and back The development of students' additive and multiplicative reasoning skills Dirk De

From addition to multiplication … and back The development of students’ additive and multiplicative reasoning skills

Dirk De Bock, Wim Van Dooren, Lieven Verschaffel HUB RESEARCH PAPER 2009/37 NOVEMBER 2009

1 From addition to multiplication

From addition to multiplication … and back

The development of students’ additive and multiplicative reasoning skills

Wim Van Dooren 1, Dirk De Bock1 2, and Lieven Verschaffel1

1 Centre for Instructional Psychology and Technology, Katholieke Universiteit Leuven,

Belgium

2 Hogeschool-Universiteit Brussel, Belgium

Author for correspondence

Wim Van Dooren, Center for Instructional Psychology and Technology, Katholieke

Universiteit Leuven, Vesaliusstraat 2, PO Box 3770 B-3000 Leuven, Belgium,

[email protected], Phone +3216325755, fax +3216326274.

mailto:[email protected]


Abstract

This study builds on two lines of research that so far developed largely separately: the use of

additive methods to solve proportional word problems and the use of proportional methods to

solve additive word problems. We investigated the development with age of both kinds of

erroneous solution methods. They key question is whether and how an overall additive

approach to word problems develops into an overall multiplicative approach, and how the

transition from the first kind of errors to the second occurs.

We gave a test containing missing-value problems to 325 third, fourth, fifth, and sixth

graders. Half of the problems had an additive structure and half had a proportional structure.

Moreover, in half of the problems the internal and external ratios between the given numbers

were integer while in the other cases numbers were chosen so that these ratios were

noninteger.

The results indicate a development from applying additive methods “anywhere” in the

early years of primary school to applying proportional methods “anywhere” in the later years.

Between these two stages many students went through an intermediate stage where they

simultaneously used additive methods to proportional problems and proportional methods to

additive problems, switching between them based on the numbers given in the problem.


From addition to multiplication … and back

The development students’ additive and multiplicative reasoning skills

Introduction

Since several decades, a lot of research has focused on the development of

multiplicative reasoning, and more particularly, on the transition from an additive to a

multiplicative way of thinking (Clark & Kamii, 1996). As repeatedly argued by mathematics

education researchers, multiplication and division are more than just a different set of

arithmetic operations that are taught after addition and subtraction, and multiplicative thinking

ranges much further than merely a faster way of doing repeated addition. Although repeated

addition often remains an “implicit, unconscious, and primitive model” for multiplication

(Fischbein, Deri, Nello, & Marino, 1985, p. 4), scholars have stressed that the repeated

addition model for multiplication is incomplete, and that a significant qualitative change is

required to get from additive to multiplicative thinking (e.g., Greer, 1994; Nesher, 1988;

Nunes & Bryant, 1996; Piaget, Grize, Szeminska, & Bangh, 1977). Studies (e.g., Nunes &

Bryant, 1996; Squire, Davies, & Bryant, 2004) have confirmed that children at 8-9 years old

perform well on multiplicative tasks about one-to-many correspondence (“every car has 4

wheels, so how many wheels do 6 cars have?”) – that can be conceived as and solved via

repeated addition – but fail on tasks where multiplication needs to be conceived and handled

differently, for example as a Cartesian product (“with 6 shorts and 4 t-shirts, how many outfits

can I make?”).

We will describe the results of a study on how primary school students approach

multiplicative situations described in word problems, and contrast these with the same

students’ approach of situations that are not multiplicative but additive. Our study focuses on

a specific subset of additive and multiplicative problems, namely those referring to situations

4 From addition to multiplication of co-variation. We will clarify these situations, and the difference between additive and

multiplicative ones using the following examples:

Tom and his sister Ana have the same birthday. Tom is 15 years old when Ana is 5

years old. They are wondering how old Ana will be when Tom is 75 … (= additive

situation)

Rick is at the fish store to buy tuna. The customer before him bought a piece of 250

grams of tuna and had to pay 10 euro. Rick needs 750 grams of tuna, and he wonders

what he will have to pay … (= multiplicative situation)

Both situations in a certain way deal with co-variation: The older Ana gets, the older Tom

gets, and the more fish Rick buys, the more he has to pay. But there is an important difference

between the two situations. Mathematically speaking, the first situation can be described by a

function of the form f(x) = x + a, and the second by a function of the form f(x) = bx, and the

reasoning that is required in both situations is very different. The first situation about the age

of Tom and Ana can be called additive, in the sense that the given numbers are linked by the

operations of addition and subtraction. There are two variables in the situation (the age of Ana

and the age of Tom), and the difference between these variables is invariant: Adding 10 years

to Ana’s age will always provide Tom’s age. The second situation about Rick buying tuna is a

multiplicative one, and, more specifically, a proportional situation (Nunes & Bryant, 1996).

The two variables in this problem (weight and cost) are linked by multiplication and division

with an invariant: Knowing that the tuna costs 40 euro per kilogram, multiplying the kilos of

tuna with 40 yields the price, and dividing the price by 40 yields the kilos of tuna bought.

To summarise, we will focus on students’ thinking in additive and multiplicative

situations of co-variation. Approaches where the difference between two values is considered

and added to a third value are called additive, whereas approaches where the ratio between

5 From addition to multiplication two values is considered and multiplied with a third value are called multiplicative. Two

remarks are important here. First, when classifying students’ approaches to problems, also

repeated-addition approaches will be considered multiplicative, because repeated addition

also adequately considers the multiplicative relations of a situation. Second, given our

exclusive focus on situations of co-variation, the only multiplicative situations that we

consider are proportional situations, and various other multiplicative situations (e.g. Cartesian

product situations) are left aside. In the rest of the article, we will, therefore, use the terms

“proportional” and “multiplicative” as synonyms.

We investigated the way students approach situations that are described in a word

problem and conceive them as additive or multiplicative. It may happen that students solve

word problems that refer to a multiplicative situation erroneously in an additive way, and,

inversely, solve additive word problems multiplicatively (De Bock, 2008). Students’ tendency

to approach proportional situations additively instead of multiplicatively has been amply

documented in studies of students’ development of proportional reasoning (e.g., Hart, 1981;

Lin, 1991; Tourniaire & Pulos, 1985). At the same time, research has shown that students

often tend to use multiplicative approaches beyond their applicability range (e.g., Fernández,

Llinares, & Valls, 2008; Modestou & Gagatsis, 2007; for a review, see Van Dooren, De Bock,

Janssens, & Verschaffel, 2008), including on situations with an additive structure (e.g., Van

Dooren, De Bock, Hessels, Janssens, & Verschaffel, 2005). The main goal of the present

study is to characterise the development with age of the use of additive and multiplicative

models simultaneously. We agree with Verschaffel, Greer, and De Corte (2007) that

“although separate analyses of the conceptual fields of additive and multiplicative structures

will doubtless continue, there is a strong need for a comparative analysis between or a

synthesis of these two hitherto rather separate bodies of research” (p. 588). Only by

6 From addition to multiplication investigating the simultaneous development of students’ use of additive and multiplicative

models – rather than studying them in separate studies as mostly happened in the past – it is

possible to determine whether and how students develop an understanding of the quantitative

relations underlying additive and multiplicative situations. Before going into detail on this

study, we provide some theoretical and empirical background concerning each of the

overgeneralisation phenomena. We also frame these phenomena in the broader literature on

problem solving.

