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The present cross-sectional study investigated 953 fifth to tenth grade students’ understanding of the dense structure of rational numbers. After an inductive analysis, coding the answers based on three types of items on density, a TwoStep Cluster Analysis revealed different intermediate profiles in the understanding of density along grades. The analysis highlighted qualitatively different ways of thinking: i) the idea of consecutiveness, ii) the idea of a finite number of numbers, and iii) the idea that between fractions, there are only fractions, and between decimals, there are only decimals. Furthermore, our profiles showed differences regarding rational number representation since students first recognised the dense nature of decimal numbers and then of fractions. Learners, however, were still found to have a natural number-based idea of the rational number structure by the end of secondary school, especially when they had to write a number between two pseudo-consecutive rational numbers. En este estudio transversal sobre la densidad de los números racionales participaron 953 estudiantes desde 5º curso de educación primaria hasta 4º curso de educación secundaria. Tras un análisis inductivo, codificando las respuestas a tres tipos de ítems, se llevó a cabo un análisis clúster, que reveló diferentes perfiles intermedios en la comprensión de la densidad. Se identificaron formas de pensar diferentes: i) la idea de consecutivo, ii) la idea de número finito de números, y iii) la idea de que entre fracciones solo hay fracciones y entre decimales solo hay decimales. Además, se obtuvieron diferencias con respecto a la representación de los números racionales: los estudiantes primero reconocieron la densidad en números decimales y posteriormente, en fracciones. Se destaca que los estudiantes al final de la educación secundaria todavía tenían una idea basada en el conocimiento del número natural, especialmente cuando tenían que escribir un número entre dos números racionales pseudo-consecutivos. This research was carried out with the support of Conselleria d’Educació, Investigació, Cultura i Esport (Generalitat Valenciana, Spain) (PROMETEO/2017/135), the support of the postdoctoral grant (I-PI 69-20), and with the support of the Academy of Finland (Grant 336068, growing mind GM2, PI Minna Hannula-Sormunen).

ispartof: pages:168-172 ispartof: Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education. Volume 1 pages:168-172 ispartof: 45th Conference of the International Group for the Psychology of Mathematics Education location:Alicante, Spain date:18 Jul - 23 Oct 2022 status: Published online

ispartof: Educational Studies In Mathematics vol:104 issue:3 pages:299-311 ispartof: location:Netherlands status: published

Supplemental material, sj-docx-1-qjp-10.1177_17470218221094577 for The structure of the notation system in adults’ number line estimation: An eye-tracking study by Kelsey J MacKay, Filip Germeys, Wim Van Dooren, Lieven Verschaffel and Koen Luwel in Quarterly Journal of Experimental Psychology

ispartof: Jaarboek de Aardrijkskunde vol:2020 pages:69-72 status: published

Proportional reasoning is a pivotal concept in elementary mathematics. More insight into early predictors of later proportional reasoning might be useful to guide early interventions. Research indicates that language is of major importance for mathematical thinking and learning. The present study investigated the longitudinal association between vocabulary that is specific to proportional situations and children’s ability to reason about proportional situations. Results showed that each of the five concepts measured was indeed significantly correlated with proportional reasoning ability, and this relation persisted after controlling for children’s SES. Website: https//pme44.kku.ac.th ispartof: pages:621-629 ispartof: Interim Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education pages:621-629 ispartof: 44th Conference of the International Group for the Psychology of Mathematics Education location:Khon Kaen University, Thailand (Virtual Meeting) date:21 Jul - 22 Jul 2020 status: Published online

