Proportional reasoning is a key topic both at school and in everyday life. However, students are often misled by their preconceptions regarding proportions. Our hypothesis is that these limitations can be mitigated by working on alternative ways of categorizing situations that enable more adequate inferences. Multiple categorization triggers flexibility, which enables reinterpreting a problem statement and adopting a more relevant point of view. The present study aims to show the improvements in proportional reasoning after an intervention focusing on such a multiple categorization. Twentyeight 4th and 5th grade classes participated in the study during one school year. Schools were classified by the SES of their neighborhood. The experimental group received 12 math lessons focusing on flexibly envisioning a situation involving proportional reasoning from different points of view. At the end of the school year, compared to a control group, the experimental group had better results on the posttest when solving proportion word problems and proposed more diverse solving strategies. The analyses also show that the performance gap linked to the school’s SES classification was reduced. This offers promising perspectives regarding multiple categorization as a path to overtake preconceptions and develop cognitive flexibility at school.
Thinking flexibly and finding solutions in a flexible manner is a highly prominent goal of teaching, and mathematics education is no exception (
Indeed, adaptive expertise in mathematics is akin to finding the solution to a problem in a flexible manner by selecting the most appropriate strategy, and not merely using multiple strategies (
Some studies contrast deep structures and surface features of a situation as two possible directions for categorization to take place (
To help students recognize common structural features among different situations and flexibly switch between different categories, a pedagogical approach can be based on semantic recoding (
Nevertheless, achieving cognitive flexibility can be especially difficult when a situation elicits preconceptions. In fact, children conceptualize most notions taught at school based on prior knowledge, also known as preconceptions (e.g.,
Thus, multiple categorization can be a leverage to overcome preconceptions and adopt a more relevant perspective on the problem. A domain prone to understanding what the constraints imposed by preconceptions are and how the mechanism of multiple categorization plays a role in overtaking them is proportional reasoning (
Proportional reasoning has been long time considered as an activity of high expertise (
This early proportional reasoning however also relies on several preconceptions that are constraining and impose limits for reaching an expertise and flexibility in solving proportional problems in school and in daily life. Four main preconceptions which act as initial categories used for interpreting situations regarding proportionality can be identified.
Proportional reasoning relies on the identification of multiplicative relations (
A second preconception one can associate with proportional reasoning and the concept of ratio is division. Indeed, division is not intuitively categorized as the ratio between two quantities, but as
Another difficulty in grasping proportional reasoning comes from not categorizing a
Finally, the last preconception we will develop concerns the linear property of proportions. In fact, as young as 6, students can solve some missing value proportional problems, especially with an informal proportional reasoning that amounts to a repeated addition strategy (
But even in contexts where the proportional strategy is not valid, students implement the principle of linearity (
In order to attenuate the obstacles imposed by preconceptions in the domain of proportional reasoning, we created a pedagogical intervention based on principles of multiple categorization as a way of increasing flexibility. The intervention program consisted of 12 onehour in class math lessons. These lessons were composed of different written arithmetic word problems. Each lesson focused on one key concept of proportional reasoning in relation to the relevant preconceptions (
Session  Intuitive conception  Expert conception to be build  Objectives  

