How Mathematical Representation Works | Turning Relationships Into Diagrams, Symbols, Tables and Models

Direct Answer: Mathematical representation works when a learner turns quantities and relationships into a form that makes them easier to inspect, reason about or operate on. A representation may be concrete, pictorial, tabular, graphical, verbal or symbolic. Its value is not that it looks mathematical. Its value is that it preserves the important relationships while making some part of the problem more visible.

The simplest definition

A mathematical representation is a form used to express mathematical objects, quantities or relationships so they can be understood, compared, manipulated or communicated.

In one line: Representation changes the form without changing the relationship that matters.

The problem hidden behind “draw a model”

A student is told to draw a bar model. They produce neat rectangles, copy the numbers from the question and still cannot solve the problem.

The page now contains a diagram, but no additional mathematical structure has become visible.

Representation is not the act of drawing. It is the act of mapping the right mathematical relationships into a useful form. A diagram that does not preserve the relationship is decoration. An equation that does not correspond to the situation is notation without meaning.

The practical job is to locate whether the learner can identify the relationship, choose a useful representation, map information into it accurately, and translate back out without losing meaning.

The representation mechanism

SITUATION / CONCEPT → IDENTIFY MATHEMATICAL OBJECTS → IDENTIFY RELATIONSHIPS → CHOOSE FORM → MAP → INSPECT → OPERATE / REASON → TRANSLATE TO ANOTHER FORM → CHECK EQUIVALENCE → USE

IES Mathematics guidance gives strong-evidence support to teaching students to use visual representations in problem solving and, for elementary intervention, to using well-chosen concrete and semi-concrete representations to support concepts and procedures. The phrase well-chosen matters. More representations are not automatically better.

1. Every representation highlights some information and hides other information

A table makes paired values easy to compare. A graph makes trend and rate visible. An equation can compress a relationship into a form suitable for symbolic manipulation. A bar model makes relative parts and wholes visible. A number line makes magnitude, order and distance explicit.

The same mathematical object can therefore be represented differently depending on the learning job.

The question is not “Which representation is best?” in the abstract. It is “Which representation makes the relationship we need to reason about most visible?”

2. The representation must preserve mathematical structure

If three parts are equal in the problem, the representation should not make one visibly larger unless scale is deliberately not being preserved and the learner understands that limitation.

If an equation says one quantity is twice another, the diagram should express the same relationship. If a graph uses unequal intervals as though they were equal, the representation distorts the mathematics.

Representation skill therefore includes a fidelity check: What relationship from the original situation is this mark, line, region, symbol or coordinate standing for?

3. Concrete models are useful when they reveal structure—not because concrete is always easier

Counters, fraction strips, blocks and physical objects can make quantities and operations visible. But physical materials also contain irrelevant features: colour, size, texture, arrangement.

The learner has to understand which physical features correspond to the mathematics and which do not. A manipulable becomes mathematically useful when attention is directed to the represented relationship.

This is one reason IES guidance recommends a well-chosen set of concrete and semi-concrete representations rather than unrestricted use of materials.

4. Visual representations can reduce hidden search

A word problem may distribute a relationship across several sentences. A diagram can place the quantities together so the learner can inspect the structure simultaneously.

For example, a bar model can make “A has 30% more than B” visible as a relative relationship rather than leaving the student to hold the comparison verbally while calculating.

But drawing is not the solution. The learner still has to map the sentence correctly into the model.

5. Equations are representations, not just calculation instructions

Students often treat equations as commands: “move this to the other side,” “divide by this,” “expand brackets.” Before manipulation, an equation represents equality between expressions.

When students understand what each symbol corresponds to, algebra becomes less like moving marks and more like preserving relationships while changing form.

For notation and symbol precision at higher resolution, see How Mathematical Symbols Carry Meaning.

6. Translation between representations is a stronger test than success in one form

A learner may understand a linear relationship as an equation but fail to recognise the same relationship in a graph. Another may solve a fraction problem with a bar model but not connect the model to multiplication or division.

Ask the learner to move between forms: words → diagram → equation → table → graph → explanation.

The goal is not to use every form every time. The goal is to test whether the same underlying relationship survives translation.

7. Multiple representations can expose different properties of the same object

A quadratic function can be seen through its equation, graph, table of values, factors and roots. Each form makes some properties more accessible than others.

Comparing representations helps the learner build a richer concept: What does changing the coefficient do to the graph? Where do the roots appear in the factorised form? How is the turning point reflected in another representation?

This is mathematical knowledge becoming connected rather than stored as separate procedures.

8. Representation selection is itself a problem-solving decision

A student who has been trained always to draw a bar model can become dependent on the instruction rather than learning when a bar model is useful.

Later learning should therefore ask students to choose among representations and explain why. Would a table, graph, equation or diagram reveal the structure most efficiently?

IES problem-solving guidance explicitly recommends teaching students how to select appropriate visual representations for the problem they are solving. This is representation as judgement, not compliance.

9. Representation errors often occur before calculation errors

If the diagram assigns a quantity to the wrong part, perfect arithmetic will still produce the wrong answer. If the equation reverses a ratio, flawless algebra faithfully solves the wrong model.

This is why the earliest useful weak link matters. When a Mathematics solution is wrong, inspect the first representation of the relationship before drilling calculation.

For the full downstream problem-solving chain, see How Mathematical Problem Solving Works for a Student.