Theoretical and Empirical Background

Multiplicative Reasoning and the Over-reliance on Additive methods

Because of its wide applicability, proportionality takes a pivotal role in primary and

secondary mathematics education. Full mastery of the multiplicative reasoning that is required

in proportional situations is not achieved easily, but several studies have shown that children

already at a young age can successfully handle simple proportional situations (e.g., Lamon,

1994; Spinillo & Bryant, 1999), for instance by relying on repeated addition strategies: “If 2

pineapples cost 4 euro, 6 pineapples cost 4+4+4 = 12 euro”. The actual teaching of

proportionality generally only starts in the upper elementary (or lower secondary) grades,

where students intensively practice proportional reasoning skills with missing-value

proportionality problems where three values are given and a fourth is unknown (Kaput &

West, 1994), and are confronted with various typical contexts in which proportional reasoning

is required (mixtures, costs, currency conversion, …). For example: “Grandma adds 2

spoonfuls of sugar to juice of 10 lemons to make lemonade. How many lemons are needed if

6 spoonfuls of sugar are used?”

Given the pivotal role of proportional reasoning in mathematics education, a lot of

research has focused on how students acquire proportional reasoning skills, which difficulties

7 From addition to multiplication they experience, and how it can be enhanced by instruction (e.g., Freudenthal, 1973, 1983;

Harel & Behr, 1989; Hart, 1981, 1984; Kaput & West, 1994; Karplus, Pulos, & Stage, 1983;

Lesh, Post, & Behr, 1988; Nunes & Bryant, 1996; Tourniaire & Pulos, 1985; Vergnaud, 1983,

1988). Among the strategies that students can apply in proportional situations, the literature

distinguishes correct multiplicative approaches and erroneous additive approaches. Before

going into the latter, let us first briefly explain the various correct multiplicative approaches.

Students approaching the above lemonade problem multiplicatively will most often

use a scalar approach (Vergnaud, 1983, 1988), focusing on the internal ratio of sugar to sugar

(6 spoonfuls / 2 spoonfuls), and apply this to the number of lemons (3 × 10 = 30 lemons for 6

spoonfuls of sugar). The alternative is a functional approach (Vergnaud, 1983, 1988),

focusing on the external ratio of sugar to lemon juice (10 lemons / 2 spoonfuls of sugar 6 ×

5 = 30 lemons are needed). A variant of the functional approach is the unit factor approach

(Vergnaud, 1983, 1988), which goes first to the unit value of one of the quantities (e.g., 10

lemons for 2 spoonfuls of sugar 5 lemons for 1 spoonful of sugar 5 × 6 = 30 lemons are

needed). Finally, students can approach the situation by a more elementary approach, that

could be called building up or replication: for 2 + 2 + 2 spoonfuls of sugar, 10 + 10 + 10

lemons are needed. It is clear that this approach for solving missing-value proportionality

problems is based on the repeated-addition character of multiplication, and therefore has

characteristics of additive reasoning. Nevertheless, we categorise it as multiplicative, as it

appropriately handles the multiplicative character of the problem situation.

Besides these correct multiplicative approaches, there is one erroneous approach that

has received a lot of attention in the literature: the additive one, whereby the relationship

between given values is computed by subtracting one value from another, and applying the

difference to the third one. For example, in the lemonade problem above, students reason that

8 From addition to multiplication for the second mixture there are 6 – 2 = 4 spoonfuls of sugar more, so 10 + 4 = 14 lemons are

needed. Research has identified both subject- and task-related factors that influence the

occurrence of such additive errors on proportional problems. As an example of the former,

this kind of error is more typical for younger children with limited instructional experience

with the multiplicative relations in proportional situations. But also after instruction, additive

errors still occur, particularly on more difficult proportional problems. An important task-

related factor in preventing additive errors is when the rates (external ratios) in the problem

have a dimension that is familiar to students, e.g., speed in kilometres per hour, cost in price

per unit (Karplus et al., 1983; Vergnaud, 1983). Another task-related factor strongly related to

the occurrence of additive errors – and that will be central in the present study – is when the

numbers given in the problem form non-integer ratios (Hart, 1981; Kaput & West, 1994;

Karplus et al., 1983; Lin, 1991; Tourniaire & Pulos, 1985). For instance, when the lemonade

problem mentioned before is transformed into “Grandma adds 2 spoonfuls of sugar to juice of

5 lemons to make lemonade. How many lemons are needed if 3 spoonfuls of sugar are used?”

it becomes more difficult to execute the multiplicative operations because the ratios are not

integer (5/2 and 3/2), and students will therefore more often fall back to erroneous additive

reasoning (2+1 spoonfuls of sugar in the lemonade, so 5+1 lemons).

Research on the Overuse of Proportionality

Besides the extensive body of evidence of students reasoning additively in

multiplicative situations, other lines of research have indicated that students are also inclined

to apply multiplicative methods outside their applicability range. Especially for

nonproportional problems presented in a missing-value format, students tend to erroneously

apply multiplicative methods. This has been shown in various domains of mathematics,

including elementary arithmetic, geometry, probability, or algebraic generalisation (for a

9 From addition to multiplication review, see Van Dooren et al., 2008). For instance, many students answer “2/6” to the

probabilistic problem “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?” (Van Dooren, De Bock,

Depaepe, Janssens, & Verschaffel, 2003).

Particularly relevant for the present study is the erroneous application of proportional

methods to problems with an additive structure. Van Dooren et al. (2005) gave a test

containing both proportional and various kinds of nonproportional word problems to large

groups of third to eighth graders. Among the nonproportional problems, there were additive

problems including the following:

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?

Generally speaking, and in line with the research on proportional reasoning

summarised above, their study showed that students in the early years of primary school

already could provide correct answers to proportional word problems, but performance on the

proportional problems further improved until eighth grade, with most learning gains being

made between third and fifth grade, i.e., the years in which this is instructed in classrooms.

However, already before the start of formal instruction in proportionality students also gave

proportional answers to the nonproportional word problems: In third grade, 30% of all

nonproportional word problems were answered proportionally, and this increased

considerably until sixth grade. For the additive problem mentioned above, the percentage of

wrong proportional responses (“24 laps”) increased from 10% in third grade to more than

50% in sixth grade whereas correct additive answers (“16 laps”) decreased from 60% in third

grade to 30% in sixth grade (and then increased again to 45% in eighth grade).


Building further on this observation, a recent study by Van Dooren, De Bock, Evers,

and Verschaffel (2009) with fourth to sixth graders showed that – just like the tendency to

reason additively in multiplicative situations is affected by the fact that given numbers do not

form integer ratios (e.g., Kaput & West, 1994) – the nature of the given numbers also affects

students’ tendency to use multiplicative strategies in situations that are not multiplicative.

Solutions to additive word problems where the numbers formed integer ratios, such as the

runner problem mentioned above, were compared with problems where the numbers formed

non-integer ratios. A variant of the above-mentioned runner problem above with non-integer

ratios (6/4 and 10/4) would be

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


has Kim run?

It was found that for the latter non-integer variants students were less inclined to use

proportional strategies, and therefore performed better than on the versions with integer ratios

between numbers. This was particularly the case in fourth graders. Apparently, the fifth, and

especially sixth graders had already become so skilled in doing proportional calculations

involving non-integer ratios that they therefore also overused these skills.

