One-Third is Three-Quarters of One-Half

This paper reports on part of a larger cross-sectional study of the development of students’ quantitative concepts of fractions. In total, 1676 students in Years 4–8 were asked a series of questions designed to elicit their concept images of various fractions. Three questions asked the students to construct a regional model (parts of a circle) for one-half, one-third and one-sixth. The initial analysis of the evoked concept image of one-third revealed an unexpectedly high number of responses shading one-quarter of a circle. Further analysis suggested that what appeared to be one-quarter was frequently intended to be one of three parts, confirming that drawings frequently do not “speak for themselves”. Area is not always the intended feature of regional models of fractions in students ’ concept images and it is argued that the “number of pieces ” interpretation is a common response to regional models of fractions. Mathematics teaching seeks to build on what students already know. When it comes to the teaching of fractions, the limitations of the “part of a whole ” model of a fraction (Kerslake, 1986; Kieren, 1988) have been described. Many students appear to see fractions as two unrelated whole numbers—three-quarters is the whole number three written ove