Mathematics is the science of measurement. That is why the concepts in mathematics must be carefully, logically defined. Concepts are either valid or not. Mathematical statements using concepts can be determined to be either true or false.
We need to understand what concepts and statements mean. Learning the language of mathematics – the vocabulary of concepts – is crucial.
When a student who has just learned to count encounters fractions for the first time he/she must definitely expand his/her consciousness. The student is coming to grips with a second level of abstraction – abstractions built upon abstractions.
Without care in teaching – and coaching in learning – students get lost very fast.
Often students are told that fractions are just parts of wholes, like pieces of pie. It is confusing to then instruct them to just “multiply and divide pieces of pie”. It is little wonder that many students will simply give up on understanding that kind of teaching. Other students quit on mathematics when they sense their teachers are themselves fearful or if they lack the understanding to clearly solicit and answer their questions.
Let’s be very clear – a fraction is a number. It is a count of the multiples of the number of equally divided parts of a whole. That “whole” can be a region or a collection of like items.
The “number” of equal divisions and the “count of multiples” are first level abstractions because we can point to concretes as examples. At the first level we have abstracted a count from we perceive. However, “a fraction” is a second-level abstraction in that we have related abstractly two first-level abstractions..
Like other numbers fractions can be made to correspond to a point on the “number line” – another second-level abstraction. Arithmetic with counting numbers can be justified on the number line and so can arithmetic with fractions.
When fractions are taught well they connect easily to what was taught in arithmetic. And ;earning “Fractions” like the above is an important step in introducing the even more abstract subject of “algebra”.
When will teachers simply point out that “algebra is generalized arithmetic” – and then simply relate these “new abstractions” to what students should already know well?