10. Good representations can reduce working-memory load

Externalising relationships means the learner does not have to hold every quantity and connection mentally at once. A labelled diagram, table or equation can act as an external workspace.

But poorly designed representations can increase load by adding labels, colours, arrows or features that do not clarify the mathematics.

The best representation makes important information easier to inspect while minimising irrelevant demand.

11. Abstraction is not abandoning representation

Students sometimes move from concrete objects to pictures to symbols as though the symbolic form is the final representation and all earlier forms should disappear.

Experts still use graphs, diagrams, sketches and tables. Abstraction means the learner can choose a representation appropriate to the task and reason about relationships that are no longer tied to one physical example.

The endpoint is flexible representation, not symbolic purity.

12. Students should explain the mapping, not only display the representation

Ask: What does this bar stand for? Why are these two lengths equal? What does the slope represent here? Which part of the equation corresponds to this region?

These questions reveal whether the student understands the representation as a mathematical model or has reproduced a familiar visual format.

13. Comparing two representations can reveal a misconception

Give a diagram and an equation that supposedly describe the same situation. Ask whether they are equivalent.

If the learner says yes, ask them to map each quantity and relationship. The mismatch often exposes hidden misunderstandings faster than another routine exercise.

Comparison turns representation into evidence about concept structure.

14. Representation should eventually become learner-selected and learner-generated

At first, a teacher may supply the model. Later, the learner completes part of it. Then they choose from alternatives. Eventually they generate an appropriate representation independently.

This progression is a form of scaffold fading. The educational receipt is not that the student can understand the teacher’s diagram; it is that they can create or choose a useful one when the task requires it.

What mathematical representation is not

It is not drawing for decoration.
It is not one preferred model for every problem.
It is not a stage that experts permanently outgrow.
It is not copying the teacher’s diagram without understanding the mapping.
It is not using more representations by default. Each additional form should have a job.
It is not separate from algebra. Symbols and equations are themselves representations.

The smallest useful representation test

Give the learner a simple relationship in words and ask for two different representations. For example: “A quantity is three times another, and together they total 48.” Ask the learner to:

draw or model the relationship;
write an equation;
map each part of the equation to the visual representation;
explain what would stay the same if the total changed from 48 to another value.

This tests whether the learner owns the relationship rather than one memorised picture.

What a representation problem may actually be

What adults see	Possible weak link	Useful next test
Cannot draw a model	May not understand the relationship yet	Ask learner to state quantities and relationships verbally first
Diagram looks right but equation is wrong	Translation between forms	Map each visual component to a symbol
Can use teacher model but not choose one	Representation selection	Offer two possible forms and ask which reveals the relationship better
Algebra strong, graphs weak	Cross-representation connection	Translate one equation into table and graph
Uses one model for every question	Strategy/representation rigidity	Compare cases where different representations are efficient
Beautiful diagram, no solution progress	Representation is not exposing needed structure	Ask what new information the diagram makes visible

For parents: should I insist that my child always draws a model?

No single representation should become compulsory for every problem. Models are valuable when they reveal structure or support current learning.

Ask the child what the representation is helping them see. If they can solve and explain the relationship efficiently another way, the educational goal may already have moved from supplied representation toward flexible choice.

For students: how to use representation as a thinking tool

Name the quantities and relationships first.
Choose a form that makes the difficult relationship visible.
Label what each element represents.
Check that the representation preserves the original constraints.
Use the representation to reason, not merely to decorate working.
Translate into another form when that form makes the next operation easier.
After solving, map the answer back to the original situation.

How do we know representation skill is improving?

The learner identifies mathematical relationships before choosing a form.
Representations preserve quantities and constraints accurately.
The learner can explain what each element stands for.
They can translate between visual, verbal and symbolic forms.
They choose representations for a reason rather than by ritual.
Wrong solutions are caught earlier because representation mismatches become visible.
Teacher-supplied models become less necessary.

The complete representation chain

IDENTIFY OBJECTS → IDENTIFY RELATIONSHIPS → CHOOSE FORM → MAP → CHECK FIDELITY → INSPECT → REASON / OPERATE → TRANSLATE → COMPARE → SELECT MORE INDEPENDENTLY → TRANSFER

Frequently asked questions

Are bar models only for Primary Mathematics?

No. Bar models are one representation of part–whole, comparison and multiplicative relationships. Older students may use algebra or other forms more efficiently, but the underlying representational reasoning remains relevant.

Should students use concrete materials before symbols?

Concrete and semi-concrete representations can support understanding, especially when well chosen, but there is no requirement that every concept pass through an identical sequence. Match representation to the mathematical idea and learner state.

Can diagrams create misconceptions?

Yes. A diagram can distort scale, imply a relationship that is not present or draw attention to irrelevant features. Always connect the visual form explicitly to the mathematics it represents.

Why should students use more than one representation?

Different forms reveal different properties. Translating between them can test whether the learner understands the underlying relationship rather than one surface format. More forms are useful only when they add a meaningful comparison or route.

Evidence bridges

The U.S. Institute of Education Sciences / What Works Clearinghouse Improving Mathematical Problem Solving in Grades 4 Through 8 gives strong-evidence support to teaching students to use visual representations and recommends connecting represented information to mathematical notation. The 2021 guide Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades gives strong-evidence support to a well-chosen set of concrete and semi-concrete representations. Current IES professional-learning materials continue to operationalise these recommendations. These sources support purposeful representation; they do not imply that one representation should be mandated for every learner or every problem.