Proportional/linear reasoning
Geneva, October 29 2015
Lieven Verschaffel
Dirk De Bock
Tine Degrande
Stephanie Lem
Wim Van Dooren
SECONDARY SCHOOL PUPILS’
ILLUSION OF LINEARITY
Intro
Let’s take a test!
Problem 1
This word problem is a Greek one. Try to fill in a number on the dotted line.
Problem: Ada kalka porelantora liktoun kottor.Noverganica tinestari 4 posstor
io chion anpera ton vorcon 16 staton estano tuv magcaneto.Probalenti mogronates
8 ogront o gnoston kalkono tot lindenan, nag kij nisvork sckrinon lopenado maorn
eweinst?
Answer: Gelomal lopandora rit ..................... nifj toto.
Problem 2
• 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 10 rounds, how many rounds
has Kim run?
Problem 3
• Carl needs 8 bags of seed to put grass on a square
park with sides of 50m. How many bags of grass
seed will he approximately need for a square park
with sides of 100m?
Problem 4
• John’s best time to run 100 meters is 17 seconds.
How long will it take him to run 1 km?
SECONDARY SCHOOL PUPILS’
ILLUSION OF LINEARITY
Problem 5
Can you estimate the gnome’s
weight?
Problem 6
• Draw the evolution of a person’s height in relation
to his age
height
age
Content
1.
2.
3.
4.
5.
6.
7.
Proportionality/linearity: some terminology
Proportionality/linearity: representations
Proportionality/linearity: properties
Proportionality/linearity in the curriculum
Proportionality/linearity: solution strategies
Additive reasoning errors
The over-use of proportionality/linearity in
arithmetic word problem solving
8. The over-use of proportionality/linearity in
geometry
9. Proportional/linear reasoning in school vs real
life
Content
1.
2.
3.
4.
5.
6.
7.
Proportionality/linearity: some terminology
Proportionality/linearity: representations
Proportionality/linearity: properties
Proportionality/linearity in the curriculum
Proportionality/linearity: solution strategies
Additive reasoning errors
The over-use of proportionality/linearity in
arithmetic word problem solving
8. The over-use of proportionality/linearity in
geometry
9. Proportional reasoning in school and real life
1. Some terminology
Ratio = a fractional relation between two quantities
a
b
or
a:b
Examples:
•
•
•
•
“For this recipe, use five oranges for each liter of juice”
“In that school the ratio is one teacher to five students”
“The population grows with 1000 units each year”
“The cyclist had a velocity of approximately 40 km. / hour”
1. Some terminology
Proportion = a statement that two ratios are equal
Examples:
a
b
c
d
2 / 3 = 10 / 15
8 km in 15 min = 32 km / hr
1. Some terminology (Vergnaud,
1983)
• In proportional reasoning (at least) two “measure
spaces” are involved that are modelled by a linear
function, i.e. a function of the form f(x) = ax.
•
M1 M2
When she makes strawberry jam, my
grandmother uses 2 kg of sugar for 5
•
a
b
kg of strawberries. So, for 10 kg of
•
•
•
strawberries she needs 4 kg of sugar.
c
d
Strawberry weights
Sugar weights
1. Some terminology (Vergnaud,
1983)
• In proportional reasoning (at least) two “measure
spaces” are involved that are modelled by a linear
function, i.e. a function of the form f(x) = ax.
•
M1 M2
When she makes strawberry jam, my
grandmother uses 2 kg of sugar for 5
•
a
b
kg of strawberries. For 10 kg of
•
•
•
strawberries she needs 4 kg of sugar.
c
d
Strawberry weights
Sugar weights
1. Some terminology (Vergnaud,
1983)
• In proportional reasoning (at least) two “measure
spaces” are involved that are modelled by a linear
function, i.e. a function of the form f(x) = ax.
•
M1 M2
When she makes strawberry jam, my
grandmother uses 2 kg of sugar for 5
•
a
b
kg of strawberries. For 10 kg of
•
•
•
strawberries she needs 4 kg of sugar.
c
d
Strawberry weights
Sugar weights
1. Some terminology (Vergnaud,
1983)
• In proportional reasoning (at least) two “measure
spaces” are involved that are modelled by a linear
function, i.e. a function of the form f(x) = ax.