1  From additive to multiplicative language  More and times more are similar  Times more as a ratio 
Understanding the equivalence between more and less Distinguishing between more and times more 
2  Multiplicative language  Times more as a repeated addition. Times less as a repeated subtraction  Times more / Times less as the search of a ratio 
Move from the "repeated addition" view (3 + 3 + 3 + 3) to the notion of ratio: 4 × 3 Adopt the "times more" and "times less" points of view Understand that multiplication and division correspond to the search for a ratio 
3  Conversion problems  Multiplying to get more and dividing to get less  Multiplying and dividing to find a ratio between 2 quantities 
Understand that multiplication and division are about finding a ratio 
4  Distributivity  Quantizer is not taken into account  Proceed by part (expansion) or grouping (factoring) 
Perceive a quantity as parts and perform an expansion or perceive the whole and perform a factorization 
5  Fraction  Fraction as a bipartite structure (parts/whole)  The fraction as a number of something 
Understand the fraction as a number Identify the fraction of the whole and the fraction of each part 
6  Partitive division and quotitive division  Dividing for sharing  Divide for measuring 
Understand that division addresses not only a situation of sharing but also of quotition 
7  Equivalence between division and multiplying by a fraction  Fraction as a bipartite structure: « 3/4, it is 3 divided by 4 »  Fraction as a multiplication of a fraction and an integer: “3/4 is 3 quarters, is 3 × 1/4” 
Understand that multiplying by a fraction is akin to dividing by the inverse fraction 
8  Proportion  Proportion as the conservation of difference  The proportion as a ratio to be kept 
Understand proportion as a ratio between two quantities 
9  Proportion  3 strategies  Proportion as the conservation of difference  Proportion as the conservation of ratio 
Use 3 different strategies (proportion, fraction, times less) to find the same result 
10  Proportion  3 strategies  Proportion as the conservation of difference  Proportion as the conservation of ratio 
Use 3 different reasonings (times more, times less, proportion) to solve a missing value proportional problem without performing a base unit rate 
11  Proportion  4 strategies  Proportion as the conservation of difference  Proportion as the conservation of ratio 
Use 4 different reasonings (times more, times less, proportion, base unit rate) to solve a missing value proportional problem 
12  Final session  Isomorphic problems to the previous sessions 
In our intervention, the notion of multiple categorization was made explicit to the students through the notion of point of view. For example, in order to learn the reciprocity between multiplication and division, crucial in the construction of the concept of ratio, two points of view can be taken on the following situation: “Jena has 15 marbles and Mateo has 5 marbles”. Taking Jena's point of view, labeled “times more” one can conclude: “Jena has three
Furthermore, comparing and contrasting two solution methods by their efficiency can lead to greater gains in flexibility than studying the solution methods one at a time (
The current study investigated the impact of a pedagogical intervention based on multiple categorization principles as a way of achieving flexibility. It used proportional reasoning as a tool of intervention and investigation. The general rationale was that since difficulties in understanding proportionality are rooted in preconceptions, categorizing situations in alternative ways should make it possible for students to overcome the constraints induced by preconceptions and to adopt strategies aligned with the expert conception of the mathematical concepts.
We expected students in the experimental group to better succeed than students in the active control group. Each group included subgroups created based on grade level and on the school’s SES. First, during the pretest, we expected no difference between the groups and between the different subgroups (Prediction 1). At posttest, the experimental group and subgroups were expected to score higher than the control group (Prediction 2). For each skill measured in the tests, at pretest, we expected no differences between groups and between subgroups (Prediction 3). And at posttest, the experimental group and subgroups should score higher than the control groups and subgroups for each subscore regarding the studied notions (Prediction 4).
Twentyeight French classes participated to the study. 588 students (53% female, mean age 10.5 years,
The experimental and control classes were paired according to the socioeconomic status commonly associated with the context of the participating schools (low SES, middle SES, high SES). In France, most students attend nonpriority education public schools. These are schools with a relatively mixed student population (
Subgroups  Experimental Group 
Control Group 
Total  

