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Mathematics · Foundation · Chapter 1

How many? Counting you can trust

Practice

How many: ● ●: 2
How many: ● ● ●: 3
How many: ● ● ● ●: 4
A dice pattern of 5 dots: 5
Count 6 blocks — the last number?: 6
Fingers on one hand: 5
How many: ● ● ● ● ● ●: 6
Two dots and two more dots: 4
★ ★ ★ — how many?: 3
You count up to seven. How many?: 7

Easy questions

Count the stars: ★ ★ ★. How many?: 3 — one number per star; the last word, 'three', is the total
You count 5 cups. How many cups?: 5 — the last number counted is how many
Spot it without counting: ● ●: 2 — small amounts you just see (subitising)
A dice shows 4 dots. How many?: 4 — the four-pattern tells you, no counting needed
Count: ● ● ● ●. How many?: 4

Medium questions

You count a row: one, two, three, four, five, six. How many?: 6 — the last number said is the total
Which can you see without counting: 3 dots or 9 dots?: 3 dots — subitising works only for small amounts (about 4-5)
You counted 4 blocks, then counted the same blocks again. How many now?: still 4 — counting again doesn't change how many there are
● ● ● and ● ●. How many altogether?: 5 — see 3 and 2, then count on: four, five
Count 5 toys starting from the other end. How many?: 5 — the order you count in doesn't change the total

Hard questions

While counting 5 blocks you tap one block twice. What number do you say, and how many blocks are really there?: you'd say six, but there are only five — the double-tap counted one block as two; one touch each
See ● ● ● ● and ● ● ● ● quickly, then find the total.: four and four — count on from four: eight
Someone counts 7 pencils and says 'seven'. Asked 'how many?', what is the answer without recounting?: seven — the last number counted already tells you how many
Why does counting the same toy twice give the wrong total?: it counts one toy as two, so the total comes out bigger than what is really there — count each thing exactly once
Can you subitise ● ● ● ● ● ● (six dots)?: not reliably — six is past the ~4-5 limit, so count it, or split it into 3 and 3

Lesson

Count each thing once

Counting has two rules. Rule one: touch each thing exactly once as you say the numbers in order — one number per thing, none skipped and none counted twice. Rule two is the big one: when you stop, the LAST number you said is how many there are. Count 'one, two, three, four' and there are four — done. You never have to count again to answer 'how many?'; the answer is already the last word you said. The only ways to go wrong are to skip a thing or to count the same thing twice, so keep it to one touch, one number. This idea — that the last count word tells the total — is called cardinality, and the whole of maths is built on it.

● ● ● — 'one, two, three'. How many? Three: the last word.
Count 4 spoons: one, two, three, four → there are four spoons.
Asked again straight after? Still four — counting again doesn't change what's there.
Trap: tap one spoon twice and you'd say 'five' — but there are only four. One touch each.

Lesson

See small amounts at a glance

Some amounts are so small you can see how many there are in an instant, without counting one by one. Two things, three things, four things — your eyes just know. This is called subitising. It works up to about four or five, which is exactly why dice and dominoes use dot patterns: you see the five-pattern and know it is five without counting. Bigger amounts you still have to count. Being able to spot small groups instantly makes counting faster, because you can count on from a group you already recognised instead of starting again at one.

● ● — you see 'two' at once, no counting.
⚃ on a dice — the four-corner pattern says four instantly.
Three fingers up — 'three' without counting them.
● ● ● ● ● ● ● — too many to just see; now you count.

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