times it is substantial and leads to acomputer model. The range of problems has clearly changed since Noble's day. Someproblems have evolved from earlier ones, but other new needs have arisen. Today there

��

222 Hamson

is a rich catalogue of application fields, many of which could not have been foreseentwenty years ago--or even ten years ago.

Skeptics (and there are still some around) may chime: where are the current industrialproblems? Briefly we list: security, environmental pollution, distribution, financial,provision of utilities, scheduling, logistics, food provision, oil, post, communication, andso on. Then there are all the smaller companies and software establishments. These aresome of the big areas of application but overall the underpinning nature of mathematics ismanifest throughout industry, business and commerce.

MODELLING IN SCHOOLS

Math educators have slowly woken to the importance of including modules on 'how toapply mathematics.' It is fair to point out that this has been a very slow process indeed.Many educators with responsibility for directing school mathematics curricula seem tohave been reluctant to change from the old ways. Despite the initiatives at certainvocational colleges and polytechnics both in the US and UK since as far back as 1980,where mathematical modelling has played a key part in the students' mathematicaltraining, these initiatives have been largely ignored in the schools sector, not to mentionsome of the old established universities.

We will next consider the strategic placement of problem solving and mathematicalmodelling in school curricula. In its completeness, the full modelling activity is timeconsuming as well as difficult. The demand on students obviously increases as thecontextual problems selected for consideration become more open-ended and somespecialized understanding of the content becomes necessary. Care must be taken as weintroduce modifications to the mathematics curriculum in schools, but not to the extentthat we protect students from complexity. Some might argue that the complete activity ofmathematical modelling (including group work, data collection, report writing andadvocacy of outcomes-armed with a reasonably large repertoire of math ammunition)can only be fully implemented at university level, but several other authors in this volumeoffer us convincing evidence that younger students can profitably engage in the processof mathematical modelling.

We can begin elementary modelling actrvitres in schools for 11-16 year olds andgradual1y extend what is demanded as student maturity grows. Early on we shouldacquaint school students with the nature of a mathematical model and have them analyseits use and suitability in context. Furthermore, there are plenty of simple situations inreal life that can be model1ed using only simple mathematical techniques: car followingproblems, car windscreen wipers, ladder problems, raffle prize situations, and manymore. (See Edwards & Hamson, 2001.) A key factor in doing modelling must beremembered-students can only use the mathematical techniques that they already know,understand, and are confident in using. Nevertheless we cannot dismiss the possibilitythat insight and inspiration play a role in the process and that students may actual1ydeepen their understanding of the mathematics when dealing with real problems incontext.

��


An Applied Curriculum for 16-18It is essential within an enlightened 16-18 mathematics programme to include a moduleon the use of mathematics in problem solving (perhaps better called 'Foundations ofApplied Mathematics') within which students are acquainted with models and modelling.Such a module should be offered simultaneously with a module of pure mathematics thatdeals with topics in calculus, algebra, exponential functions, series and trigonometry. Anoutline of the topics that should be included in the applied module is given below.

Foundations of Applied MathematicsCompulsory within a post-16 mathematics programme for all students intending to readmathematics, science or engineering at university.

Part 1 Mathematical ModellingThe nature of applied mathematics: a brief history of the development of mechanics,operational research and mathematical biology.Definition of mathematical modelling.Simple classroom tasks to illustrate modelling.The modelling cycle.Critical examination of a range of simple models taken from the areas of geometry,kinematics, optimisation, scheduling, simulation, finance.Modelling course work (undertaken by pairs of students) on one case study taken from:

• Drink can construction• Lamp shade cut-out• Basketball throws• Car queues at road repair scene• Price War• Single track rail line capacity• How much skin have you got?• Hire purchase• Car depreciation• Crossing the road.

There are many other possible course work examples found in Edwards & Hamson(2001) or Burghes, Galbraith, Price & Sherlock (1996).

Part 2: Topic-based StudyConventional work from mechanics, statistics and decision mathematics. Examples usingNewton's laws of motion, simulation, linear regression, route evaluation and linearprogramming.

This suggests two modules, pure and applied, for delivery side-by side for the 17-yearold student. These two courses may be terminal for some who wishes to concentrate onother interests in science or arts. But for the many we would hope their interest has beencaptured and that they would continue their mathematical study in the next academic yearwith more of the same, again from two modules in pure and applied math. The pure mathmodule should contain differential equations and difference equations, some numericalmethods, complex numbers and matrix algebra. We must not lose sight of the power ofthe software packages readily available; there is no need for the endless teaching of

��

224 Hamson

standard integrals. The applied math module perhaps called "Applying Math in Scienceand Business" can advance on each of the fronts Modelling, Mechanics, Statistics andDecision Maths. This time we can get students to carry out some further model buildingcarefully chosen not to be too time consuming and open-ended. There is plenty ofsuitable material in the literature, for example, in the journal Teaching Mathematics andits Applications. We need to provide the foundations that prepare students for the nextrung on the learning ladder: university study of math, engineering and science. It is clearthat university curricula may also have to be modified for the majority of enteringstudents who will not know about sinh(x) or conservation of angular momentum in a rigidbody!

Modelling Across the Mathematics CurriculumIn Figure 3, I present a schematic summary of the desirable content of mathematicseducation, from age 6 to (perhaps) 25, indicating the student's development ofmathematical knowledge and also the place of problem solving and modelling. Theobjective is to motivate the learning of relevant math skills and techniques at each level,this being achieved through continual emphasis on the application and use ofmathematics. (This does not mean there is no room for covering the elegance and beautyof the subject at the appropriate moment.)

Teacher PreparationIt is essential that all intending and practicing math teachers have confidence not onlywith the mathematical material they are teaching but also why that material is in thecurriculum and what role it plays in advanced mathematics and its applications. It issurprising to this author that little attention within postgraduate teacher education coursesis devoted to this point. Most effort seems to be placed on 'school practice' andclassroom management. It is surely vital that all teachers are aware of how to applymathematics through building mathematical models. All teachers should experiencemathematical modelling in their training courses.

ICTMA 11 THEME

I conclude by examining the theme of this conference, "Mathematical Modelling: A WayofLife." It is instructive to answer an the questions listed in the aims of the conference tosee where one stands!

What do we value about mathematical modelling? It trains you how to apply math. Itmotivates the learning of math techniques. Like swimming you can only be successfulby actually doing it!

Is mathematical modelling valued by society? I do not think mathematics is sufficientlyvalued in society. We have not made enough impact even with those guiding schoolwork.

How do we communicate our values to students in the teaching process? By offeringrelevant courses that give students confidence in the worth of learning mathematics for acareer.

��


AGE GROUP TIME DIRECTION MA TH APPLIED

B Recreational arithmetic,

PRIMARY A _•.............._-~ magic squares, patterns,.............._......_-_.................................S primes, tiles, shapes,6-10I language and symbols

C

CU Problem solving,

SECONDARY Rcontextual questions,

...._............._._................---R

..............__.•.~ data collection, sports11·16 data, timetables, map

I reading, Fermat's LastC Theorem, talks onU careersLUM Foundations of applied

SIXTH FORM maths, using models,HIGH SCHOOL ..__._----...................... T ._.........._--~ reading articles. doingCOLLEGE A simple modelling,

16-19 K presenting work, talks

E on careers, visits to work

S places

UNIVERSITY19-23

P1- .__ _.._ _ L _ __.__)

ACE

Modelling courses (2nd

year ), professionaltraining industry orbusiness, project work

POSTGRADUATEPROFESSIONALTRAINING

Modelling experience in-.-.- -.. -.-- - _> teacher education andprofessional development

Figure 3. Mathematics from age 6 to age 25.

��

226 Hamson

How do we recognise growth in competence in mathematical modelling? By offeringwell constructed modelling courses in universities and then getting employers to noticethe difference.

How do we ensure the survival and growth ofmathematical modelling? By ensuring thatits place and purpose is understood by all school mathematics curriculum developers andteachers.

What are the persistent issues and obstacles that we must address? Some powerfultraditionalists who cannot see further than their well worn handouts in complex variabletheory.

CONCLUSION

It is important that mathematical education in mathematical modelling is improved for allstudents. The viability of the next generation of practitioners depends upon ourwillingness to examine current practice and create needed changes. By incorporating theideas outlined here it is hoped that the present crisis can be addressed and eventuallyreversed.

REFERENCES

Andrews JG, MeLone RR (1976) Mathematical Modelling London: Butterworths.Burghes D, Galbraith P, Price N, Sherlock A (1996) Mathematical Modelling Prentice

Hall.DERIVE Computer Algebra Package, Honolulu: Soft Warehouse Inc.Edexcel GCSE Mathematics A Specification (November 2000) Publication Code

UG008976.Edexcel GCE Advanced. Mathematics, Applied Mathematics, Statistics etc. (January

2000) Publication Code UG006083.Edwards D, Hamson M J (2001) Guide to Mathematical Modelling Second Edition

London: Palgrave Mathematical Guides.Edwards D, Hamson M J (1996) Mathematical Modelling Skills, London: Macmillan

College Workout Series.Independent Newspaper, UK, 15 August 2002.Noble B (1967) Applications ofUndergraduate Mathematics in Engineering Washington,

DC: Mathematical Association of America.Sangwin C (August, 2001) 'AS: The Use of Mathematics, Curriculum 2000', MS.o.R.

Connections, YoU, No.3."Teaching Mathematics and Its Applications' Journal of the Institute of Mathematics

Cambridge: Oxford University Press.

��

20

What is Mathematical Modelling?

Jonei Cerqueira BarbosaFaculdade Integrada da Bahia e Faculdades Jorge Amado, [email protected]

In this paper, I present some theoretical ideas about modelling inmathematics education. Using examples from classrooms and puttingemphasis on socio-cultural aspects, modelling is related to problems thatare rooted in reality. The integration of modelling into the mathematicscurriculum is also discussed.

��

228

INTRODUCTION

Barbosa

For many years, mathematical modelling has been the focus of my attention. I havedeveloped modelling activities, trained other teachers, and guided many investigations.In my practice, I am frequently asked the question, "What is modelling?"

In general terms, modelling may be defined as the application of mathematics to otherareas of knowledge. In this sense, modelling is an umbrella term, sheltering from ourimmediate view, the rich, complex processes and activities entailed in modelling. Someauthors borrow the standards of applied mathematics expressed in flow charts (forexample, Edwards and Hamson, 1990) to define modelling. Modelling flow chartssuggest stages to guide the students' activity. The most difficult context in which toanswer the question, however, is in relation to modelling in schools. In this case, theobjectives, the dynamics, and the nature of mathematical discussions are different fromthe professional contexts in which modelling is used (Matos and Carreira, 1996).

In this chapter, I carry out a systematic reflection on modelling from the locus ofmathematics education. I do not suggest a separation of applied mathematics frommathematics education (indeed, I recognize the intersection of these fields), but rather,the singularisation of the object in the field of mathematics education.

WHY MATHEMATICAL MODELLING?

There are many good reasons to include modelling in the school curriculum (Blum,1995). In general, five arguments have been presented as a rationale for modelling inschools: motivation, facilitating learning, preparation for the use of mathematics indifferent areas, developing general competencies, and comprehension of the sociocultural role of mathematics. I chose to focus on the last item in Blum's list, namely, thesocio-cultural benefits of mathematical modelling.

Some studies have analysed the socio-cultural dimensions in mathematics education(Atweh, Forgasz and Nebres, 200 I; D'Ambrosio, 1996). It has been noted that peopleseek veracity and reliability from mathematical applications. This phenomenon is calledan "ideology of certainty" by Borba and Skovsmose (1997). It suggests that individualsare less comfortable debating about social issues.

