Numbers name quantities
A counting number answers how many? — not which? or how long?. When you say there are five apples, you are naming the size of a collection.
The counting words (one, two, three…) are an ordered list. Each word stands for a quantity one larger than the word before it.
One-to-one correspondence
Reliable counting needs one-to-one correspondence:
- Touch or point to each item once.
- Say the next unused counting word.
- Stop when every item has been tagged.
The last word you say is the size of the whole set (the cardinality principle).
You count a row of cups and say “one, two, three, four.” What does “four” tell you?
If two sets can be paired one-to-one with nothing left over, they have the same count.
Adding to a count
When a set grows, you do not need to recount from one. Start from the old cardinality and keep counting forward for each new item — that is the counting-on idea behind addition.
A shelf has three books. You add two more without removing any. How many books are on the shelf now?
Example: 5 · integer
What is 2 + 7?