Proposal Type:	Symposium
Domain:	Assessment and Evaluation
SIG:	Assessment and Evaluation
Type	Submitted Symposium
Title	Assessing potentials for mathematics learning and markers for dyscalculia
Abstract	The majority of the tests address the assessment the performance of specific arithmetical abilities. Not many tools are available to give us an explanation for understanding the errors people make. For young children, an instrument that is validated by a combination of theoretical models and therefore can be used for an in-depth diagnostic assessment seems to be the TEDI-MATH (Van Nieuwenhoven, Gregoire, & Noel, 2001). This multi componential instrument is based on a combination of neuropsychological (developmental) models of number processing and calculation. It has an age range form 4 to 8 years of age (kindergarten to 3rd grade) and has already been translated into a German, Dutch, Spanish and French version. The test highlights five facets of arithmetical and numerical knowledge: logical knowledge, counting, representation of numerosity, knowledge of the numerical system and computation. In this symposium researchers from the French speaking part of Belgium and France, the Dutch speaking part of Belgium, Autriche and Germany put there experiences together with this dyscalculia assessment to look for early prenumerical and numerical predictors of mathematics learning and mathematics learning disabilities. In a first paper Gregoire and Wierzbicki investigate if the Piagetian model of number is useful for assessing mathematical learning and disabilities. In a second paper Stock et al. focus on the role of logic thinking, counting and knowledge of number row on arithmetic abilities in preschools. In a third paper Krinzinger and colleages look at gender differences in acquiring the base-10 system of multi-digit numbers. In a fourth paper Desoete et al. elaborates on the predictive value of the tests all other presenters also used to predict arithmetical reasoning and numerical facility in elementary school children. Finally Gregoire and Meert describe how a developmental model can also help to understand the concept of fractions.
Equipment	PC and projector
Keywords	Assessment Learning difficulties Mathematics education

Chair list
Name	Surname	Institution	Country	E-Mail	EARLI Number
Jacques	Gregoire	Universite catholique de Louvain	Belgium	jacques.gregoire@psp.ucl.ac.be

Organiser list
Name	Surname	Institution	Country	E-Mail	EARLI Number
Jacques	Gregoire	Universite catholique de Louvain	Belgium	jacques.gregoire@psp.ucl.ac.be
Annemie	Desoete	Ghent University	Belgium	annemie.desoete@telenet.be

Discussant list
Name	Surname	Institution	Country	E-Mail	EARLI Number
Marcel	Crahay	Universite de Geneve	Switzerland	Marcel.Crahay@pse.unige.ch

