Those Troublesome Fractions

I argued here that textbooks often neglect to present fractions as division expressions, preferring instead to talk about them as ratios.

In truth, one of the reasons fractions are so difficult a topic for young learners (and a lot of adults) is that they, indeed, have at least two distinct personalities (or seem to)--they are on the one hand expressions, prescribing operations or steps to be followed, and on the other hand results, or answers.

Of course, we know these are not mutually exclusive categories. The expression 3 + 1 is by no means any less of a "result" than is 4. But there is evidence to suggest that, when it comes to fractions, students do see expressions and answers as belonging in two separate camps:

Research on the role of partitioning in the development of fractions has focused on fractions as quotients . . . and fractions as operators . . . In the latter set of studies, fractions are mapped onto the action of partitioning a set; the unit of reference for the fraction is the set. For example, to divide a set into 5 groups and consider 2 of those groups is to find two fifths of the set. In the former set of studies, . . . fractions are mapped onto the state that results from the action of partitioning a set into a given number of groups; the unit of reference for the fraction is a single item in the set. For example, to divide a set of 5 items into 4 groups is to find 1 and one fourth items per group. Although the operator and the state are mathematically equivalent, . . . they appear to be psychologically distinct for young children (Kieren, 1995) who treat their partitioning activities and their descriptions of the results of partitioning as separate entities (Empson, 1999). So, a child who partitions a candy into thirds may say that he or she "split in threes" but refer to each part as "a half."¹

Kieren, referenced above, makes the case in this way, though he too uses the language of quotients and operators in other places:

The folding activity helps students think of fractions as related to multiplicative actions in many ways and shows their multiplicative thinking to a teacher. The "talking about three-fourths" task provides an opportunity for students to think of fractions as additively combinable amounts.²

More on this fascinating stuff later.


1 Empson, S., Junk, D., Dominguez, H., & Turner, E. "Fractions as the Coordination of Multiplicatively Related Quantities: A Cross-Sectional Study of Children's Thinking". Educational Studies in Mathematics (2006) 63/1: 1-28.

2 Sowder, Judith T. (1995). Providing a Foundation for Teaching Mathematics in the Middle Grades. Albany: State University of New York Press.

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