I believe that the activities of modelling may serve to challenge the ideology of certaintyand bring a critical view to mathematics applications. Discussions in the classroom mayset important questions such as: What does it represent? What are the assumptions?Who made them? Whom do they serve? The classroom provides a forum in which todiscuss the nature of applications, the criteria by which they are judged, and their socialmeaning. Skovsmose (1990) refers to this as reflective knowledge.

I have recently had the opportunity to observe a classroom where students criticallyexamined models in light of social implications. I visited a 7th grade class in a schoollocated close to the rural zone, in a countryside town in Brazil. The teacher brought upfor discussion a newspaper article about a governmental distribution program involving

��

What is Mathematical Modelling? 229

37.5 tons of beans and com seeds, staples among the families who practice subsistenceagriculture. Each family would receive 5 kg of seeds. The teacher encouraged thestudents to consider the adequacy of this criterion for the distribution of the seeds. Thestudents finally agreed to devise another criterion, because they thought it was not fairthat families with different number of members received the same amount.

In groups, the students discussed different criteria. Some suggested that the distributionshould be based on the number of children, others, on the family's available plantingarea, and so forth. In this case, the students were devising a model to relate themathematical results, the amount each family would receive, and the criteria used fordistribution.

In addition to the modelling task itself and the mathematical knowledge, the studentsdiscussed issues that might be classified as reflective knowledge: What are the familiesneeds? Which criterion is the fairest? Is the amount of seed allocated for the distributionsufficient? Can a proposal be sent to City Hall? The students had the chance to discussthe role of mathematics in society and, to develop a critical sense of its uses. From thisperspective, mathematical modelling may serve to empower citizens in debates and indecision-making that involves mathematical applications.

I do not suggest, however, that modelling activities by themselves can achieve thesepurposes. The nature of communication between teacher and student and among thestudents themselves mayor may not stimulate reflection (Aim & Skovsmose, inprogress). Critical discussions depend, for the most part, on the organisation and theorchestration of these activities by the teacher.

Consider a hypothetical situation. In the seed distribution case already discussed, theteacher could have taken the newspaper article to the classroom, put an emphasis on themathematical content and established a communication pattem in which the students'mathematics would be corrected by the teacher on the spot. In this situation, the chanceto critically discuss the situation and the role of mathematics on the social practiceswould be lost. Experience suggests that, in the classroom, students will not achieve thisdimension of analysis on their own; it must be permitted and encouraged by the teacher.On the other hand, while proactively facilitating student discussion and reflection, theteacher must exercise some restraint to ensure that his or her personal perspective will notinterfere with the course of the modelling activities. In a previous study (Barbosa, 200 I),I found that the teacher's strongest conceptions and beliefs interfere in the design of themodelling activities.

WHAT COULD COUNT AS A MODELLING ACTIVITY?

Classroom activities provide contexts in which students are invited to act. Skovsmose(2000) calls one such context a learning milieu. In the case of modelling, someconditions and questions are introduced and these promote determined actions anddiscussions that are sometimes related to other learning milieus.

��

230 Barbosa

Modelling is strongly linked to problem posing and to investigation. The first refers tothe act of creating questions/problems while the second, to activities such as searching,selecting, organising, manipulating testing and revising information. Thus modellingactivities promote reflection.

Imagine that the teacher proposes that students study the impact of the social contributiontax. This is a tax deducted from people's salaries by the Brazilian Government for themaintenance of the social welfare. The students certainly will have to formulatequestions, search for information, organise it, draw up strategies, apply mathematics,evaluate the results, etc.

Although the situation originated in a field other than mathematics, the students areinvited to use mathematical ideas, concepts, and algorithms. Apart from applyingknowledge already acquired, there is the possibility that the students wiIl acquire newknowledge, during the work of modelling (Tarp, 2001).

Niss (2001) discussed the use of highly simplified and idealised activities for classroomuse. However, 1am interested in situations that exist naturally in the social world and arenot artificially created. Skovsmose (2000) says that such activities have reference inreality. This doesn't mean that fictitious situations serve no purpose in the classroom.These may also lead to rich discussions and maths learning. However, I believe that amaths class must have real situations that are not highly structured.

Modelling provides opportunities to use some abilities and knowledge not afforded bygames, fictitious situations, and the like. In modelling, the students deal with situationsthat are part of their day-by-day lives and part of the working world. Bringing theseactivities into the classroom prepares the students to function in society and to becomeinvolved in important decision-making. Unfortunately, there is no empirical research toconfirm this hypothesis.

Some teachers state that children cannot solve problems that are truly rooted in reality.Lamon (2001), using the notion of enculturation, points out that in work with thestudents, we share values and beliefs of our community about what is an appropriatemodelling work. The author presents results of an empirical study, which shows thatstudents in a first course were strongly resistant to the modelling culture. This is anatural consequence of the fact that the students were socialised in the traditionalteaching culture, based in exercises and structured situations. On the other hand, whenstudents are encouraged to take an active role in a modelling milieu, they are able toadapt and to demonstrate remarkable thinking (Lamon, 1998). More qualitative researchabout the process of enculturation into mathematical modelling is needed.

I return to my original question: What is mathematical modelling? My response is this:Modelling as a learning milieu where students are invited to take a problem andinvestigate a situation with reference to reality via mathematics.

��


The reader will observe that I have tried to characterise modelling in terms of the context(school), the nature of the activities (investigation), and the areas involved (mathematicsand areas with reference to reality).

WHAT IS THE PLACE OF MODELLING IN THE CURRICULUM?

There are various ways to implement modeIling in the curriculum. I typically avoid theisland approach, in which the conventional programme is interrupted by occasionalmodelling activities. This approach does not motivate the students to seriously engage ininvestigations. Araujo and Barbosa (in progress) reported on a study in which studentsintroduced fictitious information to their problem-solving activities. The hypothesis isthat this happened because their teacher made no effort to relate other school assignmentsto their real lives. This suggests the importance of there being some agreement betweenmodelIing and other school tasks.

ModeIling literature has characterised modelling activities according to the duration andthe extent of the task as it is posed by the teacher. Galbraith (1995) elaborates on severalsuch levels of modelling activity. Using an idea similar to Galbraith's, I prefer to speakabout three cases or regions ofpossibility afforded by a modeIling task as it is presentedto students.

In case I, the teacher presents a problem with quantitative and qualitative information,and the students are expected to investigate the situation. The students do not need to getout of the classroom to collect additional data or information and the activity is not veryextensive. I offer an example of this kind of activity from one of my classes in which Iasked students to investigate various payments plans for securing an internet connection.In Brazil, the price depends primarily on the time of use. I collected the prices (inBrasilian currency) from a company that provides internet service, as shown in Figure I,and I asked the students to decided which of the plans is a better value.

Monthly subscription Access time Additional cost per(R$) included (h) hour (R$)

Plan I 17,95 - 0,73Plan 2 27,95 15 0,53Plan 3 49,95 60 0,35Plan 4 75,95 150 0,35

Table 1. Pricing scheme for internet usage.

In this case, the students dealt with a true problem, because they did not know exactlyhow to proceed. It was a situation that anyone might face in everyday life. However noadditional data was necessary to accomplish the task as it was posed. The investigationtook a relatively short period of time, about 150 minutes (or, roughly 3 class periods),followed by a discussion of their results. The teacher exercised some control over thetime and depth of the investigation.

��

232 Barbosa

Case 2 provides other different possibilities for student engagement. The teacher posesan initial question to the students and they become responsible for collecting data and forpresenting their solutions. In this case, the students are mostly responsible for regulatingtheir own activities. For example, in another class term, I presented the followingquestion: "How much is it to have access to internet?" I discussed with the studentspossible questions and implications. However, they were not provided any pricing guidesand the various working groups were responsible for collecting the data that they needed.Afterwards, they had to organise and consider variables such as hours of usage. Thisactivity demanded more time than the previous case, consuming some weeks. Duringthis time, students worked outside of class and discussed their progress in class. Theproject concluded with an oral presentation by each group and more discussion. In thiscase, the teacher exercised less control upon the students' problem-solving activities, andthe students had a greater opportunity to experience all the phases of the modellingprocess.

A third case entails projects with non-mathematical themes that may be chosen either bythe teacher or by the students. The students are responsible for formulating the problem,collecting information, and solving the problem. In Brazil, this is a common approach tomodelling (Bassanezi, 1994) inspired by ethnomathematical studies (D'Ambrosio, 1996).

I used a case 3 modelling project, when I taught a mathematics course on management.The students were invited to choose any social issue that interested them.Telecommunications, hunger, inflation, marketing, and the social contribution taxmentioned above, were the issues chosen by the 5 groups of student participating in thecourse. I will focus my discussion on the group that chose the topic of the socialcontribution tax.

The students began by raising questions about the topic. In the beginning, they did nothave a clear idea about which points to pursue. As they became more familiar with theissues and variables, after some negotiation, they focused upon a single question: What isthe impact ofthe social contribution tax upon salaries? From there, they had to collectand organize data before they could attempt to answer their question. In this case, themodelling activity took considerably more time than the kinds of investigations requiredin cases I and 2, mainly because of the need to formulate the problem. As in the previouscase, the teacher led discussions in the classroom, but students were required toaccomplish major portions of their work outside the normal class period.

An added benefit of type 3 modelling projects is that the questions students choose toinvestigate become a good source of problems for use with other classes in the future. Isuggest this strategy to colleagues who complain it is too difficult for them to getmodelling problems. These problems further address the questions "What is real?" and"Can teachers decide what is real for students?" If you ask the students to formulate thequestions themselves, you will have problems rooted in reality.

From case I to case 3, the teacher's responsibility and degree of guidance of themodeling activity is progressively shifted to the students. The scheme in Table 2 conveysa shift in roles from teachers to students. When teachers admit that they feel insecure

��


about introducing modelling activities (Barbosa, 200 I), I use this classification to helpthem see that there are multiple possibilities and to choose a case that is comfortable forteacher and students alike.

Case I Case 2 Case 3Elaboration of the

teacher teacher teacher/studentproblem

Simplification teacher teacher/student teacher/studentQuality and quantity of

teacher teacher/student teacher/studentdata

Solution teacher/student teacher/student teacher/student

Table 2. Responsibility in the modelling process.

The cases are not static configurations, but rather, possible regions of involvement. Theyoffer teachers some flexibility in implementing modelling in the classroom. At certaintimes, the emphasis might be on a small project or investigation as in case I, while atanother time, longer projects might be feasible. Ultimately the teacher is required todecide which case is appropriate. Thoughtful decision-making is required to ensure that aproject affords possibilities and minimizes limitations. The teacher might consider theknowledge, preferences, and experience of the students, the purpose in incorporating themodelling activity, and a host of other factors that may affect the quality of theexperience.

A FEW FINAL WORDS

In this chapter, I reflected on the question, "What is mathematical modelling?" My goalis to stimulate discussion about the research and the meaning of modelling inmathematics education. I have put emphasis on problems and investigations based onreal situations and have introduced the notion of cases inspired by Galbraith (1995). It ismy hope that through this discussion and by critically examining our work, we canimprove the practice of modelling in schools.

REFERENCES

Aim H, Skovsmose 0 (in progress) Dialogue and learning in mathematics education:intention. reflection. critique.

Araujo J L, Barbosa J C (in progress) 'Face a face com a Modelagem Matematica: comoos alunos interpretam essa atividade?'

Atweh B, Forgasz H, Nebres N (Ed) (2001) Sociocultural research on MathematicsEducation. Mahwah, NJ: Erlbaum.

Barbosa J C (200 I) Modelagem Matematica: concepcbes e experiencias de futurosprofessors Doctoral dissertation, Universidade Estadual Paulista.