Paper Details

Title	Is the Piagetian model of number useful for assessing mathematical learning and disabilities?
Abstract	This research is related to the development of a diagnostic test assessing the development of numerical abilities from 5 to 8. This test, called TEDI-MATH (Van Nieuwenhoven, Gregoire & Noel, 2001), assess five facets of the numerical development: (1) counting, (2) representation of numerosity, (3) computation, (4) knowledge of the numerical system, and (5) logical operations. The test was standardized in France and French-speaking Belgium. This research investigated the construct validity of the tasks included in the fifth facet, which is related to the Piagetian model of number. Our results showed a very significant relationship between logical reasoning, assed by the Piagetian tasks, and arithmetical abilities at grades 1 and 2. These results emphasized the diagnostic usefulness of Piagetian tasks included in the TEDI-MATH.
Summary	Research question For nearly forty years, the Piagetian theory (Piaget & Szeminska, 1941) was an essential reference for understanding the processes underlying mathematical learning and disabilities. Piaget proposed a psychogenetic explanation of number where logic plays the role of a normative system progressively built by the child. He described the logical abilities that a child progressively acquires and co-ordinates to master the concept of number. This developmental model was increasingly discussed when researchers observed that the children’s performances of Piagetian logical tasks varied strongly according to their verbal and material characteristics (for a review, Fayol, 1990, and Bideaud, 1988). But the main challenge to the Piagetian model came from Gelman and Gallistel’s (1978) research on the counting procedure, and from the development of neuropsychological models of number processing and calculation (e.g., McCloskey, Caramazza & Basili, 1985). These new models of numerical competence have been successfully applied to study the number understanding and the numerical disabilities. Consequently, the validity and utility of the Piagetian theory applied to mathematical learning was questioned. In this research, we studied the usefulness of the Piagetian model to assess mathematical development and discriminate between high and low mathematical achievers. Should we continue to refer to this model for mathematical assessment and understanding of learning disabilities? Hypothesis 1. Children who master the logical operations underlying the number understanding should be better than those who do not master these operations to solve arithmetical additions and subtractions, and word problems. 2. The difference should be observed for each logical operation and for the global logical score. Sample Children (n = 217) were randomly selected from regular classes in France and Belgium. All of them were French-speaking: 74 at the 1st grade were tested in November, 75 at the first grade were tested in May, and 68 at the second grade were tested in November. Tasks We assed five logical abilities considered by Piaget as essential for numerical reasoning: seriation, classification, conservation, inclusion and additive composition of numbers. To assess these abilities we used some tasks created by Piaget and several new tasks we developed for the TEDI-MATH. The arithmetical abilities were assessed by 38 arithmetical items, including additions, subtractions and word problems. Results and discussion For each logical task, the children were classified as “mastery” or “non mastery”. We observed that children classified as “mastery” had, on average, better scores on all the arithmetical tasks than children classified as “non mastery”. Most of the differences were statistically significant. But the effect size varied strongly according to the logical task. This effect was small for the classification task. It was medium for the conservation and the inclusion tasks. It was high for the seriation and the additive composition tasks. These differences between tasks can be related to a lower validity of some tasks. Moreover, the differences on the arithmetical items and the word problems according to the total logical score are very important. The children can be classified in three categories: (1) those who succeeded in one, two or none of the logical tasks, (2) those who succeeded in three of the logical tasks, (3) those who succeeded in four or five logical tasks. Clearly, our results showed that the children who master all the logical operations described by Piaget have much better scores on arithmetical tasks than those who do not master these operations at all. Consequently, our hypotheses were confirmed. But the observed relationship between the logical abilities and the arithmetical tasks was not straightforward. Some children showed rather poor performances on the logical tasks but succeeded to the arithmetical tasks. We also observed the reverse profile. These observations mean that the arithmetical tasks can be solved without relying on the logical abilities (e.g., using counting or retrieving the solution from long term memory). On the other hand, being competent on the logical tasks does not automatically lead to high arithmetical performances. To succeed in arithmetical tasks, the children need some other abilities to supplement their logical ability (e.g., knowledge of the numerical system). This research was conducted on a sample of normal children, without learning disabilities. Further research is needed with children with dyscalculia in order to check the relations we observed with normal children.
Keywords	Assessment Learning difficulties Mathematics education
Appendices

Authors
Name	Surname	Institution	Country	e-mail	EARLI Number	Presenting
Jacques	Gregoire	Universite catholique de Louvain	Belgium	jacques.gregoire@psp.ucl.ac.be		*
Claudine	Wierzbicki	ECPA	France	cwierzbicki@ecpa.fr