Combining the two research lines described so far, we endorse the claim by Cramer,

Post, and Currier (1993) that “we cannot define a proportional reasoner simply as one who

knows how to set up and solve a proportion” (p. 160). Similarly, additive reasoning comprises

more than merely being able to successfully complete the required arithmetical operations. A

crucial issue is that students need to be able to identify proportional and additive situations

and distinguish them from each other and from other situations. Nevertheless, studies in the

past have usually just relied on proportional problems while the combination of various

11 From addition to multiplication problem types is needed to show whether students are able to understand the quantitative

relations properly.

The Overuse of Learned Procedures and Superficial Problem Solving Behaviour

The idea that students tend to overuse previously learned procedures is of course not

new and it has also been shown in research outside of mathematics. For instance, in the

domain of English spelling, children have to learn to use morphology and avoid sound-based

errors in spelling past-tense forms (e.g. writing “walkt” instead of walked). Nunes, Bryant,

and Bindman (1997) have found that after children managed to overcome these errors, they

also begun to overuse the –ed forms, for instance in spelling slept as “sleped”. For similar

findings in French spelling, see Fayol, Thevenin, Jarousse, and Totereau (1999). New in this

study, however, is that we investigate the use of two learned procedures and their mutual

interaction and development with age. In order to do this, we need to frame our work also in

the wider literature on (mathematical) problem solving. Students determining their solution

procedures by relying on surface-level features of problems (such as the problem formulation

or the given numbers) instead of the deep-level features is a well-known phenomenon (e.g.

Chi, Feltovitch, & Glaser, 1981; Hinsley, Hayes, & Simon, 1977). People infer a correlation

between the success of a particular strategy and characteristics of the problems for which the

strategy has been successfully implemented. Hinsley et al. (1977), for instance, showed how

students could categorize algebraic word problems by relying on merely the first few words of

a problem (e.g., linking “In a sports car race …” with distance-rate-time problems). Research

has shown that tudents very strongly tend to rely on superficial cues (e.g. key words and key

expressions, prototypical situations described in the word problem, the name of the chapter in

which a problem appears) to decide upon the arithmetical operation(s) to be performed (e.g.,

Greer, 1993; Reusser & Stebler, 1997; Verschaffel, De Corte, & Lasure, 1994; Verschaffel,

12 From addition to multiplication Greer, & De Corte, 2000). And for single-operation word problems, also the influence of the

given numbers has been shown (e.g., Bell, Fischbein, & Greer, 1984; Eckenstam & Greger,

1983; Fischbein, Deri, Nello, & Marino, 1985; Greer, 1987; Hart, 1981; Sowder, 1988).

As such, the detection of predictive correlations between a problem’s surface structure

and its solution procedure can be very effectual, allowing a fluent solution of problems

without going through laborious problem-solving steps. But as argued by Ben-Zeev and Star

(2001), the detected correlations can be spurious, and then problem-solving experience can

become ineffectual. Ben-Zeev and Star (2001) gave experienced students several sets of

algebraic equations, along with algorithms to solve them. They showed that these students

were susceptible to experimentally induced spurious-correlations between irrelevant

characteristics of the equations and the algorithm that is used to solve them. Their results also

indicated that students were not necessarily aware that they were responsive to such a

spurious correlation. It became part of their implicit knowledge: When asked directly,

students indicated that they were choosing randomly between two strategies, but actually they

were responding systematically to spurious correlations.

For proportional reasoning, the missing-value formulation of a word problem is

probably the most salient feature for students. The majority of the proportional reasoning

tasks that students encounter in the upper grades of elementary school and in the lower grades

of secondary school are formulated in a missing-value format (Cramer et al., 1993), and a lot

of attention is paid to the development of fluency in solving such problems. At the same time,

nonproportional problems stated in a missing-value format are very rare. This may explain

why Van Dooren et al. (2005) observed a considerable increase in the number of proportional

answers to nonproproportional word problems throughout primary school. A similar

explanation could be given for the fact that students can associate even the number

13 From addition to multiplication characteristics of word problems with a certain solution procedure. When students are first

introduced in proportionality, word problems usually contain numbers that allow calculations

with easy multiplicative jumps. This way, students can focus on recognizing and working

through the proportional structure of the situation, and applying the taught procedures.

Therefore, a spurious correlation association (as proposed by Ben-Zeev & Star, 2001)

also seems likely for students’ responsiveness to superficial problem characteristics when

applying additive or proportional solution strategies. The difference with Ben-Zeev and Star’s

work, however, is that in their work, students switch between algorithms in response to task

characteristics when the algorithms were actually interchangeable in the sense that they both

lead to a correct answer and, therefore, students’ algorithm switches can at best be called

unfounded, but certainly not wrong. In our work, however, students switch between one

approach that is correct for the task and another one that is clearly incorrect; therefore, we can

really speak of the overuse of approaches.

Research Questions

The previous section started with two lines of research that so far developed largely

separately, relating to the tendency to approach proportional situations additively and the

tendency to approach to additive situations proportionally. Typically, these tendencies – and

more generally even the abilities to reason additively and multiplicatively – are not studied

simultaneously in the same students. Nevertheless, it is interesting to investigate how both

abilities and both types of errors develop over age, and more importantly, whether it is

possible that both – seemingly opposite – overgeneralisations can occur at the same time in

individuals, for example in the transition phase from one kind of overgeneralisation to the

other. As explained before, only by examining additive and multiplicative reasoning

14 From addition to multiplication simultaneously, it is impossible to determine the actual reasoning abilities of students to

understand the quantitative relations that distinguish additive from proportional situations.

Based on the available literature reported in the previous section, we first of all

anticipated a development with age from an overall additive approach to missing-value

problems (consisting of the correct use of additive methods in additive situations and the

incorrect use in multiplicative situations) towards an overall multiplicative approach

(involving the correct use of multiplicative methods in multiplicative situations and the

incorrect use in additive situations). A key question, however, was how the development from

an additive to a multiplicative approach would look like, and more specifically how the

transition could be characterised. Would students in the transition tend to apply additive and

multiplicative methods appropriately to additive and multiplicative problems? If not, would

they choose randomly for additive or multiplicative methods, or would they rely on

superficial problem characteristics? The literature on additive and proportional reasoning

summarised above suggests that, if students in the transition rely on irrelevant problem

characteristics, they will most likely consider the numbers given in word problems, and use

multiplicative methods when the ratios between given numbers are integer and additive

methods when ratios between given numbers are non-integer. This would imply that students

in this intermediate stage at the same time use multiplicative methods in additive situations

(i.e., when the ratios between given numbers are integer) and use additive methods in

multiplicative situations (when the ratios between given numbers are non-integer).

Method

Participants

Students from third to sixth grade from two different primary schools in Flanders

participated in the study: 88 third graders, 78 fourth graders, 81 fifth graders and 78 sixth

15 From addition to multiplication graders. One school was situated in a middle-sized city, the other in a smaller village. Both

schools were average in size and attracted students from mixed socioeconomic backgrounds,

mainly from the immediate neighbourhood. The sample consisted of approximately equal

numbers of boys and girls.

