Math flashcards

A flashcard is about as simple as a teaching tool gets: a prompt on one side, an answer on the other, and a learner who tries to recall the answer before turning it over. Cards of this kind are often traced to the English writer Favell Lee Mortimer, whose 1834 reading course Reading Disentangled taught with printed cards. For arithmetic they became standard classroom equipment in the twentieth century, when "math facts" — the hundred or so single-digit sums and products — were drilled until they could be answered on sight.

Why the drill works was explained before the cards were common. In 1885 Hermann Ebbinghaus published Über das Gedächtnis, the first experimental study of memory. Testing himself on lists of nonsense syllables, he measured how quickly they were forgotten and how much faster they were relearned, which gave psychology the forgetting curve. In 1972 the German science journalist Sebastian Leitner turned the result into a routine: cards answered correctly move to a box that is reviewed less often, cards answered wrongly go back to the first box. The Leitner system is the ancestor of the spaced-repetition apps used today.

The other half of the explanation is the testing effect. In a study published in 2006, Henry Roediger and Jeffrey Karpicke found that students who practised recalling a passage remembered it better a week later than students who spent the same time rereading it. A flashcard is a recall test that fits in a hand. Arithmetic facts suit it well because they are small, numerous and needed constantly: a child who has to work out 7 × 8 every time has less attention left for the problem it is part of.

Key properties

• There are 100 multiplication facts from 1 × 1 to 10 × 10, but because 7 × 8 and 8 × 7 are the same fact, only 55 need learning separately.
• Up to 12 × 12 there are 144 cards and 78 distinct pairs of factors.
• Addition facts from 0 + 0 to 10 + 10 make 121 cards, and every sum is between 0 and 20.
• The subtraction cards here always put the larger number first, so no answer is negative.
• The division cards are built backwards from a multiplication fact, so every one divides exactly and none divides by zero.
• Each division fact is a multiplication fact read the other way: 56 ÷ 8 = 7 because 8 × 7 = 56.
• The deck is drawn from a seeded generator, so the same settings and seed always give the same cards in the same order.

Where they turn up

• Primary classrooms, where teachers flash cards to the whole class or run games like "Around the World", in which pupils race each other to answer.
• Leitner boxes and spaced-repetition apps such as Anki, which schedule each card according to how well it was last recalled.
• Practice for timed tests such as the multiplication tables check in English schools.
• Homeschool and tutoring sessions, where a short, varied deck makes a daily five-minute review.
• Speech and language therapy, which uses picture cards built on the same prompt-and-recall principle.