The evidence

What the research says, and what we haven’t measured

Fact fluency means answering a basic fact such as 7 × 8 accurately and without effort, from memory instead of by working it out. The Common Core standards ask children to know their single-digit products from memory by the end of Grade 3 (3.OA.C.7), and a national expert panel concluded that whole-number arithmetic rests on automatic recall of multiplication and division facts (National Mathematics Advisory Panel, 2008).

Times Machine’s practice loop combines three methods that each have published research behind them: spacing practice out, timing every answer, and retyping a missed answer from memory. We have not run our own study of Times Machine. So this page sorts every choice we made into what research supports and what is our own bet, shows where researchers disagree, and lists what we don’t know yet.

Written by David Monson. Independent educator review pending. Last updated 22 October 2026.

Three methods, and the research behind each

For each method: what the research found, then what Times Machine does about it. The second part is a description of our design, not a result.

Spaced repetition: coming back after a gap

What the research found

A review of ten study techniques rated distributed practice, which means spreading practice out over time, as high utility, along with practice testing (Dunlosky et al., 2013). A meta-analysis pooling 317 experiments found that the gap giving the best retention grows with how long you need to remember (Cepeda et al., 2006), and a later experiment with more than 1,350 people found that a gap that is too long starts to hurt (Cepeda et al., 2008).

For multiplication facts, 44 third graders retained facts almost twice as well when tests came at widening gaps as when they practiced in one block (Rea & Modigliani, 1985). And memory does fade: in 2015 a single participant repeated Ebbinghaus’s experiment with nonsense syllables and got a curve that closely resembled his original (Murre & Dros, 2015).

In Times Machine

Smart Practice picks the next fact at random, weighted: facts you just missed come up most often, and facts you’ve mastered come up rarely. A mastered fact stops coming up while other facts still need work, and once it has gone more than 7 days without practice it comes back for a check. 7 days is our choice, not a research finding.

Times Machine does not run FSRS, SM-2 or any other per-fact memory model. SM-2 keeps a separate easiness factor for each item (SuperMemo SM-2), and FSRS fits its schedule to a learner’s review history (FSRS). Times Machine uses neither, only the weighted draw above.

Timed retrieval: answering from memory, against a clock

What the research found

Retrieval practice means answering from memory instead of re-reading or being shown the answer. In two experiments with prose passages, students who were tested remembered more two days and a week later than students who re-studied, although re-studying did better on a test five minutes later (Roediger & Karpicke, 2006).

Timing adds speed to accuracy. The National Mathematics Advisory Panel describes automaticity as fast, accurate and effortless processing that frees up working memory for the harder parts of a problem (National Mathematics Advisory Panel, 2008). A U.S. Department of Education practice guide rated “strong evidence” for regularly including timed activities as one way to build fluency, based on 27 studies. The guide is written for students who struggle with math; it advises one to five minutes inside a larger program and does not recommend timed worksheets alone (What Works Clearinghouse practice guide, 2021).

In a randomized study of at-risk first graders, speeded flashcard practice produced larger gains in simple arithmetic than practice without speed (Fuchs et al., 2013). And in one study of 212 seventh graders in China, speed on reading and arithmetic tasks, but not general reaction speed, predicted math results at the end of Grade 7 and on the Grade 9 entrance exam (Cheng et al., 2021). That last result is an association, not proof that practicing speed raises later scores.

In Times Machine

Every question is timed, one fact at a time. A correct answer within 4 seconds counts as Mastered, and the timer turns yellow at that line. A correct answer after it counts as Practiced. If the timer runs out, at 6.5 seconds, the answer counts as a miss. The line is the same at every age. The numbers are a design bet: 4 seconds is our choice, not a research finding.