ispartof: School- en Klaspraktijk vol:61 issue:3 pages:6-10 status: published

Histograms are widely used and appear easy to understand. Research never the less indicates that students, teachers and researchers often misinterpret these graphical representations. Hence, the research question addressed in this paper is: What are the conceptual difficulties that become manifest in the common misinterpretations people have when constructing or interpreting histograms? To identify these conceptual difficulties, we conducted a narrative systematic literature review and identified 86 publications reporting or containing misinterpretations. The misinterpretations were clustered and—through abduction—connected to difficulties with statistical concepts. The analysis revealed that most of these conceptual difficulties relate to two big ideas in statistics: data (e.g., number of variables and measurement level) and distribution (shape, centre and variability or spread).These big ideas are depicted differently in histograms compared to, for example, case-value plots. Our overview can help teachers and researchers to address common misinterpretations more generally instead of remediating them each individually. ispartof: Educational Research Review vol:28 pages:1-23 status: published

ispartof: European Traditions in Didactics of Mathematics (ICME-13 Monographs) pages:91-91 ispartof: ICME-13 Monographs pages:91-91 status: published

ispartof: pages:351-360 ispartof: Investigación en Educación Matemática XXIII pages:351-360 ispartof: SEIEM XXIII. Investigación en Educación Matemática XXIII location:Valladolid, Spain date:4 Sep - 6 Sep 2019 status: Published online

Many people misinterpret histograms. The conjecture is that some of these misinterpretations emerge from the application of interpretation strategies associated with case-value plots. To investigate this, eye-movement data were collected from six university students solving questions on histograms and case-value plots. Analysis of gaze data and cued retrospective verbal reports showed that participants seemed to use a histogram interpretation strategy, a case-value plot interpretation strategy or an elimination strategy. Several participants appeared to use a single preferred strategy without distinguishing between the type of graphs at stake. As conjectured, some participants applied a casevalue plot strategy also to histograms. In addition, analysis of gaze data and verbal reports suggest that more experienced participants abandoned their nitial strategy when necessary.

Many people misinterpret histograms. The conjecture is that some of these misinterpretations emerge from the application of interpretation strategies associated with case-value plots. To investigate this, eye-movement data were collected from six university students solving questions on histograms and case-value plots. Analysis of gaze data and cued retrospective verbal reports showed that participants seemed to use a histogram interpretation strategy, a case-value plot interpretation strategy or an elimination strategy. Several participants appeared to use a single preferred strategy without distinguishing between the type of graphs at stake. As conjectured, some participants applied a casevalue plot strategy also to histograms. In addition, analysis of gaze data and verbal reports suggest that more experienced participants abandoned their nitial strategy when necessary.

Many people misinterpret histograms. The conjecture is that some of these misinterpretations emerge from the application of interpretation strategies associated with case-value plots. To investigate this, eye-movement data were collected from six university students solving questions on histograms and case-value plots. Analysis of gaze data and cued retrospective verbal reports showed that participants seemed to use a histogram interpretation strategy, a case-value plot interpretation strategy or an elimination strategy. Several participants appeared to use a single preferred strategy without distinguishing between the type of graphs at stake. As conjectured, some participants applied a case-value plot strategy also to histograms. In addition, analysis of gaze data and verbal reports suggest that more experienced participants abandoned their initial strategy when necessary.

Natural number bias in fraction comparison has been studied examining the role of congruency effect. However, the congruency effect has been observed in opposite directions, suggesting that an explanation may lie in the strategies used by students. One of the strategies that may be used by students is gap thinking. We have carried out a cross-sectional study from 5th to 10th grade with 438 participants examining the effect of congruency and gap thinking in students’ responses and reasoning. Results show that gap thinking has influenced students’ responses and support the claim that gap effects could explain differences between congruent and incongruent items, extending this result from primary to secondary school. However, it seems that this effect disappears (in Spanish students) at the end of Secondary Education. ispartof: pages:459-466 ispartof: Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2) pages:459-466 ispartof: Conference of the International Group for the Psychology of Mathematics Education location:Umeå, Sweden date:3 Jul - 8 Jul 2018 status: Published online