Title	The role of logic thinking, counting and knowledge of the number row on arithmetic abilities in preschoolers
Abstract	Besides the Piagetian logical operations, several other prenumerical arithmetic abilities seem to be important in the development of arithmetic. The importance of those abilities is however debated heavily and the relation with important arithmetic domains like conceptual and procedural knowledge still remains obscure. In this study, the relation between six prenumerical arithmetic abilities -knowledge of the number row, counting, seriation, classification, conservation and inclusion- and conceptual and procedural arithmetic knowledge was investigated. 242 preschoolers were assessed with different subtests of the TEDI-MATH. The results show that scores on procedural knowledge can be predicted in almost ninety percent of the children based on the achievement on the six prenumerical arithmetic abilities. For conceptual knowledge, only knowledge of the counting row, seriation and inclusion seem to be important. Longitudinal designs are needed in order to investigate causality and the sentence of the relationships between the different factors.
Summary	Research questions In this study we want to investigate the relations between the different arithmetic domains in preschoolers. We want to asses both the conceptual (knowledge of the number system) and procedural knowledge (the ability to execute simple arithmetic operations) since research has found that a disconnection between procedural competence and conceptual understanding may be characteristic of elementary school children’s developing arithmetic knowledge (Laupa & Becker, 2004). We here investigate the importance of the logical operations, counting abilities and knowledge of the counting row in the development of conceptual and procedural knowledge. We hypothesize that children who have acquired a good understanding of logical thinking, counting knowledge and the number row are better performers on conceptual and procedural knowledge tasks. Method Participants and Instruments This study was been carried out with 116 boys and 126 girls.recruited in 14 different nursery schools.. All children were tested with the TEDI-MATH (Gregoire, Noel & Van Nieuwenhoven, 2004). In this study, we made use of the items on knowledge of the number row, counting, logical operations on numbers, number knowledge and some elementary arithmetic operations. Results All obtained scores were transformed to z-scores. Two multiple regression analysis were conducted in order to evaluate how well the prenumerical arithmetic abilities predicted number knowledge and the execution of arithmetic operations. Six prenumerical arithmetic abilities were included: knowledge of the number row, counting, seriation, classification, conservation and inclusion. Approximately 18% of the variance in number knowledge in the sample could be accounted for by the linear combination of the prenumerical arithmetic abilities. Knowledge of the number row, seriation and inclusion had a significant contribution to the number knowledge. The second multiple regression analysis pointed out that approximately 44% of the variance in arithmetic operations in the sample could be accounted for by the linear combination of the prenumerical arithmetic abilities. All 6 predictors had a significant contribution to the execution of arithmetic operations. Our second research question is whether it is possible to predict the score on number knowledge and the execution of arithmetic operations based on the six predictors discussed earlier. High achieving toddlers had a z-score that was in the first quartile, the children in the group of low achievers all had a score that was in the lowest quartile for this group. A