•
M1 M2
When she makes strawberry jam, my
grandmother uses 2 kg of sugar for 5
•
a
b
kg of strawberries. For 10 kg of
•
•
•
strawberries she needs 4 kg of sugar.
c
d
Strawberry weights
Sugar weights
1. Some terminology (Vergnaud,
1983)
• In proportional reasoning (at least) two “measure
spaces” are involved that are modelled by a linear
function, i.e. a function of the form f(x) = ax.
•
M1 M2
When she makes strawberry jam, my
grandmother uses 2 kg of sugar for 5
•
a
b
kg of strawberries. For 10 kg of
•
•
•
strawberries she needs 4 kg of sugar.
c
d
Strawberry weights
Sugar weights
1. Some terminology
Linear function = an infinite series of equal ratios
f(x) = ax
a
b
c
d
e
f
g
h
…
Examples:
• 0.25 litre of paint for 1 m2, 0,50 litre of paint for 2 m2, 0,75 litre of
paint for 3m2,…
• 1 km in 5 min, 3 km in 15 min, 12 km in 60 min,…
1. Some terminology
Linear function: a straight line through the origin
Paint (in
litre)
f(x) = 0.25x
2
Area (in m )
1. Some terminology
Linear function: a straight line through the origin
Paint (in
litre)
f(x) = 0.25x
0,25
1
2
Area (in m )
1. Some terminology
Linear function: a straight line through the origin
Paint (in
litre)
f(x) = 0.25x
0,50
0,25
1
2
2
Area (in m )
1. Some terminology
Linear function: a straight line through the origin
Paint (in
litre)
f(x) = 0.25x
0,75
0,50
0,25
1
2
3
2
Area (in m )
1. Some terminology
Missing-value (proportional) problem: problem in which a
missing value in one of two ratios needs to be found
a
b
c
?
Examples:
• 10 eggs weigh 600 gram. What is the weight of 80 eggs?
• 10 eggs weigh 600 gram. How many eggs when the total
weight is 1200 gram?
In Vergnaud’s (1983) terms…
10 eggs weigh 600 gram.
What is the weight of 80
eggs?
10 eggs weigh 600 gram.
How many eggs when
the total weight is 1200
gram?
M1
M2
M1
M2
(eggs)
(gram)
(eggs)
(gram)
10
600
10
600
80
?
?
1200
Two other types of proportional
problems
• Comparison problems: problems in which the
relationship between two ratios needs to be
determined
• E.g., which ratio is the smallest: 4/8 or 12/20?
• Transformation problems: problems in which two
ratios are given but values need to be adapted to
create two equivalent ratios
• E.g., What have you to do to make the second ratio
equal to the first one: 4/16 and 14/64
Content
1.
2.
3.
4.
5.
6.
7.
Proportionality/linearity: some terminology
Proportionality/linearity: representations
Proportionality/linearity: properties
Proportionality/linearity in the curriculum
Proportionality/linearity: solution strategies
Additive reasoning errors
The over-use of proportionality/linearity in
arithmetic word problem solving
8. The over-use of proportionality/linearity in
geometry
9. Proportional reasoning in school and real life
2. Proportionality/linearity: representations
Table:
Area (m2)
Amount of paint (l)
Graph:
Formula:
f(x) = 0.25 x
1
2
3
0.25
0.50
0.75
Representational fluence vs.
flexibility
Representational fluency: being able to switch
accurately and quickly between a table, a graph and a
formula
Representational flexibility: being able to adapt the
representational form to task, subject, and context
features
Content
1.
2.
3.
4.
5.
6.
7.
