Learning Science · Reading, representing and explaining

Moving between concrete and abstract representations

Move between concrete cases and general relationships while keeping their correspondence visible.

Use the examples to connect this lesson with the earlier topics. No specialist equipment is needed.

After reading, you can

Identify the structure represented by a concrete model.
Check an abstract rule using more than one concrete case.

Concrete objects can clarify and distract

Six counters shared equally between three people make division visible: each receives two. The counters provide objects to move and a result to inspect. But the learner also needs to connect this action with 6 ÷ 3 = 2. Otherwise success may remain tied to distributing these particular counters rather than to equal sharing as a general relationship.

Concrete means that a representation retains specific perceptual or contextual features; abstract means that selected relationships are expressed with fewer of those details. These are relative descriptions. A simple drawing is more abstract than real counters but more concrete than a symbol alone. Neither end of this range is inherently superior for every task.

Keep the mapping while changing the surface

Name what each object, action or symbol stands for. Then change the context while keeping the structure: share six seeds into three equal packets, or partition a six-unit strip into three equal lengths. Ask which part corresponds to the original people, shares and total. This makes the common relationship available across different surfaces.

A gradual transition can replace objects with drawings, then with a labelled relation and symbols. However, a fixed concrete-to-abstract sequence is not a universal law. Some learners already know the notation, while some concrete materials introduce irrelevant complexity. Use evidence from explanation and new cases to decide whether a step supports understanding.

Return to meaning when symbols mislead

Abstract notation is powerful because it can express a relationship independently of one story. It is also easy to manipulate without meaning. A learner might read the division sign correctly yet swap the divisor and quotient. Returning to equal sharing can reveal which quantity counts groups and which counts items in each group.

Check both directions. Can the learner write an expression for a concrete situation? Can they invent a valid situation for the expression? Translation in only one direction leaves some misunderstandings hidden. If the model stops fitting, state why: counters work naturally for positive whole-number sharing, while negative numbers or continuous quantities may require a different representation.

Worked example

From tickets to a relationship

What does each part of 3p = 12 represent in the equal-ticket story?

Three tickets cost 12 currency units when each has the same price and no extra fee applies. Draw three equal boxes whose total is 12. Each box must represent 4. The abstract relationship is 3p = 12, where p is the price of one ticket. Dividing both sides by 3 gives p = 4.
Now ask for a different story with the same equation, such as three equal lengths totalling 12 metres. Then add a booking fee to the ticket case. The equation must change because the whole 12 is no longer only the sum of three ticket prices. This comparison teaches both the useful abstraction and its boundary.

Three equal ticket prices make the total 12, so p = 4 under the no-fee assumption. A fee requires a different equation.

Map equal parts to the equation

Three equal boxes represent ticket prices p. They total 12 currency units, so each is 4. A separate fee changes the equation to 3p + f = 12.

Try it yourself

Think through each question before opening the answer.

What correspondence is missing when counters are shown beside an equation without explanation?
Show the answer and reasoning
The learner may not know which objects and actions stand for each quantity and operation.
Point to and explain those mappings explicitly.
Does a learner who can manipulate an equation necessarily understand its context?
Show the answer and reasoning
No.
Ask for a matching situation and a reason for each operation to check the meaning behind the manipulation.
Why change the story while retaining the equation?
Show the answer and reasoning
It helps separate the shared relationship from incidental objects, making the abstraction usable across contexts.
The ticket and length stories differ in objects while preserving three equal parts with total 12.

What to carry forward

Connect concrete actions with the quantities and relations they represent.
Check translation in both directions and vary the context.

References

Sources supporting the explanations and learning checks in this lesson.