Learning from multiple representations means understanding how different forms describe the same idea—or different aspects of it—and using those connections to reason. A table, graph, equation and explanation do not automatically become one coherent model because they share a page. The learner has to identify what corresponds, preserve the units and conditions, and notice when two forms disagree. This guide follows one worked example across several forms, then shows how to test the connections without turning the lesson into a collection of disconnected exercises.
The How Learning Works hub explains the wider learning system. This article focuses on a narrower difficulty: a student can read each representation separately but cannot reliably move between them.
Four correct-looking answers can conceal four different interpretations
A student fills a table, plots a graph, writes an equation and produces a sentence. Every part looks familiar. Yet the graph begins at zero while the equation includes an initial amount. The sentence says “three litres every two minutes” while the table increases by six litres over two minutes. The page contains several representations, but they do not describe the same model.
The next teaching move should not simply be another set of graphs or another set of equations. The missing work is correspondence. Where does the initial amount appear in each form? What does the coefficient mean in the sentence? Which interval in the table determines the rate? What assumption permits a straight line between points?
These questions turn the lesson from repeated format production into meaning-preserving translation. The learner must account for the same information in each form. A change in one form should have a predictable consequence in the others.
The numerical cases in this guide are deliberately invented. They are mathematical models with stated assumptions, not measurements from a physical investigation or reports of student performance. Their purpose is to make the translation decisions inspectable.
Why more representations are not automatically better
Ainsworth’s DeFT framework considers the design of multiple representations, the functions they serve and the tasks required of the learner. It distinguishes functions such as complementing information, constraining interpretation and supporting the construction of deeper understanding. The framework does not treat the number of representations as a guarantee of benefit. The learner’s work in coordinating them remains central. See Ainsworth’s original framework.
There is also empirical evidence that the order and type of representational support matter. Rau, Aleven and Rummel studied 74 pupils in grades three to five using a fractions tutoring system. Their comparison supported sense-making activities before perceptual-fluency activities in that setting, rather than the reverse sequence. This is a specific study, not a universal age rule. Read the original research abstract.
The practical implication developed here is to ask for meaning before rewarding fast conversion. A learner who can rapidly reproduce a familiar graph may still misunderstand what its axes represent. Conversely, a learner who explains the relationship accurately but plots slowly may need fluency practice rather than a complete conceptual restart.
Do not assume that adding a graph will repair a misunderstood equation. The graph introduces its own conventions. It can help only if those conventions are available or taught. The same applies to tables, maps, timelines and symbolic notation: each form carries information through rules the learner must understand.
The shared model: a tank with a starting amount
Use this description: “At the start of observation, a tank contains 12 litres of water. For the next 10 minutes, water is added at a constant rate of 3 litres per minute. No water leaves the tank, and its capacity is sufficient for this interval.” Define t as elapsed time in minutes and V as the volume in litres.
The initial amount is 12 litres. The added amount after t minutes is 3t litres. Therefore the model is V = 12 + 3t, for 0 ≤ t ≤ 10. The time restriction belongs to the model. It should not disappear when the sentence becomes an equation.
The description contains several different kinds of information. “12 litres” is an initial state. “3 litres per minute” is a rate. “Constant” describes how that rate behaves over the interval. “No water leaves” prevents an unstated outflow from changing the calculation. “Next 10 minutes” bounds the period for which the stated rule is guaranteed.
Before drawing anything, ask the learner to identify these roles. A student who treats 12 and 3 as interchangeable numbers has not yet represented the situation. A student who can name the roles has a basis for constructing the other forms.
This first step matters because a representation is not merely a place to put numbers. It expresses a relationship. The same numbers can appear in a different model if their roles change. “12 litres per minute for 3 minutes” is not equivalent to “12 litres initially, then 3 litres per minute”.
From the description to a table
Choose times within the stated interval. At zero minutes the tank still contains the initial 12 litres. At two minutes, 6 litres have been added, giving 18 litres. At four minutes, 12 litres have been added, giving 24 litres. Continue with the same relationship.
| Elapsed time, t (minutes) | Water added (litres) | Total volume, V (litres) |
|---|---|---|
| 0 | 0 | 12 |
| 2 | 6 | 18 |
| 4 | 12 | 24 |
| 6 | 18 | 30 |
| 8 | 24 | 36 |
| 10 | 30 | 42 |
The middle column is optional in a finished table, but it is useful while diagnosing the model. It distinguishes the amount added from the total amount present. A student who writes 30 litres as the total at ten minutes has calculated the added amount but omitted the initial 12 litres.
