Actions and objects
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.
Actions Becoming Objects: the role of notation
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.
Recognising Relationships & Perceiving Properties
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
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.