4^{th} grade  5^{th} grade  Subtotal  4^{th} grade  5^{th} grade  Subtotal  
Pretest  
Middle SES  36  48  84  29  52  81  165 
Low SES  61  42  103  46  66  112  215 
High SES  54  56  110  54  52  106  216 
Total  151  146  297  129  170  299  596 
Posttest  
Middle SES  37  48  85  30  55  85  170 
Low SES  57  43  100  47  68  115  215 
High SES  55  56  111  54  54  108  219 
Total  149  147  296  131  177  308  604 
Pre and Posttest  
Middle SES  36  48  84  29  52  81  165 
Low SES  57  40  97  44  66  110  207 
High SES  54  56  110  54  52  106  216 
Total  147  144  291  127  170  297  588 
The teachers, who participated in experimental and control groups, did so on a voluntary basis. In each of the subgroups, the selection process for teachers was similar. At the beginning of the year, several projects were presented to them, including the current project. The objective was thus to control for the teacher's "motivation" effect (
The pretest consisted of 17 items for 4^{th} graders and 23 items for 5^{th} graders. The pretest differed between the two grades since at the beginning of the school year, 4^{th} graders have never been taught division, fractions, and proportionality. The posttest was identical for the two grades and consisted of 35 items. The items included in the posttest all required expert conceptions of proportional reasoning to successfully solve the problem. Four items from French national evaluations and 4 items from
Distinguishing between additive and multiplicative structures
Solving distributivity problems
Solving multiplicative problems
Decomposing and comparing fractions
Solving fraction problems
Solving proportion problems
The control and experimental groups took the pretest at the end of the first trimester and the posttest during the last month of school year. The booklets of the tests were composed of a series of problems. Each problem statement was followed by a box to indicate the calculation and a line for the answer statement. In order to control for order effects, 4 booklets were created. At pre and posttest, students were informed that the test was part of a scientific study and were instructed about the importance of completing the calculation. Each item had to be solved in a limited time (2 or 3 minutes depending on the item). The timing was determined based on pilot tests, and it was introduced to limit the total duration of the test. Once the time was up, the experimenter informed the students they should move on to the next exercise without going back to the previous ones. The pretest was administered by the first author. The posttest was divided into two testing sessions to limit the duration of each testing session for the students. Due to the high amount of testing sessions, two additional experimenters were recruited for conducting the experiments in the classrooms. Teachers were present during the administrations of pre and posttest but did not intervene and did not keep copies of the tests.
The control group followed the usual math curriculum. In France, each class has to follow an official mathematics curriculum specified for each grade (
The experimental group participated in 12 lessons of 1 hour over a 5month period. The lessons were part of the teaching hours dedicated to math teaching. The lessons in the middle SES group were entirely conducted by the first author in the presence of the teacher. For the other two groups, half of the lessons were conducted by the first author and half by the teachers. Before the beginning of the intervention, teachers from the experimental classes participated in a 2hour training on preconceptions and multiple categorization, given by the first and last authors. Before each lesson they had to teach, the teachers received a teacher's guide and the necessary material (student worksheets and slides) (
For each problem, the expert strategy – i.e., a strategy that does not rely on preconceptions but requires categorizing the situation in the expert point of view – was defined prior to collecting the data (
A global score (ranging from 0 to 18 points for 4^{th} graders and from 0 to 29 points for 5^{th} graders on the pretest and 40 points on the posttest)
A subscore associated with each studied notion (see
To compare the pretest and posttest which did not contain the same items, a
The data regarding student performance were not independent, since it was the classrooms that were recruited and not individual students Along with checking the equivalence of the two groups at pretest, this also required to check the variance explained by the hierarchical organization of the data (class clustering). At pretest, the
At posttest, the average
To study the interaction between the Time of testing (Pretest vs. Posttest) and Group (experimental vs. control), linear mixedeffects models (
Models  AIC  χ^{2}  

M0: Performance ~ (1  Participants) + (1  Classroom)  3017.5  –  –  – 
M1: Performance ~ Time + (1  Participants) + (1  Classroom)  2941.5  77.964  1  < .001 
M1: Performance ~ Time * Group + (1  Participants) + (1  Classroom)  2844.8  100.721  2  < .001 
M2: Performance ~ Time * Group + Grade + (1  Classroom)  2836.8  10.055  1  < .01 
M3: Performance ~ Time * Group + Grade + SES + (1  Participants) + (1Classroom)  2794.0  46.771  2  < .001 
To better understand the importance of the fixed factors, we then also constructed a model to investigate only the results of the posttest, which included the Group, Grade and SES as the fixed factors and classroom as random factor (
Effect  Name  Estimate  

Fixed  Intercept  0.4243  0.1439  21.6586  2.949  .0075 
Group (Experimental)  0.6893  0.1261  22.3942  5.467  < .001  
Grade (5^{th})  0.5709  0.1164  40.2340  4.905  < .001  
SES (Low)  1.2468  0.1499  20.8792  8.316  < .001  
SES (Middle)  1.0636  0.1634  20.4878  6.508  < .001  
Variance  
Random  Classroom (Interc)  0.07096  0.2664  28  604  
Residual  0.82442  0.9080 
Furthermore, the experimental conditions depending on the grade level (
Groups  