Cover-Copy-Compare: retype it from memory

What the research found

Cover-Copy-Compare is a classroom routine: a student looks at a correct answer, covers it, writes it from memory, then checks it against the original (Skinner et al., 1997). A 1989 paper in School Psychology Review presented it as a method for increasing multiplication performance (Skinner et al., 1989). A 2012 meta-analysis found that it and its variations helped students learn skills and build fluency, with the largest benefits when combined with other evidence-based instruction (Joseph et al., 2012), and a 2025 study of 57 third graders found that three versions each raised multiplication fact fluency above control problems (Dawes et al., 2025).

In Times Machine

In Smart Practice and level drills, when you miss or the timer runs out, the correct answer is shown, and then the same fact comes back and you type it from memory, with no clock. Boss fights work differently: the answer flashes and the fight moves on. Our retype step is a one-step version of the routine, not the protocol those studies tested, and we don’t know whether it has the same effect.

The evidence ledger

Each row is a claim Times Machine leans on: how strong the support is, where it comes from, and what we do with it. Where we chose something with no study behind it, the row says “Design bet”, even when a study points the same way.

ResearchA published study, or a review of studies.
ReferenceA standard or official document. It states an expectation; it is not a study.
Design betOur own choice. Nothing was measured to pick it.
ContestedResearchers disagree about it.
Our dataResults from Times Machine players. There are none yet.

Evidence ledger: each claim, how strong its support is, its source, and what Times Machine does with it
Claim	Strength	Source	What we do with it
Children are expected to know single-digit products from memory by the end of Grade 3.	Reference	Common Core 3.OA.C.7 Scope. A standard states an expectation. It is not a study.	We aim for recall, not correct-but-slow answers. The tables run to 15 × 15, and squares and cubes to 20. See the standards, state by state
Whole-number arithmetic rests on automatic recall of multiplication and division facts.	Research	National Mathematics Advisory Panel, 2008 Scope. A federal expert panel's review of the research (2008). A synthesis, not an experiment.	This is why the goal is recall rather than working it out each time.
Facts with 4s to 7s take more repetitions to learn than facts with 2s and 3s.	Research	Burns et al., 2015 Scope. 15,402 students in grades 3 to 5, from a national database.	Smart Practice does not weight by table. It weights by each learner's own answers, so a harder table gets more practice only if that learner is slower on it.
The best gap between practice sessions grows with how long you need to remember, and a gap that is too long starts to hurt.	Research	Cepeda et al., 2006 Cepeda et al., 2008 Scope. Laboratory studies of recall, and a meta-analysis of them. Not specific to multiplication facts.	Facts do come back after a gap, but our gap is one fixed number (see the next rows), which simplifies what these studies describe.
Multiplication facts tested at widening gaps were retained almost twice as well as facts practiced in one block.	Research	Rea & Modigliani, 1985 Scope. One experiment with 44 third graders, in 1985. Gaps between tests widened; they were not a single fixed gap.	We do not use widening gaps. We use one fixed check.
A mastered fact is checked again once it has gone more than 7 days unseen.	Design bet	None. Our choice. Scope. Our choice. It is not taken from a study of multiplication facts, and we have not tested other gaps.	The fact goes back into Smart Practice's pool and is weighted to come up soon.
Facts you just missed come back most often; facts you have mastered come back rarely.	Design bet	None. Our choice. Scope. Our weighting. We have not compared it with other schemes.	Smart Practice draws the next fact at random, weighted this way.
Practice testing and distributed practice earned high-utility ratings among ten learning techniques.	Research	Dunlosky et al., 2013 Scope. A review across many studies and subjects. Not specific to multiplication.	Smart Practice asks you for the answer, and brings facts back after a gap.
Being tested beat re-studying on tests two days and a week later, though not on a test five minutes later.	Research	Roediger & Karpicke, 2006 Scope. Prose passages in laboratory experiments, not math facts.	After a miss we show the answer once, then ask you to type it from memory.
Brief timed activities, as one part of a larger intervention, help build math fluency for students who struggle.	Research	What Works Clearinghouse practice guide, 2021 Scope. Rated strong evidence, from 27 studies. The guide is written for students who struggle with math, and advises activities of one to five minutes inside a larger program. The panel does not recommend timed worksheets alone.	We time every answer, one fact at a time, never a page of problems.
In at-risk first graders, speeded flashcard practice produced larger gains in simple arithmetic than practice without speed.	Research	Fuchs et al., 2013 Scope. Randomized, 16 weeks of tutoring, about 190 children per practice condition, one school district. The two conditions differed in more than speed, so speed alone cannot be isolated. A different age group and a different setting from ours.	This is the closest research to our timer, and it is not about our game.
Speed on reading and arithmetic tasks, but not general reaction speed, predicted later math results in 212 seventh graders in China.	Research	Cheng et al., 2021 Scope. One school and one test per skill. An association, not proof that practicing speed raises later scores. The authors call for research on whether fluency training helps.	We time answers to separate recall from working it out. We make no claim that speed practice here raises test scores.
4 seconds is the fast line, and 6.5 seconds is the limit.	Design bet	Standards and Testing Agency, 2024 Scope. Our choice. For scale, England's national multiplication check allows 6 seconds per question, a limit its authors set after a trial with over 1,000 children across 50 schools. That is a different test, and it does not validate our line.	A correct answer within 4 seconds is Mastered. Correct but slower is Practiced. Running out of time at 6.5 seconds is a miss.
One line for every age.	Design bet	None. Our choice. Scope. Chosen so a grade means the same thing to a first grader and to a parent. We have not tested whether the same line suits every age equally.	The clock stays the same; what changes with the learner is which facts come up next.
Cover-Copy-Compare (look at the correct answer, cover it, write it from memory, check it) helps students learn skills and build fluency.	Research	Joseph et al., 2012 Skinner et al., 1989 Dawes et al., 2025 Scope. A meta-analysis of studies of spelling and math, with the largest benefits when combined with other instruction; and a 2025 study of 57 third graders on multiplication facts.	Our retype step borrows from it (next row).
A miss is followed by a retype from memory, with no clock.	Design bet	None. Our choice. Scope. An adaptation of the routine, not the protocol those studies tested. We do not know whether it has the same effect.	In Smart Practice and level drills, a miss shows the answer once and then asks for it again. In boss fights, the answer flashes and the fight moves on.
Whether timed practice causes math anxiety.	Contested	Boaler, 2015 Maki et al., 2024 Tsui & Mazzocco, 2007 Barshay, 2023 Scope. Researchers disagree, and the direct tests we could find are few and small. See the section on timed practice and anxiety.	We claim nothing either way. We describe what we chose and what we do not claim.
Times Machine improves multiplication fluency or test scores.	Our data	None yet Scope. No study yet.	We make no such claim. See what we do not know yet.