Proportionality/linearity: some terminology
Proportionality/linearity: representations
Proportionality/linearity: properties
Proportionality/linearity in the curriculum
Proportionality/linearity: solution strategies
Additive reasoning errors
The over-use of proportionality/linearity in
arithmetic word problem solving
8. The over-use of proportionality/linearity in
geometry
9. Proportional reasoning in school and real life
3. Some properties of
linear/proportional relations
Area (m2)
1
Amount of paint (l) 0.25
2
3
0.50
0.75
f(kx) = k f(x)
(e.g., 0,50 = 2 x 0,25)
3. Some properties of
linear/proportional relations
Area (m2)
1
Amount of paint
(l)
+
0.25 +
2
0.50
=
=
3
0.75
f(x + y) = f(x) + f(y)
(e.g., 0,75 = 0,25 + 0, 50)
3.Some properties of
linear/proportional relations
Area (m2)
Amount of paint
(l)
1
2
3
0.25
0.50
0.75
Same external ratio:
1
0.25
2
0.50
3. Some properties of
linear/proportional relations
Area (m2)
Amount of paint
(l)
1
2
3
0.25
0.50
0.75
Same internal ratio:
1
0.25
2
0.50
Content
1.
2.
3.
4.
5.
6.
7.
Proportionality/linearity: some terminology
Proportionality/linearity: representations
Proportionality/linearity: properties
Proportionality/linearity in the curriculum
Proportionality/linearity: solution strategies
Additive reasoning errors
The over-use of proportionality/linearity in
arithmetic word problem solving
8. The over-use of proportionality/linearity in
geometry
9. Proportional reasoning in school and real life
4. Proportional reasoning in the
curriculum
• Proportionality is among the most important topics
in the school math curriculum.
• Proportionality is the capstone of elementary
arithmetic, number and measurement concepts, but
at the same time one of the most elementary
understandings one needs for more advanced
mathematics (geometrical similarity, probability…),
and it is most useful for everyday life.
• The development of proportional reasoning is a
complex and multi-dimensional process that
progresses gradually over many years.
4. Proportional reasoning in the
curriculum
Grade 4-6
10 eggs weigh 600 gram. What is the weight of 20 eggs?
“k times A, k times B”
4. Proportional reasoning in the
curriculum
Grade 1-2
1 pineapple costs 3 euro. How much do 4 pineapples cost?
k times A, k times B
4. Proportional reasoning in the
curriculum
Grade 1-2
1 pineapple costs 3 euro. How much do 4 pineapples cost?
“k times A, k times B”
M1
M2
(# pineapples)
(euro)
1
3
4
?
Foundations laid in early years
• Learners at the beginning of primary school use
informal strategies (e.g. repeated addition) to deal
with these elementary multiplicative problems.
• The understanding and use of these informal
strategies is based on the fundamental idea of oneto-many correspondence.
• One-to-many correspondence = the first stepping
stone towards understanding proportionality
One-to-one vs. one-to-many
One-to-one
Pete has 3 apples.
Ann has 4 apples.
How many more?
One-to-many
There are 4 hourses. In
every house there are 3
cats. How many cats
altogether?
4. Proportional reasoning in the
curriculum
Grade 6 - university
Formalisation: linear functions (tables, graphs, formulas)
Numerous applications
Relation diameter – perimeter circle
Travelling time – travelled distance
Mass – volume of a substance
Falling speed – falling time of object
Linear models in statistics, calculus
,…
Content
1.
2.
3.
4.
5.
6.
7.
Proportionality/linearity: some terminology
Proportionality/linearity: representations
Proportionality/linearity: properties
Proportionality/linearity in the curriculum
Proportionality/linearity: solution strategies
Additive reasoning errors
The over-use of proportionality/linearity in
arithmetic word problem solving
8. The over-use of proportionality/linearity in
geometry
9. Proportional reasoning in school and real life
5. Solution strategies
• Example: When she makes strawberry jam, my
grandmother uses 3 kg of sugar for 6 kg of
strawberries. How much sugar does she need for
18 kg of strawberries?
M1
M2
(strawberry weight)
(suger weight)
6
18
3
?
5. Solution strategies
• Example: When she makes strawberry jam, my
grandmother uses 3 kg of sugar for 6 kg of
strawberries. How much sugar does she need for
18 kg of strawberries?
How do you solve the problem?
Do you know other ways to solve the problem?
Do you remember how you learnt to solve this kind
of problems at (elementary) school?
