The current study explored individual and gendered differences in Black students’ motivation for learning mathematics using three key Situated Expectancy-Value Theory (SEVT) constructs ... Show moreThe current study explored individual and gendered differences in Black students’ motivation for learning mathematics using three key Situated Expectancy-Value Theory (SEVT) constructs (expectancies of success, interest, and importance). It also evaluated whether math motivational profiles in 6th grade or 10th grade predicted math achievement and STEM career aspirations in 10th grade among Black students while controlling for prior math achievement. Black students (n = 408, 55% female) attending schools in a metropolitan area of Tennessee, USA and mostly from families surviving economic marginalization completed surveys and math achievement assessments across middle and high school. Latent Profile Analysis identified three profiles of math motivation in 6th grade, including a profile of high motivation across constructs, and Black girls were less likely to be in the high motivational profile than Black boys. Profile membership in 6th grade predicted 10th grade math achievement. In contrast, math motivation profiles in 6th grade did not predict STEM career aspirations in 10th grade. Parallel analyses for concurrent relations in 10th grade were similar, except that there were no gender differences in profile prevalence. Overall, findings suggest that SEVT is useful for understanding motivation and academic performance among Black students when a person-centered analytic approach is used, but more work is needed to expand the theory to understand the development of Black students’ STEM career aspirations. Show less
Emilie Prast en Marian Hickendorff van de Universiteit Leiden verdiepen in hun keynote het thema differentiatie vanuit een meer vakdidactische insteek. Vertrekkend vanuit algemene principes van... Show moreEmilie Prast en Marian Hickendorff van de Universiteit Leiden verdiepen in hun keynote het thema differentiatie vanuit een meer vakdidactische insteek. Vertrekkend vanuit algemene principes van differentiatie voor verschillende doelgroepen zoomen zij in op differentiatie voor sterke rekenaars. Je krijgt inzicht in het belang van differentiatie voor sterke rekenaars, manieren waarop in de praktijk al wordt gedifferentieerd voor deze groep, en kansen om differentiatie voor sterke rekenaars te verbeteren. Daarbij krijg je tips om je rekenonderwijs systematisch af te stemmen op de onderwijsbehoeften van deze groep leerlingen. Show less
Fractions are an important but notoriously difficult domain in mathematics education. Situating fraction arithmetic problems in a realistic setting might help students overcome their difficulties... Show moreFractions are an important but notoriously difficult domain in mathematics education. Situating fraction arithmetic problems in a realistic setting might help students overcome their difficulties by making fraction arithmetic less abstract. The current study therefore investigated to what extent students (106 sixth graders, 187 seventh graders, and 192 eighth graders) perform better on fraction arithmetic problems presented as word problems compared to these problems presented symbolically. Results showed that in multiplication of a fraction with a whole number and in all types of fraction division, word problems were easier than their symbolic counterparts. However, in addition, subtraction, and multiplication of two fractions, symbolic problems were easier. There were no performance differences by students’ grade, but higher conceptual fraction knowledge was associated with higher fraction arithmetic performance. Taken together this study showed that situating fraction arithmetic in a realistic setting may support or hinder performance, dependent on the problem demands. Show less
Adapting education to students’ diverse educational needs is widely recognised as an important, but also complex aspect of effective teaching. In this chapter, we provide insight into how Dutch... Show moreAdapting education to students’ diverse educational needs is widely recognised as an important, but also complex aspect of effective teaching. In this chapter, we provide insight into how Dutch primary school teachers implement differentiation based on students’ current mathematics achievement level. We review evidence from four independent samples in which the same teacher self-assessment questionnaire was administered (N = 907 teachers in total), supplemented with qualitative data from various perspectives: external observers, students, and teachers. Based on these sources of information, we identify the following general patterns. Teachers generally implement achievement-based differentiation at least to some extent. That is, student achievement is monitored, and efforts are taken to adapt instruction or practice to students’ current achievement level. This is often organised using within-class homogeneous achievement groups. While low-achieving students regularly receive additional instruction, specific instruction for high-achieving students is uncommon. Refined, qualitative strategies to diagnose students’ individual educational needs and to adapt education to these individual needs are also used relatively infrequently. These relatively infrequently used strategies point to areas for improvement. Furthermore, the flexibility of within-class achievement groups seems to vary and deserves more attention in future research and practice. Show less