Timed practice and anxiety

Researchers disagree about whether timed practice causes math anxiety. Here is each side, then what we chose.

The concern

Jo Boaler’s paper “Fluency Without Fear” argues that timed tests are a major cause of math anxiety and that anxiety can block working memory under pressure. It attributes the figure that, for about a third of students, timed tests are where math anxiety begins to her 2014 article in Teaching Children Mathematics (Boaler, 2014). She has said some students are fine with timed quizzes (Barshay, 2023).

The response

According to The Hechinger Report, a group of researchers called Science of Math argues that the idea that timed tests cause math anxiety is a myth, and says it found only two experiments that tested it, neither of which concluded that timed tests produce anxiety. Boaler replied that the field is contested and the claim cannot be called a myth. A member of the group said its work was about whether timing causes anxiety, not about how timing affects performance (Barshay, 2023).

What direct tests found

The direct tests we could find are few and small. In a 2024 study, 113 fourth and fifth graders did simple and complex problems under overt and covert timing, and timing made no significant difference to the anxiety they reported. Students with medium to high baseline math anxiety were more likely to report more anxiety on complex problems timed covertly (Maki et al., 2024). An earlier study of mathematically gifted sixth graders found lower scores on a timed test than on an untimed one only when the timed test came first, with a smaller gap for children with higher math anxiety; its authors call it preliminary (Tsui & Mazzocco, 2007).

