Drawing on a Theoretical Model to Study Students’ Understandings of Fractions

Abstract

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.

Teaching of Division of Fractions through Mathematical Thinking

“…The research by Charalambous, et al (2007) indicated that the concept of equivalence fractions helps students to work with the subconstructs of fractions and manage the operation of fractions more easily. Using the concept of ratio and table to teach division of fraction is shared by many researchers (Sharp 1998;Sharp 2002;Tirosh, 2000).…”

Cognitive Factors' Influence on Fraction Development in Students of Varying Ability Levels

“…Each of these sub-constructs represents a different function of fractions. Some evidence suggests that this theoretical model lines up with the way in which students conceive fractions (Charalambous & Pitta-Pantazi, 2007), and the five sub-constructs have been widely accepted (Behr et al, 1993;Charalambous & Pitta-Pantazi, 2007;Kieren, 1976).…”

“…Fractions do not represent one monolithic construct, rather researchers have identified five interrelated fraction sub-constructs: part-whole, ratio, operator, quotient, and measure (Behr, Harel, Post, & Lesh, 1993;Charalambous & Pitta-Pantazi, 2007;Kieren, 1976;Pantziara & Philippou, 2012). Each of these sub-constructs represents a different function of fractions.…”

“…Each of these sub-constructs represents a different function of fractions. Some evidence suggests that this theoretical model lines up with the way in which students conceive fractions (Charalambous & Pitta-Pantazi, 2007), and the five sub-constructs have been widely accepted (Behr et al, 1993;Charalambous & Pitta-Pantazi, 2007;Kieren, 1976).Each of the five sub-constructs highlights certain functions and qualities of fractions.Students not understanding one or more of the constructs will have an incomplete conception of fractions. To understand the difficulty students have when attempting to learning about fractions, it is first necessary to explore the five different fraction sub-constructs.…”

“…That is, the numbers in a numerator and denominator of a ratio only have meaning in relation to each other. Key to understanding ratios is the covariance-invariance property (Charalambous & Pitta-Pantazi, 2007). This property states that the two values of a ratio must change together so that the relationship remains the same.…”