Intuitions play an important role in mathematical reasoning. Stavy and Tirosh pro-posed the intuitive rules theory and showed how various tasks are incorrectly solved on the basis of intuitive rules triggered by salient task characteristics. In this study we wanted to replicate and extend the results of Stavy and Tirosh. Furthermore, we wanted to test two different ways of making intuitive reasoning more likely (textual and graphical saliency). We found that all tasks tested in this study showed similar patterns of accuracy rates and reaction times as in the studies of Stavy and colleagues. How-ever, we were not able to replicate the result that more salient task elements lead to more intuitive reasoning. We propose explanations for these different results and discuss implications for further research and educational practice. ispartof: pages:299-306 ispartof: Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3) vol:3 pages:299-306 ispartof: Conference of the International Group for the Psychology of Mathematics Education location:Umeå, Sweden date:3 Jul - 8 Jul 2018 status: Published online

No isbn ispartof: Educational Practices Series vol:30 status: Published online

This one page paper describes four misinterpretations with histograms identified in the literature.

This one page paper describes four misinterpretations with histograms identified in the literature.

In this study we focus on the integration of mathematical learning content (i.e. proportional reasoning) in a GBLE. More specific two kinds of GBLEs are set up; an extrinsically integrated GBLE and an intrinsically integrated GBLE. In the former environment, the mathematical content is not part of the core mechanics and structure of the gaming world. In the latter environment, the mathematical content is delivered through the parts of the game that are the most fun to play and embodied within the structure of the gaming world and the players interactions with it. Fifty-eight vocational track students participated in the study, all working in one of the two versions of the self-developed game-based learning environment ‘Zeldenrust’. The results of this study provide evidence for the effect of this different way of implementing learning content in an educational math game. More specific, students playing in an extrinsically integrated game show higher learning gains, motivational gains and perceived usefulness than students who played with a game in which the content was intrinsically integrated

In this contribution, we present a game-based learning environment for 12–16-year-old vocational students in which they can practice proportional reasoning problems. The learning content and goals, as well as the specific game features are discussed. We can conclude that developing a serious game implies many choices and decisions led by theoretical foundations, as well as by practical limitations and pragmatic considerations.

Externe representaties van dataverdelingen, zoals histogrammen en boxplots, zijn alomtegenwoordig. Vooral histogrammen komen we vaak tegen, bijvoorbeeld in kranten en nieuwsprogramma’s op televisie. Boxplots worden dan weer vaak gebruikt in technische rapporten en wetenschappelijke publicaties. Toch blijkt uit recent onderzoek dat deze (en andere) externe representaties heel wat interpretatiemoeilijkheden met zich meebrengen. In dit artikel bespreken we achtereenvolgens histogrammen en boxplots. Voor beide representaties geven we eerst een korte inleiding op de representatie, haar opbouw en geschiedenis. Vervolgens stellen we de belangrijkste misinterpretaties voor. Tot slot bespreken we telkens exemplarisch één of twee onderzoeken naar een specifieke misinterpretatie van de betreffende representatie. ispartof: Wiskunde en Onderwijs vol:40 issue:157 pages:19-32 status: published

In Game Based Learning (GBL) environments addition, players act and see how the system responds, which may lead to intuitive learning. GBL learning lacks the deep and integrated structure that is characteristic for sustainable knowledge. Cognition-based instructional scaffolds and support may facilitate the acquisition of sustainable knowledge in GBL environments. How can we implement such support effectively without hampering the motivating character of the GBL environment? We introduce a game that implements two cognition-based principles in an integrated way. First, Integration, suggests that cognitive activities used for game play also improve the learning of arithmetic and enhance intrinsic motivation. Second, Reflection, enables players to explicate their knowledge. In a pilot study, we tested a basic version of the game and measurement instruments (using only Integration). Motivation and perception questionnaires proved to be reliable (N = 81). However, a small-scale pretest-posttest group showed no increase in performance after playing the game (p > .05). After revisions, 3 additional game versions were created: a structured reflection version (information about math problem solving strategies and then reflection prompts), an open reflection version (only reflection prompts) and an information-only version (information about math problem solving strategies). Currently pretest-posttest experiments are testing the hypothesis that the combination of Integration with Structured Reflection will outperform the other conditions on performance.