Bassanezi R C (1994) 'Modelling as a teaching-learning strategy' For The Learning ofMathematics 14 (2), 31-35.

Blum W (1995) 'Applications and Modelling in mathematics teaching and mathematicseducation - some important aspects of practice and of research' in Sloyer C, Blum

��

234 Barbosa

W, Huntley ID )Eds) Advances and perspectives in the teaching ofmathematicalmodelling and applications. Yorklyn, DE: Water Street Mathematics, 1-20.

Borba M, Skovsmose 0 (1997) 'The ideology of certainty in mathematics education' Forthe Learning of Mathematics 17(3), 17-23.

D' Ambrosio U (1996) Educacdo Matematica: da teoria aprdtica. Campinas: Papirus.Edwards D, Hamson M (1990) Guide to Mathematical modelling. Boca Raton: CRC

Press.Galbraith P (1995) 'Modelling, teaching, reflecting - what I have learned' in Sloyer C,

Blum W, Huntley I. Advances and perspectives in the teaching of Mathematicalmodelling and Applications. Yorklyn, DE: Water Street Mathematics, 21-45.

Lamon S J (1998) 'Algebra, modelling, and achievement' in Galbraith P, Blum W,Booker G, Huntley ID (Eds) Mathematical modelling: Teaching and assessmentin a technology-rich world Chichester: ElIis Horwood, 307-315.

Lamon S J (2001) 'Enculturation in mathematical modelIing' in Matos J F, Blum W,Houston SK, Carreira SP (Eds) Modelling and mathematics education.Chichester: Ellis Horwood, 335-341.

Matos J F, Carreira S (1996) 'The quest for meaning in students' mathematical modelIingactivity' Proceedings ofthe 2(/h Conference for the PME Valencia: Universitat deValencia, pp. 345-352.

Niss M (2001) 'Issues and problems of research on the teaching and learning ofapplications and modelling' Matos J F, Blum W, Houston SK, Carreira SP (Eds)Modelling and mathematics education. Chichester: ElIis Horwood, 72-88.

Skovsmose 0 (1990) 'Reflective knowledge: its relation to the mathematical modelIingprocess'Int. 1. Math. Educ. Sci. Techno!., 21 (5),765-779.

Skovsmose 0 (2000) 'Cenarios de investigacao' Bolema - Boletim de Educaciiomatemdtica, 14, 66-91.

Tarp A (2001) 'Mathematics before or through applications: Top-down and bottom-upunderstandings of linear and exponential functions' in Matos JF, Blum W,Houston SK, Carreira SP (Eds) Modelling and mathematics education.Chichester: ElIis Horwood, 119-129.

��

21

Beyond the Real WorId: How Mathematical ModelsProduce Reality

Susana CarreiraUniversidade do Algarve;Universidade de Lisboa-ClEFUL, [email protected]; [email protected]

There is a growing concern with the need to prepare students to criticallyexamine the ways in which mathematics affects human lives. Taking thisas a starting point, I suggest looking at mathematical modelling as a way ofcreating possible worlds where similar problems are posed andcorresponding concepts are used. The exploration and inquiry onmathematical models intends to highlight the power of mathematics intranscending the real world while showing mathematical modelling as aform of reasoning about reality.

��

236 Carreira

MATHEMATICS AND THE (CONCEIVABLE) REAL WORLD

One of the relevant ways in which mathematics mediates and indeed colonises our dailyand real world is connected with the formatting power of mathematics. Such powerstands on the capacity of mathematical models to produce rules, norms, and prescriptionsand to dictate the forms of social, economical, and political systems (Skovsmose, 1992,1994; Keitel, 1993; Niss, 1994; Frankenstein, 1998).

Even in science and other domains where the scientific method prevails, some wouldargue that mathematics is also imposed upon the real world. Whenever we make choices,establish assumptions, define variables and quantify, select what is considered relevant,and dismiss elements of a particular real situation, we are introducing criteria that acquirelegitimacy in virtue of the adoption ofa mathematical point of view. Therefore, we forceour mathematical knowledge and thinking upon real problems. Proportional reasoning isone of the many examples of imposed mathematics; it is often seen as a fair criterion toaddress life problem situations dealing with quantities.

To look at mathematical modelling from a critical perspective entails taking into accountthe effects and repercussions of mathematical modelling and applications in our everydaylives. In other words, when creating and applying mathematical models we are neverdealing with a neutral external reality; we are also conceptual ising that reality and actingupon its structure while shaping it. In that sense, we are not just modelling the world butrather modelling a conceivable world (Onnell, 1991a, 1991b; Levy, 1995; Campbell,2001).

CHALLENGING THE LIMITS: MODELLING THE IMAGINARY

Along with representing a fertile ground for the development of independent thinking andcritical citizenship, mathematical modelling affords an opportunity to encounter andexplore mathematical realities. Such realities are often redrawn from the most tangibleand earthly referents and can only be encountered through mechanisms of idealisation,imagination and abstraction.

Most of the contemporary curricular trends and pedagogical orientations convey the ideathat mathematics teaching and learning ought to be more rooted in real experiences, morepractical, more concerned with the process of mathematisation, more focused onpromoting investigative attitudes and problem solving abilities.

Those directions can be pursued in a number of ways, one of which is related toexperiencing mathematical modelling as conceptual modelling. The importance ofconceptual or mental models within the process of mathematisation has been stressed(Freudenthal, 1991; Mason and Davis 1991; Matos and Carreira, 1997; Mason, 200 I) as acentral stage of the modelling process without which mathematical models cannot begrasped. The focus here is to analyse mathematical modelling in terms of the productionand investigation of conceptual models where the connection to the real, concrete worldis mainly a question of seeing how mathematics can unfold routes to new possibleworlds. The possibility of stepping out of our real world and yet continuing to model

��

How Models Produce Reality 237

mathematically reveals much of the power of mathematics in shaping even the worldsthat escape our physical experience.

ONE MATHEMATICAL CONCEPT, THREE (MATHEMATICAL) REALITIES

Possible worlds: The blue (b), the pink (p), and the violet (v)Instead of a situation we are given a setting. It involves three possible worlds that share anumber of common characteristics. All the worlds are inhabited by intelligent beingswho live their normal lives between houses, schools and workplaces, who travel by roads,railways or cross rivers, who use cars, bicycles, and planes. They have maps, compasses,and systems of orientation. They make drawings, have rulers, use geometry, calculus,computers, and organise much of their thinking in terms of time and distance. Amongother fundamental things, they have also feelings, doubts, questions, rules, concepts, andthoughts.

One distinction among the three worlds is the way that distance is defined. In all theworlds there is a kind of rectangular system of coordinates for the representation ofpositions on the plane. As shown in Figure I, people in the three worlds conceivedistance between two points in diverse ways.

• • • •• • • •• • • •

B

y

A Jl

122"Blue world: db =VX + y

Pink world: d p =Ixl +IylViolet world: d, =max ~xl,lyl)

Figure l. Distance between points A and B in the three worlds.

One initial observation is that people from the different worlds do not find the samedistance between A and B. In the blue world, the points A and Bare 5 units apart, in thepink world they are 7 units apart, and in the violet world only 4 units apart. Apparently,although people from the three worlds share certain assumptions when defining distance,they do not conceive it in the same way. Is it legitimate to conclude that we are lookingat three distinct realities?

One possible way to search for an answer is to investigate how other concepts may beaffected by the variation of the definition of distance. Our next step is to analyse howstudents from each of the worlds describe the mediator of a line segment. Indeedstudents from all the worlds learn in their mathematics classes that a mediator(perpendicular bisector) of a line segment is the locus of points that are equidistant from

��

238 Carreira

the end points of the segment. The students we know from the three worlds have someexperience and knowledge about geometric loci and are familiar with computational toolsthat enable the visualisation and interpretation of geometrical objects in a dynamic way.Such tools are actually regular resources in mathematics classrooms and we would easilyexpect them to make use of it when asked to show us the object that is called the mediatorof[AB].

Interestingly, we may observe that in each of the worlds, students start to give anexplanation by speaking of the set of points that are equidistant from a fixed point. Theyjustify their reasoning by saying that we should look at two sets of points and identifytheir intersections. In their accounts they make a reference to the notion of circumferenceand they define it as the set of points located at a constant distance from a fixed point ofthe plane. Then they imagine two circumferences with equal radii centred on A and B,each with radius greater than the length of the segment. They identify the intersectingpoints of the two circumferences, as belonging to the mediator. Figure 2 shows theseinterpretations.

(I (II

· · · · · · ·· · · · · · ·· · · · · · ·· · V ~ · ·· · · · · · ·· · • · · · ·· · e · · · ·

(III)

Figure 2. Circumferences of equal radii in: (I) blue world, (II) pink world, (Ill) violetworld.

One of the exciting capacities of their computer software is the fact that students canmanipulate their representations by enlarging the radius of the circumferences in order tosee all the points that lie at the same possible distance from the two endpoints of thesegment. Furthermore, they can change the position of their line segment on the grid, andlook for possible modifications to the resulting set of points.

Indeed, the students from the blue world do several experiments and come to theconclusion that regardless of the position of the segment [AB], the mediator isperpendicular to the line segment and crosses it at the middle point. As shown in Figure3, the pink world's students notice some particularities in a complex world where thingsdepend strongly on the slope ofthe line segment. There are situations where the mediatorcoincides with the one in the blue world, but other times it can be a segment and twohalf-lines or a segment and two quadrants.

��


+

Figure 3. Some representations of the mediator in the pink world.

According to the description of mediator given by students in the violet world we suspectthat theirs is a peculiar world where the mediator is not always the same. As shown inFigure 4, the mediator also depends on the slope and it looks more similar to the case ofthe pink world: sometimes a line, other times a segment and two half-lines, other times asegment and two quadrants. Sometimes it can also coincide with the blue world'smediator.

+ +

+ •

. ·~ I~

.~ A7 "1/ r-,

. •

Figure 4. Some representations of the mediator in the violet world.

Solving the same problem in the blue (b), the pink (p), and the violet (v)At this stage we are becoming convinced that people from the different worldsexperience mathematical objects in diverse ways. Someone from the blue world askssomeone in the violet world: "Can you draw a line like my circumference?" A plausibleanswer could be: "Ofcourse, we can. But we do not call it a circumference!"

In all the three worlds, people solve problems by creating mathematical models. Considerthis problem that was posed to the three worlds:

The distance between Anna's home and the school is 3 units. The distancebetween Brian's home and the school is 5 units. What is the distancebetween Anna's home and Brian's? (de Lange 1993, p. 13)

��

240 Carreira

The first approach to the problem in the three worlds was based on the assumption thatthe houses and the school could be represented as points on a plane. Of course, thisassumption reflects a particular mathematical point of view, namely that this is a planegeometry problem. One simple instance is to consider the three points as collinear points.In this case, the conclusions drawn are identical in the three worlds, that is, the distancebetween A and B is either 8 units or 2 units, depending on whether S lies on the segment[AB] or not.

Afterwards, students set out to consider another hypothesis: what if A, Band S are notcollinear points? A suggestion given by one of the worlds is to imagine the school as thecentre of two concentric circumferences of radii 5 and 8. Then, getting all the possibledistances between A and 8 is a matter of placing Anna's house on one point of the innercircumference and let Brian's house take all the positions on the outer circumference.Soon, all the students were aware of the fact that the distance between A and 8 would

change with the angle ASB, as it varies from 00 to 1800 (no need to consider the otherhalf-circumference, since everything is symmetric). Both the students from the blueworld and pink world also noticed that the distance could be described as a function ofthe x-coordinate of point B. Students in the violet world replied that such a functionalrelationship would not hold. Hence, another difference between the three worlds wasacknowledged.

