General Introduction to Teaching the Four Operations

General introduction to teaching the four operations

The four operations ( + - × ÷ ) are central concepts in teaching maths. They are best taught in a way that applies unchanged to small numbers, large numbers, vulgar fractions, decimal fractions, negative numbers, algebra and much else. This avoids the multitude of rules generally taught in maths (count the noughts; move the point; find the common denominator; multiply tops and bottoms; change ÷ to × and turn upside down!).

In Making Maths Make Sense (MMMS), + - × ÷ and = are given much more significance than is generally given. They are the first five Big Ideas (of a total of twelve Big Ideas) designed to ensure rigorous maths is accessible by all pupils. The manner in which they are taught is crucial to all subsequent learning.

There is an essential feature that must be in the teacher’s mind in using the MMMS Scripts to teach the first five Big Ideas. Each of the symbols + - × ÷ = is to be thought of as an imperative verb; it is an instruction; it can be enacted without reference to the symbol (usually a number) that follows it. When we see 3 + 2 we do not say ”3 [pause] add 2” but “3 [pause] add [pause] 2”. Each symbol is treated separately. It is for that reason the script below insists that + means ‘get ready to get some more’.

You can see that 3 + 2 = 5, and all similar expressions, have the form:

noun; imperative verb; noun; imperative verb; noun. They have this form because cups are used as a proxy for the indeterminate object 3. The cups enable you to think about the maths.

This important noun; imperative verb; noun; imperative verb; noun form is universal. It is achieved by ensuring that novices without exception learn that

+ means ‘Get ready to get some more’
- means ‘Get ready to take some away’
× means ‘Do the same thing lots of times’
÷ means ‘Look at it and wonder about …’
= means ‘Look at the Maths Table and count …

It is absolutely vital that each operation is thought of as an imperative verb provoking a specific action without reference to the noun that follows. In essence 3 + 2 = is thought of as ’what you start with; the type of job; the size of the job; look at the maths table; …’. This practised set of actions ensures that young learners inspect each symbol carefully (rather than assume any two numbers on the page are inviting addition) and that this enables older learners to deal logically (rather than impulsively) with negative numbers, algebraic symbols etc.

Basic Script: Preparation for teaching Division

Before showing the actions for division you need to establish the important idea of

‘Same Value; Different Appearance’ by using a version of this script:

Is this

the same as this?

Yes it is (because there is still six cups). No it is not (because these are on the table and these are piled up). So the answer is: yes it is; no it is not; yes it is; no it is not; yes it is; no it is not; yes it is; no it is not.

The answer that mathematicians give is

Same Value: Different Appearance.

This is, of course, an important aspect of equality (i.e. the ‘equals’ sign).

Although it is true that 3 = 3 it is boring (same value; same appearance). On the other hand 2 + 1 = 3 is interesting (Same Value: Different Appearance).

The phrase and the idea of Same Value: Different Appearance is in fact used much earlier than this (in Reception) … but it needs specific attention in relation to division

Basic Script: Division

This is my Maths Store: This is the Maths Table:

it is where I keep all my cups. this is where we do maths.

Maths Store Maths Table

Neutral position

I need a helper. Stand in the middle here [places pupil in neutral position]. My job is to tell you what to do. You are the helper. Your job is to do what I tell you. I want you to

Start by putting six cups on the maths table.

[Helper moves to the Maths Store, picks up 6 cups, walks to the maths table, places them on the maths table and moves to the Neutral position].

Good. Now you know where the Maths Table is … I want you to

Look at it and wonder about [Helper stands at the maths table and scratches her head to act a mock confusion] piles oftwo cups [ensure helper makes piles of two cups]

Count the piles! How many piles of two cups can you see?Three.

This process is repeated many times so that ‘helpers’ physically respond to the spoken words:

Start by putting [number] cups on the maths table
Look at it [the maths table] and wonder about piles of
[Number] cups
Count the piles of two cups …
How many piles can you see?
[number]

Now I am going to pretend I have lost my voice: I cannot talk. My job is still to tell you what to do. This time I am going to tell you what to do by writing. When I write [teacher writes the figure 6]

you Start by putting six cups on the maths table. When I write [teacher inserts the ‘divided by’ sign]

6 ÷

you Look at it and wonder about piles of … . When I write [teacher inserts the figure 2

6 ÷ 2

you Make piles of two cups. When I write [teacher inserts the equals sign]

6 ÷ 2 =

you Look at the maths table and count the piles. Then you say “three” and I write

6 ÷ 2 = 3

The numbers and signs have been written one by one (so that each is acted by the helper) to build up the expression 6 ÷ 2 = 3

Learners need to understand expressions like 6 ÷ 2 = 3 as a series of instructions that can be interpreted as physical actions and eventually visualised as images of those physical actions. Notice that the first number is what you start with: it is cups. The number after the division sign is the ‘much-ness’ in each pile (e.g. two cups). The number after the equals sign is an indication not of ‘much-ness’ but of ‘many-ness’ – it is a pure number response to ‘How many piles of [for instance] two cups can you see?