Ask what changes from one row to the next. The elapsed time increases by two minutes and the total volume increases by six litres. Dividing the volume change by the time change gives three litres per minute. The learner should not infer a rate of six litres per minute merely because six is the visible difference between adjacent volume entries.
Then choose unequal time steps: zero, one, four and ten minutes. The volume increases between rows will no longer be equal, although the rate remains constant. This is a useful test of whether the student understands rate as change relative to an interval rather than as a repeated difference in an arbitrary table layout.
A good table translation preserves labels as well as values. Without units and variable names, a row containing 4 and 24 could be misread. The notation should make clear that four is elapsed time and twenty-four is total volume, not the reverse.
From the table to a graph
Place elapsed time on the horizontal axis and total volume on the vertical axis. Label both axes and their units. The point (0, 12) represents the starting amount. The point (10, 42) represents the total after ten minutes. Under the stated constant-rate model, the intervening values lie on a straight line segment.
Three translation questions are worth asking before the student completes the plot. Where is the initial amount? Where is the rate? Where is the limit of the stated interval? The answers should point to the vertical intercept, the change in volume relative to the change in time, and the segment ending at ten minutes.
The visual steepness of the printed line is not the rate by itself. Changing the physical scale of either axis changes how steep the line appears on the page without changing three litres per minute. The rate must be read using the numerical scales. This follows from what the axes represent, not from how dramatic the picture looks.
Now supply a graph with the correct shape but a missing vertical-axis label. Can the learner tell whether it shows volume, added volume or water level? Not reliably from shape alone. In the hypothetical model, total volume begins at twelve; added volume begins at zero. Water level would require information about the tank’s geometry before it could be identified with volume.
This is a useful boundary lesson. A graph that rises is not enough to establish the quantity represented. Labels and assumptions are part of the information. A student should be able to say “the graph does not tell us yet” rather than guessing from a familiar visual pattern.
From the graph and equation back to words
Give the equation V = 12 + 3t without the original paragraph. Ask the learner to write a description, including the units and the time interval supplied with the equation. A defensible description identifies twelve as the initial volume and three as the volume added per minute.
The equation alone does not contain every feature of the original story. It does not specify the tank’s colour, shape or location. It does not reveal how the water is added. Those details should not be invented merely to make the description more vivid. The representation establishes a relationship under a stated interpretation; it is not a complete photograph of a situation.
Ask for a second story with the same algebraic form. For example, a fictional collection could begin with twelve tokens and receive three additional tokens per round. The arithmetic relationship is the same, but the variable is now a count of rounds rather than continuous time. That difference changes which input values are meaningful.
This comparison shows why translation must preserve more than the visual form of an equation. In the tank model, a time such as 2.5 minutes is meaningful under the stated continuous-rate assumption. In a model counting completed rounds, 2.5 completed rounds may not be an admitted input. The formula can look identical while its domain differs.
The learner should therefore translate symbols together with definitions, units and conditions. A bare formula is an incomplete description until those choices are supplied.
Reverse the question to test the connection
Instead of asking for the volume after six minutes, ask when the volume reaches thirty litres. From the equation, 30 = 12 + 3t, so 18 = 3t and t = 6. In the table, the corresponding row contains six minutes and thirty litres. On the graph, the horizontal line at thirty litres meets the model at six minutes.
The three methods should agree. Agreement is not three independent measurements, because all three representations were generated from the same assumptions. It is an internal consistency check. This distinction is important: copying one erroneous model into several formats does not create several independent pieces of evidence for it.
Now ask when the volume reaches fifty litres. Continuing the algebraic rule gives t = 38/3 minutes, which is beyond ten minutes. The original description only guarantees the rule for the next ten minutes. Therefore the proposed time is a conditional extrapolation, not an answer established by the stated model interval.
A strong response says what additional assumption would be needed: the same net filling rate must continue beyond ten minutes, with adequate capacity and no other relevant change. The student need not refuse to calculate. They should distinguish a calculation under an added assumption from a conclusion already licensed by the information.
This is where multiple representations can make boundaries visible. The graph segment ends at ten minutes. The table ends at ten minutes. The equation can still be evaluated numerically outside that range, but the written domain limits what the result means. The most flexible representation is not automatically the one with the widest valid interpretation.