The educational standards in Flanders (Ministerie van de Vlaamse Gemeenschap,

1997) indicate that by the end of sixth grade students should be able to compare the equality

of two ratios and calculate the missing value when confronted with a missing-value

proportionality problem. Even though schools in Flanders use a variety of textbooks, the

general instructional approach and the timing for the teaching of proportional missing-value

problems is very similar. In second and third grade, the focus is on solving simple

multiplication word problems (e.g., “1 pineapple costs 2 euro. How much do 3 pineapples

cost?”). In fourth grade, this focus gradually shifts toward solving proportional missing-value

problems, typically referring to contexts such as unit/price, weight/price, and time/distance

(e.g., “12 eggs cost 2 euro. What is the price of 36 eggs?”). These missing-value proportional

problems are further rehearsed in fifth and sixth grade, and some new application contexts are

introduced as well (e.g., currency exchanges, mixtures in recipes or paints). In sixth grade,

some attention is also paid to tackling word problems with larger and/or rational numbers

(and the use of a pocket calculator to do so), and students learn how to solve problems with

noninteger ratios, usually by means of a unit factor approach. Mathematics textbooks for

primary school (and secondary school) do not pay attention to contrasting proportional and

nonproportional missing-value problems.

Materials

All students solved four experimental word problems. The design of these word

problems is explained and illustrated in Table 1.


Table 1

Design and Examples of Experimental Items

Problem

type

Number type Example

Integer (I) Evelien and Tom are ropeskipping. They started together, but Tom

jumps slower. When Tom has jumped 4 times, Evelien has jumped

20 times. When Tom has jumped 12 times, how many times has

Evelien jumped?

Prop

ortio

nal

Non-integer

(N)

A motor boat and a steam ship are sailing from Ostend to Dover.

They departed at the same moment, but the motor boat sails faster.

When the steam ship has sailed 8 km, the motor boat has sailed 12

km. When the steam ship has sailed 20 km, how many km has the

motor boat sailed?

Integer (I) 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?

Add

itive

Non-integer

(N)

Lien and Peter are reading the same book. They read at the same

speed, but Peter started earlier. When Lien has read 4 pages, Peter

has read 10 pages. When Lien has read 6 pages, how many has Peter

read?


Two of the word problems were proportional problems, for which proportional

calculations (i.e. finding the value of x in b/a = x/c) lead to the correct answer. The other two

were additive word problems, for which additive calculations (i.e. finding x in b – a = x – c)

are required. As can be seen in Table 1, proportional and additive problems were formulated

similarly. The crucial difference between proportional and additive situations lies in the

second sentence. For example, for the integer additive problem in Table 1, the additive

character of the situation lies in the fact that both girls run at the same speed, but one started

later, implying that the difference in laps between both girls remains constant. The problem

can be easily turned into a proportional one by changing the second sentence into:

Ellen and Kim are running around a track. They started together, but Kim runs faster.


has Kim run?

We also experimentally manipulated the number characteristics of the word problems.

For two of the problems (i.e., one additive and one proportional) the given numbers were

chosen so that when doing proportional calculations, one has to work with integer ratios (I-

version). For the other two problems (again, one additive and one proportional), the numbers

formed non-integer ratios (N-version). In the latter case, care was taken, however, that the

outcome of proportional calculations still would be integer. This way, we wanted to avoid that

students would start to doubt about the correctness of their calculations just because they

obtain a non-integer outcome.

Eight different test variants were constructed. One test variant included the

experimental items as they are shown in Table 1. The other variants were created by

reformulating the additive problems as proportional problems and vice versa, reformulating

the non-integer variants as integer variants and vice versa, or combinations of these. This way,

18 From addition to multiplication any uncontrolled variance in our results (e.g., due to the different contexts dealt with in the

word problems) would be cancelled out.

Procedure

In order to be able to detect individual student profiles, students should solve at least

one variant of each of the four experimental items. At the same time, it was very important

that students would not become aware of the goal of the study and that their response on one

experimental item would not influence their behaviour on another one. Therefore, we limited

ourselves to offering only four experimental word problems per student. The four

experimental items were moreover embedded in two larger tests. Each test contained 15

problems on a wide variety of mathematical topics that were related to the students’ school

curriculum. Mixed among these 15 buffer items, each test had two of the experimental items:

one proportional word problem (I- or N-version) and one additive word problem (I- or N-

version). The other test then contained the other two experimental items, also mixed among

15 buffer items.

Both tests were administered with one week in between. Students were told that the

tests were meant to assess their progress in mathematics in general. No further instructions

were given as to how to solve the problems, except for the fact that a pocket calculator could

be used, and that we explicitly asked students to record their calculations on the answer

sheets.

Results

In a first stage, we will look at the responses to the four experimental items separately, and

consider the extent to which they are affected by students’ age, the additive or proportional

character of the experimental word problems, and the number characteristics (integer/non-

19 From addition to multiplication integer) of the word problems. In a second stage, we will look at students’ solution profiles to

the four experimental items together.

General Results

Responses to the experimental word problems were classified as

- Proportional answer, when proportional operations were executed on the given

numbers (i.e. calculating x in the expression b/a = x/c)

- Additive answer, when additive operations were executed (i.e. finding x in b – a = x –

c), or

- Other answer, when the given numbers were combined in another way with arithmetic

operations than specified above, or when the problem was left unanswered.

When purely technical calculation errors (e.g., 8 × 2 = 14) were committed, the answer was

not necessarily scored as other: If the calculations were clearly proportional or additive, we

labelled them as such.

Table 2 shows the solutions given by students to the four experimental items. Taken as

a whole, the results confirm those of earlier studies by Van Dooren et al. (2005, 2009). First

of all, the proportional word problems elicited significantly more proportional responses

(32.3%) than the additive word problems (26.2%), χ²(1) = 293.535, p < 0.0001, and the

additive problems elicited significantly more additive responses (56.9%) than the proportional

problems (44.2%), χ²(1) = 42.081, p < 0.0001. The differences were not very large, however,

and the results presented in Table 2 clearly indicate an overgeneralisation in both directions:

Students often used proportional strategies on additive problems and additive strategies on

proportional problems.


Table 2

Overview of solutions given by students (in %) (Correct solutions are indicated in bold)

Proportional problems Additive problems Total

Grade P A O P A O P A O

3 (n = 88) 19.3 48.9 31.8 17.0 54.5 28.4 18.2 51.7 30.1

4 (n = 78) 29.5 43.6 26.9 35.9 46.2 19.9 32.7 44.9 22.4

5 (n = 81) 70.4 19.8 9.9 48.1 34.6 17.3 59.3 27.2 13.6

6 (n = 78) 84.6 10.3 5.1 67.9 19.2 12.8 76.3 14.7 9.0

Inte

ger p

robl

ems

Total (n =

325) 50.2 31.1 18.8 41.5 39.1 19.4 45.8 35.1 19.1

3 (n = 88) 0 59.1 40.9 1.1 79.5 19.3 0.6 69.3 30.1

4 (n = 78) 2.6 70.5 26.6 1.3 80.8 17.9 1.9 75.6 22.3

5 (n = 81) 16.0 60.5 23.5 8.6 80.2 11.1 12.3 75.6 22.3

6 (n = 78) 41.0 38.5 20.5 33.3 57.7 9.0 37.2 48.1 14.7

Non

-inte

ger p

robl

ems

Total (n =

325) 14.5 57.2 28.3 10.8 74.8 14.5 12.6 66.0 21.4

3 (n = 88) 9.7 54.0 36.4 9.1 67.0 23.9 9.4 60.5 30.1

4 (n = 78) 16.0 57.1 26.9 18.6 63.5 17.9 17.3 60.3 22.4

5 (n = 81) 43.2 40.1 16.7 28.4 57.4 14.2 35.8 48.8 15.4

6 (n = 78) 62.8 24.4 12.8 50.6 38.5 10.9 56.7 31.4 11.9 Tota

l

Total (n =

325) 32.3 44.2 23.5 26.2 56.9 16.9 29.2 50.5 20.2


Second, students’ solutions were clearly affected by the number characteristics of the

word problem: Regardless of the problem type, problems in which the numbers form integer

ratios elicited significantly more proportional responses (45.8%) than problems in which the

numbers form non-integer ratios (12.6%), χ²(1) = 258.736, p < 0.0001, whereas the opposite

was the case for additive responses (35.1% for problems with integer ratios and 66.0% for

problems with non-integer ratios), χ²(1) = 139.600, p < 0.0001. The impact of numbers was

even stronger than the effect of the proportional or additive character of the situation

described in the word problem. The proportional problems elicited only 6.1% more

proportional answers than the additive problems, and only 12.7% less additive answers. These

differences are much larger when contrasting the different number variants: Problems with

integer ratios elicited 33.2% more proportional strategies and 30.9% less additive strategies

than problems with non-integer ratios.