People from the blue world created an animation of the point B, rotating on itscircumference, to illustrate how the segment [AB] changed in length. It became perfectlyclear that the length takes all values between 2 and 8. The students from the other twoworlds followed the example.Figure 5 shows the work of the students in the blue world. They used trigonometry andalgebra to get a model of the length variation.Taking a as the independent variable:db(a)= .J34- 30cosa 0 s a s IT·

Taking the x-coordinate of 8 (Brian's house) as the independent variable:db(x)=.J34-6x -5~x~5.

B

Figure 5. Modelling distance variation in the blue world.

��


Students from the pink world had a more complicated task using trigonometry.Nevertheless, based on Figure 6, they successfully represented the distance as both afunction of the angle a and a function ofthe x-coordinate ofB.

Figure 6. Modelling distance variation in the pink world.

Taking a as the independent variable:

112- 3tan 0.1+ 5 tan 0. 0 :0;0. < 2:

dp(o.) = I + tan 0. 2

8 2::O;o.:O;1t2

Taking the x-coordinate of B as the independent variable:

dp(x)=lx-~+15-lxll -5::;x ::;5·

Finally there was the violet world's answer. It was already known that it would beimpossible to consider a function of the x-coordinate of B. Therefore, students from theviolet world concentrated only on the angle a, as shown in Figure 7.

~B

B I-, ,

-, I-,

-, I-,

-, /\-, ..IS po

Figure 7. Modelling distance variation in the violet world.

��

242

Taking a as the independent variable:

2 o~a<arctan(i)

5 tan a arc tan (i)s a <~

dv(a)= 5 ~~a<arccot(-i)

3 - 5 cot a arc cot ( - i)s a < 341t

31t8 -~ a ~ 1t

4

Carreira

This is equivalent to the more condensed representation:

max (2,15 tan aJ) 0 ~ a < 2:-4

max (8,1- 5 tan aD

1t 31t-~a<-4 4

31t-~a~1t

4

Our legitimate reaction when faced with such different mathematical models is toperceive the three worlds as rather dissimilar. In fact, they are so distinctly unlike that wetend to see totally different pictures of three surprising realities. Yet all the three worldsshare a common notion of distance in such a way that their mathematicians candemonstrate that there is a sense for which all the three worlds are perfectly equivalent.Still, Anna and Brian are not always the same distance apart in every world, though theirdistances from the school they go to are actually the same in every world.

As we go further with the exploration of these realities, we observe that the distancevariation in the three worlds has a certain common feature. Distance changes in acontinuous way in all the worlds. However it is not a smooth change in some of them. Inthe cases of the pink and violet worlds, distance is not differentiable. Will this mean thatpeople from the blue world are more stationary in their forms of experiencing change andmovement when compared with the others? That would possibly mean pushingmathematical modelling too far and risking other more deceptive accounts of what reallyis in each of these possible worlds.

TO CONCLUDE: EXPANDING MATHEMATICAL MODELLING

One of the purposes of the story above is to reflect on the power of mathematics to createand produce certain aspects of our real world. Taking a very immediate and spontaneousconcept such as the notion of distance we can see how mathematics, by means ofabstraction and imagination, is able to amplify the scope and meaning of our dailyexperience. Furthermore, it is possible to try out its application through the use ofmathematical artefacts in apparently ingenuous objects and situations. It offers a way to

��


get a sense of the capacity mathematics can have in shaping not only our real world butour imaginary worlds as well.

Finally, it also means an attempt to conceive mathematical modelling and applications asa fundamental kind of mathematical thinking that can be wider and more impregnated inour mathematics teaching and learning than it is often accepted. The development ofmathematical concepts is one of the significant aspects of adopting a mathematicalmodelling perspective where conceptual models come to the foreground.

REFERENCES

Campbell SR (200 I) 'Enacting possible worlds: Making sense of (human) nature' inMatos JF, Blum W, Houston SK, Carreira SP (Eds) Modelling and mathematicseducation: Applications in science and technology Chichester: HorwoodPublishing, 3-14.

de Lange J (1993) 'Innovation in mathematics education using applications: Progress andproblems' in de Lange J, Keitel C, Huntley I, Niss M (Eds) Innovation in mathseducation by modelling and applications Chichester: Ellis Horwood, 3-17.

Frankenstein M (1998) 'Reading the world with maths: Goals for a critical mathematicalliteracy curriculum' in Gates P (Ed) Proceedings of the First InternationalMathematics Education and Society Conference Nottingham: Centre for the Studyof Mathematics Education, Nottingham University, 180-189.

Freudenthal H (1991) Revisiting mathematics education Dordrecht: Kluwer AcademicPublishers.

Keitel C (1993) 'Implicit mathematical models in social practice and explicit mathematicsteaching by applications' in de Lange 1, Keitel C, Huntley I, Niss M (Eds)Innovation in maths education by modelling and applications Chichester: EllisHorwood, 19-30.

Levy T (1995) 'Em que Mundo(s) Vivemos Nos?' in Matos 1F, Amorim I, Carreira S,Mota G, Santos M (Orgs) Matematica e Realidade: Que Papel na Educaciio e noCurriculo? Lisboa: Seccao de Educacao Matematica, Sociedade Portuguesa deCiencias da Educacao, 7-24.

Mason 1 (2001) 'Modelling modelling: Where is the centre of gravity of-for-whenteaching modelling?' in Matos 1F, Blum W, Houston SK, Carreira SP (Eds)Modelling and mathematics education: Applications in science and technologyChichester: Horwood Publishing, 39-61.

Mason 1, Davis 1 (1991) Modelling with mathematics in primary and secondary schoolsVictoria: Deakin University Press.

Matos 1F, Carreira S (1997) 'The Quest for Meaning in Students Mathematical ModellingActivity' in Houston SK, Blum W, Huntley I, Neill NT (Eds) Teaching andlearning mathematical modelling: Innovation. investigation and applicationsChichester: Albion Publishing, 63-75.

Niss M (1994) 'Mathematics in Society' in Biehler R, Scholz RW, Stral3er RandWinkelmann B (Eds) Didactics of mathematics as a scientific disciplineDordrecht: Kluwer Academic Publishers, 367-378.

��

244 Carreira

Onnell C (199Ia) 'A Modelling View of mathematics' in Niss M, Blum W, Huntley I(Eds) Teaching of mathematical modelling and applications Chichester: EllisHorwood, 63-69.

Onnell C (1991b) 'How ordinary meaning underpins the meaning of mathematics' Forthe Learning ofMathematics II( 1),25-30.

Skovsmose 0 (1992) Towards a philosophy ofcritical mathematics education Dordrecht:Kluwer Academic Publishers.

Skovsmose 0 (1994) 'Democratic competence and reflective knowing in mathematics'For the Learning ofMathematics 12(2),2-11.

��

22

Reconnecting Mind and World: Enacting a (New) Wayof Life

Stephen R. CampbellSimon Fraser University, CanadaUniversity of California, Irvine, [email protected]; [email protected]

A common assumption in teaching mathematical modelling andapplications is that mind and world are ontologically distinct. This dualistview give rise to an explanatory gap as to how these two realms connect.An alternate view where mind and world are ontologically identical isexplored here. This alternate view, grounded in a monist ontology ofembodied cognition, undermines and attempts to fill this explanatory gap.Embodied cognition presents challenges of its own, but it also presents newpedagogical opportunities.

��

246

INTRODUCTION

Campbell

Mathematical modelling helps us to understand the world as it is and to envision how itmight be. Whether one is concerned with tracking inventory and the transactions ofgoods, establishing boundaries, predicting earthly and celestial events, or charting acourse through spacetime, mathematical modelling has enhanced and extended humanpotential in virtually all areas of human endeavour, and virtually everyone has benefitedto one extent or another from these profound developments, whether they are aware ofthese benefits or not.

Even if one is aware of the value of mathematical modelling and applications, teachingand learning how to engage in this kind of activity is another matter altogether. It is arelatively easy thing to become motivated and enthralled with the practical applicationsof mathematics. It is exceptionally difficult, however-particularly for those teachingmathematical modelling and applications to others, and even for those most adept atdoing it-to understand what mathematics is apart from those applications, and to fathomand explain why it is that it can be applied to the world so effectively. Perhaps if we hada better way of understanding these persistent difficulties, mathematical modelling andapplications would be easier and more enjoyable to learn, and the value thereof morewidely accessible to and appreciated by society at large.

The standard view in teaching mathematical modelling and applications is to assume thatmathematics provides an idealised way of thinking about reality. The model exists in themind, and the application is embedded in the world. In this dualist view, mathematicalmodelling and applications is a matter of connecting mind and world through a variety ofactivities such as formulating, solving, interpreting, and comparing-applied iterativelyto some practical end (e.g., Mason, 1984). As helpful as this approach quite evidently isfor teaching and learning, however, it leaves persistent philosophical problems as to therelation between mind and world, the nature of mathematics, and the mysterious efficacyof applying mathematics unresolved. An alternate, non-dualist, ontological assumption isexplored in this chapter-one based on a monist notion of embodied cognition thatundermines this dualist separation between mind and world and renders the distinctionbetween them more transparent.

RECONNECTING MIND AND WORLD

The standard view typically assumes that mind and world are ontologically distinct. Thisontological separation derives from Descartes' infamous mindlbody dualism based onwhat is, prima facie, a clear distinction between mind (res cogitans) and matter (resextensa). This assumption is widely accepted and rarely questioned in teachingmathematical modelling and applications-despite the fact that it leaves a recalcitrantexplanatory gap as to how the ideal mental world of mathematics connects with the realmaterial world of situations.

Some argue (e.g., Lokhorst & Kaitaro, 2001) that this explanatory gap can be filled, asDescartes himself attempted to do, by admitting that these distinct ontological realmsmust interconnect in some way. The history of philosophy and science, however, is

��

Reconnecting Mind and World 247

replete with failed or unsatisfactory accounts as to just how this can possibly be (e.g.,Armstrong, 1968; Hegel, 1964; Popper & Eccles, 1977). Mind and world are, after all,assumed to be absolutely distinct. Be this as it may, mathematical model1ing andapplications, much like language, appears to bridge mind and world somehow. Despitethe perennial conundrums, if a mathematical model can be effectively applied,pragmatically speaking-who cares why it works?

Adopting such a position is akin to evaluating learners on the basis of answers tomathematical questions obtained by rote, independently of any meaningful understandingas to why those answers happen to be correct. I argue against Cartesian dualism and formonist theses elsewhere (Campbell, 2002; Campbell & Dawson, 1995). Accordingly, Iwill only assert here, from the perspective of embodied cognition, that mind and worldare onto logically identical. I will admit, albeit for non-Cartesian reasons, that in thecourse of human development, mind and world only appear to become separated. Fornow it suffices to note that this foundational shift in assumptions avoids persistentphilosophical problems associated with Cartesian dualism.

The theoretical task in this chapter, then, is to expound a non-standard view ofmathematical modelling as embodied cognition-a view that gives rise to new ways ofthinking about mathematical modelling. The practical task is to draw out someeducational implications of this view.

THEORETICAL CONSIDERATIONS

An immediate implication of ontological monism is that dualist difficulties associatedwith an ontological separation between mind and world dissolve. One is now faced witha more tractable set of problems, which in turn require a new framework from withinwhich they can be addressed and resolved. First, one should provide a philosophicalaccount as to how mind and world can be seen as ontologically identical. Secondly, onemust provide an account for the apparent separation, or dualism, between mind andworld. This latter problem, I maintain, turns out to be more an educational outcome thana philosophical problem. These two theoretical problems comprise the basic needs for anew theoretical framework for understanding mathematical model1ing from an embodiedperspective.

