5 Big Ideas about the Development of Fraction and Division Knowledge

by AJ Barnes

1. For both students and teachers, fractions can be frustrating.  Yet fractions are crucial for success in mathematics.   

2. Students are often taught that a fraction represents a “part” of a “whole.”  While this is not an inappropriate way to explain fractions to children, it is insufficient and may lead to later misconceptions. Students also need to develop knowledge of important sub-constructs of the concept of fraction: ratio, operator, quotient, and measure. (Kieren, 1976)

3. A common misconception among students is that fractions are not numbers in and of themselves.  Children may see fractions as being composed of multiple whole numbers instead of numbers unto themselves; this is known as the whole number bias and is behind many of the errors and mistakes children make when working with fractions. (Gabriel, et al. 2013)

4. Even when secondary students may know how to perform an operation with fractions by hand, the majority of them reach for a calculator to assist in their work or verify their answer.   

5. One of the main reasons that division is believed to be difficult for children is the distinction between partitive and quotative division.  Partitive (sharing) division is often more difficult for students to do since the dividend and divisor do not have the same units.  Young children will often solve partitive division problems by dealing.  While this may seem a primitive and inefficient strategy, it can be made more sophisticated for more complex problems, such as those involving remainders. 

References:

Gabriel, F., Coché, F., Szucs, D., Carette, V., Rey, B., & Content, A. (2013). A componential view of children's difficulties in learning fractions. Frontiers in psychology, 4, 715.

Kieren, T. E. (1976, April). On the mathematical, cognitive and instructional foundations of rational numbers. In Number and measurement. Papers from a research workshop (Vol. 7418491, p. 101).