Repeated addition
This approach relies on the one-to-many
correspondence idea and comes down to the buildingup approach
Problem: When she makes
strawberry jam, my
grandmother uses 3 kg of
sugar for 6 kg of
strawberries. How much
sugar does she need for 18 kg
of strawberries?
Strategy: For the first 6 kg of
strawberries, I need 3 kg of
sugar. I also need 3 kg of
sugar for the next 6 kg of
strawberries, and another 3
kg for the last batch of 6 kg
of strawberries. Thus, I need
3 + 3 + 3 = 9 kg of sugar
Within-strategies (internal ratio;
scalar relation)
One determines the factor of change within one
measure space first, and then applies this factor to the
other measure space
Problem: When she makes
strawberry jam, my
grandmother uses 3 kg of
sugar for 6 kg of
strawberries. How much
sugar does she need for 18 kg
Strawberries
of strawberries?
(kg)
Sugar (kg)
Strategy: From 6 to 18 kg of
strawberries, I have to
multiply by 3, so I will also
multiply 3 kg of sugar by 3,
which is 9
6
12
18
24
30
36
3
6
9
12
15
18
Within-strategies (internal ratio;
scalar relation)
One determines the factor of change within one
measure space first, and then applies this factor to the
other measure space
Problem: When she makes
strawberry jam, my
grandmother uses 3 kg of
sugar for 6 kg of
strawberries. How much
sugar does she need for 18 kg
Strawberries
of strawberries?
(kg)
Sugar (kg)
Strategy: From 6 to 18 kg of
strawberries, I have to
multiply by 3, so I will also
multiply 3 kg of sugar by 3,
which is 9
6
12
18
24
30
36
3
6
9
12
15
18
Between-strategies (external ratio,
functional relation)
One searches the factor by which one has to multiply
(or divide) the strawberry weight in order to obtain
the sugar weight, and one applies this factor to the
second strawberry weight that is provided.
Problem: When she makes
strawberry jam, my
grandmother uses 3 kg of
sugar for 6 kg of
strawberries. How much
sugar does she need for 18 kg
of strawberries?
Strategy: The weight of sugar
is obtained by halving the
weight of strawberries, so for
18 kg of strawberries, one
needs 18/2 = 9 kg of sugar
Strawberri
es (kg)
Sugar (kg)
6
12
18
24
30
36
3
6
9
12
15
18
Between-strategies (external ratio,
functional relation)
One searches the factor by which one has to multiply
(or divide) the strawberry weight in order to obtain
the sugar weight, and one applies this factor to the
second strawberry weight that is provided.
Problem: When she makes
strawberry jam, my
grandmother uses 3 kg of
sugar for 6 kg of
strawberries. How much
sugar does she need for 18 kg
of strawberries?
Strategy: The weight of sugar
is obtained by halving the
weight of strawberries, so for
18 kg of strawberries, one
needs 18/2 = 9 kg of sugar
Strawberri
es (kg)
Sugar (kg)
6
12
18
24
30
36
3
6
9
12
15
18
Rule of three (unit ratio approach)
• Taking an intremediate step to find out the value
of the second measure space when the value of the
first measure space is 1.
Problem: When she
makes strawberry jam,
my grandmother uses 3
kg of sugar for 6 kg of
strawberries. How much
sugar does she need for
18 kg of strawberries?
Strategy:
1. To make 6 kg of stawberries,
grandma needs 3 kg of sugar
2. To make 1 kg of strawberries,
she needs 3 / 6 kg of sugar
3. To make 18 kg of
strawberries, grandma needs
(3 / 6) x 18 = 9 kg of sugar
Rule of three (unit ratio approach)
• Taking an intremediate step to find out the value
of the second measure space when the value of the
first measure space is 1.
Problem: When she
makes strawberry jam,
my grandmother uses 3
kg of sugar for 6 kg of
strawberries. How much
sugar does she need for
18 kg of strawberries?
