Diss. (sammanfattning) Stockholm : Karol. inst., 2002. / Härtill 4 uppsatser.

Diss. (sammanfattning) Stockholm : Karol. inst., 2005. / Härtill 4 uppsatser.

Mémoire (M.Ed.)--Université du Québec à Chicoutimi, 1998. / Bibliogr.: f. 122-134. Document électronique également accessible en format PDF. CaQCU

Made available in DSpace on 2015-03-03T15:36:11Z (GMT). No. of bitstreams: 1 AbraaoEBRA_DISSERT.pdf: 2596166 bytes, checksum: 8d00c61db09f9a300447defa62ee568b (MD5) Previous issue date: 2013-04-08 / The objective of this work if constitutes in creation a proposal for activities, in the discipline of mathematics, for the 6th year of Elementary School, that stimulates the students the develop the learning of the content of fractions, from the awareness of the insufficiency of the natural numbers for solve several problems. Thus, we prepared a set with twelve activities, starting by the comparison between measures, presenting afterward some of the meanings of fractions and ending with the operations between fractions. For so much, use has been made of materials available for use in the classroom, of forma ludic, for resolution of challenges proposed. Through these activities, it becomes possible students to recognize the necessity of using fractions for solve a amount larger of problems / O objetivo deste trabalho se constitui na cria??o de uma proposta de atividades, na disciplina de matem?tica, para o 6? ano do Ensino Fundamental, que estimule os estudantes a desenvolver a aprendizagem do conte?do de fra??es, a partir da conscientiza??o da insufici?ncia dos n?meros naturais para resolver diversos problemas. Desta forma, elaborou-se um conjunto com doze atividades, partindo do processo de compara??o entre medidas, depois apresentando alguns dos significados de fra??es e finalizando com as opera??es entre as fra??es. Para tanto, fez-se uso de materiais dispon?veis a utiliza??o em sala de aula, de forma l?dica, para resolu??o dos desafios propostos. Atrav?s destas atividades, torna-se poss?vel aos discentes reconhecer a necessidade do uso das fra??es para solucionar uma maior quantidade de problemas

De nombreux chercheurs se sont intéressés à la Combinatoire des mots aussi bien d'un point de vue théorique que pratique. Pendant plus de $100$ ans de recherche, de nombreuses familles de mots ont été découvertes, certaines sont infinies et d'autres sont finies. Dans cette thèse, on s'intéresse aux mots de Christoffel. On aborde aussi les mots de Lyndon et les mots Strumians standards. Dans cette thèse, nous donnons de nombreuses propriétés sur les mots de Christoffel et on approfondit l'étude de la notion d'équilibre. Il est connu que les mots de Christoffel sont des mots équilibrés sur un alphabet binaire et sont formés par la discrétisation de segments de droite de pente rationnelle. Les mots de Christoffel sont aussi retrouvés dans l'étude de la synchronisation de k processus dirigé par k mots équilibrés. Pour k=2, on retombe sur les mots de Christoffel, tandis que pour k>2, la situation est plus compliquée et nous amène à la conjecture de Fraenkel qui est ouverte depuis plus de 40 ans. Comme c'est difficile d'atteindre cette conjecture, alors nous avons cherché à construire des outils qui nous aide à s'approcher de cette conjecture. On introduit ainsi la matrice d'équilibre B_w où w est un mot de Christoffel et la valeur maximale de cette matrice est l'ordre d'équilibre du mot binaire utilisé. Comme les mots de Christoffel sont équilibrés alors la valeur maximale dans ce cas là sera égale à 1 et chaque ligne de cette matrice sera formée des mots binaires. Cela nous pousse à tester de nouveau l'ordre d'équilibre de chaque mot obtenu et une nouvelle matrice est obtenue qui s'appelle matrice d'équilibre du second ordre . Cette matrice admet de plusieurs propriétés et de symétries et a une forme particulière comme on est capable de la partager en $9$ blocs où c'est suffisant de savoir 3 parmi eux pour construire le reste. Ces trois blocs correspondent à des matrices de mots de Christoffel qui se trouvent dans des niveaux plus proches de la racine de l'arbre des mots de Christoffel. La valeur maximale de cette nouvelle matrice U_w est appelée équilibre du second ordre. En regardant les chemins qui minimisent cette valeur tout au long de l'arbre, on remarque que le chemin suivi par les fractions obtenues du rapport des nombres consécutifs de la suite de Fibonacci, appelé chemin de Zig-zag est l'un des chemins minimaux. On retrouve ces chemins géométriquement sur le chemin de Christoffel en introduisant une nouvelle factorisation pour les mots de Christoffel