How can I find the point where a pupil's maths understanding breaks down?

Look for the barrier

Work backwards with a few examples: counting, comparing quantities, explaining tens and ones, estimating, then the current operation. Ask the pupil to show their thinking with objects or a drawing. Look for the first point where the explanation becomes uncertain.

A wrong answer at the end of a calculation may reflect a problem much earlier: number meaning, place value, a representation, the language of the question or an operation. Ask the pupil to show how they reached the answer using objects, drawings or talk. Stop at the first point where their explanation becomes uncertain instead of practising the whole procedure again.

A correct answer does not always show secure understanding. A pupil might memorise a procedure that works on familiar questions while failing to recognise when to use it in a word problem. Ask them to represent the quantity with objects or a drawing, explain a choice and estimate whether the answer is sensible. Change one feature of the problem at a time. This helps identify whether the obstacle is number knowledge, language, the representation, the written method or recalling a fact.

What to try in school

Teach that small prerequisite explicitly with concrete materials, then drawings and finally symbols. Ask the pupil to explain why a method works rather than memorising only the steps. Keep access to the current lesson by offering an appropriate representation and worked example.

Teach the small prerequisite explicitly. Show a concrete example, draw it and then connect it to the written symbols, explaining what stays the same. Ask the pupil to explain a fresh example rather than copy the model. Keep them connected to the class topic through an appropriate representation while addressing the missing concept separately. A correct answer produced by following prompts is not yet evidence of independent understanding.

Use a short conversation about a real error. Rather than correcting the full page, ask the pupil to show what each number represents and why they chose the operation. A manipulative, drawing or number line can reveal a relationship that a memorised algorithm conceals. If the pupil understands with objects but cannot connect them to written notation, teach that link explicitly and practise moving between forms. If they understand the calculation but not the problem wording, address the language in the lesson. Keep the next teaching goal narrow enough that both teacher and pupil can recognise success in a new example.

A worked example

A pupil repeatedly gets 32 minus 8 wrong. With tens and ones, they can build 32 but say that taking eight away is impossible because there are only two ones. The teacher demonstrates exchanging one ten for ten ones, asks the pupil to build and explain a second number, then links that action to the written method. A later unfamiliar subtraction checks whether the idea transfers beyond the first example.

Review what changes

Recheck the same underlying idea in a fresh problem without the model. If success stays tied to one format, vary the representation before moving to harder calculations.

Record the explanation and representation the pupil can use independently, not merely the number of completed sums. If success disappears as soon as objects are removed, strengthen the connection between objects, drawings and symbols. If the pupil understands the exchange but miscounts, teach and monitor that more specific skill.

Keep a record of a few short diagnostic questions and what the pupil says while solving them. After focused teaching, revisit an equivalent problem and one slightly different application. If a representation helps in the teaching session but is never available in class, arrange for its use in both places. When errors persist, look again at the earliest uncertain step instead of repeating the entire procedure with more speed or larger numbers.

What does the evidence tell us?

The EEF's mathematics guidance recommends using manipulatives and representations purposefully and attending to misconceptions. It does not suggest keeping every pupil on concrete materials indefinitely. The goal is to make the mathematical structure clear and help the pupil use it with increasing independence.