Strawberri
es (kg)
Sugar (kg)
6
1
18
24
30
36
3
0,5
9
12
15
18
Strategy:
1. To make 6 kg of stawberries,
grandma needs 3 kg of sugar
2. To make 1 kg of strawberries,
she needs 3 / 6 kg of sugar
3. To make 18 kg of
strawberries, grandma needs
(3 / 6) x 18 = 9 kg of sugar
Formally manipulating the
proportion: creating equivalent
fractions
Starting from the proportion
Problem: When she makes
strawberry jam, my
grandmother uses 3 kg of
sugar for 6 kg of
strawberries. How much
sugar does she need for 18 kg
of strawberries?
6 = 18
3
x
• Strategy: I need to multiply
the numerator of the left
fraction (6) by 3 to obtain the
numerator of the right fraction
(18), so I multiply the
denominator of the left
fraction (3) by 3 too
Formally manipulating the
proportion: creating equivalent
fractions
Starting from the proportion
Problem: When she makes
strawberry jam, my
grandmother uses 3 kg of
sugar for 6 kg of
strawberries. How much
sugar does she need for 18 kg
of strawberries?
6 = 18
3
x
• Strategy: I need to multiply
the numerator of the left
fraction (6) by 3 to obtain the
numerator of the right fraction
(18), so I multiply the
denominator of the left
fraction (3) by 3 too
Formally manipulating the
proportion: Cross-multiplication
Starting from the proportion
Problem: When she makes
strawberry jam, my
grandmother uses 3 kg of
sugar for 6 kg of
strawberries. How much
sugar does she need for 18 kg
of strawberries?
x = 18
3
6
Strategy: x can be obtained by
multiplying the numerator of the
right fraction (18) by the
denominator of the left fraction (3)
and dividing it by the numerator of
the left fraction (6). So, (18 x 3) / 6
=9
Formally manipulating the
proportion: Cross-multiplication
Starting from the proportion
Problem: When she makes
strawberry jam, my
grandmother uses 3 kg of
sugar for 6 kg of
strawberries. How much
sugar does she need for 18 kg
of strawberries?
x = 18 . 3
6
Strategy: x can be obtained by
multiplying the numerator of the
right fraction (18) by the
denominator of the left fraction (3)
and dividing it by the numerator of
the left fraction (6). So, (18 x 3) / 6
=9
Choosing flexibly between strategies
• The (additive) building-up strategy is less
sophisticated, more meaningful for the novice
learner, but also more limited in applicability
• Example 1: When I need 3 kg of sugar for 6 kg of
strawberries, how much sugar do I need for 18 kg
of strawberries?
Choosing flexibly between strategies
• The building-up strategy is less sophisticated,
more meaningful for the novice learner, but also
more limited in applicability
• Example 1: When I need 3 kg of sugar for 6 kg of
strawberries, how much sugar do I need for 18 kg
of strawberries?
• Example 2: When I need 3 kg of sugar for 6 kg of
strawberries, how much sugar do I need for 8 kg
of strawberries?
Choosing flexibly between strategies
• The building-up strategy is less sophisticated,
more meaningful for the novice learner, but also
more limited in applicability
• Example 1: When I need 3 kg of sugar for 6 kg of
strawberries, how much sugar do I need for 18 kg
of strawberries?
• Example 2: When I need 3 kg of sugar for 6 kg of
strawberries, how much sugar do I need for 8 kg
of strawberries?
Choosing flexibly between strategies
• Between- and within-strategies are more
sophisticated, less limited, but less accessible to
the novice learner
• In many cases the rule-of-three is unnecessary
complicated for the task at hand
• Cross-multiplication is an efficient method, but is
not transparent to the learner
• This algorithmic procedure consists of the blind
manipulation of numerical symbols according to
formal rules that have no transparent relation with
the original problem context
•
•
•
•
•
Strategies: Implications for
educators
Awareness of different strategies is essential to
understand students’ approaches to proportional
problems.
All strategies (can) lead to a correct answer, but they
are very different in nature, advantages and
disadvantages.
All strategies are valuable at certain times, for certain
purposes...
Algorithmic methods should be taught only after
other methods of solving proportional problems.
Students need to understand the relations between
these strategies and develop strategy flexibility
Content
1.
2.
3.
4.
5.
6.
7.