Third, the above effects were strongly affected by students’ age. In third grade, still

30.1% of the answers to the four problems were other answers, whereas this was only 10.9%

in sixth grade. Furthermore, whereas third graders – regardless of the problem type – gave a

lot of additive responses (60.5%), this strongly decreased towards sixth grade (31.4%), χ²(3)

= 41.290, p < 0.0001. A significant opposite trend could be observed for the number of

proportional responses, χ²(3) = 298.267, p < 0.0001. In third grade, only 9.4% of all answers

were proportional, but this increased up to 56.7% in sixth grade. It can be seen in Table 2 that

the decrease in additive answers and the increase in proportional answers were present both

for the proportional and for the additive problems. So, the number of correct answers to

proportional problems increased with age whereas on additive problems it decreased.

A last observation is that while there was an effect of the numbers in the problem at all

age levels, the effect strongly interacted with age, χ²(1) = 148.645, p < 0.0001. The number

22 From addition to multiplication effect was the smallest in third grade (with a decrease of 17.4% in proportional answers and

an increase of 17.6% in additive answers on items with non-integer ratios); it was largest in

fifth grade (with a decrease of 47% and an increase of 48.4%, respectively), and then

somewhat smaller again in sixth grade (an increase of 39.1% and a decrease of 33.4%,

respectively). So, as expected, students’ sensitivity to the number characteristics showed a

curvilinear shape, with the strongest impact in fifth grade.

Answer Profiles

Besides the analysis of the items separately – which confirmed findings of previous

research in proportional reasoning (Hart, 1981; Kaput & West, 1994; Karplus et al., 1983;

Lin, 1991; Tourniaire & Pulos, 1985) on the one hand and nonproportional reasoning (Van

Dooren et al., 2009) on the other hand – we also looked at students’ individual solution

profiles, i.e., the patterns of students’ solutions to the integer and non-integer versions of the

proportional and additive problems. Using our classification of responses (proportional,

additive, and other) as explained above, theoretically speaking there are 34 or 81 ways to solve

the four experimental word problems. However, for our research questions, only a few

profiles were of importance, namely:

- correct reasoners, who solved the two proportional problems proportionally and the

two additive problems additively

- additive reasoners, who solved both the two proportional and the two additive

problems additively

- proportional reasoners, who solved both the two proportional and the two additive

problems proportionally

- number-sensitive reasoners, who solved problems with integer ratios proportionally

and problems with non-integer ratios additively


An analysis of students’ profiles showed that already 119 out of 325 students (i.e.,

36.6%) could be characterised as perfectly fitting to one of these four profiles. An exploration

of the remaining answer patterns revealed that another 32 students (9.8%) all had an identical

profile, for which on second thought a sensible interpretation could be given too. These

students could be called “correct/number-sensitive reasoners”, since they responded correctly

to the two integer problems (i.e. additively to the additive problem and proportionally to the

proportional problem), but reasoned additively to both non-integer problems, suggesting that

they experienced difficulties calculating the proportional answer when non-integer ratios are

involved.

A way to identify additional students with interesting profiles was to tolerate, in the

four answer profiles listed above, one “other answer”. According to this less strict

categorisation, students who gave three proportional responses to the experimental word

problems, and one other response (but not an additive response) were still considered as

proportional reasoners, and students giving three additive responses and one other response

(but no proportional response) were still coded as additive reasoners. A similar rule was used

for correct reasoners and number-sensitive reasoners. This less strict categorisation allowed us

to characterise an additional 93 students (i.e., 28.6%) so that in total 212 of the 325 students

(75.1%) fitted in one of the above categories. The remaining 24.9% of the students were

considered as a “Other” category.

This last – less strict – categorisation will be taken as the basis for the further

discussion of the results, but for the interested reader, we provide the results of the strict

categorisation too (see Table 3). Before going into the results, we want to remark that a

cluster analysis confirmed that our less strict classification also provides the statistically best

description of the data. A two-step cluster analysis using Schwarz’s Bayesian Criterion

24 From addition to multiplication indicated that a solution with six clusters described the data best. The cluster centre of the first

cluster (n = 90) had an additive profile, the second (n = 76) a number-sensitive profile, the

third (n = 41) a correct/number-sensitive profile, and the fourth (n = 36) a proportional

profile. The fifth cluster (n = 33) referred to students giving mostly other answers, and the

sixth cluster (n = 39) to a remainder category comprised of various other profiles. Because

there were so few students with a “correct” profile, a cluster of students responding correctly

was not identified. In the cluster analysis, these students were included in the correct/number-

sensitive cluster, but given the particular theoretical importance of this group, we treated them

as a separate group. Overall, there was a very strong agreement between the less strict

categorisation that we applied (using the five categories as shown in table 3) and the cluster

membership as identified in the cluster analysis (Kappa = 0.86, with a 95% confidence

interval of 0.82 – 0.90)1.

Table 3 shows the results of our categorisation, providing details on the percentage of

students who perfectly fitted the profiles and the percentage that still fits the profile but where

one other answer was given. As revealed in this table, distribution of students over the various

categories strongly differs according to the grade level, χ² (115) = 113.4, p < .00015,

Cramer’s V = 0.341.


Table 3

Overview of solution profiles of individual students (in %)

Grade

3

(n = 88)

4

(n = 78)

5

(n = 81)

6

(n = 78)

Total (n =

325)

Perfect fit 0.0 0.0 1.2 3.8 1.2

One other 1.3 5.1 4.9 5.1 4.0 Correct

Total 1.3 5.1 6.2 9.0 5.2

Perfect fit 30.7 17.9 11.1 3.8 16.3

One other 15.9 19.2 6.2 2.6 11.1 Additive

Total 46.6 37.2 17.3 6.4 27.4

Perfect fit 0.0 1.3 7.4 19.2 6.8

One other 0.0 1.3 2.5 12.8 4.0 Proportional

Total 0.0 2.6 9.9 32.1 10.8

Perfect fit 3.4 9.0 18.5 19.2 12.3

One other 5.7 11.5 14.8 6.4 9.5 Number-

sensitive Total 9.1 20.5 33.3 25.6 21.8

Correct/number

-sensitive 5.7 5.1 16.0 12.8 9.8

Other 37.5 29.5 17.3 14.1 24.9

26 From addition to multiplication With respect to these age-related differences, a first observation is that the category “Other”

was relatively large in third grade (37.5%), but gradually decreased towards sixth grade

(14.1%). Second, the number of correct reasoners was very low, at all age levels. Altogether,

only 1.2% of the students (i.e., one fifth grader and three sixth graders) responded correctly to

all four experimental problems. Another 4.0% responded correctly to three of them and gave

one other answer (meaning that they still succeeded in not committing a proportional error to

an additive problem nor an additive error to a proportional problem).

