Math // Mathematical Thinking

Real-World to Maths Connections

Move between a real-world situation and a mathematical representation using concrete objects, drawings, diagrams, tables, number sentences, or bar models

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

Quick check

If the learner is working out a real-life maths problem — like sharing 8 sweets between 2 people — can they use objects or draw a picture to show what's happening before writing any numbers?

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.
  • Given a story about combining groups, represent it with counters or cubes and find the total
  • Given a set of objects, tell a simple addition or subtraction story to match
  • Connect a physical action (putting together, taking away) to the matching operation

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

2 topics
  • Addition and subtraction word problemsHelpful

    Addition & Subtraction // Ages 4 to 6

    Solving word problems within 10 with objects requires translating between situation and representation

  • Representing Addition and SubtractionHelpful

    Addition & Subtraction // Ages 4 to 6

    Representing addition/subtraction with objects and drawings is the core exercise of early quantitative reasoning

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Real-World to Maths Connections

Ages 5 to 6 // Thinking skill

Leads to

1 topic
  • Connecting RepresentationsRequired

    Mathematical Thinking // Ages 6 to 7

    Age 6-7 quantitative reasoning builds on age 5-6 concrete-to-representation skills

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, quantitative reasoning is a two-way translation between a real situation and a concrete/pictorial model: acting out 'five ducks and two more swim up' with cubes, AND going the other way — given cubes, telling a matching story. The heart (CCSS MP2) is attending to the MEANING of quantities, not just producing a total: knowing THIS cube stands for a duck, that 5 and 2 are the parts and the whole is the total, and that 'putting together' is what makes it addition. Taught through story-acting and the concrete→pictorial bridge NCETM describes for the youngest bar model: line objects up, stand a drawn bar beside them, and ask 'what's the same, what's different?' so the picture is understood as an abstraction of the real thing.

Common misconceptions

  • Treats the model as a picture of two separate amounts, not a relationship — lays out 5 and 2 but cannot say which is the whole or show the joining
  • Loses the exact quantities when telling a story from objects ('some ducks and some more') instead of preserving 5 and 2
  • Produces the total by counting all but cannot point to the parts or the action inside the model
  • Picks the operation from a word ('more' → add) rather than from what happens in the situation

Useful representations

  • Counters/cubes for acting out and combining/separating
  • Five-frame and ten-frame for structuring the quantities
  • Part-whole picture: two groups lined up with a drawn bar beside them (the bar model's earliest form)
  • Simple number sentence (with +, −) as the symbolic end of the concrete→pictorial→abstract bridge

Teaching practices

  • Act stories out with concrete objects first, then draw alongside, so the child sees the drawing as standing FOR the objects (NCETM's line-up→cubes→bar sequence with 'what's the same, what's different?')
  • Insist the child name the referent of each quantity ('the 5 is the ducks already there') — MP2's 'attend to the meaning of quantities, not just how to compute'
  • Work translation in BOTH directions: situation→model AND model→story, so the child is not just illustrating answers but reading meaning off a representation
  • Pair children (one moves cubes, one draws/records) so the mapping between concrete and pictorial is talked aloud
  • Connect the physical action to the operation explicitly — putting together = addition, taking away = subtraction — rather than letting the child guess from words

Activities to try

  1. 1

    Build the story, then flip it

    Why it works: The model carries meaning in both directions: numbers stand for ducks, the joining stands for adding, and the child can travel from world to symbol and back — the model is not a picture of an answer but a thing that holds a relationship.

    1. Listen to or watch a short story: 'Five ducks bob on a pond, then two more paddle in.'
    2. Drag/place five cubes into the left of a part-whole frame and two into the right to match the story.
    3. Watch a bar form across the top and the sentence 5 + 2 = 7 appear, with the '5' marking the first cubes and the '2' marking the newcomers.
    4. Point to which cubes are the 5, which are the 2, and show the whole group.
    5. 'Flip it': take a pile of cubes already split (e.g. 4 and 3) and invent a story the picture could tell — then notice the SAME cubes could be 'four birds and three more' OR 'seven birds, three flew off'.
  2. 2

    Line up, cubes, bar — what's the same, what's different?

    Why it works: The bar/picture is an abstraction OF the real objects, not a decoration — and the two parts combine into one whole (part-part-whole), not two unrelated piles.

    1. Line real objects (toy ducks, counters) up to match a told situation.
    2. Replace them with linking cubes, then stand a drawn bar alongside the cubes.
    3. Compare: 'What's the same between the cubes and the bar? What's different?'
    4. Push the two groups together under one bar to show the whole being made of parts.
  3. 3

    Talk it through with a story partner

    Why it works: Naming what each quantity refers to, and the action that joins them, keeps the numbers attached to meaning; saying it aloud makes the concrete→pictorial mapping explicit.

    1. Listen to a partner (an adult, peer, or in-role puppet) tell a story and build it with cubes.
    2. Answer 'where's the 5 in your cubes? which bit is the 2? what happened to make it the whole group?'
    3. When the partner builds a model, say what story it could be — keeping the exact numbers in.
    4. In pairs, one child moves the cubes while the other draws/records, talking the mapping aloud.

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