How Students Connect Descriptions
of Realistic Situations
to Mathematical Models
in Different Representational Modes
Wim Van Dooren, Dirk De Bock, and Lieven Verschaffel
ICTMA 15, Melbourne, 14-19 July 2011
Introduction
Phenomenon
under
investigation
Understanding
Situation
model
Modelling
Communication
Mathematical
analysis
Evaluation
Report
Interpreted
results
Mathematical
model
Interpretation
Derivations
from
model
Verschaffel, Greer, and De Corte, 2000
Introduction
Students’
situation model
Mathematical
model
Key step in a mathematical modelling cycle
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?
100
80
60
correct
40
proportional
20
0
(Van Dooren et al., 2005)
3rd
4th
5th
6th
7th
8th
Introduction
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?
2
3
6
8
3
24
Stacey (1989)
Introduction
We focus on two elements that can hinder
that transition
• 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)
Introduction
Students’ (lack of)
representational
fluency was the
main focus of
several recent
empirical studies
In this study we
systematically insert
the representational
aspect in our
“linearity” line of
research
2010
Introduction
We focus on the relation between models and
representations
E.g. Graphical environments seem to be more likely
to evoke linear patterns (through the origin)
• 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)
• Sketch the graph of the
height of a person from
birth to the age of 30
Leinhardt, Zaslavsky, and Stein (1990)
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
x
y
x
0
8
0
0
0
1
-4
1
12
1
2
-16
2
24
3
-28
3
4
-40
4
y
x
y
0
8
12
1
20
2
6
2
32
36
3
4
3
44
48
4
3
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?
50000
9000
45000
40000
8000
90000
80000
7000
35000
30000
5000
20000
15000
4000
1
2
3
4
40000
40000
30000
20000
20000
1000
0
50000
30000
2000
0
50000
60000
3000
10000
5000
60000
70000
6000
25000
70000
10000
10000
5 0
0
1
2
3
4
5
0
0
0
1
2
3
4
5
0
1
2
3
4
5
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
• For linear relations: all representations are quite
good
• 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?
• Formula seems to be misleading for affine relations
Situations 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 nonlinearity 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)
• ...