nrpages: 31 status: published

nrpages: 33 status: published

HUBrussel Research Paper, 2010/32 nrpages: 24 status: published

Le problème d’Isis, qui emprunte son nom à une connexion mythique avec le culte de la déesse Isis dans l’Egypte antique, s’énonce ainsi : « Quels rectangles, avec des nombres entiers comme côtés (mesurés avec une unité donnée), possèdent la propriété que leur aire et leur périmètre (considérés comme nombres) sont égaux ? » Bien que le problème d’Isis puisse se résoudre sans mathématiques trop techniques ni avancées, ce n’est pas pour autant un exercice de routine. On peut trouver des démonstrations à partir de points de vues divers (allant de raisonnements mathématiques pré-formels jusqu’à l’algèbre, la géométrie ou l’analyse). Certaines démonstrations sont plus convaincantes que d’autres : elles démontrent non seulement que mais aussi pourquoi les rectangles trouvés sont uniques. Dans le cadre d’une recherche récente à Leuven, nous avons demandé à des dizaines de futurs professeurs de mathématiques de trouver tous les rectangles ayant la propriété mentionnée ci-dessus et de démontrer qu’il n’en existe pas d’autres. Après cela, nous leur avons présenté cinq démonstrations alternatives avec la consigne de les évaluer et de les commenter. Les résultats nous permettent de mieux comprendre les critères conceptuels et esthétiques utilisés par les futurs professeurs dans leur jugement d’une démonstration. ispartof: Losanges. Périodique bimestriel édité par la Société Belge des Professeurs de Mathématiques d’expression française. vol:7 pages:15-23 status: published

This study builds on two lines of research that so far developed largely separately: the use of additive methods to solve proportional word problems and the use of proportional methods to solve additive word problems. We investigated the development with age of both kinds of erroneous solution methods. They key question is whether and how an overall additive approach to word problems develops into an overall multiplicative approach, and how the transition from the first kind of errors to the second occurs. We gave a test containing missing-value problems to 325 third, fourth, fifth, and sixth graders. Half of the problems had an additive structure and half had a proportional structure. Moreover, in half of the problems the internal and external ratios between the given numbers were integer while in the other cases numbers were chosen so that these ratios were noninteger. The results indicate a development from applying additive methods “anywhere” in the early years of primary school to applying proportional methods “anywhere” in the later years. Between these two stages many students went through an intermediate stage where they simultaneously used additive methods to proportional problems and proportional methods to additive problems, switching between them based on the numbers given in the problem. https://lirias.hubrussel.be/bitstream/123456789/2844/1/09HRP37.pdf nrpages: 48 status: published

See also http://140.122.140.1/~icmi19/files/Volume_1.pdf ispartof: pages:184-189 ispartof: Proof and proving in mathematics education. ICMI Study 19 Conference Proceedings. vol:1 pages:184-189 ispartof: ICMI Study 19 Conference: Proof and Proving in Mathematics Education location:Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan date:10 May - 15 May 2009 status: published

The Isis problem, which has a link with the Isis cult of ancient Egypt, asks: “Find which rectangles with sides of integral length (in some unit) have area and perimeter (numerically) equal, and prove the result.” Since the solution requires minimal technical mathematics, the problem is accessible to a wide range of students. Further, it is notable for the variety of proofs (empirically grounded, algebraic, geometrical) using different forms of argument, and their associated representations, and it provides an instrument for probing students’ ideas about proof, and the interplay between routine and adaptive expertise. A group of 39 Flemish pre-service mathematics teachers was confronted with the Isis problem. More specifically, we first asked them to solve the problem and to look for more than one solution. Second, we invited them to study five given contrasting proofs and to rank these proofs from best to worst. The results highlight a preference of many students for algebraic proofs as well as their rejection of experimentation. The potential of the problem as a teaching tool is outlined.