Pretest 
Posttest 

4^{th} Grade  
Experimental group  .07  .93  .39  1.13 
Control group  .03  1.23  .28  .94 
5^{th} Grade  
Experimental group  .11  .86  .94  1.28 
Control group  .02  .79  .21  1.00 
Posttest Group*Grade  Control4^{th} Grade  Control5^{th} Grade  Experimental4^{th} Grade 

Control5^{th} Grade  < 
–  – 
Experimental4^{th} Grade  < . 
.8311  – 
Experimental5^{th} Grade  < . 
< . 
< . 
At posttest, each control subgroup scored a lower
The performance regarding the different SES conducting were then compared, using pairwise comparison with Bonferroni correction for
At pretest, for each SES the experimental group had a similar
Groups  

Pretest 
Posttest 

High SES  
Experimental group  .60  .74  1.45  1.05 
Control group  .89  .77  .66  .89 
Middle SES  
Experimental group  .11  .83  .32  1.06 
Control group  .32  .72  .27  .82 
Low SES  
Experimental group  .49  .75  .08  1.11 
Control group  .60  .75  .43  .90 
Group*SES  SES 


ControlHigh  ControlLow  ControlMiddle  ExperimentalHigh  ExperimentalLow  
Pretest  
ControlLow SES  –  –  –  –  
ControlMiddle SES  1.00  –  –  –  
ExperimentalHigh SES  1.00  –  –  
ExperimentalLow SES  1.00  1.00  –  
ExperimentalMiddle SES  1.00  
Posttest  
ControlLow SES  –  –  –  –  
ControlMiddle SES  1.00  –  –  –  
ExperimentalHigh SES  –  –  
ExperimentalLow SES  .1467  1.00  –  
ExperimentalMiddle SES  1.00  1.00 
Furthermore, the results revealed that the performance gap between the three SES among the experimental groups was maintained at posttest. However, differences were observed in the gap among different SES subgroups between the control and experimental groups. The middle SES experimental group had a lower
Then, each proportional reasoning subscore was analyzed at pretest and posttest (
Subscores according to studied notions  Control Group 
Experimental Group 


