Big Ideas in Primary Mathematics

Practical tips for understanding and teaching maths

Big Idea: Mathematical Understanding in the Early Years

It is the five counting principles (Gelman and Gallistel, 1978) that are really key to early number work.
Children can struggle to see number symbols as having a discrete meaning unless real-world situations emphasise this.
Children are very capable of finding meaningful ways of recording real world mathematical happenings themselves. Their attempts to do so form a deep part of their understanding and they often naturally reach out for standard symbols they have been exposed to in time.

Standard use of symbols can be encouraged early so long as the initial alien nature of it is acknowledged along with children's ability to make sense of their world anyway if they are encouraged to do so.

Big Idea: Place Value

• We group in tens. Each collection of ten ones makes a 'ten'.
• The digits in a number relate to groups of tens and ones.

Later comes the significant seismic follow up to this:

• Each column represents how many groups of ten have been made from the column to the right.

Thus, a ten has the same value as ten ones; a hundred has the same value as ten tens. A thousand has the same value as ten hundreds. This BIG IDEA holds true for any link in our counting system including part wholes and across the division point of wholes and part wholes. Ten tenths make a whole unit and ten hundredths make a tenth.

The links between the second and third BIG IDEAS here relate well to work with Dienes blocks, where the column values can be easily represented.

Big Idea: Addition and Subtraction

The empty number line supports understanding grouping in tens and ones, as well as adding and subtracting in ones, tens and 100s. It allows this knowledge to be applied in situations that allow children to take control and show their understanding. It links directly to trios and to complementary addition linked to subtraction. It is a powerful, visual scaffold.

Big Idea: Division

Instead of dividing a large number into a particular group size it is possible to chunk the total into smaller pieces and do these calculations; then total the different quotients to achieve the answer that solved the original problem.

378 = 300 + 60 + 18

        = (100 * 3) + (20 * 3) + (6 * 3)

        = 126 x 3

Answer = 126

Big Idea: Multiplication

a. 3 x 2 of any group size will always make 6 of that group size. This applies to any two numbers multiplied.
b.  Multiplication and division are inverse operations; so dividing by 10 will reverse an operation of multiplying by 10. Therefore 6 * 10 = 60 and 60/10 = 6 .
So...

Big Idea: Algebra

When encouraging older primary children to represent variables emphasise the symbol is standing for whatever value is being used in this relationship.

If x boxes of eggs equal 30, then the value of y is 30.
With 6 eggs in a box the value of x would be 5.
(6 + 6 + 6 + 6 + 6) = 30

Big Idea: Fraction

A fraction has a numerator and a denominator. The denominator, numerically, is denoted by the number beneath the line; the numerator by the number above it. The denominator indicates how many equal-sized parts make up the whole. The numerator indicates how many of those equal parts make up this particular fraction.
The line itself can be seen as signifying the process of division. It can also be seen as signifying a term such as 'out of to link the numerator and denominator. So 3/4 is then seen as 3 'out of' 4 as well as 3 divided by four, or divided into four equal-sized pieces.

Big Idea: Measurement

• Young children, and older ones, can have their learning potential lowered by the need to be 'right' as opposed to being curious about why they were wrong. Therefore, mistakes are to be actively encouraged as learning opportunities. This takes real commitment by a teacher, and possibly by a whole school.
• Young children often associate being 'higher with success. Therefore, on balance scales, they may initially think the bucket that goes up has more weight because it is higher. It may be useful to discuss this as a teaching example, interpreting the balance scale alongside predicting which object is heavier or lighter.

These extracts were taken from Big Ideas in Primary Mathematics by Robert Newell. Explore the book for more big ideas and to deepen your understanding of maths.