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Pupils’ over-use of proportionality
How numbers may change solutions …
Dirk De Bock
Wim Van Dooren
Marleen Evers
Lieven Verschaffel
Center for Instructional Psychology and Technology
Catholic University of Leuven
Belgium
SECONDARY SCHOOL PUPILS’
OUTLINE
ILLUSION OF LINEARITY
Introduction
 « over-use of proportionality »?
Empirical and theoretical background
 previous study on arithmetic problems
 how numbers may change solutions
A new empirical study
 method
 main results
Conclusions and discussion
Over-use of proportionality
Students’ tendency to treat every
numerical relation between numbers as if
it were linear (or proportional)
(see, e.g., Freudenthal, 1973; Rouche, 1989; …)
Examples
Stacey (ESM, 1989):
To make a ladder with 2 rungs,
I need 8 matches.
How many matches do I need
to make a ladder with 10 rungs?
 Most frequent error: 40 matches
Examples
Aristotle:
“An object which is 10 times as heavy as another
object, will reach the ground 10 times as fast as that
other object”
SECONDARY SCHOOL PUPILS’
OUTLINE
ILLUSION OF LINEARITY
Introduction
 « over-use of proportionality »?
Empirical and theoretical background
 previous study on arithmetic problems
 how numbers may change solutions
A new empirical study
 method
 main results
Conclusions and discussion
Previous study
• Missing-value arithmetic problems
• Evolution of over-use of proportionality from 3rd to
8th grade
 already present in 3rd grade
 considerable increase until 6th grade
(~ emerging proportional reasoning skills)
 decrease afterwards
(cf. Van Dooren et al., Cognition and Instruction, 2005)
Previous study
“CONSTANT” PROBLEM
A group of 5 musicians plays a piece of music in 10 minutes.
Another group of 35 musicians will play the same piece of music.
How long will it take this group to play it?
100
80
60
correct
40
proportional
20
0
3rd
4th
5th
6th
7th
8th
Previous study
“ADDITIVE” PROBLEM
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
3th
4th
5th
6th
7th
8th
Previous study
“AFFINE” PROBLEM f(x) = ax + b
The locomotive of a train is 12 m long.
If there are 4 carriages connected to the locomotive, the train is 52
m long.
If there would be 8 carriages behind the locomotive, how long
would the train be?
100
80
60
correct
40
proportional
20
0
3th
4th
5th
6th
7th
8th
But …
Numbers may change solutions!
Literature on (acquisition of) proportional reasoning
 Frequently reported error: additive reasoning
 Esp. with « messy » numbers/non-integer ratios
aamixture
mixturewith
with21
20kg
kgsugar
sugarfor
for95
100
l water
l water
tastestastes
equally
equally
sweetsweet
as as
aamixture
mixturewith
with23
60kg
kgsugar
sugarfor
for97
140
l water
l water
20 + 40
100 + 40
(e.g., Noelting, 1980, Hart, 1984, Karplus, Pulos, & Stage, 1983)
So numbers change solutions!
• Integer ratios facilitate proportional reasoning to
proportional problems
• What with proportional reasoning to NONproportional problems?
 Earlier studies: always integer ratios
 « It remains a question for further research
whether an approach with non-seductive
numbers will prevent children from making the
multiplication error »
(Linchevski et al., 1998)
SECONDARY SCHOOL PUPILS’
OUTLINE
ILLUSION OF LINEARITY
Introduction
 « over-use of proportionality »?
Empirical and theoretical background
 previous study on arithmetic problems
 how numbers may change solutions
A new empirical study
 method
 main results
Conclusions and discussion
Method
508 students, 4th, 5th and 6th grade
Test with 8 word problems:
Non-proportional
Proportional
2
Additive
2
Constant
2
Affine
2
Nature of numbers in problems was manipulated
Method
Manipulation of numbers:
Ellen and Kim are running around a track. They run equally
fast, but Ellen started later.
When Ellen has run 16 rounds, Kim has run 32 rounds.
When Ellen has run 48 rounds, how many rounds has Kim
run?
16
x2
32
x3
II-version
48
?
Method
Manipulation of numbers:
Ellen and Kim are running around a track. They run equally
fast, but Ellen started later.
When Ellen has run 16 rounds, Kim has run 24 rounds.
When Ellen has run 36 rounds, how many rounds has Kim
run?
16
x2.25
36
x1.5
24
NN-version
?
 One version at random for each student
Results
PROPORTIONAL PROBLEMS
100
100
80
80
60
60
40
40
20
20
0
General
4th
5th
II version
6th
0
General
4th
5th
NN version
6th
Results
ADDITIVE PROBLEMS
100
100
80
80
60
60
40
40
100
20
20
50
0
General
4th
0
5th
6th
General
correct
II version
0
General
4th
5th 6th
proportional
4th
5th
other errors
NN version
6th
Results
CONSTANT PROBLEMS
100
100
80
80
60
60
40
40
100
20
20
50
0
General
4th
0
5th
6th
General
correct
II version
0
General
4th
5th 6th
proportional
4th
5th
other errors
NN version
6th
Results
AFFINE PROBLEMS
100
100
80
80
60
60
40
40
100
20
20
50
0
General
4th
0
5th
6th
General
correct
II version
0
General
4th
5th 6th
proportional
4th
5th
other errors
NN version
6th
SECONDARY SCHOOL PUPILS’
OUTLINE
ILLUSION OF LINEARITY
Introduction
 « over-use of proportionality »?
Empirical and theoretical background
 previous study on arithmetic problems
 how numbers may change solutions
A new empirical study
 method
 main results
Conclusions and discussion
Conclusions
Hypotheses generally confirmed
Proportional problems
- Integer ratios facilitate correct reasoning
 confirmation of other findings in literature
- Older students less “hindered” by non-integer ratios
Conclusions
Hypotheses generally confirmed
Non-proportional problems
- Integer ratios also facilitate over-use of proportional
reasoning
- Additive problems:
 non-integer ratios cause more correct answers
- Other non-proportional problems:
 non-integer ratios cause more other errors
- Older students ‘benefit’/‘suffer’ less from non-integer
ratios
Discussion
Theoretical implication
 Not only key words or problem formulations
 Also the (combination of) numbers in a problem
can be associated with a solution method
 Association interacts with students’ prior
knowledge
Discussion
Methodological implication
 Assessing over-use of proportionality using
items with integer ratios may strengthen the
effect
 Only / especially for younger students
Discussion
Practical implication
 During classroom teaching of proportionality:
 Explicitly discuss (validity of) criteria students
choose to apply proportional methods
 Take care to use variety of examples, not sharing
same superficial task characteristics