Published 2011

One of the powerful developments in mathematics in the 20th century was the realisation that mathematics can be seen as the study of objects through the study of actions that act upon those objects. Thus arithmetic is the study of the actions of adding, subtracting, multiplying and dividing, on numbers. What is important is the relationships between actions, not the calculating of answers as such. Similarly, geometry is the study of actions on shapes that preserve some aspects (but not necessarily all) of those shapes.

For example, to study integers, it is useful to think of actions such as 'add 3' or 'subtract 5', or 'subtract from 7'. These actions can be composed ('adding 3' can be followed by 'adding 5' and the overall result turns out to be the same as 'adding 8').

To study shapes, it is useful to think of actions which preserve orientation (translation, rotation, scaling), or which preserve distances between points (translation, rotation and reflection) or which preserve angles between lines (scaling, rotating, translating, reflecting).

Actions which preserve some property are particularly fruitful. Thus 'adding 3' preserves the order and the distance between numbers (and can be thought of geometrically as a translation). Translating any geometrical shape, rotating it about some point through some angle, scaling by some factor, and reflecting it in some mirror-line each preserves some relationships but alters others.

Very often it is useful to develop a notation for actions, and then to start treating the actions as objects worthy of study in their own right. For example, 'add 3' can be seen as an action on the number line. Denoting it by A3, and denoting 'subtract 5' by S5, then S5 is the same as A-5. Performing A3 followed by S5 (usually denoted S5 o A3) has the same effect as S2 which is the same as A-2.

Studying actions on objects involves a subtle transition in what is attended to, what is stressed and what is consequently ignored. Attending to relationships between particular objects can be referred to as recognising relationships. For example, that 3 and 5 are the same distance apart and in the same order as 7 and 9, and the same distance apart but in the opposite order to -3 and -5 (since 5 is to the right of 3, but -5 is to the left of -3). There is a different way of attending (referred to as perceiving properties) in which properties are perceived as being instantiated in particular instances. Thus the fact that 3 and 5 are a distance 2 apart is a particular instance of the property of 'being a distance two apart'. The shift between these is ever so subtle. Sometimes learners make it with ease, and sometimes they dwell in the recognising relationships in particular rather than perceiving properties as being instantiated. When this happens, it can be useful to invite learners to generate more examples of what the teacher perceives as a property, until the learners become aware of a property being instantiated.

For example, the fact that 3 + 5 is the same as 5 + 3 is not due to the fact that the answers are both 8, but rather because addition is invariant under change of order. becoming aware of this property can be approached through experiencing in quick succession a number of instances, such as

2 + 5 and 5 + 2; 3 + 5 and 5 + 3;
4 + 5 and 5 + 4;

3 + 4 and 4 + 3; 3 + 6 and 6 + 3;
3 + 7 and 7 + 3;

Here varying one number can draw learner attention to the fact that any number can appear in that place; varying the other number similarly draws attention to the generality; learners making up their own examples offers them the opportunity to be creative and adventurous. Linking these with experience of, for example, putting collections of objects into a bag, calls upon the body sense that putting one collection in the bag and then the other produces a number of objects in the bag that is independent of the order in which the collections were 'added'.

Fractions are a particular example of where confusion between action and the result of the action confuses learners. Fractions are actions on 'units' or 'wholes' (which are collections of objects). First the 'unit' is divided into a specified number of equal sized parts, and then a specified number of those parts are selected. Fractions themselves are not numbers; their effect on the unit 1 can be placed on the number line, and so are treated as numbers. Thus to multiply two fractions is to compose their actions (one followed by the other), and is much easier to comprehend than the addition or subtraction of fractions. Addition of fractions is the combining of the results of two actions on the same unit, and so is a third action (putting together of collections) each of which arose from the result of an action on the same unit or whole.

One route to algebra is through composing arithmetic actions and then treating the composition as a single action or 'function machine'. Expressing the effect of the compound action on some as-yet-unspecified number (perhaps a cloud used to represent what someone is thinking about who is not accessible and so the number remains for the present as-yet-unspecified) generates algebraic expressions (generalities). The mechanics of algebra (algebraic manipulation) are concerned with studying the effects of combining, undoing and otherwise relating expressions. These make little or no sense unless learners have themselves gained facility in generating expressions so that they know how they arise. In the process of gaining facility, they will encounter most of the algebraic manipulations needed to display fluency in algebra.

Mathematical structure is a description of properties that are preserved or maintained under a set of actions. The properties themselves, instantiated in many different contexts, constitute a structure. For example, there may be an action (sometimes called an operation) on any two objects from a set that yields a third element from that same set. Such an action or operation is called closed: closure is a structural property of that action. Similarly, order in which actions are performed may not matter: such actions are said to commute, or the action to be commutative. For example addition and multiplication are commutative when considering them as actions on whole numbers, integers, rationals, reals or complex numbers, but not when applied to subtraction or division.