discriminant analysis procedure was performed and the Fischer’s linear discriminant function was used to investigate the accurateness of the predicted classifications. A discriminant analysis was conducted to determine whether three predictors, knowledge of the number row, seriation and inclusion could predict number knowledge. Overall the predictors differentiated among the HA and LA group. Based on the standardised weights of the predictors knowledge of the number row and seriation demonstrate the strongest relationships with number knowledge. The HA for Number knowledge did better on knowledge of the number row, seriation and inclusion than the LA group. Based on the z-scores for these three predictors, 82.7% was classified correctly into the HA or LA group for number knowledge. The multiple regression pointed out that all six prenumerical arithmetic abilities accounted for the variance in the execution of arithmetic operations. A second discriminant analysis pointed out that differentiation between HA and LA on the execution of arithmetic abilities based on the six predictors was possible. Knowledge of the number row, seriation and inclusion seemed to be the strongest predictors, yet they all have a significant contribution. The HA did better on arithmetic operations on the six predictors than children out of the LA group. Based on the z-scores for these six predictors, it was possible to classify 87.4% correctly into the HA or LA group for arithmetic operations. Discussion The analysis points out that a small fifth of the variance in number knowledge is explained by the prenumerical arithmetic abilities. Only three of the six ‘markers’ show significant contributions: knowledge of the number row, seriation and inclusion. For the procedural component, the execution of arithmetic operations, almost half of the variance can be explained. In this analysis, all six prenumerical arithmetic abilities have significant influence. However, based on the scores for those prenumerical arithmetic abilities, rather good predictions of the scores on conceptual and procedural abilities can be made. Children who do perform weak on knowledge of the number row, seriation and inclusion are not good in conceptual knowledge, with more than eighty percent of the predictions classified correctly. For procedural knowledge, the prediction even gets slightly better, almost ninety percent of the toddlers is correctly classified based on knowledge of the number row, counting, seriation, classification, conservation and inclusion. Children who perform badly on those six markers, generally perform weak on the procedural task. Those results underline the importance of the described prenumerical arithmetic abilities. Counting, classification and conservation are important in the development of procedural knowledge but have no significant influence in developing a conceptual framework in arithmetic, where knowledge of the number row, seriation and inclusion seem to have influence on both domains. This study however has a few limitations. The cross sectional method used here impoverishes the conclusions that can be made from the data collected because no causal relationships can be detected. A follow-up of these children in first and second grade is planned in order to rule these things out. In conclusion the importance of several prenumerical arithmetic abilities can be underlined. Based on the scores for knowledge of the number row, counting, seriation, classification, conservation and inclusion, good predictions can be made for procedural knowledge while only knowledge of the counting row, seriation and inclusion seem to be of importance for the prediction of conceptual knowledge.
Keywords	Assessment Learning difficulties Mathematics education
Appendices

