Seeing what symbols stand for helps them make sense

A 2014 review recommends starting with concrete materials and fading to symbols, so learners read unclear symbols in terms of things they understand. A review of 55 studies found teaching with physical objects beat symbols alone. The gains were moderate to large for remembering and small for solving new problems.

What we built

When a sign inside parentheses goes wrong, Topic Tutor tests your line with a number for x. It shows what your line takes away next to what the parentheses are worth. When the exercise has a story, such as a window cut out of a banner, it draws it to scale at that number. For multiplying out parentheses, it draws squares and strips. Every size written on a drawing is checked against it.

Where the evidence stops

The 55 studies used objects you hold, not drawings or numbers, and their gains were small on new problems. An area can't be negative, so the tutor shows a minus sign with numbers. No study has tested ours.

• Fyfe, McNeil, Son & Goldstone (2014). Concreteness Fading in Mathematics and Science Instruction: a Systematic Review. Educational Psychology Review.
• Carbonneau, Marley & Selig (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology.