Math // Mathematical Thinking

Multi-Step Problem Solving

With teacher support, make sense of multi-step problems involving larger numbers or mixed operations by breaking them into parts, choosing strategies, and checking answers for reasonableness — children at this stage are developing the habit with guidance; independent strategy evaluation comes later

Ages
7 to 8
Type
Thinking skill
Builds on
5 topics
Leads to
2 topics

Quick check

If the learner gets an answer that seems way too big or too small for a maths problem, do they notice and go back to check, even without you pointing it out?

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.
  • Break a two-step word problem within 1000 into sub-problems and solve each part
  • Estimate an answer before calculating to set a reasonableness benchmark
  • Check an answer using a different method or inverse operation and revise if needed

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

5 topics
  • Guided Multi-Step Problem SolvingRequired

    Mathematical Thinking // Ages 6 to 7

    Age 7-8 problem-solving builds on age 6-7 problem-solving

  • Building sentencesHelpful

    English // Grammar & Punctuation // Ages 4 to 6

    Cross-subject: making sense of multi-step word problems requires understanding that sentences express complete thoughts (reading comprehension foundation)

  • Missing number problems (age 7+)Helpful

    Addition & Subtraction // Ages 7 to 8

    Multi-step A+S problems exercise breaking problems into parts

  • Trying a New ApproachHelpful

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

    Trying a different mathematical strategy when stuck is the maths-specific application of the universal strategy-switching habit

  • Estimating by roundingHelpful

    Addition & Subtraction // Ages 7 to 9

    Estimating and checking answers exercises problem-solving at this age

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Multi-Step Problem Solving

Ages 7 to 8 // Thinking skill

Leads to

2 topics
  • Multi-Step Problem SolvingRequired

    Mathematical Thinking // Ages 8 to 9

    Age 7-8 problem-sense-making is prerequisite to age 8-9 level

  • Choosing a StrategyHelpful

    Learning to Learn // Learning to Learn // Ages 9 to 10

    The LtL strategy evaluation skill (9-10) builds on the maths-specific checking habit developed with teacher support at 7-8

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 7-8 (with teacher support): make sense of multi-step problems within 1000 by breaking them into ordered sub-problems, choosing and sequencing strategies, estimating a reasonableness benchmark first, and checking with an inverse operation or a different method — revising when the two disagree. The central aim is to replace the keyword habit (tying a word to an operation), which research on released items reports succeeds on under 50% of single-step and under 10% of multi-step problems, with attention to problem structure, and to counter the compulsion to calculate (answering with no estimate and no check). Self-questioning aloud and visual representations (e.g. bar/strip diagrams and number lines for sequencing jumps) are the two levers cited with strong evidence in the WWC grades 4-8 guide; strategy-switching when the first approach stalls should be normalised.

Common misconceptions

  • Keyword strategy on multi-step problems — tying a word to an operation and applying operations in the wrong order
  • Computing an answer with no estimate and no check, accepting answers that are wildly off (compulsion to calculate)
  • Believing that if the first method doesn't work, you're simply stuck (no strategy-switching)
  • Doing the sub-steps but not connecting them — losing track of what has been found and what remains

Useful representations

  • Bar / strip diagram decomposing a multi-step problem into sub-parts
  • Number line for sequencing multi-step jumps
  • Estimation / rounding to a benchmark before computing
  • Inverse-operation check (work backwards to the start)
  • A monitoring prompt list across the sub-steps: plan → do a step → check → next

Teaching practices

  • Estimate first to set a reasonableness benchmark, then compute, then check by an inverse operation or a different method — revise if they disagree
  • Teach problem structure (schema), not keywords: the keyword strategy collapses on multi-step problems (match rate <10%)
  • Model self-questioning aloud and provide a prompt list for monitoring progress across the sub-steps (WWC Rec 2, strong evidence)
  • Represent the problem with a bar/strip diagram so the sub-problems and their order are visible (WWC Rec 3, strong evidence)
  • Break the problem into named sub-problems and solve each, keeping the whole plan in view
  • Normalise strategy-switching: if the first approach stalls, deliberately try a different representation or method rather than abandoning

Activities to try

  1. 1

    Estimate-first multi-step solver with inverse check

    Why it works: Decomposition is something you physically arrange and sequence, not a rule triggered by a cue word; the estimate and a backwards check that undoes the problem's actions (not merely the child's own calculation) are two separate signals that can catch a wrong-order or wrong-operation answer.

    1. Take a two-step problem, e.g. 'A shop had 320 pencils. It sold 4 boxes of 50. How many pencils are left?' and arrange it as two connected sub-problem cards in the order you will solve them (first 4×50 = 200 sold, then 320 − 200), adding more cards only if a longer problem needs them
    2. Before computing, state 'about how big' the answer should be — set an estimate benchmark (e.g. 'about 300 sold... no, 200, so a bit over 100 left')
    3. Solve each sub-step in order, keeping the whole plan visible
    4. After you have an answer, work backwards through the story (inverse) to see whether you return to the starting number: put back the pencils that were sold, 120 + 4×50 = 120 + 200 = 320. Undo the actions in the problem, not just your own calculation, because undoing a wrong calculation (e.g. 320 − 50 = 270, then 270 + 50 = 320) would still get you back to 320
    5. Compare your answer against your own estimate: if it is wildly off, treat that as a signal to recheck both the answer and the estimate
  2. 2

    Think-aloud study partner for self-monitoring

    Why it works: Self-questioning across sub-steps is a repeatable routine for monitoring progress and catching errors, and being able to explain the sequence and the check shows understanding of the structure.

    1. Work with a partner (or in-role coach) who never hands over the steps but asks you to state a plan and an estimate before computing
    2. Use a monitoring prompt list to pace yourself through the sub-steps: plan → do a step → check → next
    3. Answer monitoring questions aloud: 'What have you found so far? What's still unknown? How could you check that?'
    4. Justify the order you chose for the sub-steps and explain your reasonableness check in your own words
    5. When you are stuck, respond to the prompt to represent the problem a different way — for example a number line for the sequence of jumps — instead of being given the answer
  3. 3

    Worked-example reference with revealed self-questions

    Why it works: Seeing the decomposition and the reasonableness reasoning laid out gives a model of what good monitoring looks like before you do it independently.

    1. Study a worked multi-step problem where each sub-step can be revealed one at a time
    2. For each revealed sub-step, read the self-question the solver asked and see why the estimate rules certain answers out
    3. After studying, re-solve a fresh but similar problem yourself, sequencing the steps and doing your own check

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