appelée la factorisation standard symétrique. Nous avons, également, pu trouver une relation directe entre la matrice U_w et le mot de Christoffel initial sans passer par la matrice B_w et cela en étudiant l'ensemble des vecteurs abéliens associés. Tout ce travail nous a permis de réfléchir au sujet initial qui est la synchronisation de k mots équilibrés. Ainsi, pour le cas de 3 générateurs, nous avons pu étudier tous les cas possibles de la synchronisation et une discussion bien détaillée est faite en utilisant un nouvel élément appelé la graine qui est la première colonne de la matrice de synchronisation. La matrice du second ordre d'équilibre, avec toutes ses propriétés va être un bon outil pour étudier la synchronisation de k générateurs et cela constitut mon projet de recherche dans le futur. Nous avons aussi utilisé toutes nos connaissances autour des mots de Christoffel pour avancer dans la reconstruction de polyominoes convexes. Comme le contour d'un tel polyomino est formé des mots de Christoffel de pentes décroissantes, on a introduit un nouvel opérateur qui modifie ce chemin tout en gardant la décroissance des pentes c'est-à-dire en conservant la convexité qui est un premier pas vers la reconstruction. / Many researchers have been interested in studying Combinatorics on Words in theoretical andpractical points of view. Many families of words appeared during these years of research some ofthem are infinite and others are finite. In this thesis, we are interested in Christoffel words andwe introduce the Lyndon words and Standard sturmian words. We give numerous properties forthis type of words and we stress on the main one which is the order of balancedness. Well, itis known that Christoffel words are balanced words on two letters alphabet, where these wordsare exactly the discretization of line segments of rational slope. Christoffel words are consideredalso in the topic of synchronization of k process by a word on a k letter alphabet with a balanceproperty in each letter. For k = 2, we retrieve the usual Christoffel words. While for k > 2, thesituation is more complicated and lead to the Fraenkel’s conjecture that is an open conjecturefor more than 40 years. Since it is not easy to solve this conjecture, we were interested in findingsome tools that get us close to this conjecture. A balance matrix B w is introduced, where wis a Christoffel word, and the maximal value of this matrix is the order of balancedness of thebinary word. Since Christoffel words are one balanced then the maximal value obtained in thismatrix is equal to 1 and all the rows of this matrix is made of binary words. Testing again thebalancedness of these rows, a new matrix arises, called second order balance matrix. This matrixhas lot of characteristics and many symmetries and specially the way it is constructed since it ismade of 9 blocks where three of them belong to some particular Christoffel words appearing insome levels closer to the root of the Christoffel tree. The maximal value of this matrix is calledthe second order of balancedness for Christoffel words. From this matrix and this new orderof balancedness, we were able to show that the path followed by the fractions obtained fromthe ratio of the consecutive elements of Fibonacci sequence is a minimal path in the growth ofthis second order. In addition to that, these blocks are geometrically found on the Christoffelpath, by introducing a new factorization for the Christoffel words, called Symmetric standardfactorization. Similarly, we worked on finding a direct relation between the second order balancematrix U w and the initial Christoffel word without passing by the balance matrix B w but bystudying the set of factors of abelian vectors. All this work allow us to think about the initialtopic of research which is the synchronization of k balanced words. A complete study for the casek = 3 is given and we have discussed all the possible sub-cases for the synchronization by givingits seed, which is the starting column of the synchronized matrix. The second order balancematrix, with all its properties and decompositions form a good tool to study the synchronizationfor k generators that will be my future project of research. We have tried to use all the knowledgewe apply them on the reconstruction of digital convex polyominoes. Since the boundary wordof the digital convex polyominoe is made of Christoffel words with decreasing slopes. Hencewe introduce a split operator that respects the decreasing order of the slopes and therefore theconvexity is always conserved that is the first step toward the reconstruction.