Central elements of adaptive expertise in arithmetic problem solving are flexibility,flexibility, using multiple strategies, and adaptivity, selecting the optimal strategy. Research shows that the... Show moreCentral elements of adaptive expertise in arithmetic problem solving are flexibility,flexibility, using multiple strategies, and adaptivity, selecting the optimal strategy. Research shows that the strategies children actually use do not fully reflectreflect the strategies they know: there is hidden potential. In the current study a sample of 147 third graders from the Netherlands completed a comprehensive assessment of adaptive expertise in the domain of multidigit subtraction, designed to measure, first, the strategies students know and use to solve subtraction problems (potential and practical flexibility).flexibility). Second, it measured to what extent students know which strategy is optimal and to what extent they use the optimal strategy (potential and practical adaptivity). Findings for flexibilityflexibility showed that most students consistently used the same strategy across all problems: practical flexibilityflexibility was low. When prompted, students knew more strategies than they used spontaneously, suggesting hidden potential in flexibility.flexibility. Findings for adaptivity showed that students hardly ever spontaneously used the task-specifictask-specific strategy that is efficient for specificspecific problems since it has the fewest and easiest steps. However, almost half of the students could select this strategy from a set of given strategies at least once. Furthermore, an innovative, personalized version of the choice/no-choice method showed that the task-specifictask-specific strategy was usually not the optimal strategy (fastest strategy leading to a correct answer) for individual students. Finally, students used the strategy with which they performed best more often than the other strategies, but there is hidden potential for the adaptive use of task-specifictask-specific strategies. Show less
Hickendorff, M.; McMullen, J.; Verschaffel, L. 2022
The current paper presents an introduction to a special issue focusing on mathematical flexibility,flexibility, which is an important aspect of mathematical thinking and a cherished, but capricious... Show moreThe current paper presents an introduction to a special issue focusing on mathematical flexibility,flexibility, which is an important aspect of mathematical thinking and a cherished, but capricious, outcome of mathematics education. Mathematical flexibilityflexibility involves the flexible,flexible, creative, meaningful, and innovative use of mathematical concepts, relations, representations, and strategies. In this introduction we discuss the most relevant theoretical, methodological, and educational considerations related to mathematical flexibility,flexibility, which form the background of the empirical studies presented in the special issue. Collectively, these studies provide a broader understanding of the mathematical flexibility,flexibility, its subcomponents, influences, and malleability. Show less
The present study aims to provide a systematic understanding of how perspective-taking ability contributes to primary-school students’ cooperative behaviours and learning outcomes. The present... Show moreThe present study aims to provide a systematic understanding of how perspective-taking ability contributes to primary-school students’ cooperative behaviours and learning outcomes. The present study is frontline as we combined person-oriented (e.g., describing patterns of behaviours based on individual characteristics), process-oriented (e.g., examining factors affecting the quality of cooperative behaviours), and effect-oriented (e.g., examining the effect of cooperative learning on individual learning outcomes) analytical approaches within one research framework. In addition, we adhered to the multi-dimensional nature of perspective-taking ability and differentiated between social and cognitive perspective-taking ability while taking into account the contribution of perspective-taking ability at both the individual level and group level (i.e., heterogeneous and homogeneous perspective-taking ability groups) to cooperative behaviour profiles and learning outcomes of primary-school children. Based on transcribed episodes of interaction of 115 fifth-grade students, four different profiles of cooperative behaviours were discerned: captains, hard workers, switchers, and passive participants. We found that these profiles are related to perspective taking conceptualized at the group level, but not to individual-level perspective-taking ability. Profile membership, cognitive perspective-taking ability, and group-level perspective-taking ability could not predict students’ learning outcomes. Social perspective-taking ability and reading comprehension did positively predict learning outcomes. Our findings add to existing knowledge as they suggest that the influence of perspective-taking ability on cooperative behaviours and learning outcomes is susceptible to the conceptualization (i.e., cognitive vs. social) and measurement level (i.e., individual vs. group level) of perspective-taking ability. Show less
sing the choice/no-choice methodology we investigated Dutch fourth graders’ (N = 124) adaptive use of the indirect addition strategy to solve subtraction problems. Children solved multidigit... Show moresing the choice/no-choice methodology we investigated Dutch fourth graders’ (N = 124) adaptive use of the indirect addition strategy to solve subtraction problems. Children solved multidigit subtraction problems in one choice condition, in which they were free to choose between direct subtraction and indirect addition, and in two no-choice conditions, in which they had to use either direct subtraction or indirect addition. Furthermore, children were randomly assigned to mental computation, written computation, or free choice between mental and written computation. One third of the children adaptively switched their strategy according to the number characteristics of the problems, whereas the remaining children consistently used the same strategy. The likelihood to adaptively switch strategies decreased when written computation was allowed or required, compared to mandatory mental computation. On average, children were adaptive to their own speed differences but not to the accuracy differences between the strategies. Show less
Hickendorff, M.; Torbeyns, J.; Verschaffel, L. 2019