Change one assumption and propagate the change
Modify the situation: the tank begins with 20 litres instead of 12, while the rate and interval remain unchanged. The new equation is V = 20 + 3t. Every total in the table is eight litres larger. The graph has the same rate of increase but begins at twenty. The verbal explanation should change its initial amount and retain the other conditions.
This controlled change is more revealing than asking the student to redo four unrelated tasks. It checks whether one parameter has a consistent meaning across forms. A learner who changes the slope instead of the intercept has identified the wrong role for the initial amount.
Next, restore the initial twelve litres but change the rate to two litres per minute. The equation becomes V = 12 + 2t. The initial point remains unchanged; the later values and rate of increase change. Ask the learner to predict these consequences before recalculating every row.
For a more demanding variation, let filling stop after four minutes and let no water leave. The volume is then twenty-four litres at four minutes and remains twenty-four for the rest of the observation interval. A single straight rising line over all ten minutes no longer matches the description. The learner must represent the change in rule.
The purpose is not to rush into advanced notation. A younger learner can explain the two stages in words and a table. A more advanced learner can write a piecewise description. The underlying test is the same: does a change in the situation produce the appropriate change in every representation?
Use deliberate mismatches to locate the weak connection
Once the learner understands the baseline model, present a small inconsistency and ask them to find it. This is a proposed teaching activity, not an attempt to trick the student. State that one representation has been altered and that the task is to identify the disagreement and justify a repair.
| Mismatch | What must be compared | Repair question |
|---|---|---|
| The graph starts at zero, but the story begins with 12 litres. | Initial amount and vertical intercept. | Is the graph showing added volume or total volume? |
| The sentence says 3 litres every two minutes. | Rate, interval and table differences. | What volume increase should occur over two minutes? |
| The vertical axis says centimetres. | The measured quantity and its unit. | What additional information would connect water level to volume? |
| The line continues indefinitely. | The plotted domain and the stated observation interval. | Which part of the continuation is an extra assumption? |
| A table entry gives 30 litres at ten minutes. | Added amount and total amount. | Where has the initial 12 litres gone? |
Listen to the explanation before deciding what to reteach. If the learner understands the rate but misreads the graph scale, practise the scale. If the learner confuses added and total amounts, revisit the model. If the learner notices the mismatch but cannot express it, provide language for the relationship rather than assuming the concept is absent.
After correction, use a different mismatch. A student who remembers that “the answer is the intercept” may solve the repeated item without inspecting the new situation. Change the location of the error so the learner must compare meaning again.
Not every translation preserves all the information
A table of selected times does not show every value directly. A graph may make a trend easy to see while making a precise value harder to read. A sentence can state conditions that disappear from a simplified sketch. An equation can encode a relationship compactly while leaving its physical interpretation unspecified.
Ask what is retained, omitted or added at each translation. In the tank case, the table was calculated from a constant-rate model. A different situation might provide only a few observed measurements. Joining measured points with a straight line would then require an interpretation, not merely a change of format.
This distinction prevents a common error: treating a representation produced from assumptions as though it independently confirmed those assumptions. The graph of V = 12 + 3t shows what the equation implies. It does not demonstrate that a real tank followed that equation. An actual investigation would need observations and appropriate measurement procedures.
For learners, a useful annotation is to identify the status of information: “given”, “calculated under the model”, “observed” or “assumed”. The annotation need not appear on every routine exercise. Use it when the task could otherwise blur the difference between information and inference.
Knowing what a representation cannot establish is part of using it well. A polished graph should not make a weak conclusion feel stronger than the underlying information permits.
The same discipline applies outside Mathematics
In reading, a passage and a timeline may describe the same events. The timeline should preserve order without inventing exact times or causes. A character map should distinguish relationships explicitly stated in the text from interpretations inferred by the reader. A summary should not silently turn a possibility into a certainty.
In a scientific explanation, a labelled diagram, a results table and a written conclusion may perform different jobs. The diagram can show the arrangement; the table can contain observations; the conclusion can interpret them. They do not become interchangeable because they appear in the same question. Ask which representation supplies which part of the evidence.
In a project plan, a list of tasks and a dependency map can share the same task names while expressing different information. A list may establish what must be done. A dependency map must also establish which task requires another to be completed first. Adding arrows without specifying those relationships produces an attractive but potentially misleading plan.
These are original applications of the same checking discipline: identify the referents, define the connections, preserve the conditions and keep inference separate from supplied information. The representations differ by subject, but the questions remain useful because they concern fidelity to the task.
