4 Knowing and thinking in mathematics
4.1 From awareness to understanding
In this section the mathematical content is more obvious as we talk explicitly about what it means to know and to think in mathematics. We will also address your own personal knowledge in the subject.
Like any other activity, doing and learning mathematics involves:
using and adapting existing knowledge;
acquiring and constructing new knowledge through thinking and learning;
building up links that enable known things to be accessed when needed.
Activity 12: Calculations involving addition
Do the following calculations. As you do them:
notice how you do them;
think about the knowledge that you are using.
6 + 4 = ?
2 + ? = 10
15 + ? = 10
12 + 5 + 38 + 6 + 17 + 15 + 24 = ?
36.5 + 77.8 + 54.5 + 53.2 = ?
Write a brief description of what you did and how you arrived at your answers.
10; 8; −5; 117; 222.
You may have worked out the calculations in your head, on paper or with the aid of a calculator. You may have done different calculations using different methods; or you may have done one using one method and then have checked it using another. But to do the calculations at all, you needed to use a lot of knowledge, including:
number facts, e.g. spotting number pairs that add up to 10;
decimal number system;
different ways of writing additions;
how to use a calculator.
You probably also had some sense of what the mathematical operation of addition is all about.
You would have acquired this knowledge and the ability to use it over a number of years. Initially, you would have learnt something about addition by putting two or more sets of objects together to make a bigger set of objects. You would then have extended this understanding to summing whole numbers and making a ‘larger’ number
(e.g. 6 + 4 = 10).
Gradually, you would have learnt number facts that you can now instantly recall. You would also have learnt that subtraction ‘undoes’ (i.e. is the inverse of) addition: although you probably did not understand it in those terms, you learnt how to do it, perhaps by ‘counting back’.
It is likely that, at this stage, you thought that addition ‘makes larger’ and subtraction ‘makes smaller’; but when your knowledge of numbers was extended to include negative numbers, this notion had to be amended or reconstructed.
Learning about fractions and decimals extended your knowledge of addition further as you discovered that it could be applied to all kinds of numbers, and could even be used with ‘letters’ (variables) in algebra.
Along the way, you would have learnt several techniques such as mental strategies, ways of setting out written calculations, and how to extract a calculation from a word problem.
Learning is a gradual process, and consists of a continual building upon and modification of what has gone before. It can be thought of as a ‘See, Experience, Master’ framework (see Mason, 1999):
seeing a concept go by rather quickly, leaving one with only a taste;
experiencing an idea using previously mastered skills;
mastering by using newly acquired skills in different contexts.
The learning process involves an initial awareness, then an experiencing phase of manipulating, making sense, and practising use, before one is able to articulate understanding. Consider what happens when you learn a new word (perhaps you have encountered a few for the first time while doing this course): first, you struggle with its meaning and pronunciation; then perhaps you start to use it tentatively, gradually gaining understanding and confidence, until it becomes part of your vocabulary.
This framework is fundamentally a psychological one which summarises a view of the process of learning, which is that both the building and modification of knowledge are stimulated by the experiences that come the learner's way. Structures that have already been built by past experiences are brought to bear on any new experience. This may require no significant effort on the part of the learner: it may just involve adding together more complex pairs of numbers as the learner becomes more proficient at addition. On the other hand, the new experience may pose a problem which the existing structures struggle with – for example, when the learner first attempts to add negative numbers. This struggle is what stimulates the growth and change of knowledge. To begin with, this growth and change may be only partial in that it is specific to the problem posed. However, as the learner experiences more and more problems of the same sort, he or she becomes sufficiently able to render modifications to knowledge which are so complete and stable that applying the same kind of thinking to new situations becomes possible.
The labels ‘manipulating’, ‘making sense of’ and ‘articulating’ are an attempt to capture this process. The learner manipulates particular problems – that is, he or she ‘plays around’ with them in an attempt to discover some kind of an answer. After manipulating several examples in a rather similar way, the learner begins to make sense of the general process being used. Eventually, the process is learnt so well that it can be articulated – for example, by explaining it to someone else.