What the timing research does and doesn’t say

The evidence above that timed practice helps build fluency is not evidence about anxiety, and the studies of anxiety are not studies of fluency. None of them tested a game like Times Machine. The practice guide’s panel, which backs brief timed activities, also advises that for students who feel anxious, a large list of problems presented all at once is the thing to avoid (What Works Clearinghouse practice guide, 2021).

What we chose, and what we don’t claim

One fact at a time, never a page of problems.
The timer turns yellow at the 4-second line. A slow but correct answer counts as Practiced, not as wrong. Only running out of time, at 6.5 seconds, counts as a miss.
After a miss, the retype has no clock.
One line for every age, so a grade means the same thing to a first grader and to a parent.
Boss fights are the high-pressure part. You win a fight only by answering every question correctly, and a miss flashes the answer and moves on. We have not measured how that feels to children.

These are design choices. We are not claiming they prevent or reduce math anxiety, and we haven’t measured whether they do.

What we don’t know yet

We have not run our own study of Times Machine, and we don’t know of research on this exact combination of methods. So we are not putting a number on how much faster anyone gets, or on how long it lasts. These are the open questions and how we would test each. Before we look at any results, we will publish the question, the measure and the comparison on this page.

Does practicing with Times Machine make children faster and more accurate at multiplication facts, and does it last? We would compare children who use it with similar children who don’t, on an independent fact check that is not part of our game, with the outcome chosen in advance.
Is 7 days the right gap for re-checking a mastered fact? We would give comparable learners different fixed gaps and compare how many of those facts they still know afterwards.
Is 4 seconds the right line, at every age? We would compare answer times at each age with whether the fact is still known weeks later, and try other lines.
Does the timer change how children feel about practice? We would ask children how they feel before and after practice, and compare timed and untimed versions.
Does retyping a miss help more than only seeing the answer? We would give some learners the retype step and others the answer alone, chosen at random, then compare what they remember.

Any study that uses children’s play data will need a parent’s consent first.

Glossary

Fact fluency: Answering a basic fact accurately and with ease, from memory. The What Works Clearinghouse defines fluency as the ability to perform mathematics accurately and with ease (What Works Clearinghouse practice guide, 2021).
Automaticity: Producing answers quickly without much mental energy (What Works Clearinghouse practice guide, 2021).
Spaced repetition: Practicing something again after a gap, not all at once. Also called distributed practice.
Retrieval practice: Answering from memory instead of re-reading or being shown the answer. Also called practice testing.
Cover-Copy-Compare: Look at the correct answer, cover it, write it from memory, then check it against the original.
Smart Practice: The Times Machine mode that picks each next fact for you. Facts you just missed come up most, and facts you’ve mastered come up rarely.
Mastered, Practiced, Missed: Our three grades for an answer. Mastered: correct within 4 seconds. Practiced: correct, but slower. Missed: wrong, or the timer ran out at 6.5 seconds. A fact counts as mastered when its latest answer was Mastered and none of its last three answers was a miss.
Fast line: The 4-second mark, where the timer turns yellow.
Per-fact memory model: A system that estimates how well you know each item and schedules it from that estimate, such as SM-2 and FSRS. Times Machine doesn’t use one.
Design bet: A choice we made without a study to back it. The evidence ledger marks each one.

