Supporting students to reason about the relative size of proper and improper fractions (original) (raw)
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Teaching and learning fractions has traditionally been one of the most problematic areas in primary school mathematics. Several studies have suggested that one of the main factors contributing to this complexity is that fractions comprise a multifaceted notion encompassing five interrelated subconstructs (i.e., part-whole, ratio, operator, quotient, and measure). Kieren was the first to establish that the concept of fractions is not a single construct, but consists of several interrelated subconstructs. Later on, in the early 1980s, Behr et al. built on Kieren’s conceptualization and suggested a theoretical model linking the five subconstructs of fractions to the operations of fractions, fraction equivalence, and problem solving. In the present study we used this theoretical model as a reference point to investigate students’ constructions of the different subconstructs of fractions. In particular, using structural equation modeling techniques to analyze data of 646 fifth and sixth graders’ performance on fractions, we examined the associations among the different subconstructs of fractions as well as the extent to which these subconstructs explain students’ performance on fraction operations and fraction equivalence. To a great extent, the data provided support to the associations included in the model, although, they also suggested some additional associations between the notions of the model. We discuss these findings taking into consideration the context in which the study was conducted and we provide implications for the teaching of fractions and suggestions for further research.
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The importance of the knowledge of fractions in mathematical learning, coupled with the difficulties students have with them, has prompted researchers to focus on this particular area of mathematics. The term 'fraction proficiency' used in this article refers to a person's conceptual comprehension, procedural skills and the ability to approach daily situations involving fractions. In the area of fractions, there has been a call for more research to study how, and where, efforts should be focused in order to integrate the various aspects of fraction knowledge for students, and even for teachers, to help them develop proficiency in fractions. Thus, the article presents a theoretical synthesis of the specialized literature in the learning and teaching of fractions, with the aim of proposing a framework for developing students' fraction proficiency. The frameworks presented in the article may shed light upon the implications for the design of fraction instruction, which should focus on developing a multi-faceted knowledge of fractions, rather than simply isolating one facet from the others.