Third, a strong decrease with age in the “Additive” category was observed. In third

grade, 46.6% of the students were purely additive reasoners, and this gradually decreased to

only 6.4% in sixth grade. Apparently, with increasing educational experience, students were

less inclined to reason additively – that is, to apply additive strategies regardless of the actual

model underlying the word problems, and regardless of the numerical characteristics of the

word problems.

Fourth, there are two trends that went parallel with the strong decrease with grade in

“Other” and “Additive” profiles: First, an increase in students giving proportional responses

to all four proportional problems. While this response pattern was absent in third graders,

32.1% of the sixth graders could be characterised as purely proportional reasoners. Second,

with grade there were increasingly more students who adapt their responses to the number

characteristics of the word problems. In third grade, 9.1% of the students responded additively

to problems with non-integer numbers and proportionally to problems with integer ratios, and

this number increased up to fifth grade where 33.3% of the students fell into this category,

with a small decrease to 25.6% in sixth grade. In addition, in fifth and sixth grade,

respectively 16.0% and 12.8% of the students had the earlier mentioned correct/ number-

sensitive solution profile: They gave correct responses when the numbers in the problem form

27 From addition to multiplication integer ratios, while they recur to additive reasoning when this is not the case.

The observation that so many students (i.e. the number-sensitive ones and the

correct/number-sensitive ones) in some way adapted their solution strategy to the number

characteristics of problems – irrespective to the underlying mathematical model – confirms

that it is possible that a student at the same time overuses additive methods (i.e. apply them to

proportional problems) and proportional methods (i.e. apply them to additive problems). A

detailed analysis of students’ answers showed that 19.3% of the third graders, 35.9% of fourth

graders, 44.2% of fifth graders, and 42.2% of sixth graders, made at least one additive error to

a proportional problem and a proportional error to an additive problem.

Finally, the 24.9% of students who were categorised as having a “Other” solution

profile deserve some further consideration. A closer look at their profiles indicated that about

half of them (i.e. 40 out of the 81 students in this category) had given another than the

additive or multiplicative response to at least three out of the four experimental items. So, it

can hardly be argued that students in this subgroup randomly used additive and multiplicative

strategies to solve the experimental word problems. The other 41 students in the “Other”

category, however, may have used additive and multiplicative approaches to the experimental

word problems on an entirely random basis, not taking into account the model underlying the

problem, nor the numbers that were given in the problem. As will be discussed below, this is,

however, impossible to tell from a profile of answers to four experimental problems only.

Conclusions

This study focused on two phenomena that were so far typically studied in isolation: the use

of additive strategies in proportional situations (Hart, 1981; Kaput & West, 1994; Karplus et

al., 1983; Lin, 1991; Tourniaire & Pulos, 1985) and the use of proportional strategies in

additive situations (Van Dooren et al., 2005, 2009). We investigated how both kinds of errors

28 From addition to multiplication developed throughout primary education, and what task characteristics affected the

occurrence of both kinds of errors. This was done using a within-subject design, which

allowed to identify the use of additive and multiplicative models in the same students, and

therefore to get a better view on whether students develop an understanding of the

quantitative relations underlying additive and multiplicative situations, and of the distinctions

between them.

First of all, our results showed that students behaved similarly as reported in the two

literatures: The tendency to apply additive strategies to missing-value word problems

(including proportional problems for which these strategies are incorrect) strongly decreased

with age while the tendency to apply proportional strategies to missing-value word problems

(including additive problems for which these strategies are incorrect) strongly increased with

age. Second, in all age groups, students’ use of additive and proportional strategies depended

strongly on the numbers that were given in the problem statement. When the numbers formed

integer ratios, more proportional strategies were applied, whereas when the numbers did not

form integer ratios, the number of additive strategies increased. The results even indicated that

students – when making a choice for an additive or proportional solution method – looked

more to the numerical characteristics of a word problem than to the additive or proportional

character of the situation described in the word problem. Third, as expected, the sensitivity to

the numerical characteristics showed a curvilinear shape, with an increasing impact of number

characteristics from third to fifth grade, and then a moderate decrease.

Our within-subject design enabled to analyse students’ individual solution profiles.

This latter analysis first revealed that there were only very few students who correctly solved

the four experimental problems, even in the oldest age group. This is remarkable, because as

such, the reasoning and calculations required to solve the problems correctly were not very

29 From addition to multiplication advanced, not even for fourth graders. Almost half of the third graders reasoned additively to

all four experimental problems, but this tendency decreased rapidly and was almost gone in

sixth grade. It was replaced by two other types of reasoning. First, there was a strong increase

in the tendency to apply proportional methods to all the experimental items. While none of the

third graders gave proportional solutions to all problems, this tendency became very

prominent towards sixth grade, where almost one third of the students solved all problems

proportionally. The second trend was an increase in sensitivity to another task characteristic,

namely the numbers that appear in word problems. With increasing age, more students tended

to reason proportionally when the numbers in the word problem form integer ratios and

additively when this is not the case, regardless of the mathematical model underlying the

word problems.

Taken as a whole, these results indicated that there is a development in many students

from applying additive methods “anywhere” in the early years of primary school to applying

proportional methods “anywhere” in the later years. This development could be expected on

the basis of the literature on the development from additive to proportional reasoning on the

one hand and on the growing tendency to use proportional methods, on the other hand. It was

still an open question, however, how the transition would look like: Would students at the

moment when the idea of proportionality is introduced in the curriculum initially use additive

as well as proportional methods appropriately, and only later on – due to the extensive

instructional attention to proportional methods – abandon the additive methods and start using

proportional methods anywhere? Or would the intermediate stage be characterised by the

simultaneous inappropriate use of additive and proportional methods? Our study has shown

that the latter is the case. Between the initial stage where students over-generalise additive

methods and the later stage where they over-generalise proportional methods, students’

30 From addition to multiplication behaviour is not characterised by a correct use of additive and proportional methods. Rather,

they switch between methods, not on the basis of the model underlying the problem situation

but based on a superficial problem characteristic that should not have any impact on their

choice for a solution method, namely whether the numbers that are given in a word problem

form integer ratios or not.

This latter finding means that it is possible that students simultaneously overuse

proportional and additive methods. In almost half of the fifth and sixth graders, both types of

errors were observed. We can, therefore, not attribute students’ erroneous application of

additive methods to proportional problems and their erroneous application of proportional

methods to additive problems to their inability to successfully execute the required

arithmetical operations to obtain the proportional (or additive) response, because students

have unmistakably shown that they are able to execute them. Rather, with increasing age

students’ methods seem to become influenced by an irrelevant problem characteristic, i.e., the

numbers given in a word problem.

Taken as a whole, these results indicate that the large majority of students either relied

solely on additive or multiplicative strategies to solve the entire set of experimental problems,

or apply both strategies but based on irrelevant criteria. Moreover, unlike what is suggested in

previous research, students in our study hardly became better in reasoning proportionally: The

apparent progress that students make between third and sixth grade in applying proportional

strategies goes along with a comparable increase in the erroneous use of proportional methods

to the additive problems.

