In 1879, a 29-year-old German psychologist named Hermann Ebbinghaus began an experiment with exactly one subject, himself. He memorized lists of meaningless syllables, then measured how long it took him to relearn them later. The result: one day later, he had to redo about two thirds of the original work. That is the famous forgetting curve, and it is misquoted all the time.
It still explains something every student knows: you understood the lesson in class, you reread your notes the night before the test, and then you freeze in front of the first problem. That isn't laziness, and it isn't proof that you're "not a math person." It's how memory works, and more than a century of research shows how to make it work for you instead of against you.
What Ebbinghaus actually measured
Ebbinghaus wanted to study memory without meaning getting in the way. So he built artificial syllables, each made of a consonant, a vowel and a consonant, and learned them in lists of 13 until he could recite them without a mistake. After a delay, he relearned them and measured how much time he saved compared with the first time. He called that the savings. His results, published in 1885:
- after about 20 minutes, he saved roughly 58% of the original learning time;
- after one hour, he had to redo about half the work;
- after one day, he still saved about a third;
- after 31 days, about a fifth.
Three details change how you should read those numbers.
- It isn't a percentage of memories. The popular claim that you forget 70% of a lesson within 24 hours mixes two things up: Ebbinghaus measured the effort needed to relearn, not the share of material you can still recite.
- The material was meaningless on purpose. Ebbinghaus also memorized stanzas of Byron's poem Don Juan, about 80 syllables each. He needed barely 9 repetitions per stanza, while he estimated that a list of nonsense syllables of the same length would take 70 to 80. Meaning cut the effort by roughly a factor of ten. In math, understanding why a formula works isn't a luxury: it's what makes the formula memorable.
- Nothing disappears completely. Even after a month, relearning was faster than learning the first time. That's good news for anyone coming back to math years later: if you want to relearn math as an adult, you are rarely starting from zero.
The obvious limit is a single subject, himself. But in 2015, Jaap Murre and Joeri Dros repeated the experiment with a new subject, who spent about 70 hours learning syllable lists, and the curve came out very close to the original.
Recognizing is not the same as doing
In math, forgetting takes a sneaky form. When you reread a worked example, every line looks obvious: you recognize it. But recognizing a method and producing it yourself on a blank sheet are two different things, and the second is much harder. In a study of more than 1,350 people, Nicholas Cepeda and colleagues (2008) measured both: about a year after learning a set of obscure facts, participants could recall on their own roughly one in five in the best condition, but could pick out roughly two in three from five choices.
The trap is that rereading feels like mastery. In 2009, Jeffrey Karpicke, Andrew Butler and Henry Roediger surveyed 177 students at Washington University in St. Louis: 84% said they reread their notes or textbook, and 55% called it their number one strategy. Only 11% quizzed themselves, and just 1% made that their main strategy. In 2008, Nate Kornell and Robert Bjork had 120 students study paintings by twelve artists, either grouped by artist or mixed together. On the test, 78% did better at matching new paintings to the artists they had seen mixed, yet 78% rated grouping as at least as effective, even after taking that test.
The most cited review on the subject, published in 2013 by John Dunlosky and four colleagues, rated ten study techniques. Only two earned the top utility rating: practice testing and spreading study out over time. Rereading, highlighting and summarizing all landed in the low-utility group.
The testing effect: recalling is learning
The landmark experiment dates from 2006. Henry Roediger and Jeffrey Karpicke had 120 college students read two short science passages, one about the Sun and one about sea otters. After a first reading, students reread one passage and, for the other, wrote down everything they could remember, with no feedback. Results on the final test:
- 5 minutes later, rereading won: 81% of ideas recalled versus 75%;
- 2 days later, recalling won: 68% versus 54%;
- 1 week later, the gap grew: 56% versus 42%.
In a second experiment, one group studied a passage in four short reading sessions, while another had one reading session followed by three attempts to write it down from memory. A week later, the second group recalled 61% of the ideas and the first group 40%, even though the first group had read the passage 14.2 times on average, compared with 3.4 times. The painful detail: the rereaders were the most confident they would remember.
So a test isn't just a way to measure what you know. Pulling information out of memory strengthens it. In math, testing yourself means working without a model in front of you: stating a definition, writing a formula from memory, redoing a standard problem without peeking at the solution.
Spaced practice: review at the right time
Ebbinghaus had already noticed it: 68 repetitions of a 12-syllable list in a row left it about as well learned the next day as 38 repetitions spread over the three previous days. The effect has been confirmed hundreds of times since. In 2006, Nicholas Cepeda, Harold Pashler and colleagues pooled 839 measurements from 317 experiments in 184 published papers: the best gap between two study sessions grows with how long you need to remember.
Their 2008 study gives rough numbers. For a test one week later, the best gap between two sessions was about one to three days. For a test a year later, it was about three to four weeks. With the same total study time, a well-chosen gap raised final recall by 64% compared with two back-to-back sessions. Those experiments used facts to memorize, but math follows the same rule.