Les paramètres thermodynamiques et les cinétiques d’interactions entre des polyphénols (des procyanidines et des anthocyanes) et des fractions pectiques ont été déterminés par des méthodes physico-chimiques en solution et sur support solide. Les expériences ont été réalisées avec les anthocyanes majoritaires de cassis ou des procyanidines de type B (extraites de pomme) avec différents degrés de polymérisation et des fractions pectiques issues de pectines de pomme présentant divers degrés de méthylation et différentes chaines latérales d’oses neutres.Les interactions entre les anthocyanes et les pectines sont influencées par les substituants osidiques et le nombre de groupements hydoxyles du noyau B des anthocyanes ainsi que par la composition des fractions pectiques. Les constantes d’affinité entre les pectines et les procyanidines en solution sont les plus élevées quand le degré de polymérisation des procyanidines et le degré de méthylation des pectines sont les plus élevés. De plus, le niveau de ramification des pectines limite leur association avec les procyanidines. Ces interactions impliquent à la fois des liaisons hydrogènes et des interactions hydrophobes. Après les modifications chimiques et l’immobilisation des procyanidines sur une surface, les unités de résonance obtenues par résonance plasmonique de surface entre la (-)-épicatéchine ou le dimère DP2 et les pectines de pomme sont similaires à celles obtenues avec la protéine riche en proline IB5, mais plus faible qu’avec la sérum albumine bovine. / Thermodynamical parameters and kinetics of interactions between polyphenols (procyanidins and anthocyanins) and pectic fractions were defined by physico-chemical methods in solution and with a solid support. Experiments confronted major anthocyanins presents in blackcurrant or B-type procyanidins from apple with various degrees of polymerization to pectic fractions from apple pectins presenting different degree of methylation and different neutral sugar side chains.Interactions between anthocyanins and pectins are influenced by the glycosyl substituent and the number of hydroxyl groups in the B-ring of anthocyanins, and by the composition of pectins. The affinity constant of procyanidins – pectins interaction in solution are the highest when both the procyanidins degree of polymerization and the pectins degree of methylation are the highest. Moreover, the ramification state of pectins limits their association with procyanidins. Those interactions are due to hydrogen bonds and hydrophobic interactions.After chemical modifications of procyanidins and immobilization on a solid support, the resonance units obtained by surface plasmon resonance between (-)-epicatechin or dimer B2 and apple pectins are similar to those obtained with the prolin-rich protein IB5 and lower than with bovine serum albumin.

This thesis is submitted to the University of Manchester for the degree of Doctor of Philosophy (PhD). This study developed an experimental small group teaching method in the Realistic Mathematics Education tradition for teaching fractions using models and contexts to year 7 children in Brunei (N=89) whose effectiveness was evaluated using a treatment-control design: the E1 group was given the experimental lessons, the E2 group who was given “normal” lessons taught by the experimenter, and a whole class (E3) group which acted as the control group. The experimental teaching was video recorded and subject to semiotic analysis, aiming to describe the objectifications that realized ‘learning of fractions’ by the groups.The research addresses two research questions:1. How effective was the experimental teaching in helping learners make sense of fractions, with respect to equivalence of fractions and flexibility of unitizing?2. What were the semiotic learning and teaching processes in the experimental group of the RME-like lessons? This study used a mixed method approach with a quasi-experimental design (QED) for the quantitative side, and a semiotic analysis for the qualitative side. Quantitatively, the experimental teachings proved to be relatively effective with an effect size of 0.6 from the pre- to the delayed post-teaching test, compared to the E2 and the control groups.The basic findings pertaining to the semiotic analyses were:a. The mediation of the production of fractions in terms of length, from the production of fractions in terms of the number of parts which led to equivalence of fractions;b. The use of language and gesture help to objectify the equivalence of fractions and the flexibility of unitizing–in some case it involved gesturing to the self;c. The role of the Hour-Foot clock (HFC) as a model in a realistic context; andd. The complexity of the required chains of objectifications reflects the difficulties of the topic.