Proportionality/linearity: some terminology
Proportionality/linearity: representations
Proportionality/linearity: properties
Proportionality/linearity in the curriculum
Proportionality/linearity: solution strategies
Additive reasoning errors
The over-use of proportionality/linearity in
arithmetic word problem solving
8. The over-use of proportionality/linearity in
geometry
9. Proportional reasoning in school and real life
6. Additive reasoning errors
Before being able to reason multiplicatively,
students often approach proportional situations in
an inappropriate additive way
Problem: When she makes
strawberry jam, my
grandmother uses 3 kg of
sugar for 6 kg of
strawberries. How much
sugar does she need for 18 kg
of strawberries
Strategy: when one uses 18
kg of strawberries instead of
6 kg, this means one uses 12
kg of strawberries more.
Thus one needs 12 kg of
sugar more, hence 15 kg.
Additive errors: another example
•Problem: 10 oranges in 2 litres of water. How many
oranges in 4 liters of water?
+2
• Strategy:
“4 is 2 more than 2,
so …
I do 10 + 2 = 12”
M1:
oranges
10
?
M2: litres
of water
2
4
+2
Additive errors: determining factors
• Familiarity with the meaning of the rates (external
ratios) : e.g., speed in kilometres per hour, cost in
price per unit…
• Nature of the numbers: non-integer ratios trigger
additive errors
Additive errors
• When she makes
strawberry jam, my
grandmother uses 3 kg
of sugar for 5 kg of
strawberries. How
much sugar does she
need for 8 kg of
strawberries?
• When she makes
strawberry jam, my
grandmother uses 3 kg
of sugar for 6 kg of
strawberries. How
much sugar does she
need for 18 kg of
strawberries?
Additive errors
• When she makes
• When she makes
strawberry jam, my
grandmother uses 3 kg
of sugar for 5 kg of
strawberries. How
much sugar does she
need for 8 kg of
strawberries?
The within and between ratios are non-integer
more additive errors (e.g. 3 + 3 = 6)
strawberry jam, my
grandmother uses 3 kg
of sugar for 6 kg of
strawberries. How
much sugar does she
need for 18 kg of
strawberries?
Additive errors: Implications for
educators
• Encountering multiplication merely as “repeated
addition” imposes obstacles on learners’ reasoning.
Alternative models of multiplication (e.g. splitting,
folding) should be used.
• Pay explicit instructional attention to the differences
between additive and multiplicative reasoning from
early on: At a young age children can be brought to
understand that numerical comparisons and change
can be viewed additively as well as multiplicatively
(see next slide).
Early comparisons
• Situation: I am 6 years old and my sister is only 3
years old
• Two different views:
• I am 3 years older (= additive)
• I am twice (= 2 times) as old (= multiplicative)
Content
1.
2.
3.
4.
5.
6.
7.
Proportionality/linearity: some terminology
Proportionality/linearity: representations
Proportionality/linearity: properties
Proportionality/linearity in the curriculum
Proportionality/linearity: solution strategies
Additive reasoning errors
The over-use of proportionality/linearity in
arithmetic word problem solving
8. The over-use of proportionality/linearity in
geometry
9. Proportional reasoning in school and real life
Example
Stacey (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:
 2 / 8 = 10 / ? thus 40 matches
Example
• 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 10 rounds, how many rounds
has Kim run?
• Answer of 32 out of 33 pre-school teachers:
5 / 15 = 10 / ? thus : 10 x 3 = 30 rounds
(Cramer et al., 1993)
Example
• « Une orchestre de 44 musiciens joue une
symphonie en 36 minutes. Combien de temps cette
symphonie durera quand elle est jouée par une
orchestre de 88 musiciens? »
• Most frequent errors:
• Twice as many musicans, thus twice as long, so 72
• Twice as many musicians, thus half as long, so 18
Example
• When asked for examples of
functions, students mainly
give “linear” examples
• Task: Draw the evolution of
a person’s height in relation
to his age
Leinhardt, Zaslavsky, & Stein (1990)
Example: The slave in the dialogue
Ménon by Platon
“Si on doit dessiner un carré dont l’aire est le
double de l’aire d’un autre carré, le côté doit
être le double.”
Example: The slave in the dialogue
Ménon by Platon
“Si on doit dessiner un carré dont l’aire est le
double de l’aire d’un autre carré, le côté doit
être le double.”