Authors
Name	Surname	Institution	Country	e-mail	EARLI Number	Presenting
Pieter	Stock	Ghent University	Belgium	Pieter.Stock@UGent.be		*
Annemie	Desoete	Ghent University	Belgium	annemie.desoete@telenet.be
Herbert	Royers	Ghent University	Belgium	herbert.roeyers@Ugent.be

Title	Gender differences in acquiring the base-10 system of multi-digit numbers
Abstract	Gender differences regarding complex mathematical skills favouring males have been repeatedly reported (e.g. PISA, 2003). A popular neurocognitive explanatory hypothesis is the so-called ‘spatial cognition hypothesis’ (Casey et al., 1992) emphasizing a correlation between spatial cognition and complex computational skills. Considering the spatial orientation of the ‘mental number line’ (Dehaene, 1992) and the fact that successful retrieval of numerosities requires flawless orientation on the mental number line, gender differences should also emerge regarding basic number processing. Upon collecting normative data for the German version of the calculation test TEDI-MATH 875 children aged 4 to 8 were subjected to tasks tapping abstract counting principles, counting skills, number comprehension, computational skills and approximate number comparison. Findings revealed that males outperformed females on the following subtests: transcoding (first and second grade), number comparison (second semester of first grade and second grade), base-10 system (second semester of second grade) and some aspects of exact computation (first and second grade). On the contrary, gender differences did not reach significance in kindergarten and third grade (the latter might be explained by ceiling effects). Overall, our results suggest that males might develop an earlier understanding of the base-10 system, which is essential for the successful solving of complex computations as well. Furthermore, the results will be discussed with reference to the ‘spatial cognition hypothesis’.
Summary	Background It has been repeatedly shown (e.g. PISA, 2003) that males generally outperform females in tasks tapping complex mathematical skills like algebraic word problems. Several different theories aim at explaining these findings (e.g. ‘math fact retrieval hypothesis’; Royer et al., 1999). One popular neurocognitive explanatory hypothesis is the so-called ‘spatial cognition hypothesis’ (Casey et al., 1992). This theory emphasises the role of spatial cognition in complex mathematical skills. Considering the spatial orientation of the ‘mental number line’ (Dehaene, 1992) and the fact that successful retrieval of numerosities requires flawless orientation on the mental number line, gender differences should also emerge regarding basic number processing. The term basic number processing is used here to denote those numerical and arithmetical skills that are already acquired in preschool. Likewise, Geary’s (2000) distinguishes biologically primary (determining small numerosities, understanding ordinal relations between numerosities, counting, simple additions and subtractions) from biologically secondary (base-10 system of multi-digit numbers, arithmetic fact retrieval from long-term memory, arithmetical procedures, word problems) numerical skills, the latter of which are generally acquired during formal schooling. Up to date, no gender differences have been reported for biologically primary numerical skills (i.e., basic numerical skills) and only for some components of biologically secondary numerical skills (e.g., speed of fact retrieval, complex arithmetical skills). Aim of the study was to systematically investigate effects of gender invarious components of biologically primary and secondary numerical skills. Methods 873 children aged 4 to 8 (second semester kindergarten to third grade first semester; n>50 per semester and sex) were subjected to the German version of the dyscalculia test TEDI-MATH (Nuerk, Kaufmann, Graf, Krinzinger, Delazer, & Willmes, in preparation; Belgian original version: Van Niewenhoven, Gregoire, & Noel, 2001). The TEDI-MATH is composed of six task sets (with different subtests each), which are Abstract Counting Principles, Counting Sets of Objects, Number Comprehension, Application of Numerical Concepts, Calculation Skills, and Approximate Magnitude Comparison. Results No gender differences were observed for preschool children. In primary school children, boys outperformed girls on the following subtests: transcoding (first and second grade), number comparison of multidigit numbers (second semester of first grade and second grade), base-10 system (second grade) and subtraction as well as word problems (first semester of second grade). Gender differences did not reach significance in third grade, most probably due to ceiling effects. Discussion Boys exhibited significantly better performance levels than girls in a number of tasks including transcoding, number comparison, knowledge of base-10-system, subtraction and word problems. Interestingly, upon closer inspection of data the gender differences in subtraction and word problems in the first semester of second grade were found to be due to a performance difference in items with two-digit numbers only. We interpret this finding as an earlier mastery of the base-10 system of multi-digit numbers (which is also essential for the successful solving of complex computations) in boys compared to girls. It is important to note that gender differences regarding the mastery of the Arabic base-10 system became only significant for number ranges that have not been explicitly taught at school yet. Overall, we did not find significant performance differences for girls and boys in biologically primary numerical skills. As the processing of multi-digit numbers most likely draws upon spatial skills in order to comprehend their place-value system (e.g. hundreds, decades, and units), our results are in line with the ‘spatial cognition hypothesis’ (Casey et al., 1992) suggesting a male advantage in complex spatial skills. Further research is needed to solve the question whether the earlier mastery of the Arabic place-value system displayed by boys relative to girls is due to their better spatial abilities (Casey et al., 1992) or alternatively, due to gender differences in math related motivation and interests (Geary, 1998; Kimura, 1999) and/or sociocultural factors (Halpern, 2000) Finally, our findings can not be fully explained by the popular conceptualization of the analogue ‘mental number line’ (Dehaene, 1992). Rather, they fit nicely to the hybrid model of number representation proposing an additional decomposed processing of multi-digit numbers (adults: Nuerk, Weger, & Willmes, 2001; children: Nuerk, Kaufmann, Zoppoth, & Willmes, 2004).
Keywords	Assessment Gender Mathematics education
Appendices