References

Barshay, J. (2023). Proof Points: Do math drills help children learn? The Hechinger Report, May 30, 2023. (news report)
Boaler, J. (2014). Research suggests that timed tests cause math anxiety. Teaching Children Mathematics, 20(8), 469–474.
Boaler, J. (with Williams, C., & Confer, A.) (2015). Fluency without fear: Research evidence on the best ways to learn math facts. youcubed at Stanford University (last updated January 28, 2015).
Burns, M. K., Ysseldyke, J. E., Nelson, P. M., & Kanive, R. (2015). Number of repetitions required to retain single-digit multiplication math facts for elementary students. School Psychology Quarterly, 30(3), 398–405.
Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psychological Bulletin, 132(3), 354–380.
Cepeda, N. J., Vul, E., Rohrer, D., Wixted, J. T., & Pashler, H. (2008). Spacing effects in learning: A temporal ridgeline of optimal retention. Psychological Science, 19(11), 1095–1102.
Cheng, D., Shi, K., Wang, N., Miao, X., & Zhou, X. (2021). Examining the differential role of general and specific processing speed in predicting mathematical achievement in junior high school. Journal of Intelligence, 10(1), 1.
Common Core State Standards Initiative (n.d.). Grade 3, Operations & Algebraic Thinking, standard 3.OA.C.7. thecorestandards.org.
Dawes, J., Poncy, B. C., Solomon, B. G., Duhon, G. J., & Skinner, C. H. (2025). Evaluating and comparing three variations of cover, copy, and compare on multiplication fact fluency. School Psychology.
Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students' learning with effective learning techniques: Promising directions from cognitive and educational psychology. Psychological Science in the Public Interest, 14(1), 4–58.
Fuchs, L. S. (chair), Bucka, N., Clarke, B., Dougherty, B., Jordan, N. C., Karp, K. S., & Woodward, J. (2021). Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades (WWC 2021006). U.S. Department of Education, Institute of Education Sciences, What Works Clearinghouse.
Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D., et al. (2013). Effects of first-grade number knowledge tutoring with contrasting forms of practice. Journal of Educational Psychology, 105(1), 58–77.
Joseph, L. M., Konrad, M., Cates, G., Vajcner, T., Eveleigh, E., & Fishley, K. M. (2012). A meta-analytic review of the cover-copy-compare and variations of this self-management procedure. Psychology in the Schools, 49(2), 122–136.
Maki, K. E., Zaslofsky, A. F., Codding, R., & Woods, B. (2024). Math anxiety in elementary students: Examining the role of timing and task complexity. Journal of School Psychology, 106, 101316.
Murre, J. M. J., & Dros, J. (2015). Replication and analysis of Ebbinghaus' forgetting curve. PLOS ONE, 10(7), e0120644.
National Mathematics Advisory Panel (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education.
Open Spaced Repetition (n.d.). FSRS (Free Spaced Repetition Scheduler) for Anki. github.com/open-spaced-repetition/fsrs4anki.
Rea, C. P., & Modigliani, V. (1985). The effect of expanded versus massed practice on the retention of multiplication facts and spelling lists. Human Learning, 4, 11–18 (scanned copy of the article).
Roediger, H. L., & Karpicke, J. D. (2006). Test-enhanced learning: Taking memory tests improves long-term retention. Psychological Science, 17(3), 249–255.
Skinner, C. H., et al. (1989). Cover, copy, and compare: A method for increasing multiplication performance. School Psychology Review, 18(3), 412–420.
Skinner, C. H., McLaughlin, T. F., & Logan, P. (1997). Cover, copy, and compare: A self-managed academic intervention effective across skills, students, and settings. Journal of Behavioral Education, 7(3), 295–306.
Standards and Testing Agency (2024). Multiplication tables check: validity framework. GOV.UK (last updated 3 October 2024).
Tsui, J. M., & Mazzocco, M. M. M. (2007). Effects of math anxiety and perfectionism on timed versus untimed math testing in mathematically gifted sixth graders. Roeper Review, 29(2), 132–139.
Wozniak, P. (n.d.). SuperMemo algorithm SM-2. super-memory.com.

How we checked: we opened each source and compared the sentence we rely on with what it says. Two could be confirmed only as publication records (Skinner et al., 1989, and Boaler, 2014), so we say no more about them than their titles do. The account of the Science of Math group comes from a news report, because we could not open their paper.