Learning Fractions
It is well documented in the research literature that fractions are among the most complex mathematical abstract concepts (Gabriel et al., 2013) that children encounter and equally as important it is widely acknowledged that fractions are difficult to teach. It has also been claimed that learning fractions is probably one of the most serious obstacles to the mathematical maturation of children and that many of the ‘trouble spots’ in early school mathematics are related to rational-number ideas.
Ellerbruch and Payne (1987) investigated the role of language in facilitating mathematics learning. They report that children who, first say aloud the fraction then transcribe the oral sound before going to the symbolic form seldom make the common reversal error of writing; for example (
is seldom written as
).
There has been an emphasis on the part-to-whole sub construct in school mathematics and the verbalisation of fractions has largely been ignored. Children learning mathematics create frames and sub-frames as described by (Davis and Simmt, 2003) to build complex learning schemas and links between mathematical topics. Early exposure to arithmetical operations through shared dialogue creates powerful frameworks for the four arithmetic operators which can then usually be recalled and applied to other areas of mathematics.
Taking for example the operation of addition once the basic framework has been discussed, shared and internalise it can be applied to the addition of decimals, algebraic expressions and functions with little or no re-organisation of the framework. The application of the framework to other areas of mathematics enhances and further legitimises the framework for the learner. The difficulty arises when the learner tries to apply the framework to a topic with a different set of rules for the given operation. So, applying the addition framework to fractions gives erroneous results. This leads to confusion in the mind of the learner as the framework has to be altered, amended or even totally redesigned to take care of a ‘special case’.
The clear boundaries that exist between these higher-level frameworks although they can become blurred when dealing with fractions. For example, current teaching approaches advocate that the division of fractions by using the algorithm of inverting and then multiplying. We now have a real conflict in the mind of the learner – a division framework interacting with a multiplication framework. The muddling of these higher-level frameworks in the eyes of the young learner creates a real tension that can result in cognitive conflict.
So where should the mathematics teacher start?
The starting point must be good mathematical and pedagogical content knowledge (Olanoff et al., 2014) rather than just knowing the skills, processes and procedures of manipulating fractions. Research tends to suggest that content knowledge is relatively strong when it comes to performing procedures, but that there is a general lack flexibility in moving away from procedures and using “fraction number sense” (Newton, 2008; Yang et al., 2008). It is common practice when learning fractions to consider the equivalence of fractions early in the learning sequence once the basic notion of fraction has been internalised by the learner. The concept of the equivalence of simple fractions is normally well developed in most learners by the time they reach secondary school age. The concept is often used to deal with the addition and subtraction of fractions with differing denominators. Hence;
As teachers of mathematics we need to ask ourselves why should we do this and what is the mathematical justification? More importantly why do we not use this well understood conception when dealing with the division of fractions, so instead of
and creating a conflict in the mind of the learner therefore should we not the previously learnt skill of equivalence of fractions and teach
Thus, the division framework (schema) learnt for integers, decimals, equations etc is simply being applied to fractions.