Posttest  
Distinguishing between additive and multiplicative structures  .58  .57  .33  .72  .86  .30  32272  < .001*** 
Solving distributivity problems  .31  .25  .30  .53  .50  .35  27891  < .001*** 
Solving multiplicative problems  .54  .50  .31  .63  .67  .30  35596  < .001*** 
Decomposing and comparing fractions  .54  .48  .27  .57  .22  .27  39626  .053 
Solving fraction problems  .10  .00  .15  .23  .50  .22  26568  < .001*** 
Solving proportion problems  .14  .13  .15  .28  .25  .27  30378  < .001*** 
Pretest  
Distinguishing between additive and multiplicative structures  .48  .43  .31  .50  .57  .30  42722  .42 ns 
Solving distributivity problems^{a}  .23  .00  .29  .26  .00  .28  11654  .29 ns 
Solving multiplicative problems  .44  .40  .30  .45  .40  .30  43768  .32 ns 
Decomposing and comparing fractions  .60  .67  .29  .51  .67  .35  50750  < .01** 
Solving fraction problems  .52  .60  .24  .52  .54  .22  44986  .76 ns 
Solving proportion problems  
Proportion problem^{a}  .04  .00  .10  .03  .00  .09  12532  .78 ns 
Graphic proportional situation  .77  1.00  .28  .81  1.00  .29  40076  .02* 
^{a}Items taken by 5^{th} graders only.
Therefore at pretest: there was no difference on 4 subscores of the studied notions with a superiority of the control group for the skill "Decomposing and comparing fractions" and a superiority of the experimental group for only one item – "solve a graphical situation of proportionality". These results partially confirm the first part of Prediction 3.
At posttest, the experimental group had a significantly higher mean than the control group for 5 out of 6 subscores regarding the studied notions and with a significant trend (
Each subscore by grade and SES were also analyzed with MannWhitneyWilcoxon tests. At pretest, the results between the subgroups (by level or type of school) are similar (1242 <
At posttest, for each subscore, each experimental subgroup got better scores than the control subgroup, except for the comparison of the middle SES groups on the subscore regarding the studied notion "Decomposing and comparing fractions". On 30 comparisons, 23 comparisons are significant (1565.5 <
The present study was conducted to investigate to which extent a pedagogical intervention based on multiple categorization might improve students’ mathematical flexibility. This intervention focused on proportional reasoning, for which a wide set of preconceptions might hinder students to use an appropriate strategy to find the solution. In fact, preconceptions often lead to problems being categorized based on superficial features and precludes the possibility to consider an alternative, more adequate solving strategy, which would be consistent with the expert point of view. Therefore, teaching students to analyze the notions related to proportional reasoning from different points of views, each point of view being the hallmark of categorizing the problem in a different manner, was expected to lead students to be in position to flexibly adopt relevant strategies. Namely, it was expected that the intervention would allow students to adopt strategies that are outside the scope of the intuitive conception, but consistent with an alternative categorization in line with an expert point of view.
Fourth and fifth graders from three different social backgrounds took part in the study. The experimental classes benefited from 12 lessons based on multiple categorization to guide them in overcoming their initial point of view and build an alternative one, that they could adaptively refer to when the initial one reveals to be inadequate for finding the solution. The performance of the experimental and active control groups was compared before and after the intervention. The results revealed that the control and experimental groups had homogeneous performance at pretest. At posttest, the experimental group outperformed the control group and this was consistent among the different grades and the different SES of the schools. This suggests that the pedagogical intervention based on multiple categorization had a beneficial influence on students from the experimental group when it came to building a better understanding of proportionality. In the current study these observations were made using written word problems. Yet, multiplicative thinking and proportional reasoning is crucial in realworld situations such as financial contexts or when assessing risk taking (
Additionally, this research supports the idea that to develop flexibility on problem solving in school contexts, it helps to dispose of several solving strategies. Indeed, students were encouraged to adopt as many strategies as possible by adopting different points of view. The wide variety of possible solutions was not simply the result of pooling together the different strategies proposed by different students, but all students had to propose several strategies. As a result, at posttest, more than one third of the experimental students proposed two strategies to solve distributivity problems. It was three times more than students from the control group. Additionally, for missing value proportional problems, no students from the control group succeeded to propose two strategies, whereas one seventh of the students from the experimental group succeeded. Thus, it seems that experimental students were not restricted to the first point of view induced by the problem and developed more flexible strategies. In addition, one can note that at posttest, the experimental 4^{th} grade group reaches a similar level to the 5^{th} grade control group. Finally, although the gap between the different SES subgroups remained significant across the experimental subgroups, the performance gaps between the experimental and control groups by SES subgroups have narrowed. The process of strategy selection among several strategies has also received much attention in works about conceptual and procedural knowledge in mathematics and their relations. In the latter, flexibility has been underlined as a crucial point in
Indeed, when a problem can be solved with several strategies, it can be particularly beneficial to work on the conceptual knowledge to which each strategy is attached. This was precisely done in the current study when students were introduced to points of view reflecting the different conceptions. Only after identifying these points of view were mathematical strategies associated with each point of view. This approach is in line with other findings which stress that flexibility cannot simply refer to the smooth transition between several strategies, but that achieving flexibility mobilizes the complex relations between conceptual and procedural knowledge (
Item's types  Statements  Expert conceptions  Expert strategies  Pretest 
Posttest 
Score  

