Mathematics: Multiple representations

What you'll be able to do: Move between equation, graph, table and words freely, which is what understanding a relationship actually consists of, and what exams test by switching representation without warning.

Why this is a mathematics problem specifically

y = 2x + 3 is a line, a table of pairs, a sentence about a starting value and a rate of change, and (for some purposes) a machine that takes numbers in and puts numbers out.

They are one object. They are taught in four different weeks, assessed separately, and nothing in the curriculum says they're the same thing.

The result is students who can manipulate the equation and can't read the graph; or read the graph and can't produce the table; or recite the verbal version and connect it to neither. Each skill is real and none of them is understanding the relationship.

A large share of exam difficulty is simply the question arriving in the representation you didn't practise. The content is known; the translation isn't.

The translation drill

The skill is the movement, not the forms. Take one feature and locate it everywhere:

The gradient. In the equation it's the coefficient. On the graph it's the steepness. In the table it's the constant difference between consecutive y-values. In words it's the rate of change. In calculus it's the derivative, Those are five descriptions of one thing and most students hold them as five facts.

Do the same for the intercept, the roots, the turning point, the asymptote, the domain.

Take [feature]. Ask me where it appears in the equation, on the graph, in the table, and in the verbal description, one at a time. Don't tell me; ask.

The break-and-predict test

This is what distinguishes genuine connection from four separate familiarities:

Change [parameter] in the equation. Before showing me anything, ask me what happens to the graph, to the table, and to the verbal description.

If you can predict all three, they're connected. If you can predict the graph and not the table, you've learned graph-shapes as pictures rather than as consequences.

Across the syllabus

Linear functions. The foundational case, and where the four-way connection should be built. Everything later assumes it.

Quadratics. Factorised, completed-square and expanded forms are three representations of one function, each making a different feature obvious, roots, turning point, intercept. Students learn three procedures and don't notice they're choosing a view.

Trigonometric functions. Unit circle, graph, and formula. The unit circle explains periodicity, the graph shows it, the formula computes it. Students who only have the formula can't answer "how many solutions in this interval?"

Calculus. A derivative is a formula, a gradient function, a rate of change, and the slope of a tangent. Students routinely have the first and not the fourth, which is why "interpret this derivative" questions are feared.

Sequences and series. Recursive, closed-form, graphical and tabular. Converting between recursive and closed-form is a distinct skill worth drilling.

Statistics. A distribution as a formula, a curve, a table of probabilities, and a statement about a population. The last one is where meaning lives and where students are weakest.

Linear algebra. A matrix as an array of numbers, as a transformation of space, and as a system of equations. The transformation view is the one that makes eigenvalues comprehensible, and many students never meet it.

Complex numbers. Cartesian, polar and Argand diagram. Multiplication is opaque in one and obvious in another, a clean demonstration that choosing the representation is the technique.

The habit that pays in exams

Before solving anything, ask: which representation makes this easiest?

Finding the turning point is easy in completed-square form and tedious in expanded form. Multiplying complex numbers is easy in polar form and unpleasant in Cartesian. Deciding how many solutions a trig equation has is easy on a graph and error-prone algebraically.

Experts choose the representation first. Students take the one they were given.

Pitfalls

1. Being shown the other forms rather than producing them. Recognition again; you must make them.
2. Treating the equation as the "real" one and the graph as an illustration, They're the same object.
3. Skipping the break-and-predict step. It's the only test of whether the forms are connected.
4. Practising only the translation you're good at. Ask which direction you're worst at, usually graph-to-equation.
5. The tell: you can answer questions posed the way you practised and stall on identical content posed differently.

Try this today

Take a function you know well. Produce all four representations yourself, from memory, before checking.

Then change one parameter and predict what happens in all four. The one you found hardest to produce is the direction an exam will use.