A vast amount of research in mathematics education has shown that students of different ages have a strong tendency to apply linear or proportional models anywhere, even in situations where they are not applicable. For example, in geometry it is known that many students believe that if the sides of a figure are doubled, the area is doubled too. History of science also provides several cases of thinkers who inadequately postulated linear relations to describe situations. This article focuses on secondary school students’ over-reliance on linearity in physics. Now and then, science educators report students’ tendency to assume and impose linear relations in physics, but – as far as we know – no substantial efforts were undertaken to study this phenomenon in a systematic empirical way. To fill up this hiatus in the science education literature, we report in this article on an empirical investigation aimed at identifying the competence of 8th- and 11th-graders – before and after being taught the relevant physical topics – to qualitatively grasp situations in physics, as well as their tendency to quantify linearly that qualitative insight. The results provide an ambivalent picture of students’ overuse of linearity in physics: although linear reasoning is sometimes used as a default strategy, even after instruction that addresses the relevant physical contents, this study also indicates that context is taken more into account than is suggested by research on mathematical problem solving.

status: published

Previous research showed that primary school pupils over-use proportional methods especially when solving non-proportional missing-value word problems. The current study examines whether the numbers appearing in these word problems partly explain this phenomenon. In previous studies, the numbers in the problems formed integer ratios, which may have stimulated pupils to use proportional methods. A test containing proportional and non-proportional word problems was given to 508 4th, 5th and 6th graders. Numbers in these problems were experimentally manipulated so that the internal and external ratios were either integer or not. We found that in 4th grade, integer ratios elicited much more (inappropriate) proportional methods in pupils than non-integer ratios, while this effect was smaller in 5th grade and disappeared in 6th grade.

Fractions are fundamental in students’ mathematical development. However, for many students, they are known to be a major stumbling block. In this chapter, we examine the obstacles elementary school children face when they learn fractions, through the lens of numerical cognition. We start by discussing the discrepancy between children’s conceptual and procedural knowledge of fractions, and we review studies showing that the concept of fraction magnitude is particularly difficult to learn. This has wider implications as understanding fraction magnitude has been shown to be a strong predictor of achievement in algebra and overall mathematics achievement in later years. We then discuss the natural number bias (NNB), a well-characterised misconception linked to fraction learning whereby learners are inclined to apply natural number characteristics when reasoning about fractions without considering whether it is appropriate. The NNB is persistent, appearing early in the fraction learning process and lasting through secondary school and beyond. We conclude this chapter by describing interventions aimed at improving fraction learning and we provide suggestions on how to introduce the concept of magnitude more intentionally when teaching fractions.

In this study we investigated the effect of the request to reword the text of problematic word problems on the occurrence of realistic answers. We proposed the activity of rewording four word problems to fifth-grade pupils either working individually or in dyads. We found that for one of the problems the rewording in dyads produced a strong increase in pupils’ realistic answers, for another one a small increase, and no increase at all for the other two. We also analysed pupils’ reworded texts in order to characterize the kind of information added. For all four problems the rewording in dyads produced richer texts in comparison to the individual rewording. Moreover, for the problems where an increase in realistic answers was observed, two particular kinds of information, named descriptive and action information, were added by dyads who answered realistically.