In 1978, only 24% of 8th grade students in the United States correctly answered whether 12/13+7/8 was closest to 1, 2, 19, or 21 (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980). In 2014, only 27% of 8th grade students selected the correct answer to the same problem, despite the ensuing forty years of effort to improve students’ conceptual understanding (Lortie-Forgues, Tian, & Siegler, 2015). This is troubling, given that 5th grade students’ fraction knowledge predicts mathematics achievement in secondary school (Siegler et al, 2012) and that achievement in math is linked to greater life outcomes (Murnane, Willett, & Levy, 1995). General rational number knowledge (fractions, decimals, percentages) has proven problematic for both children and adults in the U.S. (Siegler & Lortie-Forgues, 2017). Though there is debate about which type of rational number instruction should occur first, it seems it would be beneficial to use an integrated approach to numerical development consisting of all rational numbers (Siegler, Thompson, & Schneider, 2011). Despite numerous studies on specific types of rational numbers, there is limited information about how students translate one rational number notation to another (Tian & Siegler, 2018). The present study seeks to investigate middle school students’ understanding of the relations among fraction, decimal, and percent notations and the influence of a daily, brief numerical magnitude translation intervention on fraction arithmetic estimation. Specifically, it explores the benefits of Simultaneous presentation of fraction, decimal, and percent equivalencies on number lines versus Sequential presentation of fractions, decimals, and percentages on number lines. It further explores whether rational number review using either Simultaneous or Sequential representation of numerical magnitude is more beneficial for improving fraction arithmetic estimation than Rote practice with fraction arithmetic. Finally, it seeks to make a scholarly contribution to the field in an attempt to understand students’ conceptions of the relations among fractions, decimals, and percentages as predictors of estimation ability. Chapter 1 outlines the background that motivates this dissertation and the theories of numerical development that provide the framework for this dissertation. In particular, many middle school students exhibit difficulties connecting magnitude and space with rational numbers, resulting in implausible errors (e.g., 12/13+7/8=1, 19, or 21, 87% of 10>10, 6+0.32=0.38). An integrated approach to numerical development suggests students’ difficulty in rational number understanding stems from how students incorporate rational numbers into their numerical development (Siegler, Thompson, & Schneider, 2011). In this view, students must make accommodations in their whole number schemes when encountering fractions, such that they appropriately incorporate fractions into their mental number line. Thus, Chapter 1 highlights number line interventions that have proven helpful for improving understanding of fractions, decimals, and percentages. In Chapter 2, I hypothesize that current instructional practices leave middle school students with limited understanding of the relations among rational numbers and promote impulsive calculation, the act of taking action with digits without considering the magnitudes before or after calculation. Students who impulsively calculate are more likely to render implausible answers on problems such as estimating 12/13+7/8 as they do not think about the magnitudes (12/13 is about equal to one and 7/8 is about equal to one) before deciding on a calculation strategy, and they do not stop to judge the reasonableness of an answer relative to an estimate after performing the calculation. I hypothesize that impulsive calculation likely stems from separate, sequential instructional approaches that do not provide students with the appropriate desirable difficulties (Bjork & Bjork, 2011) to solidify their understanding of individual notations and their relations. Additionally, in Chapter 2, I hypothesize that many middle school students are unable to view equivalent rational numbers as being equivalent. This hypothesis is based on the documented tendency of many students to focus on the operational rather than relational view of equivalence (McNeil et al., 2006). In other words, students typically focus on the equal sign as signal to perform an operation and provide an answer (e.g., 3+4=7) rather than the equal sign as a relational indicator (e.g., 3+4=2+5). Moreover, this hypothesis is based on the documented whole number bias exhibited by over a quarter of students in 8th grade, such that students perceived equivalent fractions with larger parts as larger than those with smaller