Stages of numeracy development: the five stages of learning numeracy from emergent counting to number facts, what each looks like in a child, the usual ages, a two-minute check to find your child's stage, and what to do at each one

Children do not learn number in one piece. They pass through a small number of stages, in the same order, and each stage has its own way of solving a problem like 5 + 3. Knowing which stage a child is at tells you what to practise, and it explains two things that puzzle parents: why a child who counts to 100 cannot add, and why a seven year old still uses fingers.

Which "five stages"?

Several descriptions exist. The one most used by schools comes from the research of Leslie Steffe, developed for classrooms by Robert Wright as the Stages of Early Arithmetical Learning. It is the basis of the Mathematics Recovery programme and of the early number frameworks used in Australian and New Zealand schools. It describes numeracy development by how a child counts to solve a problem, which is something a parent can see.

The five stages

Some systems number these 0 to 5, treating counting on and counting back as separate stages. The New Zealand Number Framework extends the same idea further, through early additive, advanced additive and multiplicative thinking, up to proportional reasoning in the upper primary years.

Three things about the table matter in practice.

The answer does not show the stage. Children at stages 2, 3, 4 and 5 can all say "eight". Watch the method: the fingers, the lips, the pause.

Stages cannot be skipped. A child at stage 3 who is drilled on number facts will recite them and still count from one when the question is new.

Reciting is not a stage. Saying the numbers to 100 is memory, like a song. It tells you nothing about which row a child is in.

Underneath stage 1: the five counting principles

Before any of the stages, a child has to learn what counting is. Rochel Gelman and Randy Gallistel set out five principles in 1978, and they are still how early-years teachers check counting:

1. One-to-one. One number word for each object, no more and no fewer.
2. Stable order. The words always come in the same order.
3. Cardinality. The last number said is how many there are.
4. Abstraction. Anything can be counted: claps, steps, ideas.
5. Order irrelevance. Count the same things in a different order and the total is the same.

A child who counts five blocks and, asked "how many?", counts them again has not got cardinality yet. That one is the gate to everything else; the preschool math skills checklist lists these early skills with ages.

A two-minute check

You need eight small objects and a cloth.

1. Put out seven objects. "How many?" Watch for one touch per object and for the answer "seven" without recounting. Trouble here: stage 1.
2. Show five, cover them. Show three more, leave them in view. "How many altogether?" If the child needs to lift the cloth: stage 2.
3. Cover both groups. "Five here, three here. How many altogether?" Counting from one, often with fingers or nods: stage 3. Starting from five: stage 4. Answering at once and saying why: stage 5.
4. For a child at stage 4 or 5: "What is 8 + 6?" Counting on six by ones is stage 4. "8 and 2 is 10, and 4 more" is stage 5.

Do it once, lightly, as a game. It is a way to choose what to play, not a test.

What to do at each stage

The move from stage 3 to stage 4 is the one that most often stalls. It needs the child to believe that "five" can be held as a whole. Dice help more than anything: a child who sees a five on one die and three dots on the other soon stops counting the five.

Fingers

Fingers are a stage 2 and 3 tool, and a good one. Do not ban them. A child drops them when counting on becomes easier, and later when facts become faster than counting. A child of eight or nine who still counts every sum by ones is usually at stage 3 or 4 with too few known facts; the answer is stage 4 work, the pairs to ten and the doubles, and not more worksheets of sums.

How this fits the other models you may read about

• Concrete, pictorial, abstract. From Jerome Bruner: children learn a new idea first with objects, then with pictures, then with symbols. It describes how to teach any one idea; the five stages describe where a child is overall.
• Piaget's stages. Jean Piaget described general stages of thinking; the one that matters for number is conservation, understanding that seven counters are still seven when spread out, which usually arrives between five and seven.
• School curricula set out what should be taught each year. A stage tells you whether a particular child is ready for it.

When to ask for help

Mention it to the teacher if a child of six cannot count ten objects accurately, or a child of seven and a half is still counting everything from one, or a child of nine has almost no number facts despite practice. Most often the cause is a stage that was hurried. Sometimes it is dyscalculia, and an early look is worth having.

A 1-on-1 online class for this age is short, 25 to 30 minutes, built from games on a shared screen with a parent alongside; the tutor finds the stage in the first class and works one stage ahead. What it covers is on the kindergarten math page. The free first class is a real lesson, and you will be told plainly if your child does not need more.

Questions parents ask

See how we teach this, 1-on-1 online →