Theoretical and Methodological Implications

When judging the implications of our findings, one needs to bear in mind the possible

methodological limitations of the study. First of all, our data were gathered within a

31 From addition to multiplication particular, namely Flemish, primary school context. It is possible that in countries where the

topic of proportionality is taught or sequenced differently in the curriculum, the observed

trends are completely absent, less prominent, and/or occur at a different age. So, it would be

interesting to investigate whether our results can be replicated in countries with other

curricula and educational regimes, to see to what extent the observed development from

additive to multiplicative reasoning and the number-sensitivity in the course of this

development are an artefact of particular mathematics curricula and educational regimes. In

this respect, it has recently been found that also in Spain, students are inclined to use

proportional methods to solve additive missing-value word problems and vice versa, and that

this development occurs similarly as in Flemish students, but – remarkably – this

development was found between the first and fourth year of secondary school (age 12-16)

(Fernández, Llinares, Van Dooren, De Bock, & Verschaffel, 2009) while in Flanders it

happened between the ages of 8 and 12.

A second limitation relates to the individual student profiles. In our instrument, we

only used one item for each problem type. As explained in the Methods section, we did this to

avoid that students would learn and adapt their behaviour throughout the test. The drawback,

however, is that the systematic answering profiles could in some cases have been produced

merely by chance, while in other cases students with a systematic answering profile may not

have been detected due to measurement errors. This methodological choice also made it

impossible to detect students who responded completely randomly, giving additive and

multiplicative responses to the experimental word problems regardless of any problem

characteristics. A related issue is that our study was cross-sectional instead of longitudinal.

Our goal was to study the transition from additive to multiplicative approaches in solving

missing-value word problems, but a cross-sectional design cannot gain the rich data on

32 From addition to multiplication individual development that can be gathered from longitudinal studies. A longitudinal design

would, for example, allow to investigate the transition in individual children and, for example,

indicate whether all children go through an intermediate stage of giving number-sensitive

responses or not, and whether there are children who go through an intermediate stage in

which they appropriately use additive and multiplicative methods to additive and

multiplicative problems, before starting to overuse the multiplicative methods.

Third, it needs to be stressed that we used a specific kind of task, namely word

problems dealing with proportionality (ignoring other kinds of multiplicative reasoning),

which were formulated as missing-problems and which were included in a paper-and-pencil

test. In the interpretation of the results, we should also realise that students’ problem-solving

behaviour takes place in a particular socio-cultural c.q. scholastic setting (Lave, 1992). As

already stressed in the theoretical and empirical background section, research has indicated

that students often approach word problems quite mindlessly, and rely on superficial cues to

decide upon the arithmetical operation(s) to be performed (e.g., Greer, 1993; Reusser &

Stebler, 1997; Verschaffel et al., 1994). As argued by Verschaffel et al. (2000), one of the

origins of this tendency to approach word problems mindlessly and superficially is the

impoverished and stereotyped diet of word problems that students encounter day by day in

their mathematics lessons. This stereotyped diet, along with students’ sensitivity to

correlations between irrelevant, surface-level features of problems and the methods used to

solve them (Ben-Zeev & Star, 2001) may then explain the results that we observed. An

interesting pathway for further research may be to investigate whether the transition from

additive to multiplicative reasoning and the sensitivity to number characteristics of problems

occurs similarly in other proportional reasoning tasks than missing-value problems, such as

comparison tasks where four values are given and a judgment on proportionality is asked.

33 From addition to multiplication Studies could also use tasks that are not formulated as word problems, but for instance as

performance tasks. It would moreover be interesting to investigate in such studies how

consistently students reason across these various kinds of tasks.

A final important issue for further research is the degree to which students are aware

of the task characteristics that determine their problem-solving behaviour. When prompted to

do so, would students be able to make the task characteristics that determined their choice for

an additive or a multiplicative approach of the word problems explicit or not? And would

there be any doubt in students between an additive and a multiplicative model, or do they not

even consider the alternative? The answer to these questions is not only of theoretical and

methodological importance but also bears upon the educational implications of our work.

These will be discussed in the next section.

Educational Implications

Even though our study has certain limitations and there are several open questions, our

study has some important educational implications. We strongly agree with Nunes and Bryant

(1996, p. 182-183), that “the problems that are used at school in mathematics exercise books

for teaching children about proportions are often more an excuse to use the arithmetic than a

content for the youngsters to think about”. The majority of word problems related to

proportional reasoning that students encounter in the upper grades of elementary school and

the lower grades of secondary school are very similar. They are typically formulated in a

missing-value format (Cramer et al., 1993), while nonproportional problems stated in a

missing-value format are very rare or even completely absent. When these proportional

problems are treated in classroom, the focus often is on the fluent execution of certain

arithmetic procedures to tackle them, without – at that moment – explicitly and systematically

questioning whether they are applicable. A similar case can be made for the numbers that

34 From addition to multiplication appear in word problems. At the moment when students are introduced in proportionality the

numbers that appear in word problems allow calculations with easy, integer multiplicative

jumps. From an educational perspective, this is understandable: The intention is to focus

students to the recognition of the proportional structure of the situation and to the fluent

application of related procedures, and this could be hindered by doing (too) complex

calculations involving non-integer multiplicative jumps. Students were never explicitly taught

that problems with a missing-value formulation need to be solved multiplicatively, or that

when the numbers in a problem do not form integer ratios the problem should be solved

additively. Rather, these messages may have been conveyed implicitly to students through the

limited range of examples – and the lack of counterexamples – that they have been confronted

with. Fischbein (1987, 1993) introduced the term “figural concept” to refer to the learning of

geometrical concepts where – besides logical features of the examples learners are confronted

with – also perceptual features may play a role in concept formation. This may in turn lead to

misunderstanding as learners may include dominant but irrelevant perceptual features in their

extension of the concept beyond the examples. Herein lies the danger of prototypical

examples. For instance, if learners never encounter another case, they may start to think that if

the sides of a square are not parallel to the sides of the page, this is not a square but a

rhombus. Similarly, a learner’s conception of word problems can be distorted if in the

numerous examples and exercises that they encounter, the problem formulation or the

numbers appearing in the problem are (spuriously) associated with the problem solving

method.

Given the implicit character of the learning process as described above (Seger, 1994),

students most likely are not (entirely) aware of the task characteristics that determine their

choice for an additive or multiplicative approach. A teaching approach would therefore

35 From addition to multiplication probably have to be subtle and take a long term perspective, relying on an appropriate

variation of examples and exercises in order to prevent – rather than remedy – the implicit

learning to occur. Inspiration to achieve this can be found in Mason et al.’s (Goldenberg &

Mason, 2008; Watson & Mason, 2006) work on exemplification. This work builds on Marton

and Booth’s (1997) variation theory and starts from the assumption that just as with natural

language acquisition, meaning of mathematical objects (concepts, theorems, techniques)

arises from the experience of particular instances that learners encounter, rather than (merely)

from defining these mathematical objects. Examples (and exercises alike) can therefore act as

mediating tools between the learner and the mathematical object. Therefore, Goldenberg and

Mason (2008, p. 184) argue that “variation in examples can help learners distinguish essential

from incidental features, and, if well selected, the range over which that variation is

permitted”. They discern two important dimensions to scrutinize the “example space”:

“dimensions of possible variation” and “range of permissible change” (p. 187). The first

dimension relates to the features of an example/exercise that we expect learners to recognize

as eligible for change, without necessarily loosing its examplehood (i.e. without changing the

essential mathematical characteristics). The second dimension relates to how each dimension

of possible variation can be changed.