Authors
Name	Surname	Institution	Country	e-mail	EARLI Number	Presenting
Helga	Krinzinger	University Hospital RWTH Aachen	Germany	krinzinger@neuropsych.rwth-aachen.de		*
Liane	Kaufman	Innsbruck Medical University	Austria	liane.kaufmann@uibk.ac.at
Hans-Christoph	Nuerk	University of Salzburg	Austria	hc.nuerk@sbg.ac.at
Klaus	Willmes	University Hospital RWTH Aachen	Germany	willmes@neuropsych.rwth-aachen.de

Title	Can we predict arithmetical reasoning and numerical facility in grade 3, 4 and 5 from the (pre)numerical skills two years earlier?
Abstract	In this study in a longitudinal design the predictive validity of TEDI-MATH on arithmetical reasoning and numerical facility two years later is investigated in 240 elementary school children. The study reveals that we can predict the numerical facility and the arithmetic reasoning for 43 and 48% respectively in grade 3, for 49 and 46% respectively in grade 4, and for 22 and 44% respectively in grade 5. Assessing decomposition and estimation skills in grade 1, 2 and 3 can predict to some extend the development of early arithmetical reasoning but has no added value in the prediction of numerical facility in grade 3, 4 and 5, whereas the assessment of calculation skills and verbal and Arabic knowledge of numbers in grade 1, 2 and 3 seems important to predict both domains in middle and upper elementary school children. The study emphasizes the value of the tasks included in TEDI-MATH.
Summary	Research questions In this study we want to investigate the predictive validity of the tasks of TEDI-MATH. We here investigate the importance in the development of arithmetical reasoning and numerical facility in elementary school children. Method Participants and Instruments This study was been carried out with 240 children in 24 at random selected different elementary schools.. All children were tested with the TEDI-MATH (Gregoire, Noel & Van Nieuwenhoven, 2004) in grade 1, 2 or 3. They were tested again two year later (in grade 3, 4 or 5) with a validated test on arithmetical reasoning (KRT-R) and a test on numerical facility (TTR). TEDI-MATH (Gregoire et al., 2004 Flamish adaptation) is a test designed for the diagnostic assessment of arithmetical disabilities from preschool till grade 3. The psychometric value has been demonstrated on a sample of 550 Dutch speaking Belgian children. The Kortrijk Arithmetic Test Revision (Kortrijkse Rekentest Revision, KRT-R) (CAR Overleie,2005) is a Belgian test on mathematical reasoning which requires that children solve mental arithmetics and number knowledge tasks. The psychometric value of the KRT-R has been demonstrated on a sample of 3,246 Dutch-speaking children in total. In the study the standardized total percentile based on Flanders norms was used. The Arithmetic Number Facts test (Tempo Test Rekenen, TTR; de Vos, 1992) is a numerical facility test which requires that children in grade 1 solve as many number fact problems as possible within 2 min. The test has been standardized for Flanders on 10,059 children (Ghesquiere & Ruijssenaars, 1994). Results Grade 1-3 From the 80 children that were tested in grade 1 only 56 (34 girls, 22 boys) could be retraced in grade 3. A regression analysis revealed that we could predict for 43% the numerical facility results in grade 3 from the results of TEDI-MATH two years earliers. Calculation skills predicted 38% of the variance, whereas the verbal component of knowledge of the numerical system added 5% on the prediction. A second regression analysis revealed that we could predict for 48% the arithmetical reasoning skills from the results on the TEDI-MATH two years earlier. The representation of numerosity or estimation skills predicted 41% and the decomposition skills added 6% to the prediction. Grade 2-4 From the 70 children that were tested in grade 2 only 40 (21 girls, 19 boys) could be retraced in grade 4. A regression analysis revealed that we could predict for 49% the numerical facility results in grade 4 from the results of TEDI-MATH two years earliers. Calculation skills predicted 44% of the variance, whereas the verbal component of knowledge of the numerical system added 5% on the prediction. A second regression analysis revealed that we could predict for 46% the arithmetical reasoning skills from the results on the TEDI-MATH two years earlier. The computation skills predicted 30%, respresntation of numerosity added 7%, knowledge of the Arabic numerical system added 7.6% to the prediction. Grade 3-5 From the 90 children that were tested in grade 3 only 62 (30 girls, 32 boys) could be retraced in grade 5. A regression analysis revealed that we could predict for 22% the numerical facility results in grade 4 from the results of TEDI-MATH two years earliers . Computation skills predicted 15% of the variance, whereas the knowledge of the Arabic numerical system added 6.6% on the prediction. A second regression analysis revealed that we could predict for 44% the arithmetical reasoning skills from the results on the TEDI-MATH two years earlier. The computation skills predicted 34.5%, the knowledge of the Arabic numerical system added 5.4% and the knowledge of the verbal component of the numerical system added 4.4% to the prediction. Discussion For the numerical facility component in grade 3 and 4, almost half of the variance can be explained by the results of the assessment in grade 1 or 2 with TEDI-MATH. In this analysis, especially computation and the knowledge of the verbal component of the numerical system have significant influence. In grade 5 about one fifth of the variance in numerical facility scores could be predicted by the results of the assessment in grade 3 with TEDI-MATH. Here again computation has a significant influence. In this dataset not the verbal but the Arabic component of the knowledge of the numerical system has an additional predictive value.
Keywords	Assessment Learning difficulties Mathematics education
Appendices

Authors
Name	Surname	Institution	Country	e-mail	EARLI Number	Presenting
Annemie	Desoete	Ghent University	Belgium	annemie.desoete@telenet.be		*
Pieter	Stock	Ghent University	Belgium	Pieter.Stock@UGent.be