4^{th} grade  5^{th} grade  4^{th} and 5^{th} grades  
Comparison problem  Times more  There are 4 apples on the table and there are 5 times more oranges. How many oranges are there?  Distinguish more and times more  v  x  v  1  
Comparison problem  Ratio  Maria's train has 3 wagons and Lucas' train has 15 wagons. Which one has more? How many times more?  v  x  v  1  
Comparison problem  More  There are 6 cookies and 18 napkins on the table. But there are also apples and candies. There are 3 more apples than cookies. There are 3 times more candies than cookies. 1) How many apples are there?  v  v  v  1  
Comparison problem  Times more  There are 6 cookies and 18 napkins on the table. But there are also apples and candies. There are 3 more apples than cookies. There are 3 times more candies than cookies. 2) How many candies are there?  v  v  v  1  
Comparison problem  Ratio  There are 6 cookies and 18 napkins on the table. But there are also apples and candy. There are 3 more apples than cookies. There are 3 times more candies than cookies. 3) Are there more cookies or napkins? How many times more?  v  v  v  1  
Comparison problem  Difference  Amin has 11 marbles. Julian has 22 marbles. Julian has 7 more marbles than Leo. 1) How many more marbles does Julian have than Amin?  v  v  v  1  
Comparison problem  More with unknown reference set  Amin has 11 marbles. Julian has 22 marbles. Julian has 7 more marbles than Leo. 2) How many marbles does Leo have?  More indicates a difference  v  v  v  1 
Item's types  Statements  Expert conceptions  Expert strategies  Pretest 
Posttest 
Score  

4^{th} grade  5^{th} grade  4^{th} and 5^{th} grades  
Distributivity problem  Variable "Distance"  A team of 4 athletes participated in a rally: each athlete ran an 8 km loop, then a 2 km straight line and finally a 3 km loop. How many km did the team run in total?  Understanding multiplicative quantifiers: expansion or factoring  x  v  v  1  
x  v  v  1  
Distributivity problem  Variable "Duration"  A school director has been keeping a list of purchases list over the past 6 years. Each year, he purchased 2 computers, 4 printers, 7 screens. How many items has the school director purchased in total?  x  v  1  
x  v  1 
Item's types  Statements  Expert conceptions  Expert strategies  Pretest 
Posttest 
Score  

4^{th} grade  5^{th} grade  4^{th} and 5^{th} grades  
French evaluation's item: Multiplicative problem  A farmer puts 12 eggs in each box. When he finishes, he counts his boxes and finds 5. How many eggs did he put away?  Product  v  v  v  1  
Multiplicative problem  A seller puts 6 chocolates in each box. When he finishes, he counts his boxes and sees there are 13 boxes. How many chocolates did he put away?  v  v  v  1  
Partitive division problem  There are 8 pieces of cake and there are 4 people. How many pieces of cake will each person get?  Division  v  v  v  1  
Quotitive division problem  We have 90 envelopes. We are making piles of 15 envelopes. How many piles can we make? 
Quotitive division  v  v  v  1  
Quotititive division problem  With a package of 90 pictures, we are making piles of 6 images. How many piles can we make? 
v  v  v  1  
Division problem with remainder  90 students must be transported with 40seat buses. How many buses are needed to transport all the students?  Partitive division with remainder  x  v  v  1 
Item's types  Statements  Expert conceptions  Expert strategies  Pretest 
Posttest 
Score  

4^{th} grade  5^{th} grade  4^{th} and 5^{th} grades  
Fraction comparison  I am filling small bags of flour from a large bag of flour. A. The green bag weighs 4/13 of the large bag. B. The blue bag weighs 8/9 of the large bag. C. The red bag weighs 3/7 of the large bag. 
Analysing the ratio (the magnitude of a fraction)  x  v  v  1  
Fraction comparison  Here are three fractions. A. The fraction: 7/8 B. The fraction 3/14. C. The fraction: 2/5. 
Analysing the ratio (the magnitude of a fraction)  x  v  v  1  
TIMMS 2015  M041298: Geometric figures  Part of a whole composed of equal parts  v  v  v  1 
Item's types  Statements  Expert conceptions  Expert strategies  Pretest 
Posttest 
Score  

4^{th} grade  5^{th} grade  4^{th} and 5^{th} grades  
TIMMS 2015  M041065: Fraction of a geometric figure  TIMMS: Which of these circles has 3/8 of its area colored in?

