problem solving by analogy / problem solving as analogy
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____ THE ______ MATHEMATICS____
_________ EDUCATOR_____ Volume 17 Number 2
Fall 2007 MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA
Editorial Staff Editors Kelly W. Edenfield Ryan Fox Associate Editors Tonya Brooks Allyson Hallman Soo-Jin Lee Diana May Kyle Schultz Susan Sexton Catherine Ulrich Advisor Dorothy Y. White
MESA Officers 2007-2008 President Rachael Brown
Vice-President Nick Cluster
Secretary Kelly Edenfield
Treasurer Susan Sexton
Colloquium Chair Dana TeCroney
NCTM Representative Kyle Schultz
Undergraduate Representative Dan Davis Kelli Parker
A Note from the Editor Dear TME Reader, Along with my co-editor, Ryan Fox, and the rest of the editorial team, I am happy to present a new issue of The Mathematics Educator, the second and final issue of Volume 17. In it, I hope you will find pieces that are educational, thought provoking, and promote continued dialogue on issues of interest to the mathematics education community.
As is a tradition of TME, this issue presents a variety of viewpoints on topics in mathematics education. However, the first half of this issue does have a common focus: problem solving. This focus on problem solving is apropos, especially in the state of Georgia as our K-12 mathematics curriculum is in the process of changing to a performance standards based curriculum that challenges both students and teachers thinking about mathematics and what it means to do mathematics. Steve Benson, in a guest editorial, starts off by presenting his view that mathematics is learned through problem solving and continues by sharing teaching methods that have helped his students become better problem solvers. In his theoretical analysis, Jamin Carson argues for the importance of a knowledge base and transferability of knowledge in the teaching of problem solving. Jos Contreras completes the problem solving section by presenting his problem-posing framework, a framework for guiding the modification of given problems to create other interesting, worthwhile mathematics problems. The final three pieces provide glimpses into two research studies and into the Turkish teacher education curriculum. In the first, Brian Evans examines the attitudes, conceptions, and achievement of college students enrolled in undergraduate statistics courses, revealing some unexpected findings. Janet Frost and Lynda Wiest continue by studying how a girls mathematics and technology camp affected the confidence of the participants, paying particular attention to girls of color and those of low socioeconomic status. The final article by Mine Isiksal, Yusuf Koc, Sarure Bulut, and Tulay Atay-Turhan outlines the new elementary mathematics teacher education in Turkey, including the motivations for the changes and the present status of the revisions.
The production of TME requires the help and support of numerous groups of people. As a student-run and student-produced journal, TME is responsible for recruiting and training reviewers, associate editors, and editors, primarily from the ranks of present graduate students. I greatly appreciate the time and energy put forth by these members of our staff. I would like to thank all the reviewers, authors, and faculty members whose work and advice have enabled us to present the readers with this issue. In particular, I extend my gratitude to our training seminar participants and to the editorial staff for all of their hard work.
Kelly W. Edenfield 105 Aderhold Hall firstname.lastname@example.org The University of Georgia www.coe.uga.edu/tme Athens, GA 30602-7124
About the Cover This issues cover features two GSP sketches from the article by Jos Contreras. Dr. Contreras invites us into his problem-posing framework through an example; he takes a straightforward problem about isosceles triangles and poses additional interesting problems about triangles along with other geometric shapes.