parts (Braithwaite & Siegler, 2018b). If middle school students are unable to perceive equivalent values within the same notation as equivalent in size, it seems probable that they might also struggle perceiving equivalent rational numbers as equivalent across notations. This is especially true in light of evidence that many teachers often do not use equal signs to describe equivalent values expressed as fractions, decimals, and percentages (Muzheve & Capraro, 2012). Chapter 2 underscores the importance of highlighting the connections among notations by discussing the pivotal role of notation connections in prior research (Moss & Case, 1999) and the benefit of interleaved practice in math (Rohrer & Taylor, 2007). Finally, I propose a plan for improving students’ understanding of rational numbers through linking notations with number line instruction, as an integrated theory of numerical development (Siegler et al, 2011) suggests that all rational numbers are incorporated into one’s mental number line. Chapter 3 details two experiments that yielded empirical evidence consistent with the hypotheses that students do not perceive equivalent rational numbers as equivalent in size and that this lack of integrated number sense influences estimation ability. The findings identify a discrepancy in performance in magnitude comparison across different rational number notations, in which students were more accurate when presented with problems where percentages were larger than fractions and decimals than when they were presented with problems where percentages were smaller than fractions and decimals. Superficially, this finding of a percentages-are-larger bias suggests students have a bias towards perceiving percentages as larger than fractions and decimals; however, it appears this interpretation is not true on all tasks. If students always perceive percentages as larger than fractions and decimals, then their placement of percentages on the number line should be larger than the equivalent fractions or decimals. However, this was not the case. The experiments revealed that students’ number line estimation was most accurate for percentages rather than the equivalent fraction and decimal values, demonstrating that students who are influenced by the percentages-are-larger bias are most likely not integrating understanding of fractions, decimals, and percentages on a single mental number line. Furthermore, empirical evidence provided support for the theory of impulsive calculation defined earlier, such that many students perform worse when presented with distracting information (“lures”) meant to elicit the use of flawed calculation strategies than in situations without such lures. Importantly, integrated number sense, as measured by the composite score of all cross-notation magnitude comparison trials, was shown to be an important predictor of estimation ability in the presence of distracting information on number lines and fraction arithmetic estimation tasks, often above and beyond number line estimation ability and general math ability. The experiments reported in Chapter 3 also evaluated whether Simultaneous, integrated instruction of all notations improved integration of rational number notations more than Sequential instruction of the three notations or a control condition with Rote practice in fraction arithmetic. The experiments also evaluated whether the instructional condition influenced fraction arithmetic estimation ability. The findings supported the hypothesis that a Simultaneous approach to reviewing rational numbers provides greater benefit for improving integrated number sense, as measured by more improvement in the composite score of magnitude comparison across notations. However, there was no difference among conditions in fraction arithmetic estimation ability at posttest. The experiments point to potential areas for improvement in future work, which are described subsequently. Chapter 4 attempts to explore further students’ understanding of the relations among notations. For this analysis, a number of data sources were examined, including student performance on assessments, interview data, analysis of student work, and classroom observations. Three themes emerged: (1) students are employing a flawed translation strategy, where students concatenate digits from the numerator and denominator to translate the fraction to a decimal such that a/b=0.ab (e.g., 3/5=0.35). (2) percentages can serve as a useful tool for students to judge magnitude, and (3) students equate math with calculation rather than estimation (e.g., in response to being asked to estimate addition of fractions answers, a student responded, “I can’t do math, right?”). Moreover, case studies investigated the differential effect of condition (Simultaneous, Sequential, or