Detailed Notes

1) It is important for pupils to work extensively with basic examples.

2) When 6 ÷ 2 = 3 (for instance) has been acted, pupils need to be taught to “Read what it says ” [Six divided by two equals three] and to “Read what it means” [Six cups divided by two cups equals three] The form of the language, the places where ‘cups’ is spoken, is absolutely crucial!!

3) When the teacher writes 6 ÷ 2 = 3 (this is called a ‘Maths Story) pupils must be taught to act the ‘Real Story’ (i.e. to act it with the cups). So … they must respond to:

I will write the Maths Story. You act the Real Story.

4) When the teacher acts the ‘Real Story’ (i.e. acts it with the cups) pupils must be taught to write 6 ÷ 2 = 3 (this is called a ‘Maths Story). Detailed attention must be given to ensuring they form their numbers and symbols correctly. It is extremely important for pupils be made to see they start by putting six cups on the maths table; they then look and wonder; and they wonder about piles of two cups. They then count how many piles of cups they can see … and the answer is ‘three’ (a pure number). So … they must respond to:

I will act the Real Story. You write the Maths Story.

5) The whole point of 3) and 4) above is to create an instinctive relationship between the maths story and the real story. … but they must be taught carefully to interpret the symbols.

Practise with examples like these:

6 ÷ 2 = 3 6 ÷ 3 = 2 6 ÷ 6 = 1 6 ÷ 1 = 6

3 ÷ ½ = 6 1 ÷ ½ = 2 ½ ÷ ½ = 1 ½ ÷ ¼ = 2

Important points about pedagogy

MMMS employs detailed pedagogy in order to make maths accessible to all pupils. At first sight, it may seem that the use of ‘cups’ is either a gimmick or unnecessary or both – it does need careful study fully to appreciate its significance. Similarly, there could easily be a tendency to dismiss references to the Maths Story and the Real Story as optional. It quite simply is not optional. It is a significant contribution to mature problem solving (when additional work is done with the third story: the Real-Life Story).

Let me provide just a small part of the background. The Maths Story is abstract – it is pure maths. In order to make these symbols accessible most maths courses use real-life objects (pizzas, sandwiches, pencils and so on) so that the pure maths has a real-life meaning. This is perfectly sensible but simply poor pedagogy – because it complicates what is really quite straight-forward. Although the Maths Stories need to be made accessible, when this is done in a powerful way it is clear just how easy it is to deal with Maths Stories. MMMS uses cups so often that pupils start to think with them rather than think about them. The use of cups is a pedagogical device for teaching maths – they are associated with classroom life rather than real life - and they are a vehicle for the attention we give later to Real-Life Stories. It is when the distinction is made between Maths Stories, Real Stories and Real-Life Stories that we can begin to deal sensibly with problem solving. Maths Stories (abstractions) are understood in relation to Real Stories (concrete representations) in order to use Maths Stories to model Real-Life Stories (which are very complex, using language rather carelessly – the vernacular of real life is harder to interpret than the precise and logical language of maths).

The use of cups and the MMMS vocabulary of Maths Story / Real Story / Real-Life Story combine to make abstractions accessible and applicable. It is not a gimmick; it is not a passing fancy; it is not ‘fun’ or a novelty. It is in fact as close as it is possible to get to a principled philosophy of maths and a science of teaching. And this same principled approach underpins the complete MMMS Scheme from Nursery through to Advanced level. Incidentally, the Script outline above for division is used in Year 1 when MMMS is embedded in the school.

MMMS: The twelve Big Ideas in Mathematics
1) Addition / 5) Equals / 9) Ratio
2) Subtraction / 6) “The symbols speak to you” / 10) Infinity
3) Multiplication / 7) “The logic of the language tells you the answer” / 11) Proof
4) Division / 8) Denomination / 12) Sample / Population