Example: The slave in the dialogue
Ménon by Platon
“Si on doit dessiner un carré dont l’aire est le
double de l’aire d’un autre carré, le côté doit
être le double.”
Example
• The chance of getting a six when rolling a fair die
is 1/6. What is the chance of getting at least one
six when you roll the die twice?
• Most frequent error:
2 x 1/6 = 2/6 = 1/3
1
2
1
1
Example
2
3
4
5
• The chance of getting a six when rolling a
fair die is 1/6. What is the chance of
getting at least one six when you roll the
die twice?
6
2
1
2
3
4
5
6
• Most frequent error:
2 x 1/6 = 2/6 = 1/3
3
…
4
…
5
…
6
1
2
3
• Correct answer: 5 + 6 = 11 / 36
4
5
6
Example
An object which is 10 times as heavy as another
object, will reach the ground 10 times as fast as that
other object”
Aristotle
7. The over-use of proportionality in
arithmetic problem solving
• Examples of a widely occurring phenomenon
• A study with elementary school children (Van
Dooren et al., 2005)
A Leuven study: method
• Participants: + 1000 students from grades 3 to 8
• Material: Solving a paper-and-pencil test containing
proportional and non-proportional problems
• Proportional problems:
• e.g., 4 packs of pencils cost 8 euro. Our teacher wants to buy a
pack for every pupil. He has to buy 24 packs. How much does he
have to pay?
• Non-proportional problems: 3 types (see next slide)
3 types of non-proportional items
• Additive: 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?
• Affine: In the hallway of our school, 2 tables are standing
in a line. 10 chairs fit around them. Now the teacher puts 6
tables in a line. How many chairs fit around these tables?
• Constant: Mama puts 3 towels on the clothesline. After 12
hours, they are dry. The neighbours have 6 towels on their
clothesline. After how many hours are they dry?
Results: proportional items
4 packs of pencils cost 8 euro. Our teacher wants to buy a pack
for every pupil. He has to buy 24 packs. How much does he
have to pay?
100
90
80
70
60
50
40
30
20
10
0
Correct
3rd
4th
5th
6th
7th
correct other
8th
Results: additive item
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
90
80
70
60
50
40
30
20
10
0
Proportional
5
3th
4th
other
5th
15
5
30
6
3
Correct
6th
3
+ 10
7th
90
correct 6 proportional
30
+ 25
+ 10
8th
15
+ 25
40
Results: affine item
In the hallway of our school, 2 tables are standing in a line. 10 chairs fit
around them. Now the teacher puts 6 tables in a line. How many chairs
fit around these tables?
100
90
80
70
60
50
40
30
20
10
0
Proportional
2
3th
4th
5th
10
other
correct
2
6
3
5
Correct
4+2
5
6th
7th
30
 3proportional
6
4+2
8th
10
26
Results: constant item
Mama puts 3 towels on the clothesline. After 12 hours, they are dry. The
neighbours have 6 towels on their clothesline. After how many hours are
they dry?
100
90
80
70
60
50
40
30
20
10
0
Proportional
3
2nd
3th
other
4th
4
5th
12
correct
3
6
2
4
Correct
6th
7th
24
proportional
2
8th
Cte
12
6
Cte
12
Task features
More proportional
errors
• Ellen and Kim are running around a track. They
run equally fast but Ellen started later. When Ellen
has run 4 laps, Kim has run 8 laps. When Ellen
has run 12 laps, how many has Kim run?
• Ellen and Kim are running around a track. They
run equally fast but Ellen started later. When Ellen
has run 4 laps, Kim has run 6 laps. When Ellen
has run 10 laps, how many has Kim run?
Content
1.
2.
3.
4.
5.
6.
7.
Proportionality/linearity: some terminology
Proportionality/linearity: representations
Proportionality/linearity: properties
Proportionality/linearity in the curriculum
Proportionality/linearity: solution strategies
Additive reasoning errors
The over-use of proportionality/linearity in
arithmetic word problem solving
8. The over-use of proportionality/linearity in
geometry
9. Proportional reasoning in school and real life