In 2007, Doug Rohrer and Kelli Taylor taught college students to count the ways to arrange a string of letters like "abbccc" (there are 60). Students who split their practice into two sessions a week apart scored 74% on the final test, versus 49% for students who did the same problems in one sitting. Doing twice as many problems in a row added nothing: 49% versus 46%.
Over the long run, the effect is striking. In 1991, Harry Bahrick and Lynda Hall tested the algebra of more than a thousand people who had finished high school anywhere from weeks to decades earlier. As cognitive scientist Daniel Willingham sums it up, people who went on to take math courses beyond calculus showed no loss of their algebra, even 50 years after their last math class. The explanation: those courses made them keep using and reviewing their high school algebra for several more years.
Interleaving: mix up your problems
In most textbooks, every problem in a section uses the method you just learned, so you know in advance which formula to grab. On a test, or on the SAT, you have to figure out on your own what kind of problem you're looking at.
In a second experiment, Rohrer and Taylor had 18 college students find the volumes of four unusual solids. One group practiced in blocks (four problems on the first solid, then four on the second, and so on), the other did the same problems shuffled. During practice, the blocked group did better: 89% versus 60%. On the test a week later, it flipped: 63% for the mixed group, 20% for the blocked group. The blocked students could still do each calculation; they just couldn't tell which formula to use.
Eighteen students is a small sample. But the effect held up in real classrooms: in a trial with 787 seventh graders in 54 Florida math classes, Doug Rohrer and colleagues (2020) found scores of 61% with mostly mixed practice versus 38% with mostly grouped practice, on a surprise test given one month later.
The generation effect: produce, don't just read
In 1978, Norman Slamecka and Peter Graf showed that a word you come up with yourself from a clue, such as a synonym when you are given its first letter, sticks better than a word you simply read. A meta-analysis by Sharon Bertsch and colleagues (2007), pooling 86 studies, put the advantage at almost half a standard deviation, a solid effect by psychology standards. In math, that means predicting the next step before you read the solution, making up your own example, or restating a rule in your own words, even clumsily.
How to make math stick
For students
- The blank page. After class, take five minutes to write down from memory the day's definitions, formulas and methods, then compare with your notes. Whatever was missing is what you need to review.
- Redo, don't reread. Cover the solution of a worked example and solve it again. Use the solution only to check at the end.
- Space it out. Review a unit a day or two later, then a week later, then before the test, instead of one long session the night before.
- Mix it up. When you review, pull problems from several units, and name the method before you start computing.
- Distrust the easy feeling. If everything looks simple when you reread, test yourself. Struggling a bit to retrieve an answer is often a sign that it's sinking in.
- Keep the basics warm. Short, regular practice across your math courses beats cramming.
For parents
- Replace "Did you study?" with "Explain it to me" or "Show me without your notes."
- Offer five minutes of quick questions on the unit from two weeks ago, not just tomorrow's quiz.
- Help break test prep into short sessions spread over several days.
- Don't worry when your child gets answers wrong while self-testing: the effort of recalling is part of learning, as long as they check the right answer afterward.
The bottom line
You don't forget math because you're "bad at it." You forget it because you mostly reread it, in one block, right before the test: exactly the conditions that create a feeling of mastery without lasting memory. The methods that work have been known for a long time and don't take more time: test yourself instead of rereading, space it out, mix it up, produce your own answers. They feel harder in the moment, and that is precisely why they work.
That's the thinking behind Mathflow, the app built with Inria and used by more than 50,000 students: every topic comes in three levels, Explore, Explain and Solve, which ask you to do and justify rather than reread, and Alphi, the AI coach, gives hints instead of the answer. It's free to start.
Sources
- Hermann Ebbinghaus, Memory: A Contribution to Experimental Psychology (1913 English translation), 1885
- Jaap Murre and Joeri Dros, Replication and Analysis of Ebbinghaus' Forgetting Curve, PLOS ONE, 2015
- Henry Roediger and Jeffrey Karpicke, Test-Enhanced Learning: Taking Memory Tests Improves Long-Term Retention, Psychological Science, 2006
- Nicholas Cepeda, Harold Pashler, Edward Vul, John Wixted and Doug Rohrer, Distributed Practice in Verbal Recall Tasks: A Review and Quantitative Synthesis, Psychological Bulletin, 2006
- Nicholas Cepeda, Edward Vul, Doug Rohrer, John Wixted and Harold Pashler, Spacing Effects in Learning: A Temporal Ridgeline of Optimal Retention, Psychological Science, 2008
- Doug Rohrer and Kelli Taylor, The shuffling of mathematics problems improves learning, Instructional Science, 2007
- John Dunlosky, Katherine Rawson, Elizabeth Marsh, Mitchell Nathan and Daniel Willingham, Improving Students' Learning With Effective Learning Techniques, Psychological Science in the Public Interest, 2013
- Jeffrey Karpicke, Andrew Butler and Henry Roediger, Metacognitive strategies in student learning: Do students practise retrieval when they study on their own?, Memory, 2009