An obvious place to begin for establishing an ontological identity between mind andworld, and for reconnecting and accounting for what is now taken as an apparentseparation between the two, is the lived body. Merleau-Ponty's (1968) ontologicalprimitive ofjlesh as "an ultimate notion ... not the union or compound of two substances"(p. 140) provides a philosophical account for how mind and world are ontologicallyidentical. Egan's (1997) theory of education as the recapitulation of different kinds ofunderstanding, beginning with and based upon somatic understanding, provides anaccount for the apparent separation between mind and world.

At the Nexus of Mind and WorldThe ground of all that is human, no matter how vicarious or how far removed, is theliving body-it is a substance that can both touch and be touched, that can both see and

��

248 Campbell

be seen-it is flesh, Merleau-Ponty's ontological primitive, which Dillon (1997) aptlydescribes as "the world sensing itself through that part of itself which we are" (p. xiii).Each of us, in other words, is a part of the world within itself. Flesh is not reducible tomind or matter. For the humanistically softened and the scientifically hardened, I takethis view to entail that the 'flesh of the world' is not pure ideation, nor inert material.Rather, call it what you will, something conscious, aware, feeling, something yet moreprimitive variously manifesting as these, is innate in this foundational substance of theworld.

The main point to note here is that being of the world and in the world, flesh can beassumed to have, at least tacitly, embodied knowledge ranging from the depths ofcreation to the ends of eternity. More radically, and this follows in a strictly absolute, nota relative sense, all knowledge can only be knowledge of the world embodied in theworld. As Merleau-Ponty puts it " ... there is a logic of the world to which my body in itsentirety conforms" (1962, p. 326). This is to acknowledge that principles at play instructuring the world are at play in structuring the body and, if I may go so far as to add,vice versa. This shared logic between living bodies and the world from which weemerge, provides a more profound understanding of the nature of mathematics and therole of mathematical modelling. It is a view that presupposes an intimate relationbetween mind and world, and consequently serves to clarify, rather than mystify, whymathematics can be applied with such efficacy.

While the philosophical problem of how mind and world relate can at least begin to beaddressed and accounted for through Merleau-Ponty's theory of embodiment, animportant educational problem emerges. How does knowledge of the world latent in theflesh ofthe world that is the lived human body become manifest as thought and language,and more to the point, as mathematical thinking and formal mathematics? The history ofhuman culture provides an existence proof that such things are possible.

Education as Cultural RecapitulationEgan (1997), in a manner compatible with Merleau-Ponty's notion of flesh, maintainsthat the lived body provides an understanding that " ... remains fundamental to our graspon the world throughout our lives" (p. 35). Education, both culturally and individually,involves the emergence of different kinds of understanding from somatic understanding.From this somatic basis, according to Egan, educational development is mediated througha variety of linguistic and cognitive tools associated with orality, literacy, theory, andreflexivity, which have emerged through the course of cultural history. Some theoreticaland practical considerations regarding mathematical modelling will now follow fromblending Egan's educational theory with Merleau-Ponty's ontological monism of flesh asembodied consciousness.

How do we account for an apparent separation between mind and world? If theseparation between mind and world is only an apparent separation, then what conditionsprecipitated it in cultural history, and how does this phenomenon come about inindividuals? Comprehensive treatment of these questions is not possible here, but someconsideration is in order. A cluster of elemental distinctions are bound up with thedualist separation of mind and world: sky (or heaven) and earth; intellect and sense;

��


subject and object; to note but a few. Perceptions emerge in the formation of objectsfrom the heterogeneous milieu of somatic experience (Campbell, 1999). Merleau-Ponty(1962) considers the problem in terms of objective thought, noting that: " ... a crucialmoment in the genesis of the objective world" is in the "constitution of our body asobject" (p. 72).

The experience of body as a body that can touch and be touched, see and be seen, beginsto place the lived body in apparent contrast with other things that we feel and see.However, affiliated with the onset of orality, Egan's mythic understanding naturallyposits other beings in objects separate from our own body. To the mythic mind, the bodyas both an experienced and experiencing object is not yet Descartes' separation, but it isan important step along the way to the cogito, the realisation of self.

For Egan (1997), the signifying ground for binary terms is the body:... our bodies are our primary 'mediators' of meaning, and some of theearliest discriminations we make are in terms of our bodies-so 'wet'means wetter than my body and 'dry' means drier than my body, 'hard'means harder than my body and 'soft' means softer than my body, 'big'means bigger than my body, 'small' means smaller, and so on... (p. 40).

The onset of orality along with conceptual spaces opened by binary terms such as these,in conjunction with other memorable experiential contrasts, such as appearing anddisappearing, walking on earth and flying in air, both constitute and fuel the mythicimagination of the lived body. Sprites, nymphs, angels, dragons, cherubs, and moreregister to a now speaking and listening human being as very personal expressions of aliving world.

The spoken word is fluid and transient-washing over the ear and but for a lingeringtrace in the memory of time past, it is gone-but its impression in memory can bereinforced through recurrent use. The written word is more solid and fixed. As with therecurrent use of spoken words, written expressions etched once, allows them to bepondered at will. Egan argues that culturally, the advent ofwritten language can, as it didin the case of the ancient Greeks, lead to a cognitive shift away from mythical thinkingtoward a more fixed and permanent representational view of the world. As written wordsfix thoughts that give them expression, this invariance allows for identification anddistillation of patterns and principles through various theoretical processes such asdefining, generalising, abstracting, symbolizing, and inferring that we collectively refer toas reason. Somatic experience, imaginatively traversed and expressively intertwinedthrough speech, eventually becomes ordered and rendered into the standard dualist viewof our mental experience existing separately from the physical world.

EDUCATIONAL IMPLICATIONS

In the standard view of teaching mathematical modelling and applications, it is assumedthat the mind of the modeller is ontologically separate from the world that is to bemodelled. Consequently, the relation between mind and world is intrinsicallyrepresentational. Theoretical considerations in the previous section suggest that the

��

250 Campbell

standard view is only apparent, emerging from our embodied reality, and thus is aderivative of a much more intimate relation between mind and world than typicallythought. What are the educational implications of taking an embodied view both asphilosophical1yand psychologically prior to the standard dualist view?

Educational implications considered in this section focus first on pre-school andelementary mathematics education. This focus is warranted by the fact that the youngerchildren are, the more grounded in, and less alienated from embodiment they are. Thepractical objective of teaching mathematics and mathematical modelling in these earlyyears from an embodied perspective is to help cultivate and consolidate the grounding ofabstract mathematical concepts in the lived experience of the child's body.

Embodied mathematics: An immediate implication of embodiment is the foundationalrole of being in and of the world in cultivating the experiential and imaginative groundfor developing mathematical awareness and understanding. Playing with fingers andtoes, and feeling the beating of one's heart embodies a primal sense of numeracy, rhythmand counting. A child on a merry-go-round, a swing, or a slide both enacts and embodiesthe sense of going around in a circle or moving at an angle or along an arc. Sliding alongice, or jumping off a diving board embodies the sense of plane and parabola. In theseways, embodiment provides the experiential and imaginative ground for profoundly deepand meaningful understandings of mathematics.

There has been recent interest in metaphor as a generative model for the development ofembodied mathematics (Lakoff and Nunez, 2000), that for limited space is only notedhere. In terms of binary opposites, however, whose meaning is primarily mediated by achild's body, such as hot/cold, big/small, up/down, moving/stationary, forward/backward,fast/slow, and so on-all of these ostensibly abstract concepts are clearly based on andemerge from lived embodied mathematical spaces. These binary spaces both can andshould be enacted by, and expressly calibrated and related to the child's embodiedexperience as early as possible. Of course, all children naturally have these embodiedexperiences. What the embodied view does is alert us to how fundamental theseexperiences are in laying the conceptual ground for mathematical thinking andunderstanding.

Imaginative and linguistic shifts, from transient expressions of orality to the more lastingrepresentations of literacy, alert us to yet another kind of pedagogical opportunity: .....we[can] teach children to move from perceptually based discriminations like 'hot' and'cold' to abstract means of referring to the world... " (Egan, 1997, p. 72). For example, athermometer provides learners with a means of grounding the concept of temperature insomatic experience. A young child can relate to a thermometer's reading, because thechild has experienced changes in body temperature. Similarly, when a child's growth ismarked on a wall or doorframe, the marks are, for the child, a means of expressinggrowth in height.

Competing or complementary views? Does the embodied monist view proposed herecompete with or complement the standard dualist view? Although this embodied monistview claims ontological and epistemological priority over the standard dualist view, it

��


does not necessarily aim to replace it. The emergence of the standard view from theembodied view may, after all, entail some psychological necessity or provide somebenefit. But is the standard view necessary, and what exactly are the benefits that itprovides? These are open questions. Those ostensibly corroborating the standard view,such as de Lange (1993), Vershaffel (2002), and many others have contributed much toour understanding of teaching mathematical modelling and applications. I wouldemphasise that alternate monist views such as the one presented here are not necessarilyincommensurable with the standard view.

Given that the dualist separation evident to everyone is only apparent, that separationmay simply be a reflection of attitudes and beliefs that have long been perpetuated in thehistory of educational practice. It is not evident that literacy necessitates a dualistrepresentationalist view-thus opening the possibility we may be able to educate or beeducated in such a way that such a separation need not occur. Educating differently,though, does not imply educating for the better. There may very well be somepsychological or mathematical necessity or benefit involved in making an apparentseparation between mind and world. Whether or not this is in fact the case, given thatsuch a separation does occur in learners, a further task at hand to consider from anembodied view is how mathematical modelling can be understood in a way thatreconnects mind and world.

Proponents of the standard view might argue that I have used mathematics to model earlychildhood activities in the aforementioned examples. But that would be to put the dualistcart before the monist horse. The embodied view expounded here is a more radicalPythagorean view in that the generative principle(s) inherent in the ontological primitive,and thus the very structure of the world, is taken to be mathematical. That is to say,mathematics is not being viewed here solely as what is currently defined as formalaxiomatic mathematics. Rather, the latter is being viewed as a contemporary humanexpression of the former.

ENACTING A (NEW) WAY OF LIFE

I have been laying philosophical groundwork for this enactivist embodied view elsewhere(Campbell, 2002; Campbell & Dawson, 1995). Suffice to say here that this view entailsthat to think mathematically is to be actively engaged in reflecting on organisingprinciples of the world, or what amounts to basically the same thing, on organisingprinciples of our own experience as embodied beings embedded within the world.Thinking mathematically then, is de facto to be engaged in mathematical modelling. Ofcourse, there are different levels ofmathematical thinking. Mathematicians of the highestcaliber, I suggest, are pluming the depths of the world either as it actually is, or as itpotentially might be. The philosophical relevance of this view is that it accounts for themysterious efficacy of applying mathematics-reminding us of our intimate relation withthe world; inviting us to enact mathematical implications thereof.

The educational relevance of this embodied view of mathematical modelling andapplications is largely dispositional and motivational. It provides an alternative lens forreconsidering our relationship with the world of which we are a part and from which we

��

252 Campbell

have emerged. According to this view, our relationship with the world is much moreintimate than can be the case with constructivist and materialist ontologies that prioritiseone pole or another of the Cartesian ontological divide, i.e., res cogitans and res extensarespectively, or dualist views that are left to account for how onto logically separate anddistinct realms interconnect.

Although this monist-expressivist embodied view of cognition undermines and seeks toreplace the dualist-representationalist Cartesian view as fundamental ontologyepistemology, it does not deny the emergence of the latter view from the former. Nordoes it question or claim to invalidate research in teaching mathematical modelling thatmay assume the standard dualist view, implicitly or otherwise. Rather, it calls attentionto a potentially deeper embodied sense of reality in which mind and world are intimatelyand inextricably bound, not onto logically separate. As such, a deeper understanding as towhat mathematics is and how it can be more effectively taught, learned, and applied mayvery well be at hand.