In the literature, the need for high-quality activities, projects and professional initiatives to engage primary school pupils and teachers in (learning of and teaching in) science and technology (S&T) is widely recognised. Learning environments in which students are given high levels of autonomy, among them project- (Barak & Raz, 1998; Barak & Doppelt, 2000), inquiry- (Furtak, Seidel, Iverson, & Briggs, 2012) and design-based learning environments (Fortus, Dershimer, Krajcik, Marx, & Mamlok-Naaman, 2004), can contribute to pupils’ engagement (Wurdinger, Haar, Hugg, & Bezon, 2007). Despite the promising effects of these ‘open-ended’ S&T learning environments, until now the conditions for their effectiveness were not thoroughly investigated. It cannot be assumed that the teacher and pupils immediately find their new role when implementing such learning environments. This doctoral dissertation wanted to disentangle the decisive factors for a successful implementation of such an environment in Flemish and Dutch primary classrooms. In particular, the aims of this doctoral research were twofold. Firstly, with two studies (studies 1 and 4) the research aimed to investigate the existing literature on instruments to assess the quality of S&T learning environments, particularly project-based S&T environments, as well as to contribute to instrument development. Study 1, a review study with regard to the instruments in the field, revealed that a variety of scales, items and questions exists. The review study helps researchers in the field to make a choice when they aim particular aspects of the S&T learning environment, and stimulates to compose new instruments with scales of which the operationalisation fits best with their goals. In the intervention research as conducted in studies 2 and 3, we even went a step further. As in the profound literature search to instruments an instrument that comprehensively evaluates the teacher’s role was not found, a high-quality observation tool, the Classroom Assessment Scoring System (CLASS) (Pianta et al., 2012), was selected to measure teacher style – operationalized as the quality of the interactions. At the same time, the use of this instrument provided the opportunity to optimise the Adult Style Observation Schedule (ASOS) (Laevers & Heylen, 2013), a tool developed at the Centre for Experiential Education to measure teacher style (Study 4). The results of the concurrent validity analyses showed that the relationships of the CLASS dimensions with the ASOS dimensions that were expected on the basis of a profound theoretical analysis of the dimensions and their operationalisation, were not univocally found. In particular, the ASOS dimension Stimulation showed congruence with the CLASS dimension Content Understanding, and the ASOS dimension Sensitivity correlated with the CLASS dimension Positive Climate. No evidence emerged for the convergence of the Giving Autonomy dimension of the ASOS with similar CLASS dimensions. Secondly, in two empirical studies (studies 2 and 3), the effectiveness of the implementation of an open-ended S&T learning environment, the Village@School project, and factors related to this effectiveness and the implementation, were investigated. In Study 2, pupils’ evolution in engagement was studied and possible explaining factors in teachers’ competence profile – their attitudes towards S&T (teaching) and teacher style – for the differences between schools and/or classes with regard to this evolution were explored. The main findings of this study indicated that (a) pupils grew in their engagement throughout the implementation of the Village@School project and (b) the growth in engagement, as measured before and after the project, was positively related to Teacher Sensitivity, but negatively to Positive Climate and Content Understanding when controlling for other CLASS dimensions. In Study 3 at first the way in which the teacher’s attitudes and style before the project determined the style during Village@School, was explored. The results revealed a positive association between the initial emotional support provided and the quality of emotional support during Village@School when controlling for the teacher’s attitudes and the other CLASS- domains. Under the same conditions a negative relation was found with Classroom Organisation and the attitude towards inquiry learning. Finally, the evolution in teachers’ competence profile throughout and after the intervention was investigated. After the project, teachers didn’t grow in their attitudes in comparison to before, but classes showed better interactions involving Regard for Student Perspectives and Quality of Feedback. Surprisingly, a growth in Negative Climate was detected. The findings of studies 2 and 3 were discussed in the light of the challenges involved in the implementation of the Village@School project. status: published

Lesliemodellen beschrijven de evolutie van een populatie die verdeeld is in leeftijdsklassen. Doorrekenen toont dat er op de lange termijn twee interessante fenomenen optreden: de populatie groeit exponentieel aan met een zekere langetermijngroeifactor en de bevolkingspiramide krijgt een vaste vorm. Als we de wiskunde uit deze context distilleren, komen we uit bij de begrippen eigenwaarde en eigenvector. We onderzoeken vervolgens hoe we eigenvectoren en eigenwaarden van een (Leslie)matrix kunnen bepalen zonder massief rekenwerk. no ISBN ispartof: pages:95-108 ispartof: Wiskundeonderwijs in Vlaanderen en Europa: een stand van zaken en perspectieven voor de toekomst pages:95-108 ispartof: Wiskundeonderwijs in Vlaanderen en Europa: een stand van zaken en perspectieven voor de toekomst location:Leuven date:23 Aug - 24 Aug 2007 status: published