Teaching Fractions
Owens (1980) examined the relationship between a pupil’s concept of area and their ability to learn fraction concepts finding a positive connection and he found a positive relationship between success on area tasks and success in fractions tasks based on geometric regions. The teaching for transfer from area tasks aids a pupil’s ability to learn fraction concepts. In contrast the findings from a study by Novillis-Larson (1980, p. 423) demonstrated that pupils working “with tasks involving the location of fractions on number lines” gained an imprecise and inflexible notion of fraction.
Since pupils actively construct knowledge, teachers must actively help them dismantle their misconceptions. Often everyday misconceptions are learnt from the wider community and these can cause cognitive conflict which takes time for pupils to reconstruct. An active classroom discussion, with the teacher serving as guide, helps pupils express their misconceptions and overcome them. Misunderstandings in the use of mathematical language, both by teachers and pupils, can result in difficulties when learning mathematics. Inexact usage, by mathematics teachers, of language in defining mathematical concepts cause problems for pupils which are difficult to rectify. For example, ‘When you multiply by 10 you add a zero’ or ‘All parallel lines are straight’ or ‘multiplying by a number makes the answer bigger’ once these incorrect ‘rules’ are learnt they are seldom, in my experience, ever unlearnt.
The link between inexact use of language as a component of the design features of a mathematical lesson given to pupils and their resulting misconceptions is poorly understood, and hence the implication for mathematics teachers in in teaching of conceptually difficult mathematics topics is huge. For example, when a learner first encounters algebra the topic is often related to “x’s and y’s” rather than being the generalisation of all the schemes learnt for number and fractions. Hence learners trying to comprehend the meaning of
and more importantly how they might perform the required manipulation creates massive cognitive conflict.
