The development of students' understanding of the numerical value of fractions

An experiment is reported that investigated the development of students' understanding of the numerical value of fractions. A total of 200 students ranging in age from 10 to 16 years were tested using a questionnaire that required them to decide on the smallest/biggest fraction, to order a set of given fractions and to justify their responses. Students' responses were grouped in categories that revealed three main explanatory frameworks within which fractions seem to be interpreted. The first explanatory framework, emerging directly from the initial theory of natural numbers, is that fraction consists of two independent numbers. The second considers fractions as parts of a whole. Only in the third explanatory framework, students were able to understand the relation between numerator and denominator and to consider that fractions can be smaller, equal or even bigger than the unit. © 2004 Elsevier Ltd. All rights reserved

Extending a digital fraction game piece by piece with physical manipulatives

This paper reports results from an ongoing project that aims to develop a digital game for introducing fractions to young children. In the current study, third-graders played the Number Trace Fractions prototype in which they estimated fraction locations and compared fraction magnitudes on a number line. The intervention consisted of five 30 min playing sessions. Conceptual fraction knowledge was assessed with a paper based pre- and posttest. Additionally, after the intervention students’ fraction comparison strategies were explored with game-based comparison tasks including self-explanation prompts. The results support previous findings indicating that game-based interventions emphasizing fraction magnitudes improve students’ performance in conceptual fraction tasks. Nevertheless, the results revealed that in spite of clear improvement many students tended to use false fraction magnitude comparison strategies after the intervention. It seems that the game mechanics and the feedback that the game provided did not support conceptual change processes of students with low prior knowledge well enough and common fraction misconceptions still existed. Based on these findings we further developed the game and extended it with physical manipulatives. The aim of this extension is to help students to overcome misconceptions about fraction magnitude by physically interacting with manipulatives.acceptedVersionPeer reviewe

The mental representation of the magnitude of symbolic and nonsymbolic ratios in adults

This study mainly investigated the specificity of the processing of fraction magnitudes. Adults performed a magnitude-estimation task on fractions, the ratios of collections of dots, and the ratios of surface areas. Their performance on fractions was directly compared with that on nonsymbolic ratios. At odds with the hypothesis that the symbolic notation impedes the processing of the ratio magnitudes, the estimates were less variable and more accurate for fractions than for nonsymbolic ratios. This indicates that the symbolic notation activated a more precise mental representation than did the nonsymbolic ratios. This study also showed, for both fractions and the ratios of dot collections, that the larger the components the less precise the mental representation of the magnitude of the ratio. This effect suggests that the mental representation of the magnitude of the ratio was activated from the mental representation of the magnitude of the components and the processing of their numerical relation (indirect access). Finally, because most previous studies of fractions have used a numerical comparison task, we tested whether the mental representation of magnitude activated in the fraction-estimation task could also underlie performance in the fraction-comparison task. The subjective distance between the fractions to be compared was computed from the mean and the variability of the estimates. This distance was the best predictor of the time taken to compare the fractions, suggesting that the same approximate mental representation of the magnitude was activated in both tasks