One idea, many ways: Connect representations to deepen understanding

One of the most effective ways to build understanding isn't repetition. It's showing students the same idea through more than one lens.

When students connect different representations of a single idea, they're not just picking up an alternate method. They're building a picture of how mathematics fits together — the kind of understanding that doesn't come from repeating one correct procedure until it sticks.

Take fractions. One whole can be two halves. It can also be a half, a third, and a sixth, all at once. When students build the same whole multiple ways, they start to see how fractions relate to each other — and why two fractions that look completely different can represent the exact same amount.

That understanding deepens further when a fraction shows up in more than one form: as a shape, a point on a number line, a collection of objects, a decimal, a symbol. Each representation on its own teaches a procedure. Connected together, they teach a relationship.

The research on this is clear: understanding grows when students visualize ideas, build connections between them, and approach the same math from more than one angle. Our co-founder Jo Boaler pushes this further — move past questions that just ask students to "solve," and ask them to visualize, represent, and explore instead.

That's the design principle behind every Struggly task. Add one to a lesson and students get more than a problem to solve. They get the same math from multiple angles, real connections between representations, and understanding that goes deeper than any single method could reach on its own.

Every connection a student makes is one more piece of how mathematics fits together.