Teach the correspondence before making the task faster
Begin with two forms when four would obscure the relationship. For the tank example, a sentence and a table may be enough. Once the learner can explain each column, introduce the graph. Then ask how the equation expresses the same initial amount and rate. The sequence should follow what the learner can currently coordinate.
Keep the relevant parts available for comparison. Ask the learner to point from a phrase to a table column, from a row to a plotted point, or from a coefficient to a described rate. These actions make the correspondence explicit rather than leaving it as an assumption in the teacher’s explanation.
As understanding develops, practise the same connection with less prompting and a different example. The fractions study by Rau and colleagues provides one reason to distinguish sense-making from fluency rather than treating fast visual matching as sufficient. Its findings support that distinction in the studied tutoring context, not an inflexible sequence for every subject. Rau, Aleven and Rummel (2017).
Do not reward speed that removes checks of units or conditions. A rapid conversion that changes the quantity is not fluent understanding. Once the correspondence is sound, speed can become a useful practical goal where the performance requires it.
A student-led checking routine
Start by naming the object represented: a quantity, an event sequence, a relationship or a set of observations. Then identify the roles of the parts. In the tank case, distinguish the starting amount, the added amount, the total and the elapsed time. This prevents the student from translating marks before deciding what they mean.
Next, make one explicit correspondence. “This twelve is the amount at zero minutes.” “This six-litre increase corresponds to two minutes.” “This endpoint marks the last time included in the stated rule.” A sequence of such explanations builds an inspectable connection between forms.
Check a known case. Substitute zero, or inspect an event whose order is stated directly. Boundary and starting cases often reveal mismatches quickly. Then test one changed case. A representation should respond to the altered condition for a reason the learner can explain.
Finally, state what remains unknown. Is the graph based on a model or measurements? Does the table cover every relevant interval? Does the equation’s domain permit the requested input? This last step keeps a successful calculation from becoming an unjustified conclusion.
The routine is not a compulsory worksheet to fill in forever. Use it to make the checking decisions visible, then reduce the prompts when the learner initiates those decisions reliably.
How parents and teachers can distinguish three kinds of difficulty
A learner may have a representation-reading difficulty: the graph scale, symbol or table heading is unfamiliar. Teach the convention directly. Repeating the whole topic may be unnecessary if the relationship itself is understood.
A learner may have a connection difficulty: each form can be read separately, but the learner cannot explain which parts correspond. Use paired comparisons and translation in both directions. Ask where the same quantity or condition appears in each representation.
A learner may have a conceptual difficulty: the underlying distinction between initial, added and total amount is wrong. Changing format will not automatically repair it. Return to a simple case, make the relationship explicit and then rebuild the representations.
These are working hypotheses about the observed task, not clinical categories. Test them with a small change. For example, explain the graph scale and see whether the interpretation improves. Supply the equation’s variable definitions and ask the learner to translate again. A targeted follow-up is more informative than a global label such as “weak at graphs”.
The final test: choose, translate and defend
Give a fresh situation without telling the learner which representation to use. Ask them to choose a form and explain why it is useful. Then request one alternative representation and ask what it makes easier or harder to see. This checks selection as well as conversion.
Do not insist that every representation be produced for every problem. A table may answer an exact-value question efficiently. A graph may make a pattern easier to compare. An equation may support a calculation under stated conditions. The learner should choose according to the question rather than perform a ritual of generating four formats.
For the tank model, the strongest response is not simply four correct products. It is an explanation that ties them together: the initial twelve appears at time zero, the rate of three controls the change per minute, the total includes both initial and added water, and the model is bounded by the stated interval.
Multiple representations become useful when they constrain and clarify one another. The learner should be able to move between them without quietly changing the meaning, and should be able to stop when a requested conclusion exceeds what the information supports.
Research, limitations and next routes
Ainsworth (2006), DeFT: A Conceptual Framework for Considering Learning with Multiple Representations, supplies the theoretical framework. It is not itself a trial of the tank exercise. Rau, Aleven and Rummel (2017) provides an empirical comparison of sense-making and fluency-building support in a specific fractions-learning setting.
The worked model, mismatch table and checking routine in this article are original teaching illustrations. Their mathematical conclusions follow from the stated assumptions. Their educational effectiveness should be judged through actual learner responses, fresh tasks and later checks rather than assumed from the number of representations used.
Use Working Memory when coordinating the forms becomes overwhelming, Cognitive Flexibility when the learner cannot change approach, and Self-Explanation when a correspondence remains unspoken. Return to How Learning Works for the complete mechanism map.