This publication is supported by the College of Education at The University of Georgia
____________THE________________ ___________ MATHEMATICS________
______________ EDUCATOR ____________
An Official Publication of The Mathematics Education Student Association
The University of Georgia Fall 2007 Volume 17 Number 2
Table of Contents
2 Guest Editorial Problem Solving by Analogy / Problem Solving as Analogy STEVE BENSON
7 A Problem With Problem Solving: Teaching Thinking Without Teaching
Knowledge JAMIN CARSON 15 Unraveling the Mystery of the Origin of Mathematical Problems: Using a
Problem-Posing Framework With Prospective Mathematics Teachers JOSE CONTRERAS 24 Student Attitudes, Conceptions, and Achievement in Introductory
Undergraduate College Statistics BRIAN EVANS
31 Listening to the Girls: Participant Perceptions of the Confidence-Boosting
Aspects of a Girls Summer Mathematics and Technology Camp JANET HART FROST & LYNDA R. WIEST
41 An Analysis of the New Elementary Mathematics Teacher Education Curriculum
in Turkey MINE ISIKSAL, YUSUF KOC, SAFURE BULUT, & TULAY ATAY-TURHAN 52 Submissions information 53 Subscription form
2007 Mathematics Education Student Association
All Rights Reserved
The Mathematics Educator 2007, Vol. 17, No. 2, 26
2 Problem Solving and Analogy
Guest Editorial Problem Solving by Analogy / Problem Solving as Analogy
Everyone talks about how important it is for a young quarterback to sit on the bench and watch the game. But instead of learning how to play, all they learn is how to sit and watch. (A paraphrase of Mike Ditka on ESPNs Sunday NFL Countdown, November 18, 2007)
If current frameworks, standards, and assessments are any indication, there is international consensus that students should be able to solve new (to them) mathematical problems (real-world and otherwise) in addition to knowing specific facts and performing basic calculations. Problem solving as a part of the mathematics curriculum has gone in and out of favor for several decades, perhaps due to the range of ways it has been approached in textbooks and classrooms. Too often, problem solving is taught very algorithmically and, as mentioned in another article in this issue, is seen as independent of mathematical content. In fact, I believe that mathematics is learned through problem solving, so when taught well, mathematical content and problem solving cant really be separated.
A number of problem solving habits of mind are taught explicitly in mathematics courses at all levels. Many of these ways of thinking can be traced to suggestions from How to Solve It and other publications by the father of modern problem solving, George Plya (1945, 1954). (In fact, in Volume I of his Mathematics and Plausible Reasoning, he wrote extensively about analogy in mathematics.) I wont restate these suggestions here since most have become part of the present day mathematical lexicon, but I would like to present some methods and ideas that I have found promising in helping my students become more successful problem solvers.
After spending 7 years at Education Development Center in Newton, MA, creating and facilitating content-based professional development materials, Steve Benson is now an associate professor of mathematics at Lesley University in Cambridge, MA. He earned his Ph.D. at the University of Illinois and has taught at St. Olaf College, Santa Clara University, University of New Hampshire, and University of WisconsinOshkosh. He has been a co-Director of the Master of Science for Teachers program in the UNH mathematics department since 1997 and was lead author on "Ways to Think About Mathematics: Activities and Investigations for Grade 6-12 Teachers," published by Corwin Press.
Analogy in Problem Solving A common trait of expert problem solvers is their
ability to recognize connections between two or more problem situations and their solution methods. That is, by re-posing a problem in another context, the problem is often made more tractable.
The Handshake Problem: The twenty members of the math club met last Tuesday to plan next months annual banquet. A tradition of the club is to start each meeting by having the members shake hands with each other. How many handshakes will occur? There are two common strategies students (and others) use to approach this problem. The first is motivated by the observation that we could arrange the club members in some order (from first to twentieth). The counting of the handshakes usually starts something like this: The first member shakes hands with each of the other 19 members, while the second member shakes hands with 18 others, the third members shakes hands with 17 people, and so on, until the 19th person shakes one persons hands and the 20th person doesnt shake any. Therefore, the total number of handshakes is
19 + 18 + 17 + + 2 + 1 = 190. (Of course, the 2nd member is involved in a total of 19 handshakes, like all the others, but the handshake with the first member has already been counted, so it would be more correct to say that the 2nd member was involved in 18 more handshakes, the 3rd member (who had already shaken hands with members 1 and 2) was involved in 17 more handshakes, and so on.) The second strategy usually goes something like the following: Each of the 20 math club members would have shaken the hands of the other 19 members, for a total of 2019 = 380 hands being shaken. But each handshake requires two hands (or each handshake gets counted twice), so there are 380/2 = 190 handshakes in all.
Steve Benson 3
Later, when asked to determine a closed form solution for 1 + 2 + 3 + + n (determining the