Control) on students’ strategy use. The findings suggest that the Simultaneous approach facilitated a more developed schema for magnitude, which is crucial given that a student’s degree of mathematical understanding is determined by the strength and accuracy of connections among related concepts (Hiebert & Carpenter, 1992). Chapter 5 concludes the dissertation by discussing the contributions of this work, avenues for future research, and educational implications. Ultimately, this dissertation advances the field of numerical cognition in three important ways: (1) by documenting a newly discovered bias of middle school students perceiving percentages as larger than fractions and decimals in magnitude comparisons across notations and positing that a lack of integrating notations on the same mental number line is a likely mechanism for this bias; (2) by demonstrating that students exhibit impulsive calculation, as measured by the difference in performance between situations where students are presented with distracting information (“lures”) meant to elicit the use of flawed calculation strategies and situations that do not involve lures; and (3) by finding that integrated number sense, as measured by the composite score for magnitude comparison across notations, is a unique predictor of estimation ability, often above and beyond general mathematical ability and number line estimation. In particular, students with higher integrated number sense are more than twice as likely to correctly answer the aforementioned 12/13+7/8 estimation problem than their peers with the same number line estimation ability and general math ability. This finding suggests that integrated number sense is an important inhibitor for impulsive calculation, above estimation ability for individual fractions and a general standardized test of math achievement. Finally, this dissertation advances the field of mathematics education by suggesting instruction that connects equivalent values with varied notations might provide superior benefits over a sequential approach to teaching rational numbers. At a minimum, this dissertation suggests that more careful attention must be paid to relating rational number notations. Future work might examine the origins of impulsive calculation and the observed percentages-are-larger bias. Future research might also examine whether integrated number sense is predictive of estimation ability beyond general number sense within notations. From these investigations, it might be possible to design a more impactful intervention to improve rational number outcomes.

Soil organic matter (SOM) is a well known indicator of soil quality due to its direct influence on soil properties such as structure, soil stability, water availability, cation exchange capacity, nutrient cycling, and pH buffering and amelioration. Study sites were selected in the Valley region of Virginia with the study objectives to: i) compare the efficiency of density solutions used in recovering free-light fraction (FLF) organic matter; ii) compare different soil organic fractions as sensitive indices of short-term changes in SOM due to management practices; iii) investigate on-farm effects of tillage management on soil organic carbon (SOC) and soil organic nitrogen (SON) stocks; and iv) evaluate the role of SOM in controlling soil available nitrogen (N) for corn uptake. The efficiency of the density solutions sodium iodide (NaI) and sodium polytungstate (SPT) in recovering FLF was the same at densities of 1.6 and 1.8 g cm⁻³, with both chemicals presenting less variability at 1.8 g cm⁻³. The sensitivity of SOM fractions in response to crop and soil management depended on the variable tested with particulate organic matter (POM) being the most sensitive when only tillage was tested, and FLF being the most sensitive when crop rotation and cover crop management were added. The on-farm investigation of tillage management on stocks of SOC and total soil N (TSN) indicated significant increases at 0–15 cm depth by increasing the duration (0 to 10 years) of no-tillage (NT) management (0.59 ± 0.14 Mg C ha⁻¹ yr⁻¹ and 0.05 ± 0.02 Mg N ha⁻¹ yr⁻¹). However, duration of NT had no significant effect on SOC and TSN stocks at 0–60 cm depth. Soil available N as controlled by SOM was modeled using corn (<i>Zea mays</i> L.) plant uptake as response and several soil N fractions as explanatory variables. The final model developed for 0–30 cm depth had 6 regressors representing the different SOM pools (active, intermediate, and stable) and a 𝑅² value of 65%. In summary, this study provides information about on-farm management affects on SOM levels; measurement of such effects in the short-term; and estimation of soil available N as related to different soil organic fractions. / Ph. D.

