In mathematics education, an artificial line is commonly drawn between rote memorisation and application. Yet mathematics shares many of the same properties as language, which we have never thought to teach by isolating memory work from practical meaning. The basic symbols and concepts and rules of mathematical grammar must be memorised, it is true – in absolutely the same manner as the basic symbols and concepts and rules of language. If mathematical concepts are not simultaneously translated and revisited in the context of practical usage, the child may well struggle with “abstract” thinking all their life.
When we idealise abstract thinking, usually what we are really thinking of is symbolic thinking: the ability to separate out layers of a problem in order to develop and cross-apply a general rule. The first and easiest layer for a child to access is its hands-on representation: how many apples? do I take an apple away or do I add another apple? The second level introduces symbolism, as the number in isolation begins to replace the necessity for concrete objects. This process is no different than the child’s growing ability to separate out the colour “red” from the object “apple”. Both numbers and adjectives are descriptors which initially are linked in with specific types of objects, but which are not inherently a part of that object. As the child’s conceptual skills grow, the child gradually learns to separate out the descriptor from the object, and then to manipulate that descriptor separately from the object.
Life itself continually reinforces spoken language, grounding us each day in the interactive practical examples which allow us to develop and refine the general rules. The practicality is important to introduce the primary layer upon which symbolism is built; but even more important is the continual give and take of interaction as the child continually learns new connections to determine what does and doesn’t work. Rote memorisation only allows a baseline “A is A” type of learning, a series of isolated factoids ultimately without sense or context – and does that ever show in average writing skills! In contrast, continual interactive examples of how to use spoken and written language allows for continual refinement of the general rule – which, after all, is not an absolute.
Absolutely the same is true for mathematics as well: which originally developed as a way of explaining and applying specific functions found in everyday life. For very young children play is learning; and to some extent this never changes. Construction blocks, games, even imitative play can all be used to give the child those interactive practical examples. To this day decimal percentages are second nature to me because I learned them as the banker in playing Monopoly. Shopping trips, whether at home or as school projects, are particularly rich in potential practical examples, the possibilities limited only by imagination.
Building the foundations of a continual learning style requires far more initial work than rote memorisation, but in time those foundations will make all practical uses of mathematics much, much more accessible. For those who pursue higher mathematics, they will further allow the personal flexibility essential to those leaps of insight which allow us to recognise that what we had thought the general rule is itself a special application.