REFERENCES

Armstrong DM (1968). A materialist theory of mind. London: Routledge.Campbell S R (2002) 'Constructivism and the limits of reason: Revisiting the Kantian

problematic' Studies in Philosophy and Education: An International Journal21(6),421-445.

Campbell SR (\ 999) 'The problem of unity and the emergence of physics, mathematics,and logic in ancient Greek thought' in Proceedings of the 4th InternationalHistory and Philosophy of Science and Science Teaching Conference Calgary:University of Calgary, 143-152.

Campbell SR, Dawson AJ (1995) 'Learning as embodied action' in Sutherland R, MasonJ (Eds.) Exploiting mental imagery with computers in mathematics educationBerlin: Springer, 233-249.

de Lange 1. (\ 993) 'Innovation in mathematics education using applications: Progressand problems' in de Lange J, Huntley I, Keitel C, Mogens N (Eds) Innovation inmaths education by modelling and applications Chichester: Ellis Horwood, 3-17.

Dillon MC (1997) Merleau-Ponty's ontology (2nd ed.) Evanston: Northwestern UniversityPress.

Egan K (1997) The embodied mind: How cognitive tools shape our understandingChicago: University of Chicago Press.

Hegel GWF (1964) Phenomenology ofmind New York: Humanities Press.Lakoff G, Nunez RE (2000) Where mathematics comes from: How the embodied mind

brings mathematics into being NewYork: Basic Books.Lokhorst GJC, Kaitaro T (200 I). 'The originality of Descartes' theory about the pineal

gland.' Journal for the History ofthe Neurosciences, IO(I), 6-18.Mason J H (1984) 'Modelling: What do we really want students to learn?' in Berry JS,

Burghes DN, Huntley \D, James DJG, Moscardini A 0 (Eds) Teaching andapplying mathematical modelling Chichester: Ellis Horwood Ltd., 215-234.

Merleau-Ponty M (\ 962) The phenomenology ofperception (Smith C, Trans.) Evanston:Northwestern University Press.

��


Merleau-Ponty M (1968) The visible and the invisible (Lingis A, Trans.) Evanston:Northwestern University Press.

Popper KR, Eccles JC (1977) The selfand its brain. Berlin: Springer International.Verschaffel L (2002) 'Taking the modeling perspective seriously at the elementary

school level: Promises and pitfalls' in Cockburn AD, Nardi E (Eds) Proceedingsof the 26th Annual Conference of the International Group for the Psychology ofMathematics Education, (Vol 1),64-80.

��

23

ICTMA: The First 20 Years

Ken HoustonUniversity of Ulster, N. [email protected]

In this paper I discuss the ongms of the mathematical modeIlingmovement, and the ICTMA conferences and their development through thefourth quarter of the 20th century. It draws on the author's personalexperiences and the biennial ICTMA publications.

��

256

THE BEGINNING

Houston

It all began with the MeLone Report (MeLone, 1973). This work surveyed what recentmathematics graduates were doing in their employment, how relevant their education wasto their work, and how satisfied their employers were with their performance. MeLonedescribed the average mathematics graduate:

Good at solving problems, not so good at formulating them, the graduatehas a reasonable knowledge of mathematical literature and technique; hehas some ingenuity and is capable of seeking out further knowledge. Onthe other hand the graduate is not particularly good at planning his work,nor at making a critical evaluation of it when completed; and in any eventhe has to keep his work to himself as he has apparently little idea of how tocommunicate it to others (p, 33).

Notwithstanding twenty years of ICTMA, this is still a problem. Recently Challis,Gretton, Houston and Neill (2002) wrote:

The SIAM (Society for Industrial and Applied Mathematics) report onMathematics in Industry (SIAM, 1995) contained data from a survey ofPhD graduates working in industry. The report indicated that modelling,communication and teamwork skills together with a willingness to beflexible are important traits in employees. However the PhD graduatesthemselves indicated that they felt inadequately prepared to tackle diverseproblems, to use communication effectively and at a variety of levels, or towork in teams.

I suspect the reason for this is that too few universities have as yet embraced themathematical modelling ideal, or even the mathematical model-ing ideal!

There were other influences. The UK-based Association of Teachers of Mathematicswas advocating courses for sixth formers (UK years 12 and 13; US grades II and 12)which included "some general awareness about the nature of mathematical activity andthe kinds of situations in which it may be usefully applied" and which would develop"the ability to make effective use of both individual study and of working in a group"(ATM 1978, p.31). The Cockcroft Report (Cockcroft, 1982) said that it was importantfor sixth formers to "pursue independent investigations and to discuss and communicatetheir ideas." [Cockcroft was the Chair of the Committee of Enquiry into the teaching ofmathematics in schools in the UK. The Report had a long lasting influence onmathematics education, initiating the development of the National Curricula in thecountries of the UK.] In the USA, the Mathematical Association of America through itsCommittee on the Undergraduate Program in Mathematics, recommended that "Studentsshould have an opportunity to undertake 'real world' mathematical modelling projects,either as term projects, in an operations research course, as independent study, or as aninternship in industry" (Mathematical Association of America, 1981, p. 13). That reportgoes on to add that all mathematical sciences majors should experience mathematicalmodelling early in their studies.

��

ICTMA: The First 20 Years 257

There were many influences from many directions, and things began to happen. In theUK David Burghes, who could well be described as the Father of ICTMA, decided to tryto enliven the school mathematics curriculum by working with teachers to produceinteresting modelling investigations for pupils at secondary level. In 1978, while atCranfield University, he started publishing the Journal of Mathematical Modelling forTeachers, which in 1981 became Teaching Mathematics and its Applications, a journal ofthe UK-based Institute of Mathematics and its Applications. Burghes continued to editthis journal until 2000. He held modelling workshops for teachers up and down thecountry, and he formed the Spode Group, so called because it met in the SpodeConference Centre in Staffordshire. His co-conspirators in this venture were John Berry(then at the Open University) and Ian Huntley (then at Paisley College in Scotland). TheSpode Group published three books on the theme of solving real problems withmathematics (Spode, 1981, 1982, 1983). The articles in the Spode publications allfollowed a similar format: a problem statement, followed by teaching notes, possiblesolutions and related problems. More publications by this group would follow later.

Other people were getting involved in the movement to enhance the teaching ofmathematical modelling in schools and universities. These included John McDonald,(Paisley) and Glynn James (Coventry Polytechnic). James organised a workshop forlecturers in higher education and published the proceedings (James and McDonald,1981). Others were Hugh Burkhardt, for many years Director of the Shell Centre forMathematical Education at Nottingham University, George Hall, also at Nottingham, andDick Clements at Bristol. In Northern Ireland the Further Mathematics Project wasgetting under way. Working on the principle WYAIWYG (what you assess is what youget), this project launched an unprecedented effort to introduce an A-Level examinationin modelling and investigations (Fitzpatrick and Houston, 1987). Furthermore, theCouncil for National Academic Awards encouraged mathematics departments in thepolytechnic sector of UK higher education to include mathematical modelling in theircourses. This resulted in a marked difference between the polytechnics and theuniversities in teaching applied mathematics. The universities tended to delivertraditional lecture courses on mechanics, fluid dynamics, etc, while the polytechnicsactually got their students to do some modelling, to write reports and to givepresentations. [The polytechnics in the UK came into existence in the I970s to furtherthe government's plan to expand opportunities for tertiary level education. They wereelevated to university status, each with its own charter to award degrees, in 1992. TheUlster Polytechnic was an exception. It became a constituent part of the University ofUlster in 1984, through a merger with the New University of Ulster.]

In all of the above initiatives, the emphasis was on the teaching of mathematicalmodelling. For many years there had been conferences at which various mathematicalmodels were discussed, but there had not been any special attention to teachingmodelling. The time was ripe for such a conference.

THE EXETER CONFERENCES (1983 AND 1985)

John Berry (Open University), Burghes (who moved to Exeter University), Huntley (whomoved to Sheffield City Polytechnic), Glynn James (Coventry Polytechnic), and Alfredo

��

258 Houston

Moscardini (Sunderland Polytechnic) organized the first "International Conference on theTeaching of Mathematical Modelling." The aspiration that "this will become a bi-annualevent" [they meant biennial - every two years, not twice a year] was expressed in theconference literature. And it did so become! [The words "and Applications" were notadded to the title until ICTMA 3]. The conference was held at Exeter University in July1983, and the summer of 1983 was exceptionally hot in the UK. The meeting roomswere at the top of a hill on the campus while the living rooms were at the bottom. Therewas much perspiring while walking between the two!

ICTMA I attracted 125 delegates from 23 different countries. Not unexpectedly, themajority of the delegates (76) were from the UK. There were two plenary lectures and 39lectures in the parallel sessions. These are reported in the first of the books in theICTMA series. The majority of the presentations addressed higher education, with onlyfive reporting on developments in schools. This imbalance is understandable given that itwas much easier to change the curriculum in a polytechnic than in a school (or even in auniversity!).

At this first conference people were thinking and talking about issues that are still verypertinent, issues such as assessment and group work. Many of the talks (and the papersin the resulting book) were of the war story variety: "This is how we teach modelling" or"This is our curriculum," and many were useful descriptions of models that the author(s)had used in their teaching. There were a few papers describing research or presentingphilosophical perspectives. Four of the people who would organise subsequent ICTMAconferences were present: Werner Blum, Jan de Lange, Chris Haines and Ken Houston.The tradition of publishing a book of selected papers from the conference began withICTMA 1, and at the same time, a long association with Ellis Horwood and thepublishing companies he managed. He has published the proceedings of all the ICTMAconferences except one. A complete list of the ICTMA books is given at the front of thisvolume.

If my memory serves me rightly, there were no workshops at ICTMA 1. There we allwere, sitting in rows in lecture rooms, listening to each other expound on the virtues ofmodelling and the benefits of group work, instead of getting into groups ourselves, toexperience this way of working firsthand.

As an aid to comparisons across the years, I note that my plane fare to London was £112and the conference fee plus accommodation was £170. The conference outing was acoach excursion to Dartmoor, with dinner in a delightful country pub at Fingle Bridge.Talented members of the conference entertained us.

ICTMA returned to Exeter in 1985 as ICTMA 2. Again it was well attended. I no longerhave a copy of the list of delegates or a note of the total attending, but there were 73presentations, including two plenaries, by delegates from ten different countries, with themajority again from the UK. It was decided to publish 48 of the papers and, for the onlytime in ICTMA history, two volumes were produced. The first dealt with "ModellingMethodology, Models and Micros," while the second was titled "MathematicalModelling Courses." The preface to the second volume opens thus: "Courses in

��


mathematical modelling in higher education are now quite commonplace, especialIy inpolytechnics." There was a feeling abroad that the battle was being won, at least in thatsector of higher education. There was a great variety in the courses being offered, andthe volume containing descriptions of courses was written in the hope that "this text wilIhelp both newcomers and experienced teachers to learn from others." (Berry, Burghes,Huntley, James and Moscardini, 1987). Ten of the 25 papers were relevant to schoolmathematics---eonsiderably greater number than at ICTMA I. There were descriptions ofinnovative attempts to enhance and change what were seen as conservative nationalcurricula.

The first volume was arranged in three sections with the first section reviewingmethodology in the process of problem solving and proposing new ideas in formulatingmodels. Research reported in this ICTMA II book (Izard, Haines, Crouch, Houston andNeill, this volume) identifies model formulation as crucial skilI, but one at which studentsdo not do very well. This problem in the teaching of modelIing has not yet been solved.The second section of the volume contained a selection of models, which had alreadybeen used in teaching to serve as a resource for teachers. The third section wasinnovative for its time. It described teachers' successes using microcomputers to exploreand evaluate models, and to do simulation model1ing.