If we want students to come to grips with the differences between additive and

proportional situations, it may be beneficial to scrutinize and systematically redesign all

examples and exercises related to proportional reasoning that students are confronted with

throughout primary school in terms of these two dimensions. Proportional problems should

regularly appear in various formats (e.g., besides missing-value formats also the comparison

of two ratios in quantitative and qualitative ways, constructing several new ratios equal to a

given one, multiple proportion problems), and various nonproportional problems should also

36 From addition to multiplication be presented regularly in a missing-value format. The same goes for the numbers that appear

in word problems: Numbers that do not form integer ratios can be easily inserted in

proportional problems and numbers that form integer ratios can be inserted in various

nonproportional problems. Even students who are not yet able to do proportional calculations

involving noninteger multiplicative jumps could be confronted with such a variety of

problems. The focus then would be on what calculations should be done with the given

numbers, without the need to actually perform the calculations. Inspired by Greer’s work on

the non-conservation of numbers to counter the “multiplication makes bigger, division makes

smaller” idea in students (Greer, 1987, 1994), an interesting activity could also be to start a

lesson series with a small set of proportional and additive word problems in which particular

numbers are given, and to solve them. The numbers in the problems could then be replaced by

other numbers, and again be solved, and so on. This way, students may notice that given

numbers are not relevant to the way in which a problem should be solved, and that other

characteristics need to be considered. Another promising approach would be to probe and

extend students’ example space (Watson & Mason, 2002) by letting them construct examples

themselves. This would mean that students are engaged in problem posing activities (e.g.,

English, 1997; Silver, 1994). Several national curricular documents plea for (more)

instructional attention to the acquisition of problem-posing skills (e.g., Ministry of Education

of Peoples’ Republic of China, 2001; National Council of Teachers of Mathematics, 2000;

Ministerie van de Vlaamse Gemeenschap, 1997), the most frequent motivation being that

students as a consequence also become better problem solvers. By letting students construct

word problems, compare their constructions, and alter given word problems to fit with

particular solution methods, they might come to see the mathematically relevant features of

37 From addition to multiplication such problems, and appreciate the dimensions of possible variation and ranges of permissible

change.

In addition to all of the above educational implications which are aimed at the

creation of a “wider example space” in students via the selection and sequencing of a well-

considered problem set, students’ conceptual understanding of the difference between additive

and multiplicative can of course be addressed more directly and explicitly at certain points in

the curriculum. A possible approach may be to introduce word problem classification

activities in the classroom. Students could be given a set of (proportional and various

nonproportional) word problems that they have to classify according to self-chosen criteria.

Afterwards, a classroom discussion could deal with the variety of classification criteria used

by students, and particularly with the idea that word problems can be distinguished with

respect to the mathematical model (e.g. multiplicative, additive, …) underlying the problem

situation. A recent study has already shown that merely giving sixth graders a set of

proportional and nonproportional word problems along with the instruction to classify them

(but without any classification criteria being classified), rather than to solve them, has some

positive effects on the performance on parallel problems later on (Van Dooren, De Bock,

Vleugels, & Verschaffel, 2008, in press). Given these results, using a classification task with

more specific instructions and a classroom discussion explicitly directed at the model

underlying word problems seems very promising.


Acknowledgements

This research was partially supported by Grant GOA 2006/01 “Developing adaptive expertise

in mathematics education” from the Research Fund K.U.Leuven, Belgium.

Footnote

1 Disagreements between our categorisation and the cluster analysis categorisation mostly

concerned students that we identified as correct reasoners (a category not identified and used

by the cluster analysis) and minor disagreement on the number-sensitive and correct/number-

sensitive group when students also gave one other answer.


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Table 1

Design and Examples of Experimental Items

Problem

type

Number type Example

Integer (I) Evelien and Tom are ropeskipping. They started together, but Tom

jumps slower. When Tom has jumped 4 times, Evelien has jumped

20 times. When Tom has jumped 12 times, how many times has

Evelien jumped?

Prop

ortio

nal

Non-integer

(N)

A motor boat and a steam ship are sailing from Ostend to Dover.

They departed at the same moment, but the motor boat sails faster.

When the steam ship has sailed 8 km, the motor boat has sailed 12

km. When the steam ship has sailed 20 km, how many km has the

motor boat sailed?

Integer (I) 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?

Add

itive

Non-integer

(N)

Lien and Peter are reading the same book. They read at the same

speed, but Peter started earlier. When Lien has read 4 pages, Peter

has read 10 pages. When Lien has read 6 pages, how many has Peter

read?


Table 2

Overview of solutions given by students (in %) (Correct solutions are indicated in bold)

Proportional problems Additive problems Total

Grade P A O P A O P A O

3 (n = 88) 19.3 48.9 31.8 17.0 54.5 28.4 18.2 51.7 30.1

4 (n = 78) 29.5 43.6 26.9 35.9 46.2 19.9 32.7 44.9 22.4

5 (n = 81) 70.4 19.8 9.9 48.1 34.6 17.3 59.3 27.2 13.6

6 (n = 78) 84.6 10.3 5.1 67.9 19.2 12.8 76.3 14.7 9.0

Inte

ger p

robl

ems

Total (n =

325) 50.2 31.1 18.8 41.5 39.1 19.4 45.8 35.1 19.1

3 (n = 88) 0 59.1 40.9 1.1 79.5 19.3 0.6 69.3 30.1

4 (n = 78) 2.6 70.5 26.6 1.3 80.8 17.9 1.9 75.6 22.3

5 (n = 81) 16.0 60.5 23.5 8.6 80.2 11.1 12.3 75.6 22.3

6 (n = 78) 41.0 38.5 20.5 33.3 57.7 9.0 37.2 48.1 14.7

Non

-inte

ger p

robl

ems

Total (n =

325) 14.5 57.2 28.3 10.8 74.8 14.5 12.6 66.0 21.4

3 (n = 88) 9.7 54.0 36.4 9.1 67.0 23.9 9.4 60.5 30.1

4 (n = 78) 16.0 57.1 26.9 18.6 63.5 17.9 17.3 60.3 22.4

5 (n = 81) 43.2 40.1 16.7 28.4 57.4 14.2 35.8 48.8 15.4

6 (n = 78) 62.8 24.4 12.8 50.6 38.5 10.9 56.7 31.4 11.9 Tota

l

Total (n =

325) 32.3 44.2 23.5 26.2 56.9 16.9 29.2 50.5 20.2


Table 3

Overview of solution profiles of individual students (in %)

Grade

3

(n = 88)

4

(n = 78)

5

(n = 81)

6

(n = 78)

Total (n =

325)

Perfect fit 0.0 0.0 1.2 3.8 1.2

One other 1.3 5.1 4.9 5.1 4.0 Correct

Total 1.3 5.1 6.2 9.0 5.2

Perfect fit 30.7 17.9 11.1 3.8 16.3

One other 15.9 19.2 6.2 2.6 11.1 Additive

Total 46.6 37.2 17.3 6.4 27.4

Perfect fit 0.0 1.3 7.4 19.2 6.8

One other 0.0 1.3 2.5 12.8 4.0 Proportional

Total 0.0 2.6 9.9 32.1 10.8

Perfect fit 3.4 9.0 18.5 19.2 12.3

One other 5.7 11.5 14.8 6.4 9.5 Number-

sensitive Total 9.1 20.5 33.3 25.6 21.8

Correct/number

-sensitive 5.7 5.1 16.0 12.8 9.8

Other 37.5 29.5 17.3 14.1 24.9

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