How long to leave between maths reviews
The best-measured answer comes from Cepeda, Vul, Rohrer, Wixted and Pashler (2008), published in Psychological Science. They taught more than 1,350 people a set of facts, left a gap of up to three and a half months, gave them a single review, then tested them at delays of up to a year. The best gap depended on how long the material had to last: roughly 20 to 40 per cent of a one-week horizon, falling to roughly 5 to 10 per cent of a one-year horizon.
Turned into a planning rule: if a topic only has to survive until a test next week, review it after a day or two. If you want it next year, the study points to a gap of a few weeks to a month. The study used facts, not maths methods, so treat the percentages as a guide to the order of magnitude rather than a prescription for algebra.
It also means a fixed weekly slot for “maths review” suits almost nothing. A topic you met yesterday wants another look within a day or two, while one you have been solid on since spring can wait a month, and both gaps shift after every review.
A doubling schedule you can keep on paper
Without software, this is the version that fits on an index card. The worked topic is computing the slope of a line from two points, which builds directly on the idea of slope as rise over run. Each review is three problems you have not seen before, done from a blank page with the notes closed.
| Day | Gap since last | What to do |
|---|---|---|
| 1 | none | Learn it. Solve three, e.g. the slope through (2, -3) and (5, 3). |
| 2 | 1 day | Three new pairs of points, notes closed. |
| 4 | 2 days | Three more, at least one where x decreases from the first point to the second. |
| 8 | 4 days | Three more, including a horizontal line, where the slope is 0. |
| 16 | 8 days | Three more, with one answer that is a fraction. |
| 32 | 16 days | Three more. Keep doubling while reviews go cleanly. |
| Any miss | back to 2 days | Rework the method once, review in two days, then double again from there. |
You can shift the exact days. What matters is that the gap grows after each clean review, and that every review is a problem you have to solve rather than a page you reread. Two habits make the paper version last. First, write the working out in full, because for a procedure the act of writing it is the retrieval. Second, give formulas and methods separate clocks: the gradient formula is a fact you can test with a card, while choosing which point to subtract from which, and keeping the signs straight, only stays sharp if you keep doing it.
Where flashcards and maths part company
Flashcards suit facts, such as vocabulary. Most of what you need to keep in maths is a method, and a card that shows the same quadratic every time trains you to recognise that quadratic. After a few weeks you know that card’s answer, which is not the same as being able to factorise the next one. A proper maths review hands you a fresh problem at the right level and tells you whether you got it right, which is very hard to produce by writing cards.
There is also the labour. You decide what deserves a card while you are still new to the topic, then write each one before any review can happen. Card apps also keep scheduling while you are away, so a fortnight off returns you to a long queue of overdue reviews. The spacing still works; a lot of your study time just goes on keeping the deck in order.
What our scheduler does, rule by rule
Consiliae schedules each of its 1,424 maths concepts separately. The rules below are read from the scheduler itself. Your first review falls 2 days after you pass a lesson. After each review the gap doubles if your estimated recall is 85% or higher, grows by half from 70%, stays the same from 50% and halves below that (to no less than 1 day). A review you fail never lengthens the gap, and the longest gap is 180 days. For a learner whose reviews all go strongly and on time, the gaps run 2, 4, 8, 16, 32, 64 and 128 days, then stay at 180.
A review taken before its gap is up earns only part of the growth, in proportion to how much of the gap had passed. Remembering a method after two days tells you little about whether you will remember it after eight, so an early pass should not be credited as if it were a late one.
Every review is a newly generated problem. For computing slope, the generator picks two fresh points each time, so one review might ask for the slope through (2, -3) and (5, 3), which is 2, and the next for the slope through (-4, 1) and (-1, -1), which is -2/3. The prompt does not show the formula, because the point is to check that you can recall it.
Lessons also lean on what came before them. In the curriculum, “Computing slope from two points” lists “Slope of a line” and “Subtracting integers” as its prerequisites, and some of its quiz blocks are marked as exercising “Slope of a line”. If you get one of those blocks wrong while “Slope of a line” is on your review schedule, its next review is brought forward to 2 days from then, unless it was already due sooner. The link is written by hand, one prerequisite at a time, so only some quiz blocks carry it so far. It only works in that direction: a correct answer does not yet count as a review of the earlier concept, which still comes round on its own date.
When spacing is the wrong tool
Two situations call for something else. The first is an exam in five days. There is no room for an expanding schedule at that horizon, so spread the work across the five days instead of putting it all into one, and test yourself rather than reread. The second matters more: a topic that never made sense in the first place. The cause is usually a missing piece underneath it, and reviewing the topic itself will just repeat the same failure at longer and longer intervals until that piece is learned.
So before setting up any schedule, find out which concepts you actually have. If you are coming back to maths after years away, work out where you stand and learn forward from there. For the wider study routine that spacing fits into, see how to study maths, and if keeping up a manual system is the part that defeats you, the guide to learning maths with ADHD looks at taking that load off.
For the research on the testing half of this, the review by Dunlosky, Rawson, Marsh, Nathan and Willingham (2013) is the most readable starting point: it rates ten study techniques on the evidence, and practice testing and distributed practice are the two it rates most useful.