My fare to London was about the same as in 1983, but the cost of the conference haddropped to £126. It was not so hot in the summer of 85. The excursion was by coacharound the beautiful countryside of south Devon and included a ride on a restored steamtrain and a walk along a seaside promenade during a very high tide. Some people gottheir feet wet, and some, who attempted a detour, came very close to a fast locomotive.We were walking along the promenade at the seaside. There was a high spring tide andthe waves were crashing up onto the promenade. Some just walked on, oblivious to thewater in their shoes, but others took a detour along an adjacent railroad track. Alocomotive flew past, coming within a meter of the walkers, and angrily sounding itswhistle. Sensible people like me took a detour over a bridge, along a dry footpath, andover another smalI bridge to rejoin the group. AlI survived the experience.

FURTHER AFIELD IN EUROPE - KASSEL (1987), ROSKILDE (1989) ANDNOORDWIJKERHOUT (1991)

The next three ICTMA conferences were held in continental Europe, in Germany in1987, in Denmark in 1989 and in the Netherlands in 1991. The chairs of the organisingcommittees were, respectively, Werner Blum (Kassel University), Mogens Niss(Roskilde University) and Jan de Lange (Freudenthal Institute, Utrecht University).Given the wide mathematical education interests of these three and the institutions wherethey work, it wilI be no surprise to learn that discussion of teaching at secondary schoollevel was much more prominent than before. This is highlighted particularly in thestructure of the books from the Kassel and Roskilde conferences, in which distinctsections addressed the lower secondary, upper secondary and tertiary sectors.

The name of a conference was now the Nth International Conference on the Teaching ofMathematical Modelling and Applications. The phrase applications and modelling now

��

260 Houston

occurs very frequently in conversation and literature. This was a good developmentbecause it highlights the subtle but significant differences between a study ofmathematical models and the process of mathematical model1ing. In his plenary lectureat ICTMA 3, Mogens Niss (Roskilde University) said that an application of mathematicswas the result of the process of applying mathematics to any area of extra-mathematicalreality-a mathematical model-whereas mathematical modelling is the process (Niss,1989). Another term entered my vocabulary at this time - mathematisation. This isdefined by Niss to be the translation stage of the modelling process, when the modellermoves from the physical model created through the making of simplifying assumptions tothe mathematical model created by the introduction of variables and their mathematicalrelationships to one another.

Probably due to the fact that we invited venues outside the UK, these three had moreparticipants than the Exeter conferences. In Kassel, for example, there were five plenarylectures, 83 other presentations run in 7 or 8 parallel sessions and a choice of fiveworkshops. There were 70 papers in the book. The introduction of workshops was avery welcome innovation. We all knew that group work was important for the learningof modelling, and we practised it with our students, but so far we had not engaged ingroup work ourselves at the ICTMAs.

Philip Davis (Brown University) gave a plenary at Roskilde on "Applied Mathematics asa Social Instrument" (Davis, 1991). The goals of applied mathematics include"description, prediction and prescription - tell me what is, tell me what will be, tell mewhat to do about it." In social and economic modelling he warned that we should beware,because "mathematical descriptions tend to drive out all others," and we all know thatmodels are only as good as their assumptions.

At ICTMA 5, Christine Keitel (Berlin Technical University) observed that while societywas becoming more and more mathematised through intelligent technology, individualswere becoming de-mathematised. Mathematics at work was becoming implicit in themachinery and we no longer saw it. School education was the time to make the implicitexplicit (Keitel, 1993). At the same conference Cyril Julie (Western Cape University)described how he saw mathematics education as an agent for emancipation andempowerment. "Peoples' mathematics" is not just about mathematical empowerment butalso about political and intellectual empowerment (Julie 1993).

The cost of flights to Frankfurt and Schipol were not much more than to London.Copenhagen was more expensive. Conference fees and costs kept going up. Conferencesocial programmes were now able to offer choices. I remember visits to the Edersee Dam(where I was surprised by the high price of a cup of coffee), an organ recital in RoskildeDomkirke and visits to Copenhagen and Utrecht. There were receptions by universityrectors and civic dignitaries.

ACROSS THE ATLANTIC (1993) AND BACK AGAIN (1995)

I suppose no Conference series could claim to be truly international until there had beenat least one meeting in the United States of America. And so the Executive Committee of

��


ICTMA (more about this later) accepted Cliff Sloyer's offer to organise ICTMA 6 and tohost it at his institution, the University of Delaware - Newark in the summer of 1993.That was the tenth anniversary of ICTMA and now in 2003 we are back in the USA forthe twentieth.

The conference was organised along familiar lines with three plenary lectures, 28 parallelpresentations and 13 workshops of varying length organised in series or in paraIleldepending on the length of each. The book has 24 chapters and, in a break with tradition,was published by an American publisher. The three plenaries were in the nature ofsurvey papers. Werner Blum (Kassel University) looked at important aspects of practiceand research. He gave examples to illustrate what exactly is meant by "applications andmodelling" (that phrase again!), and answered the question "What is the use of A&M inmathematics teaching?" by giving five arguments. He wrote "A&M is one way to makethe learning and teaching of mathematics more meaningful and to supply students withorientational knowledge as well." He proposed practical measures to promote theteaching of A&M and listed several, as yet unanswered research questions (Blum, 1995).While the community has wrestled with these questions now for ten more years, andwhile we have found out much and understand more than we did in 1993, we have not yetcracked the problem!

Peter Galbraith (University of Brisbane), soon to be the chief organiser of ICTMA 8,spoke on "What I have learned." His summary notes that he felt that, even after ten years,"questions and needs continue to emerge." There was a continuing need for case studies,right through from primary to tertiary, to minimize re-invention of wheels. There was aneed to design, implement and study experiments located in a system of institutions.Much remained to be learned about ways to make group learning more effective, andhere the mathematics community could learn from other communities. Assessmentpersists as an issue of importance and controversy (Galbraith, 1995).

Ian Huntley (Sheffield Hallam University) gave a "state of the (UK) nation" report,teIling of the move in the UK towards a mass higher education system that would bringmany problems in its wake (which today are still rumbling), and he described theinitiatives in technology-based training that were then being funded. He spoke of theinitiative to embed information technology in the whole school system. He noted thatICTMA should have been ICTMMA - there are, after all, two words beginning with M inour title-and he asked if we should consider becoming the International Conference onModelling and Applications, thus broadening the scope beyond mathematics, and lettingthe teaching develop the acquisition of key personal skiIls and competencies (Huntley,1995). This is an interesting proposition, but I think it is important for this community tocontinue to concentrate on mathematical modelling. We are all aware that our studentsdevelop other key skills in the process of mathematical modelling activity, thus givingthem a double endowment. After all, "mathematical modeIling is a way of life!"

There was an excursion to see the Liberty Bell in Philadelphia, followed by a crab bakesupper near Chesapeake Bay. An excess of crabs was baked for our benefit-about 20per person! Only the locals could do justice to this feast.

��

262 Houston

At the last minute, due to the absence of a fourth plenary speaker, I was asked to fill thatslot with my previously prepared talk on Comprehension Tests. This was not because thetalk had suddenly become so important, but it was merely to let people see who KenHouston was, since his offer to organise and host the next ICTMA had just beenaccepted! This was to keep me very busy for the next four years - two up to theconference and two beyond, while editing the book. A knock-on effect has kept me busyright through to today!

I had a good local organising committee, the conference accommodation on theJordanstown Campus of the University of Ulster was brand new, there was an IRA ceasefire throughout 1995, and over 100 delegates came. People said it was a happyconference and that was my measure of success.

Being an ICTMA conference organiser is a non-trivial affair, as all those who have heldthis office will attest. Forward planning involves advertising, booking accommodation,raising sponsorship, inviting plenary speakers, setting the costs, negotiating a bookcontract, organising the conference timetable and social programme, and praying thatpeople will come. The five days pass quickly and then there is great anticlimax when allhave gone home. The after-conference work started within a month as papers came in forreview. Then came the editing and preparation ofCRC (camera ready copy).

The conference themes were not dissimilar from the past. In the preface we wrote, "It isnow widely accepted that mathematical modelling is 'the way of life' of an appliedmathematician." "Nevertheless, despite this widespread recognition and practice, thereare still needs to be met." We listed these as the need to convert some hearts and minds,the need to devise, implement and evaluate new methods of teaching and learning, theneed to conduct research and the need to disseminate (Houston, Blum, Huntley and Neill,1997). At one level, nothing changes, but at another as a community we become wiserand more aware of how to do this great task of educating the world's young people. Inthe plenaries, Sue Lamon (Marquette University) talked about modelling and what isknown about the way our minds work (Lamon, 1997), Leone Burton (University ofBirmingham) about assessment (Burton, 1997), Mark Cross (University of Greenwich)about industrial modelling, Catherine Coxhead (Northern Ireland Council for theCurriculum, Examinations and Assessment) about the Northern Ireland schools'curriculum (Coxhead, 1997), and Sally McClean (University of Ulster) about statisticalmodelling (McClean, 1997). We had an excursion to the Giant's Causeway where thesun was shining gloriously, and to the Old Bushmill Distillery where the distillingprocess was on holiday but the free samples were flowing!

TRAVELS TO BRISBANE (1997), LISBON (1999) AND BEIJING (2001)

The preface to the book of the Brisbane conference, for which Peter Galbraith (Universityof Queensland) was Chair of the organising committee, draws attention to the trulyinternational nature of ICTMA in 1997 (Galbraith, Blum, Booker and Huntley, 1998).All continents of the earth except Antarctica were represented. Furthermore, for the firsttime the word "technology" appears in the title, acknowledging the fact that the use of

��


technology is highly desirable in the teaching of applications and modelling. The prefacegoes on:

The vertical scope of the content extends from secondary schoolmathematics through tertiary undergraduate levels, so recognising that theability to apply mathematics is important whatever the level of techniquesavailable. The horizontal scope of the book reflects issues that are currentlychallenging existing theory and practice in the teaching an application ofmodelling skills.

Beside the use of technology, assessment was still a major issue as was the developmentof students' communication skills. There is a good collection of models for use inteaching, and accounts from several countries of national developments in the teaching ofmodelling. The world is now getting the message!

We travelled to the Australian Gold Coast, ending with a barbie. There were some meatdishes that, to this Irishman's palate at least, were rather exotic, but nice - kangaroo,crocodile, emu, and camel.

Two years later, in 1999, the conference was back in Europe, in Lisbon, Portugal. Thechair of the organising committee was Joao Filipe Matos of the University of Lisbon. Itseemed to me that there was a greater philosophical air about the conference this time.Yes, there were the customary selections of papers describing innovative uses oftechnology, innovative uses of mathematical methods, innovative curricula, andpedagogical issues, but three of the plenaries opened up new directions.

Stephen Campbell (University of California-Irvine) gave his audience a livelyintroduction to his philosophy of mathematical modelling. He proposed a radicalenactivist view of life in general and modelling in particular (Campbell, 200 I). SusanaCarreira (New University of Lisbon) spoke on the metaphorical nature of mathematicalmodels (Carreira, 2001) and developed the "psychological argument" for learningapplications and modelling proposed by Blum (1991). She argues that "mathematicalmodels and applied problems are a powerful context for the emergence of metaphoricalthinking." And John Mason (Open University, UK) did some meta-modelling (Mason,200 I). He used the mechanical idea of centre ofgravity as a metaphor for the focus ofattention when modelling is performed. Where is the student's centre of gravity or focusof serious attention when he or she is modelling? He writes, "exposing students tomathematical modelling is not just a form of teaching applications of mathematics, nor ofilluminating the mathematics being applied. It is equipping students with the power toexercise a fundamental duty." The way oflife idea again!