Title	A developmental model of the understanding of the concept of fraction
Abstract	Learning fractions is difficult for a lot of students. Step by step, they understand and structure different meanings of fraction: (1) fraction as a division operator, (2) faction as a ratio, and (3) fraction as a number. This evolution reflects a progressive improvement of the understanding of the concept of fraction, related to an in-depth reorganization of the child’s concept of number. The goal of the current study was to assess the validity of this developmental model, and to enlighten the conceptual understanding of fraction associated to each step. This developmental model should be used as a reference for the diagnostic of mathematical disabilities.
Summary	Research question Learning fractions is difficult for a lot of students. They have to consider the ratio between two quantities, but also to reorganize their previous knowledge about natural numbers. Before discovering fractions, children consider that natural numbers are the only numbers, and their properties are seen as absolute. Learning fractions, the children progressively understand that natural numbers are only a subcategory of a much larger universe of numbers. Step by step, they understand and structure different meanings of fraction: (1) fraction as a division operator, (2) faction as a ratio, and (3) fraction as a number. This evolution is related to a progressive improvement of the understanding of the concept of fraction, leading to an in-depth reorganization of the child’s concept of number. The goal of the current study was to assess the validity of this developmental model, and to enlighten the conceptual understanding of fraction associated to each step. Hypothesis We postulated a slow improvement of the understanding of the concept of fraction from the 4th grade of the primary school to the 3rd grade of the secondary school. At grade 4, we expected that fractions would be only considered as a division operator (e.g., “1/4 means we should divide the pie in four pieces and take one piece”). From grade 5, children should start considering fractions as a ratio, being able to compare factions and to understand that two different fractions are equal if they represent the same ratio. At the beginning of the secondary school, the students should understand that fractions are numbers that can be located on the numerical line. Some conceptual knowledge should only be mastered at this last step (e.g. the density of the rational numbers, i.e. there is an infinity of rational numbers between any couple of numbers) Sample A total of 832 students were randomly selected in primary and secondary schools of the French-speaking region of Belgium: around 130 (65 boys and 65 girls) in each of the following grades: 4th, 5th and 6th grades of the primary school, and 1st and 2nd grades of the secondary school. All these students had to complete a paper-pencil test on factions. A subsample of 279 students was drawn into the larger sample, around 45 at each grade. This subsample was used for an in-depth individual assessment on the conceptual understanding of fractions. Tasks The paper-pencil test, completed by all the students, included 26 items measuring the understanding of fractions. One third of these items only required the understanding of fraction as a division operator to be correctly solved (e.g. shading 1/4 of a square). Another third required the understanding of fraction as a ratio (e.g. circling 6/15 a set of 5 dotes). And the last third required the understanding of fraction as a number (e.g. ordering factions and natural numbers). The individual test included items assessing the conceptual understanding of fractions (e.g. “how many numbers are there between 2/7 and 6/7?”) and the calculation with fractions. Each student first completed the paper-pencil test. Then the students of the subsample completed the individual test. Results and discussion As expected we observed a strong developmental pattern between the 4th grade of the primary school and the 3rd grade of the secondary school. At the 4th grade, most of the students easily solved the items requiring the understanding of fraction as a division operator (success > 90%), but less than 40% solved correctly the items requiring the understanding of fraction as a ratio, and less than 10% solved correctly the items requiring the understanding of fraction as a number. On the other hand, nearly 100% of the students at grade 3 correctly solved the items requiring the understanding of factions as a division operator and as a ratio. And 80% correctly solved the items requiring the understanding of fractions as number. Between these two boundaries, we observed a steady progress across the degrees, with four stages: - 4th grade (primary school): students see faction mainly as a division operator, - 5th grade (primary school): students see fraction mainly as a ratio, but have some difficulties when the format is unusual (e.g., the figure is divided in triangles, instead of the more traditional squares). - 6th grade (primary school) and 1st grade (secondary school): students see fraction mainly as a ratio, but start to see it as a number. - 2nd and 3rd grades (secondary school): most of the students see fractions as numbers, having some specific properties.
Keywords	Assessment Learning difficulties Mathematics education
Appendices

Authors
Name	Surname	Institution	Country	e-mail	EARLI Number	Presenting
Jacques	Gregoire	Universite catholique de Louvain	Belgium	jacques.gregoire@psp.ucl.ac.be		*
Gaelle	Meert	Universite catholique de Louvain	Belgium	gaelle.meert@psp.ucl.ac.be