This chapter provides an overview of the current findings about (the obstacles in) primary school children’s strategy use in the domain of multi-digit arithmetic. This involves addition,... Show moreThis chapter provides an overview of the current findings about (the obstacles in) primary school children’s strategy use in the domain of multi-digit arithmetic. This involves addition, subtraction, multiplication, and division tasks in which at least one of the operands contains two or more digits. For both the additive and multiplicative domains, we provide a comprehensive framework for the classification of strategies, with two dimensions: (1) the operation that underlies the solution process and (2) the way the numbers are dealt with in computing the outcome (manipulating whole numbers or single digits). Empirical findings of children’s strategy use in the additive and multiplicative domain show that children use a variety of number-based strategies efficiently and adaptively before the introduction of the digit-based algorithms. The introduction of the digit-based algorithms seems a critical instructional event: children show a large tendency to use the digit-based algorithms once they are instructed, and they do so rather efficiently. The major obstacles children encounter in developing, selecting, or executing these strategies are their conceptual understanding, procedural fluency, and adaptive/flexible strategy selection. Show less
Fagginger Auer, M.F.; Hickendorff, M.; Putten, C.M. van 2018
This study aimed to identify reading behavior profiles in nine-to-eleven year old children based on their think-aloud responses while reading narrative and expository texts. Three profiles emerged... Show moreThis study aimed to identify reading behavior profiles in nine-to-eleven year old children based on their think-aloud responses while reading narrative and expository texts. Three profiles emerged while reading narratives: Literal Readers, who stay close to the literal text by predominantly repeating it; Paraphrasing Readers, who extract meaning from the text by paraphrasing it; and Elaborating Readers, who use background knowledge to explain the text by generating inferences. The three profiles also emerged while reading expository text. Children generally exhibited the same profiles across the two text genres, however, expository texts elicited fewer correct inferences but more invalid inferences than did narratives, suggesting that children are influenced by text demands. Elaborating Readers had better word decoding skills, reading comprehension ability, and non-verbal reasoning ability than readers of the two other profiles, indicating a positive relation between inference generation and language abilities and cognitive resources. Show less
Herein, we provide an introduction to the special issue "Latent variable mixture models in research on learning and individual differences". Latent variable mixture models are argued to be a... Show moreHerein, we provide an introduction to the special issue "Latent variable mixture models in research on learning and individual differences". Latent variable mixture models are argued to be a powerful tool for capturing non-linear and qualitative individual differences in learners' knowledge, characteristics, and development. The current special issue provides an overview of the use of these analytical tools in investigations of learning and individual differences by presenting a wide-range of empirical studies utilizing the methods. A practical non-technical introduction and discussion are also included in the special issue. Show less
Analogical reasoning is essential for acquiring and integrating new knowledge and skills. Although much research has focused on this important skill, children's paths from non-analogical to... Show moreAnalogical reasoning is essential for acquiring and integrating new knowledge and skills. Although much research has focused on this important skill, children's paths from non-analogical to analogical reasoning remain unclear. In this study, 388 children (ages 4–10 years) solved a series of figural analogies within a pretest-training-posttest design, with training comprising either multiple tries (N = 196) or tutoring feedback (N = 192). Working memory tasks were also administered. Latent transition analyses identified five phases with qualitative individual differences in children's analogy solving: duplication, idiosyncratic, beginner analogical, intermediate analogical and advanced analogical reasoning. Children's paths through these phases were not sequential; there was great variability between children and how they progressed through these phases. Working memory was related to children's reasoning phase at pretest, but not to their rate and path of change. Age and the type of feedback received during training were the clearest indicators of children's learning paths and rates of change. Show less
Strategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that... Show moreStrategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that primary school students use such shortcut strategies rather infrequently. The aims of the present study were to analyze the extent to which Dutch sixth graders (12-year-olds) use shortcut strategies in solving multidigit addition, subtraction, multiplication, and division problems, to what extent student factors and task instructions affected this frequency of shortcut strategy use, and to what extent the strategies differed in performance. A sample of 648 sixth graders from 23 Dutch primary schools completed a paper-and-pencil task of 12 multidigit arithmetic problems, designed to elicit specific shortcut strategies such as compensation. Based on the students’ written work, strategies were classified into whether a shortcut strategy was used or not. Results showed that the frequency of shortcut strategies ranged between 6 and 21% across problem types, and that boys and high mathematics achievers were more inclined to use shortcut strategies. An explicit instruction to look for a shortcut strategy increased the frequency of these strategies in the addition and multiplication problems, but not in the subtraction and division problems. Finally, the use of shortcut strategies did not yield higher performance than using standard strategies. All in all, spontaneous as well as stimulated use of shortcut strategies by Dutch sixth graders was not very common. Show less
