Math // Mathematical Thinking

Hands-On Problem Solving

Select and use familiar tools (concrete objects, fingers, ten frames) to help solve a mathematical problem

Ages
5 to 6
Type
Thinking skill
Builds on
1 topic
Leads to
1 topic

Quick check

When the learner is working out a maths problem, do they reach for helpful tools — like using their fingers, counters, or a number line — to support their thinking?

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.
  • Choose cubes, counters, or fingers to help solve an addition or subtraction problem
  • Use a ten frame to organise objects for counting or comparing
  • Explain why a particular tool (e.g. cubes rather than fingers) was chosen for a given problem

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

1 topic
  • Counting objects to 20Helpful

    Counting & Cardinality // Ages 5 to 6

    Counting objects requires choosing appropriate tools (cubes, ten frames)

You are here

Hands-On Problem Solving

Ages 5 to 6 // Thinking skill

Leads to

1 topic
  • Numbers on a number lineRequired

    Mathematical Thinking // Ages 6 to 7

    Age 6-7 strategic tool selection builds on age 5-6 familiarity with basic tools

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 5-6 year-olds, strategic tool use begins as CHOOSING among familiar concrete tools (cubes, counters, fingers, ten frame, five frame) and being able to say WHY that tool fits THIS problem. The learning lives in the reason, not fluent handling. Offer a genuine choice rather than prescribing a tool, let a tool's limitation be felt in action (using fingers to show both parts of 8+7 runs past ten), and press for a fit-reason tied to the numbers. Keep tools as temporary scaffolds and model solving a small known fact without a tool so reliance does not set in.

Common misconceptions

  • Reaches for the same familiar tool (usually fingers) every time regardless of whether it fits — e.g. fingers for 8+7
  • Uses the tool to get an answer but cannot say why it helped — treats it as a ritual, not a thinking aid (EEF)
  • Becomes reliant on the tool and cannot attempt even a small, known fact without it (EEF)
  • Thinks the ten frame is just somewhere to put counters, missing that its rows of five and ten show structure

Useful representations

  • Ten frame (rows of five/ten expose the gap to ten)
  • Cubes / multilink (joinable and splittable)
  • Counters
  • Fingers
  • Five frame

Teaching practices

  • Offer a real choice of tools rather than handing one over, then ask 'why did you pick that?' so the rationale becomes visible and assessable (EEF: clear rationale for the tool)
  • Use the ten frame to make 'how many to ten' structural, not just a mat to put counters on — the aim is to reveal structure, not merely reach an answer
  • Name each tool's limitation in play — 'fingers run out past ten', 'a ten frame only holds ten' — so children begin weighing fit rather than habit
  • Keep tools as thinking aids and model solving a small, known fact WITHOUT a tool afterwards, so reliance does not set in (EEF: manipulatives are a temporary scaffold)
  • Value the explanation as much as the answer — a child who solved it but cannot say why the tool helped has not yet met this practice

Activities to try

  1. 1

    The tool tray: 8 + 7 and 3 + 2

    Why it works: Choose a tool and strategy because they help with the particular problem. Showing all15 objects at once on one finger each exceeds ten fingers, but counting-on with fingers can still solve8+7. A ten frame offers another useful structure: the gap to ten. Accept a valid strategy when the learner explains how it helps.

    1. Face the problem 8 + 7 with a tray offering cubes, counters, a ten frame, and fingers; pick a tool and use it to solve.
    2. If you choose fingers and try to show 8 and then 7 more, notice you pass ten and run out of fingers before you reach the answer.
    3. Try the ten frame instead: place 8 (nearly filling one frame), then split the 7 — 2 completes the first ten, and the remaining 5 go beside the full frame (or into a second frame), giving 'fill to ten, then five more — fifteen.'
    4. Say why your chosen tool helped: e.g. 'it shows the gap to ten' or 'the fingers ran out'.
    5. Now solve 3 + 2 and decide which tool fits — discover that fingers work perfectly here, so the best choice depends on the numbers, not on a favourite.
  2. 2

    The puzzled friend who always uses fingers

    Why it works: Putting the fit-reason into words — distinguishing choosing a tool because it suits the numbers from grabbing a favourite out of habit — is the heart of the practice at this age, and a tool can be set aside once a fact is secure.

    1. Talk with an in-role friend or puppet who says 'I always use my fingers — would that work here?' for 8 + 7.
    2. Justify or correct the friend: explain why fingers run out when you try to show every object for a total over ten, and which tool or strategy fits better (e.g. a ten frame, or counting on from 8 with fingers).
    3. Respond when the friend poses a contrasting small problem (like 3 + 2) to test whether your reason is about fit or just habit.
    4. Answer probing questions: 'What does the ten frame show you that a loose pile of counters doesn't?' and 'If you had 12 + 5, what would you reach for instead?'
    5. Answer 'Can you do 8 + 7 now without any tool — what happened in your head?'; if you know it or can work it out mentally, that shows the tool can be set aside once the fact is secure, and if not, it is fine to still use a tool.
  3. 3

    Meet the tools (warm-up)

    Why it works: Each tool has an affordance and a limit; knowing these in advance primes children to weigh fit when they later choose for a real problem.

    1. Explore a shelf of tools — cubes, counters, ten frame, five frame, fingers — and reveal for each what it is 'good for' and when it 'runs out'.
    2. Name a purpose and a limit for each approach: fingers can show up to ten objects at once, and can also track counting-on steps; a single ten frame holds ten counters and shows the gap to ten; cubes can be joined and split.
    3. Then move on to choosing and using a tool on a real problem.

Math // Mathematical Thinking

More in Mathematical Thinking

All 48 topics