Math // Mathematical Thinking

Explaining Mathematical Reasoning

With teacher prompting, explain and justify mathematical reasoning using drawings, number sentences, or words

Ages
6 to 7
Type
Thinking skill
Builds on
3 topics
Leads to
2 topics

Quick check

If you ask the learner "how did you work that out?", can they show you with a drawing or explain their steps in simple words?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Explain a solution strategy using a drawing or number sentence (e.g. 'I made ten first, then added the rest')
  • Justify why an answer is correct by showing an alternative method that gives the same result
  • Identify and explain an error in a worked example or a peer's solution

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

3 topics
  • Showing Your WorkingRequired

    Mathematical Thinking // Ages 5 to 6

    Age 6-7 explaining with diagrams/logic builds on age 5-6 showing and telling with objects

  • What the equals sign meansHelpful

    Addition & Subtraction // Ages 6 to 7

    Determining whether equations are true/false exercises evaluating and justifying

  • Addition in any orderHelpful

    Addition & Subtraction // Ages 6 to 7

    Justifying why a+b=b+a exercises logical mathematical argument

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Explaining Mathematical Reasoning

Ages 6 to 7 // Thinking skill

Leads to

2 topics
  • Justifying mathematical reasoningRequired

    Mathematical Thinking // Ages 7 to 8

    Age 7-8 arguing/justifying builds on age 6-7

  • Teaching It BackHelpful

    Learning to Learn // Learning to Learn // Ages 7 to 8

    The universal self-explanation habit (LtL 7-8) builds on the maths-specific practice of explaining reasoning when prompted (MT 6-7)

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

For ages 6-7, with adult prompting, children move from showing a method to EXPLAINING and beginning to JUSTIFY it — using drawings, number sentences, or words to say not just what they did but WHY the answer is right. The characteristic gap is between convincing (confident but with an incomplete chain, 'I reckon it's right') and justifying (a correct, complete because-chain). Teaching supplies logical connectives and sentence stems, uses alternative-method agreement as a form of justification, and treats finding and explaining errors in worked examples as reasoning practice.

Common misconceptions

  • Confident assertion ('I reckon it's right') offered as justification, with an incomplete or incoherent chain (convincing mistaken for justifying)
  • Describing the method but stopping short of saying WHY it gives the right answer unless heavily prompted
  • Spotting that a worked example is wrong but being unable to explain the flaw
  • Treating 'justify' as 'do the sum again' rather than giving a reason

Useful representations

  • Drawings and number sentences as the medium of explanation
  • Alternative-method check (two routes to the same answer)
  • Sentence stems with connectives (because / so / therefore)
  • True-or-false equation judgements
  • The explain → convince → justify steps of the NRICH ladder

Teaching practices

  • Prompt 'why does that work?' and 'how do you know it's right?' — not just 'what did you do?'
  • Teach logical connectives and sentence stems ('because', 'so', 'therefore', 'this can't be right because…') to move confident asserting toward complete justification
  • Use an alternative method that gives the same result as a justification ('the other way also gives 13, so it makes sense')
  • Use worked examples containing errors and ask children to find and EXPLAIN the mistake
  • Keep concrete/visual support (drawings, number sentences) as the backbone of the explanation
  • Distinguish, for yourself and the child, confident-but-incomplete (convincing) from a correct complete chain (justifying), and push gently toward the latter

Activities to try

  1. 1

    The 'convince me' partner

    Why it works: A confident answer is not the same as a justification; a reason that works can be built by linking steps with 'because' and 'so', and an alternative method that agrees is supporting evidence that backs up the reason, though it does not replace explaining why.

    1. Pose the child a within-20 calculation (e.g. 8 + 5) and ask them to explain HOW they worked it out.
    2. Then ask 'why does that work?' and 'how do you know it's right?' — not just 'what did you do?'.
    3. When the child stalls or only asserts ('I just know', 'I reckon it's right'), hand over a sentence stem: 'Start with because…' and prompt for the missing link.
    4. Have the child upgrade a confident hunch into a chain, e.g. 'because 5 is 2 and 3, and 8 and 2 make 10, so 10 and 3 more is 13'.
    5. Ask the child to show the same calculation a different way and check whether it also gives the answer ('the other way also gives 13 — what does that tell us?').
  2. 2

    Build an explanation you can test

    Why it works: Justification is something you assemble and test, not just a confident feeling: steps linked with connectives form a chain; a second method checks whether the answer agrees, and checking each step tests whether the chain really holds.

    1. Give the child step cards (number sentences / method steps) and connective tiles ('because', 'so', 'therefore') to arrange into an argument chain for why a calculation is right.
    2. Have the child read the assembled chain aloud to check it actually hangs together and establishes the claim.
    3. Check the answer a second way (an alternative method) and compare — do the two routes agree?
    4. In an error-spotting task, give a worked example and have the child point to the wrong step AND say why it is wrong.
    5. Keep drawings and number sentences as the backbone of the explanation throughout.
  3. 3

    Model a complete because-chain (scaffold)

    Why it works: A worked justification shows the TARGET: every step has a reason, and the reasons connect to establish the answer.

    1. Show a justified worked example where each step is paired with its 'because' reason, e.g. reveal one step at a time so the child sees what a full chain looks like.
    2. After viewing, ask the child to re-tell the chain in their own words, then produce one for a fresh calculation themselves.

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