There was a full social programme including an excursion to Sintra to see the buildingthat for many centuries served as the summer palace of the Portuguese royal family.

And so to China in 200 I! Particularly during the 1990s, China had opened up to theWest. There was much more coming and going, and Chinese colleagues interested inteaching modelling had been attending recent ICTMAs. These included Qi-Xiao Ye

��

264 Houston

(Beijing Institute of Technology) who was appointed to organise ICTMA 10 and QiYuan Jiang (Tsinghua University, Beijing) who was also on the local organisingcommittee for ICTMA 10. There were over 150 participants, with the vast majority fromthe host nation, but only about 40 or so of the "regulars" attending. But it was awonderful experience to meet so many new teachers of modelling, so eager to tell theworld what they were doing. And there is much going on in China. Both secondary andtertiary institutions were teaching modelling, and a number of case studies are included inthe book of the conference, which, as I write in February 2003, has just gone to EllisHorwood for publishing (Ye, Blum, Houston and Jiang, 2003).

The academic programme included five plenaries. Gabriele Kaiser (University ofHamburg) spoke on "The Role of Figurative Context in Realistic Tasks" (Busse andKaiser, 2003). Zhang Jingzhong (Chinese Academy of Sciences) spoke on "How toTeach Mathematics by Using the 'Intelligent Platform for Education.'" Bai Fengshan(Tsinghua University, Beijing) spoke on "Reforming University MathematicsEducation." Avner Friedman (University of Minnesota) spoke on "IndustrialMathematics: Course Development." And Sue Lamon (Marquette University) not onlyspoke on "Modelling Rates of Change: Assessing Student Progress in DifferentialCalculus" but also invited us all to attend ICTMA 11 in 2003 at her University inMilwaukee.

The social programme included a Chinese banquet at the Summer Palace and anexcursion to the Great Wall.

THE INTERNATIONAL EXECUTIVE COMMITTEE

Before we complete our journey, we should discuss the changes that have occurred in ourorganization. In the beginning was the organising committee for ICTMA I and 2. Thencame ICTMA 3 and the tradition began to include the chief organiser of each conferencein the executive committee. Thus it grew and grew. Somehow Ian Huntley remained theperpetual chair of the committee. His principal task was to determine the venue and chieforganiser of the next conference. At each ICTMA, the venue for the next conference waschosen from the sites that had bid for it. Apart from giving advice to organisers, that wasall this committee did. It did not exercise any power to influence the programme at aconference; the design of the programme was entirely at the discretion of the localorganising committee.

By 1998 the executive committee was beginning to recognise that it was not a terriblydemocratic body. We wrote a constitution and established rules for membership andvoting. We recognized the growing number of women members and elected the firstwomen to serve on the executive committee. We held elections for President and for theexecutive committee members. The three conference organisers, those of the lastconference, the current one and next one, would remain on the committee ex officio. Ihad the honour to be elected president in 1999 and to serve in that capacity until 2003.We had moved from a (fairly) benign dictatorship to a democracy!

��

ICTMA: The First 20 Years

TWENTY YEARS TO ICTMA 11

265

We are almost there! I am looking forward to American hospitality and an excellentacademic programme as ICTMA 11 convenes in Milwaukee, Wisconsin. This is the firsttime that the book will be published before the conference. It is the first time that we canfully register for the conference on line. Let it not be said that we cannot change with thetimes!

Reflecting on this twenty-year journey, I can see that it is undoubtedly the case thatmodeIling and applications has caught the imagination of teachers and curriculumdevelopers all over the world. Many courses have been written and re-written; manyassessment schemes have been devised, evaluated and changed. Many books and articleshave been published-and not just the twelve emanating from the ICTMA series. (Bythe way, these books are tremendously fertile ground for many classes of people. Theyare a good source of teaching ideas and resources for teachers; there are many researchpapers to keep educational researchers busy and an educational historian will be able totrace the development of applications and modelling through this twenty-year period.)

Applications and Modelling has been represented on the agenda of the ICME conferencessince ICME 6 in Hungary in 1988 (Blum, Niss and Huntley, 1989). Now, an ICMEStudy is underway, with the Study Conference planned for February 2004 in Dortmund,Germany. Werner Blum is the chair of the International Programme Committee. Muchprogress has been made on many fronts, not the least of which includes winning overmany hearts and minds. But, as the Preacher once said, "What has been will be, what hasbeen done is what will be done; and there is nothing new under the sun." (Ecclesiastes, I,9, The Bible, RSV) Our themes recur-eurriculum, teaching, learning, group work,assessment, and so on. But they are evolving themes and there are new things under thesun-new ideas, new understanding, new pupils, new teachers. Mankind's quest forunderstanding and beauty will never be finished, and our community, the Community ofTeachers of Mathematical Modelling and Applications, will continue to seek innovativeways of teaching our subject to the next generation and will continue to seekunderstanding of how our pupils learn.

Some of us have been in this movement for a long time. After all, 20 years is half of anacademic's working life. Two of us remain who have attended all of the ICTMAconferences so far-Chris Haines and Ken Houston. But only one of us has presentedwork at all eleven conferences, and that is Chris Haines. His lifetime achievement awardhas already been given- the job of organizing ICTMA 12 in 2005 in London. Be there!

REFERENCES

ATM (1978) Mathematics for sixth formers Derby: Association of Teachers ofMathematics.

Berry JS, Burghes DN, Huntley 10, James DJG, Moscardini AO (Eds) (1987)Mathematical modelling courses Chichester: Ellis Horwood.

Blum W (1991) 'Applications and modelling in mathematics teaching: A review ofarguments and instructional aspects' in Niss M, Blum W, Huntley I (Eds)

��

266 Houston

Teaching ofmathematical modelling and applications Chichester: Ellis Horwood,10-29.

Blum W (1995) 'Applications and modelling in mathematics education: Some importantaspects of practice and research' in Sloyer C, Blum W, Huntley 10 (Eds) Advancesand perspectives in the teaching of mathematical modelling and applicationsYorklyn, Delaware: Water Street Mathematics, 1-20.

Blum W, Niss M, Huntley I (1989) Modelling. applications and applied problem solvingChichester: Ellis Horwood.

Burton L (1997) 'The assessment factor - by whom for whom, when and why' inHouston SK, Blum, W, Huntley 10, Neill, NT (Eds) Teaching and learningmathematical modelling Chichester: Albion Publishing, 95-107.

Busse A, Kaiser G (2003) 'Context in application and modelling - an empiricalapproach' in Ye Q, Blum W, Houston SK, Jiang Q (Eds) Mathematical modellingin education and culture: ICTMA IO Horwood Publishing: Chichester, 3-15.

Campbel1 SR (2001) 'Enacting possible worlds: Making sense of (human) nature' inMatos JF, Blum W, Houston SK, Carreira SP (Eds) Modelling and mathematicseducation ICTMA 9: Applications in science and technology Chichester: HorwoodPublishing, 3-14.

Carreira S (200 I) 'The mountain is the utility - On the metaphorical nature ofmathematical models' in Matos JF, Blum W, Houston SK, Carreira SP (Eds)Modelling and mathematics education ICTMA 9: Applications in science andtechnology Chichester: Horwood Publishing, 15-29.

Challis N, Gretton H, Houston K, Neill N (2002) 'Developing transferable skills:Preparation for employment' in Khan P, Kyle J (Eds) Effective learning andteaching in mathematics and its applications London: Kogan Page, 79-91.

Cockcroft WH (1982) Mathematics counts. Report of the committee of inquiry into theteaching ofmathematics in schools, London: Her Majesty's Stationery Office.

Coxhead C (1997) 'Curriculum development and assessment in Northern Ireland' inHouston SK, Blum, W, Huntley 10, Neill, NT (Eds) Teaching and learningmathematical modelling Chichester: Albion Publishing, 3-22.

Davis PJ (1991) 'Applied mathematics as a social instrument' in Niss M, Blum W,Huntley 10 (Eds) Teaching of mathematical modelling and applicationsChichester: Ellis Horwood, 1-9.

Fitzpatrick M, Houston SK (1987) 'Mathematical modelling in further mathematics' inBerry JS, Burghes DN, Huntley 10, James DJG, Moscardini AO (Eds)Mathematical modelling courses Chichester: Ellis Horwood, 188-208.

Galbraith P (1995) 'Modelling, teaching, reflecting: What [ have learned' in Sloyer C,Blum W, Huntley 10 (Eds) Advances and perspectives in the teaching ofmathematical modelling and applications Yorklyn, Delaware: Water StreetMathematics, 21-45

Galbraith P, Blum W, Booker G, Huntley 10 (1998) (Eds) Mathematical modelling:Teaching and assessment in a technology-rich world Chichester: HorwoodPublishing.

Houston SK, Blum, W, Huntley 10, Neill, NT (1997) (Eds) Teaching and learningmathematical modelling Chichester: Albion Publishing

��


Huntley I (1995) 'Modelling: A UK perspective' in Sloyer C, Blum W, Huntley ID (Eds)Advances and perspectives in the teaching of mathematical modelling andapplications Yorklyn, Delaware: Water Street Mathematics, 47-58.

James DJG, McDonald JJ (1981) Case studies in mathematical modelling Cheltenham:Stanley Thomes.

Julie C (1993) 'People's mathematics and the applications of mathematics' in de Lange J,Keitel C, Huntley ID, Niss M (Eds) Innovation in maths education by modellingand applications Chichester: Ellis Horwood, 31-40.

Keitel C (1993) 'Implicit mathematical models in social practice and explicitmathematics teaching by applications' in de Lange J, Keitel C, Huntley ID, Niss M(Eds) Innovation in maths education by modelling and applications Chichester:Ellis Horwood, 19-30.

Lamon SJ (1997) 'Mathematical modelling and the way the mind works' in Houston SK,Blum, W, Huntley ID, Neill, NT (Eds) Teaching and learning mathematicalmodelling Chichester: Albion Publishing, 23-37.

McClean S (1997) 'Modelling patient flow through hospitals' in Houston SK, Blum, W,Huntley ID, Neill, NT (Eds) Teaching and learning mathematical modellingChichester: Albion Publishing, 331-341.

MeLone RR (1973) The training of mathematicians Social Sciences Research CouncilReport, London: SSRC.

Mason J (2001) 'Modelling modelling: Where is the centre of gravity of-for-whenteaching modelling' in Matos JF, Blum W, Houston SK, Carreira SP (Eds)Modelling and mathematics education ICTMA 9: Applications in science andtechnology Chichester: Horwood Publishing, 39-61.

Mathematical Association of America, Committee on the Undergraduate Program inMathematics (1981) Recommendations for a general mathematical sciencesprogram Washington DC: Mathematical Association of America, 13.

Niss M (1989) 'Aims and scope of applications and modelling in mathematics curricula'in Blum W, Berry JS, Biehler R, Huntley ID, Kaiser-Messmer G, Profke L (Eds)Applications and modelling in learning and teaching mathematics Chichester:Ellis Horwood, 22-31.

SIAM (1995) The SIAM Report on Mathematics in Industry,<http://www.siam.org/mii/miihome.htm> (accessed 28 January 2003).

The Spode Group (1981) Solving real problems with mathematics, Volume 1 Cranfield:Cranfield Press.

The Spode Group (1982) Solving real problems with mathematics, Volume 2 Cranfield:Cranfield Press.

The Spode Group (1983) Solving real problems with CSE mathematics Cranfield:Cranfield Press.

Ye Q, Blum W, Houston SK, Jiang Q (2003) (Eds) Mathematical modelling in educationand culture: ICTMA 10 Horwood Publishing: Chichester.

��

mathematical modelling.pdf

Documents