Abstract Many students still have not developed a robust understanding of rational number concepts at the end of primary school, despite several years of instruction on the topic. The present study aims to examine the patterns, predictors, and outcomes of the development of rational number knowledge in lower secondary school. Latent transition analysis revealed that rational number development from primary to lower secondary school (N = 362) appears to follow similar patterns as in younger students. In particular, a majority of students had poor knowledge of the density of the rational number set. Whole number magnitude knowledge appeared to be an important predictor of the development of rational number size knowledge, but not density knowledge. Finally, fraction density knowledge appeared to be related to concurrent algebra knowledge. Together these results point to an important role for density knowledge in mathematical development.

In this paper we focus on the development of rational number knowledge and present three research programs that illustrate the possibility of bridging research between the fields of cognitive developmental psychology and mathematics education. The first is a research program theoretically grounded in the framework theory approach to conceptual change. This program focuses on the interference of prior natural number knowledge in the development of rational number learning. The other two are the r...

#1Maureen E. Gray(UCLA: University of California, Los Angeles)H-Index: 2

#2Melissa DeWolf(UCLA: University of California, Los Angeles)H-Index: 11

Last. Keith J. Holyoak(UCLA: University of California, Los Angeles)H-Index: 86

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ABSTRACTThe present study examined whether a dissociation among formats for rational numbers (fractions, decimals, and percentages) can be obtained in tasks that require comparing a number to a non-symbolic quantity (discrete or else continuous). In Experiment 1, college students saw a discrete or else continuous image followed by a rational number, and had to decide which was numerically larger. In Experiment 2, participants saw the same displays but had to make a judgment about the type of rat...

Many students’ knowledge of fractions is adversely affected by whole number bias, the tendency to focus on the separate whole number components (numerator and denominator) of a fraction rather than on the fraction's magnitude (ratio of numerator to denominator). Although whole number bias appears early in the fraction learning process and under speeded conditions persists into adulthood, even among mathematicians, little is known about its development. Performance with equivalent fractions indic...

: Recent research suggests that fraction understanding is predictive of algebra ability; however, the relative contributions of various aspects of rational number knowledge are unclear. Furthermore, whether this relationship is notation-dependent or rather relies upon a general understanding of rational numbers (independent of notation) is an open question. In this study, college students completed a rational number magnitude task, procedural arithmetic tasks in fraction and decimal notation, an...

Last. Wim Van Dooren(Katholieke Universiteit Leuven)H-Index: 27

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Abstract In this study we longitudinally followed 201 upper elementary school learners in the crucial years of acquiring rational number understanding. Using latent transition analysis we investigated their conceptual change from an initial natural number based concept of a rational number towards a mathematically more correct one by characterizing the various intermediate states learners go through. Results showed that learners first develop an understanding of decimal numbers before they have ...

Last. Kelly Trezise(University of Melbourne)H-Index: 5

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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 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 respo...

Last. Nancy C. Jordan(UD: University of Delaware)H-Index: 42

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: The present study investigated the development of fraction comparison strategies through a longitudinal analysis of children's responses to a fraction comparison task in 4th through 6th grades (N = 394). Participants were asked to choose the larger value for 24 fraction pairs blocked by fraction type. Latent class analysis of performance over item blocks showed that most children initially exhibited a "whole number bias," indicating that larger numbers in numerators and denominators produce la...

#2Jake McMullen(UTU: University of Turku)H-Index: 13

Last. Erno Lehtinen(UTU: University of Turku)H-Index: 33

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ABSTRACTDifficulties with rational numbers have been explained by a natural number bias, where concepts of natural numbers are inappropriately applied to rational numbers. Overcoming this difficulty may require a radical restructuring of previous knowledge. In order to capture this development, we examined third- to fifth-grade students' understanding of the size of rational numbers across a 1-year period. On three occasions, students answered a test about the size of decimals and fractions. Lat...

#1Melissa DeWolf(UCLA: University of California, Los Angeles)H-Index: 11

#2Miriam Bassok(UW: University of Washington)H-Index: 25

Last. Keith J. Holyoak(UCLA: University of California, Los Angeles)H-Index: 86

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Abstract Recent work has identified correlations between early mastery of fractions and later math achievement, especially in algebra. However, causal connections between aspects of reasoning with fractions and improved algebra performance have yet to be established. The current study investigated whether relational reasoning with fractions facilitates subsequent algebraic reasoning using both pre-algebra students and adult college students. Participants were first given either a relational reas...

Last. Erno Lehtinen(UTU: University of Turku)H-Index: 33

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Many people have serious difficulties in understanding rational numbers, limiting their ability to interpret and make use of them in modern daily life. This also leads to later difficulties in learning more advanced mathematical content. In this study, novel tasks are used to measure 263 late primary school students’ spontaneous focusing on quantitative relations, in situations that are not explicitly mathematical. Even after controlling for a number of known predictors of rational number knowle...

#2Jake McMullen(UTU: University of Turku)H-Index: 13

Last. Michelle Hurst(U of C: University of Chicago)H-Index: 5

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Understanding fractions and decimals requires not only understanding each notation separately, or within-notation knowledge, but also understanding relations between notations, or cross-notation knowledge. Multiple notations pose a challenge for learners but could also present an opportunity, in that cross-notation knowledge could help learners to achieve a better understanding of rational numbers than could easily be achieved from within-notation knowledge alone. This hypothesis was tested by r...

Last. Wim Van Dooren(Katholieke Universiteit Leuven)H-Index: 27

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Although a good rational number understanding is of crucial importance for learners’ general maths achievement, many learners have misconceptions about fractions. An often described misconception i...

Last. Wim Van Dooren(Katholieke Universiteit Leuven)H-Index: 27

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Students often show difficulties in understanding rational numbers. Often, these are related to the natural number bias, that is, the tendency to apply the properties of natural numbers to rational...

School mathematics comprises a diversity of concepts whose cognitive complexity is still poorly understood, a chief example being fractions. These are typically taught in middle school, but many students fail to master them, and misconceptions frequently persist into adulthood. In this study, we investigate fraction comparison, a task that taps into both conceptual and procedural knowledge of fractions, by looking at performance of highly mathematically skilled young adults. Fifty-seven Chilean ...

Students' mathematics learning achievement is influenced by many factors, one of which is the learning media used. The textbook is one alternative learning media. The textbook used previously was not focused on problem-solving, so that this development research aims to produce a textbook based on problem-solving in algebra that is valid, effective, and practical. This research uses Gall & Borg development research which is limited to seven steps namely research and information collecting, planni...