Did you know that a child can explain a mathematics idea correctly aloud while writing a solution that does something else? Conversely, a neat answer can hide reasoning copied from a model. A useful check looks at the spoken explanation, the written steps and a fresh independent question together, rather than declaring the pupil capable or incapable from a single performance.
Comparing spoken reasoning with written working means checking whether each verbal mathematical claim corresponds to a valid operation on the page. Ask the pupil to point to the line where an operation appears, explain why it is allowed, and then solve a new but related problem without prompts. This separates a presentation slip from a misconception and from knowledge that is not yet reliable.
The point is not to make every child speak continuously while doing mathematics. It is to locate the first difference between an idea they seem to understand and the steps they record—and then teach that specific step.
The six-minute check: what to look for first
- Choose one question the learner has already attempted. Keep the original working visible; do not tidy it first.
- Invite a calm explanation: “Tell me what this line means.” Avoid “You should have distributed the three, right?”
- For each arrow or equals sign, ask what changed on both sides of the equation.
- Compare the spoken claim with the ink, symbol by symbol. Mark the first mismatch, not every later consequence.
- Offer one short correction with a visual or numerical check, then remove the example.
- Give a parallel question with different numbers. Let the pupil complete and check it independently.
Three different problems can produce the same wrong answer
A concept gap means the pupil does not yet understand a rule, such as how a bracket is expanded. A recording gap means the pupil can describe a correct operation but writes the wrong symbol or omits a step. A retrieval gap means the pupil can follow a demonstrated procedure but cannot select or complete it once support is withdrawn. The remedies differ: explanation, recording discipline and independent retrieval practice should not be treated as interchangeable.
Spoken reasoning is only one piece of evidence. A shy learner, a pupil learning in an additional language or someone who needs thinking time may understand mathematics without giving a fluent oral account on demand. Offer pointing, annotation, drawing or quietly writing an explanation as valid alternatives. A supportive check asks what the learner can show, not how confident they sound.
Worked example: the bracket that disappeared
Illustrative invented pupil work, not a real case. A learner receives the equation 3(x − 2) = 18. Their written lines are 3x − 2 = 18, then 3x = 20, then x = 20 ÷ 3. The final answer is not valid because the first expansion failed to multiply the −2 term by 3.
What the pupil might say
Asked to explain the first line, the learner says, “Three times the bracket means I multiply everything inside by three.” That verbal rule is correct, but the written line has −2 where it should have −6. We should not infer that the pupil has permanently mastered expansion; we have evidence of a mismatch to investigate.
Repair only the first unsupported step
Expand correctly: 3(x − 2) = 3x − 6. Therefore 3x − 6 = 18; adding 6 to both sides gives 3x = 24; dividing both sides by 3 gives x = 8. Substitute: 3(8 − 2) = 3 × 6 = 18. The original question balances, so the corrected answer passes the check.
The original x = 20 ÷ 3 fails substitution: 3(20/3 − 2) = 20 − 6 = 14, not 18. This independent check is useful because it does not rely on trusting the same incorrect expansion again. Encourage the child to write a one-sentence error diagnosis: “I multiplied x by 3 but forgot to multiply −2 by 3.”
Check transfer with a new case
Now close the worked example and ask the learner to solve 4(y + 1) = 28. The correct sequence is 4y + 4 = 28, 4y = 24, y = 6; substitution gives 4(6 + 1) = 28. If they solve it independently and explain the expansion, that is stronger evidence than repeating the repaired original equation.
A tiny diagnostic that protects learner confidence
- If speech is correct but writing is not: slow down the recording step; circle each bracket term and draw arrows from the multiplier. Ask the learner to compare symbols after each line.
- If writing is correct but speech is incomplete: ask for a diagram, substitute values or give thinking time. Do not automatically reclassify a correct method as guessing.
- If both are wrong: return to concrete examples such as three equal packets each containing x counters and two removed counters; move to symbols after the meaning is clear.
- If both are correct on the familiar question but fail on a new one: alternate examples with different signs or structures and require independent reasoning.
- If performance fluctuates: record the type of problem, time, prompts and conditions. One successful item is not proof of stable mastery.
Progressive practice: model, explain, transfer
Stage 1 — guided comparison
Present a deliberately wrong line, 2(a − 5) = 2a − 5, beside the correct line, 2(a − 5) = 2a − 10. Ask which one represents two copies of (a − 5), and why. Use a numerical trial: if a = 9, the left side is 2 × 4 = 8, while the wrong right side is 13. The numerical contradiction identifies the missing multiplication.
Stage 2 — learner-led annotation
Give 5(b + 3) = 35. Ask the pupil to annotate each transformation with the operation used. A valid solution is 5b + 15 = 35, then 5b = 20, then b = 4; check 5(4 + 3) = 35. In a correction, ask which original line needs changing and why, rather than asking the child to copy an entire model.
Stage 3 — fresh independent check
On another day, give 2(c − 4) = 10 with no arrows drawn in advance. Expected: 2c − 8 = 10; 2c = 18; c = 9; check 2(9 − 4) = 10. Vary the numbers and bracket signs while keeping the core principle comparable; record whether help was required.
A practical observation log for home and school
Write down the date, problem structure, first unsupported line, kind of help, independent transfer result and next review date. Instead of “weak in algebra”, write “expanded x but not the constant in two of three fresh questions; corrected after one prompt; next check without hints on Friday”. This turns feedback into an actionable learning decision.
For parents, a helpful prompt is “Which line would you check first?” rather than “Why did you get this wrong?” For teachers, a short whole-class discussion can compare two anonymous worked solutions, followed by individual practice. For learners, the target is not perfect commentary; it is the ability to justify an operation and confirm the final answer independently.
What this method cannot establish
A conversation does not identify a learning disability, measure general intelligence or guarantee an exam result. Noise, language demands, tiredness, test anxiety and unfamiliar vocabulary can affect performance. Sometimes the correct response is to change the mode of explanation, not to insist on more speech. Preserve dignity and adjust the task before drawing a broad conclusion.
For a wider approach to diagnostic assessment, see the Education Endowment Foundation’s checking-for-understanding resource. The classroom method here is an illustrative practical routine, not a claimed validated diagnostic instrument.
Frequently asked questions
What if the student can solve the problem silently but cannot explain it aloud?
Accept a labelled diagram, written note, comparison of two lines or substitution check. Demonstrate understanding through more than one mode before concluding there is a conceptual problem.
Should every equal sign have a spoken explanation?
No. Select a step where the reasoning is uncertain. Excessive narration can distract from doing the mathematics. The goal is a justifiable transformation, not a performance.
How many successful questions show mastery?
There is no universal magic number. Look for repeated, unprompted success across more than one day and an unfamiliar variation, while keeping a record of task difficulty.
Should a parent correct the first wrong line immediately?
Give the pupil one opportunity to check it; then offer a targeted cue or model. Do not let an incorrect rule become the only remembered example.
Explore related eduKate learning resources
Continue with checking every equal sign, asking diagnostic questions without leading, the Education Hub, and the Secondary 1 Mathematics transition reference. Each offers a complementary way to move from a visible answer to actual mathematical understanding.
The core aim: a pupil should be able to make a valid mathematical move, show it faithfully, and check it on a different question. A short conversation becomes valuable when it helps them do all three without depending on someone else’s answer.