Fraction as a ratio of 2 numbers  v  v  x  1  
Additional question for 5th graders: Which of these circles has 2/3 of its area colored in? __________  Fraction as a ratio of 2 numbers  x  v  x  1  
French national evaluation item: from drawings to fractions  The fraction of one unit  x  x  v  1  
French national evaluation item: Fraction decomposition  5/4 = 1 + __________  A unit is a fraction such as a/a  x  x  v  1  
Several numerical representations for the same fraction  A fraction has an infinite number of representations  x  x  v  1 
Item's types  Statements  Expert conceptions  Expert strategies  Pretest 
Posttest 
Score  

4^{th} grade  5^{th} grade  4^{th} and 5^{th} grades  
TIMMS 2011 M041299: Fraction addition problem  Tom ate 1/2 of the cake and Jane ate 1/4 of the cake. How much of the cake did they eat altogether?  Fraction as a number  x  x  v  1  
Fraction decomposition problem  How many quarter hours are there in 1 hour and 15 minutes? Justify your answer.  A unit can be written in the form of a fraction  x  x  v  1  
Fraction decomposition problem  Tom ate 1/2 of the cake. And Jane ate 1/4 of the cake. Between them, what fraction of the cake did they eat?  A fraction can be written as a number multiplied by a fraction (a x 1/b)  x  x  v  1  
Fraction multiplication problem  18 people eat a third of a pizza each. How many pizzas are there? 
Multiplication is a product between two numbers (integers and/or fractions)  x  x  v  1  
Fraction division problem  easy version  There are 4 pieces of pizza and there are 8 people. How many slices can each person have?  The result of a division can be a fraction  v  v  v  1  
Fractional division problem  difficult version  18 people want to share 3 cakes. How much of the cake will each of them get? 
x  x  v  1 
Item's types  Statements  Expert conceptions  Expert strategies  Pretest 
Posttest 
Score  

4^{th} grade  5^{th} grade  4^{th} and 5^{th} grades  
Fraction division problem  At snack time, there are 2 cakes: one with nuts and one with grapes. Julia, Ylies and Mylan want to share the cakes. But Julia is allergic to nuts. They all 3 want to eat the same number of slices. How can they do that?  Fraction of each part  x  x  v  1  
Half of  Circle half of the stars.

Fraction of a whole and fraction of each part  v  v  v  1  
A quarter of  Circle a quarter of the hats. 
v  v  v  1 
Item's types  Statement  Expert conceptions  Expert strategies  Pretest 
Posttest 
Score  

4^{th} grade  5^{th} grade  4^{th} and 5^{th} grades  
Drawing  MCQ  Reasoning about ratio  v  x  x  1  
Missing value proportion problem  16 pens cost 48€. The pens are all identical. They weigh 20 grams each. How much do 4 pens cost? Can you suggest a second way to solve the problem?  Conservation of ratio  x  v  v  1  
x  v  v  1  
Missing value proportion problem  12 kg is the weight of 36 chairs. The chairs are all identical. They cost 20€ each. How many chairs are there if the weight is 4kg?  x  v  v  1  
x  v  v  1 
Item's types  Statement  Expert conceptions  Expert strategies  Pretest 
Posttest 
Score  

4^{th} grade  5^{th} grade  4th and 5th grades  
French national evaluation item: missing value proportion problem  6 identical objects cost 150€. How much do 9 of these objects cost? Can you suggest a second way to solve the problem?  Conservation of ratio  x  v  v  1  
x  v  v  1  
TIMMS 2011  M031183: Complete a recipe  Division of a fraction  x  v  v  1  
Proportion problem  90 students must be transported in 40seat buses. If the first buses are all full, which proportion of the last bus will be full?  Proportion is a ratio  x  x  v  1 
The work carried out by the first author has been financed by a doctoral contract from the Université Paris Lumières.
The empirical work has been carried in accordance with the relevant ethical principles and standards of University Paris 8.
The authors have declared that no competing interests exist.
The authors have no additional (i.e., nonfinancial) support to report.