Hickendorff, M.; Torbeyns, J.; Verschaffel, L. 2018
We aimed to investigate upper elementary children's strategy use in the domain of multidigit division in two instructional settings: the Netherlands and Flanders (Belgium). A cross‐sectional sample... Show moreWe aimed to investigate upper elementary children's strategy use in the domain of multidigit division in two instructional settings: the Netherlands and Flanders (Belgium). A cross‐sectional sample of 119 Dutch and 122 Flemish fourth to sixth graders solved a varied set of multidigit division problems. With latent class analysis, three distinct strategy profiles were identified: children consistently using number‐based strategies, children combining the use of column‐based and number‐based strategies, and children combining the use of digit‐based and number‐based strategies. The relation between children's strategy profiles and their instructional setting (country) and grade were generally in line with instructional differences, but large individual differences remained. Furthermore, Dutch children more frequently made adaptive strategy choices and realistic solutions than their Flemish peers. These results complement and refine previous findings on children's strategy use in relation to mathematics instruction. Show less
Strategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that... Show moreStrategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that primary school students use such shortcut strategies rather infrequently. The aims of the present study were to analyze the extent to which Dutch sixth graders (12-year-olds) use shortcut strategies in solving multidigit addition, subtraction, multiplication, and division problems, to what extent student factors and task instructions affected this frequency of shortcut strategy use, and to what extent the strategies differed in performance. A sample of 648 sixth graders from 23 Dutch primary schools completed a paper-and-pencil task of 12 multidigit arithmetic problems, designed to elicit specific shortcut strategies such as compensation. Based on the students’ written work, strategies were classified into whether a shortcut strategy was used or not. Results showed that the frequency of shortcut strategies ranged between 6 and 21% across problem types, and that boys and high mathematics achievers were more inclined to use shortcut strategies. An explicit instruction to look for a shortcut strategy increased the frequency of these strategies in the addition and multiplication problems, but not in the subtraction and division problems. Finally, the use of shortcut strategies did not yield higher performance than using standard strategies. All in all, spontaneous as well as stimulated use of shortcut strategies by Dutch sixth graders was not very common. Show less
Hickendorff, M.; Edelsbrunner, P.A.; McMullen, J.; Schneider, M.; Tresize, K. 2017
This article gives an introduction to latent class, latent profile, and latent transition models for researchers interested in investigating individual differences in learning and development. The... Show moreThis article gives an introduction to latent class, latent profile, and latent transition models for researchers interested in investigating individual differences in learning and development. The models allow analyzing how the observed heterogeneity in a group (e.g., individual differences in conceptual knowledge) can be traced back to underlying homogeneous subgroups (e.g., learners differing systematically in their developmental phases). The estimated parameters include a characteristic response pattern for each subgroup, and, in the case of longitudinal data, the probabilities of transitioning from one subgroup to another over time. This article describes the steps involved in using the models, gives practical examples, and discusses limitations and extensions. Overall, the models help to characterize heterogeneous learner populations, multidimensional learning outcomes, non-linear learning pathways, and changing relations between learning processes. The application of these models can therefore make a substantial contribution to our understanding of learning and individual differences. Show less
Torbeyns, J.; Hickendorff, M.; Verschaffel, L. 2017
This study examined the contributions of developmental changes in social-cognitive ability throughout adolescence to the development of narrative comprehension. We measured the effects of... Show moreThis study examined the contributions of developmental changes in social-cognitive ability throughout adolescence to the development of narrative comprehension. We measured the effects of sensitivity to the causal structure of narratives and of sensitivity to differences in social-cognitive processing demands on narrative recall by children (8–10 years old), adolescents (13–15 years old), and adults (19–21 years old). Generalized mixed-effects models for dichotomous variables revealed that social-cognitive processing demands of story elements predicted differences in narrative recall between the age groups, over and above the causal importance of story elements. Children's and adolescents' recall of the narrative differed from that of adults, and these differences were most apparent for social-cognitive aspects of the narrative. These findings suggest that immature social-cognitive abilities limit narrative comprehension in childhood and adolescence and, in doing so, contribute to our understanding of the interaction between reader characteristics and text characteristics in the development of narrative comprehension. Show less
