Wim Van Dooren, Dirk De Bock, and Lieven Verschaffel

ICTMA 15, Melbourne, 14-19 July 2011

How Students Connect Descriptionsof Realistic Situations

to Mathematical Modelsin Different Representational Modes

Introduction

Verschaffel, Greer, and De Corte, 2000

Key step in a mathematical modelling cycle

Students’ situation model

Mathematicalmodel

But ... far from an obvious one!

Introduction

Ellen and Kim are running around a track. They run equally fast, but Ellen started later.

When Ellen has run 5 rounds, Kim has run 15 rounds. When Ellen has run 30 rounds, how many rounds has

Kim run?

0

20

40

60

80

100

3rd 4th 5th 6th 7th 8th

correct

proportional

Introduction

(Van Dooren et al., 2005)

With 8 matches, I can make a ladder with 2 rungs like this.

How many matches are needed to make a ladder with 6 rungs?

Stacey (1989)

2

6 24

83 3

Introduction

We focus on two elements that can hinder that transition

Introduction

• Students’ overreliance on the linear model (even when they have already met several other kinds of models)

• Students’ lack of representational fluency (i.e. their unability to switch between representational modes of a function)

Introduction

Our group in Leuven has a long research tradition in studying the “illusion of linearity” in various areas of mathematics

“Explanatory factors were found in (1) the intuitive, heuristic nature of the linear model, (2) students’ experiences in the mathematics classroom, and (3) elements related to the mathematical particularities of the problem situation in which the linear error occurs” 2007

Introduction

The math education literature emphasizes the (stimulating) role of “multiple” representations

(NCTM, 1989, 2000)

“Different representations of problems serve as different lenses through which students interpret the problems and the solutions. If students are to become mathematically powerful, they must be flexible enough to approach situations in a variety of ways and recognize the relationships among different points of view”

(NCTM, 1989)

Students’ (lack of) representational fluency was the main focus of several recent empirical studies

Introduction

In this study we systematically insert the representational aspect in our “linearity” line of research 2010

We focus on the relation between models and representations

Introduction

E.g. Graphical environments seem to be more likely to evoke linear patterns (through the origin)

• Sketch the graph of the height of a person from birth to the age of 30

Leinhardt, Zaslavsky, and Stein (1990)

• When students were asked to draw a graph of a function that passes through two given points, they typically drew straight lines

Markovits, Eylon, and Bruckhaimer (1986)

Introduction

Research questions

• How accurate are students in connecting descriptions of realistic situations to linear and “almost linear” models?

• Does accuracy and model confusion depend on the representational mode in which a function is given?

Method

• 64 participants (first year Educational Sciences)

• Written multiple-choice test

12 verbal descriptions of realistic situations

Task: connect each situation with appropriate

mathematical model

4 types of models

Linear (y = ax)

Inverse linear (y = a/x)

Affine with positive slope (y = ax + b with a > 0)

Affine with negative slope (y = ax + b wit a < 0)

Method

• 64 participants (first year Educational Sciences)

• Written multiple-choice test

12 verbal descriptions of realistic situations

Task: connect each situation with appropriate

mathematical model

4 types of models

These models were given either in a graphical,

tabular or formula form (each representation was

given in one third of the cases)

Method

Example 1

During the war, butter was rationed. Each week butter was delivered and fairly shared amoung the people. Which formula properly represents the relation between the number of people who wants butter and the amount of butter everybody receives?

y = 150 x

y = 150/x

y = 150 x + 30

y = -150 x + 30

Method

Example 2

Jennifer buys minced meat at the butcher's shop. Which table properly represents the relation between the amount of minced meat that Jennifer buys and the price she has to pay?

x y

0 8

1 -4

2 -16

3 -28

4 -40

x y

0 0

1 12

2 24

3 36

4 48

x y

0

1 12

2 6

3 4

4 3

x y

0 8

1 20

2 32

3 44

4 56

Method

Example 3

A chemical concern has a big citern with hydrochloric. This morning they started to pump with a constant pace all hydrochloric out of this citern. Which graph properly represents the relation between the time elapsed and the amount of hydrochloric that is still in the citern?

Method

Analysis

Data were analysed by a repeated measures logistic regression analysis followed by multiple pairwise comparisons

Results

Accuracy

Main effect of model:

Linear (94% correct matches) > inverse linear (77%) = affine with negative slope (70%) = affine with positive slope (67%)

No main effect of representation:

Graph (83% correct matches)= Table (75%)= Formula (74%)

Results

Accuracy

Model × representation interaction effect (!):

Formula Table Graph

y = ax 89 95 98 Formula < table and graph

y = a/x 92 69 70 Formula > table and graph

y = ax+b (a < 0)

48 81 81 Formula < table and graph

y = ax+b(a > 0)

66 55 81 Formula and table < graph

• Percentages of correct matches• Best result(s) in bold

Results

Accuracy

• Graph is best representation in all cases, except for inverse linear relationships. For that kind of relations: the formula is more supportive

Straight line or “equal distances” stereotype?

• For linear relations: all representations are quite good

• Formula seems to be misleading for affine relationsSituations were described in “y = a ± bx” order, while formulas were given in “y = ax + b” form?

Results

Accuracy

• Students can interpret all representations (correct matches for all representations between 74% and 83%, no main effect of representation),

• they also can detect underlying mathematical models (more than 80% of the students detect the underlying model in at least one of the three representations), but

• some representations support the underlying model, while others put them on the wrong track

To get a better understanding of these findings, an error analysis was done

Results

Error analysis

Situations with underlying y = a/x model

Linear errors were frequently made (15%), especially in the tabular (27%) and graphical (14%) representational mode

Finding confirms results of previous studies on the “predominance of the linear model” for an “almost linear model”. Students are most likely misled by characteristics of representions of linear model (straight line and equal distances)

Results

Error analysis

Situations with underlying y = ax + b (a < 0) model

Inverse linear errors are most frequent (15%), especially in the formula representational mode (23%)

Both models are decreasing. For many students, the independent variable in the denominator is more appealing than the negative sign in the numerator

Attractivity of “doubling/halving” prototype in situations of decrease

Results

Error analysis

Situations with underlying y = ax + b (a > 0) model

Linear errors are most frequent (30%), especially in the formula (31%) and in the tabular representational mode (41%)

This model comes closest to the linear model. In the graphical mode one can see the Y-intercept (which is more difficult for the other representations)

Conclusion

Conclusions and discussion

• Study confirms “default” role of the linear model

• Linear model and various “almost linear” models are confused

• This confusion is representation-dependent: In some representations, particular aspects of non-linearity are more easily noticed than in other representations

Conclusion

Educational implication

• Mathematics education should highlight representations, explicitly discuss differences between linear and “almost linear” models (e.g. by using “elementary modelling tasks” as in the current study)

• ...