Fractions are very hard to learn. As the joke goes, “Three out of two people have trouble with fractions”. Yet the invention of a notation for fractions is very ancient, dating back to Babylonians and Egyptians. Moreover, it is thought that ratio representation is innate. And obviously, fractions are part of our everyday life. We read them in recipes, we need them to estimate distances on maps or rebates in shops. In addition, fractions play a key role in science and mathematics, in probabilities, proportions and algebraic reasoning. Then why is it so hard for pupils to understand and use them? What is so special about fractions? As in other areas of numerical cognition, a fast-developing field in cognitive science, we tackled this paradox through a multi-pronged approach, investigating both adults and children.<p>Based on some recent research questions and intense debates in the literature, a first behavioural study examined the mental representations of the magnitude of fractions in educated adults. Behavioural observations from adults can indeed provide a first clue to explain the paradox raised by fractions. Contrary perhaps to most educated adults’ intuition, finding the value of a given fraction is not an easy operation. Fractions are complex symbols, and there is an on-going debate in the literature about how their magnitude (i.e. value) is processed. In a first study, we asked adult volunteers to decide as quickly as possible whether two fractions represent the same magnitude or not. Equivalent fractions (e.g. 1/4 and 2/8) were identified as representing the same number only about half of the time. In another experiment, adults were also asked to decide which of two fractions was larger. This paradigm offered different results, suggesting that participants relied on both the global magnitude of the fraction and the magnitude of the components. Our results showed that fraction processing depends on experimental conditions. Adults appear to use the global magnitude only in restricted circumstances, mostly with easy and familiar fractions. <p>In another study, we investigated the development of the mental representations of the magnitude of fractions. Previous studies in adults showed that fraction processing can be either based on the magnitude of the numerators and denominators or based on the global magnitude of fractions and the magnitude of their components. The type of processing depends on experimental conditions. In this experiment, 5th, 6th, 7th-graders, and adults were tested with two paradigms. First, they performed a same/different task. Second, they carried out a numerical comparison task in which they had to decide which of two fractions was larger. Results showed that 5th-graders do not rely on the representations of the global magnitude of fractions in the Numerical Comparison task, but those representations develop from grade 6 until grade 7. In the Same/Different task, participants only relied on componential strategies. From grade 6 on, pupils apply the same heuristics as adults in fraction magnitude comparison tasks. Moreover, we have shown that correlations between global distance effect and children’s general fraction achievement were significant.<p>Fractions are well known to represent a stumbling block for primary school children. In a third study, we tried to identify the difficulties encountered by primary school pupils. We observed that most 4th and 5th-graders had only a very limited notion of the meaning of fractions, basically referring to pieces of cakes or pizzas. The fraction as a notation for numbers appeared particularly hard to grasp. <p>Building upon these results, we designed an intervention programme. The intervention “From Pies to Numbers” aimed at improving children’s understanding of fractions as numbers. The intervention was based on various games in which children had to estimate, compare, and combine fractions represented either symbolically or as figures. 20 game sessions distributed over 3 months led to 15-20% improvement in tests assessing children's capacity to estimate and compare fractions; conversely, children in the control group who received traditional lessons improved more in procedural skills such as simplification of fractions and arithmetic operations with fractions. Thus, a short classroom intervention inducing children to play with fractions improved their conceptual understanding. <p>The results are discussed in light of recent research on the mental representation of the magnitude of fractions and educational theories. The importance of multidisciplinary approaches in psychology and education was also discussed. <p>In sum, by combining behavioural experiments in adults and children, and intervention studies, we hoped to have improved the understanding how the brain processes mathematical symbols, while helping teachers get a better grasp of pupils’ difficulties and develop classroom activities that suit the needs of learners.<p> / Doctorat en Sciences Psychologiques et de l'éducation / info:eu-repo/semantics/nonPublished