What the evidence says about learning math(s)

Rohrer, D., Dedrick, R. F., Hartwig, M. K., & Cheung, C.-N. (2020). A randomized controlled trial of interleaved mathematics practice. Journal of Educational Psychology, 112(1), 40–52. doi:10.1037/edu0000367

Preregistered cluster RCT (classes randomised). Who: 787 students analysed (856 enrolled), 54 classes, 15 teachers, 5 middle schools, grade 7, one Florida school district, USA.

Finding. Four months of mostly interleaved (mixed problem types) or mostly blocked assignments, then an interleaved review for all. On an unannounced test one month later the interleaved group scored 61% versus 38%, d = 0.83. Positive for every one of the 15 teachers (ds 0.23 to 1.48).

One district; researcher-written test; one-month delay; teachers knew their condition.

Rohrer, D., Dedrick, R. F., & Stershic, S. (2015). Interleaved practice improves mathematics learning. Journal of Educational Psychology, 107(3), 900–908. doi:10.1037/edu0000001

Classroom experiment, within-subjects (each student learned some skills by interleaved and some by blocked practice). Who: n = 126, grade 7, Hillsborough County Public Schools, Florida, USA.

Finding. Same problems, different arrangement, over three months, then a review and an unannounced test 1 or 30 days later. Interleaved practice beat blocked practice on both tests: d = 0.42 at one day and d = 0.79 at 30 days.

Single school; researcher-designed materials and test; the benefit grew with delay, so short-delay studies will understate it.

Rohrer, D., Dedrick, R. F., & Burgess, K. (2014). The benefit of interleaved mathematics practice is not limited to superficially similar kinds of problems. Psychonomic Bulletin & Review, 21(5), 1323–1330. doi:10.3758/s13423-014-0588-3

Classroom experiment (interleaved versus blocked practice). Who: n = 140, grade 7, USA (school setting).

Finding. Nine weeks of blocked or interleaved practice, then an unannounced test two weeks later. Interleaved 72% versus blocked 38%, d = 1.05. Earlier studies used near-identical problem types; here the types looked different from one another, and the benefit still appeared.

Two-week delay only; researcher-written test; same research group as the other interleaving studies.

Emeny, W. G., Hartwig, M. K., & Rohrer, D. (2021). Spaced mathematics practice improves test scores and reduces overconfidence. Applied Cognitive Psychology, 35(4), 1082–1089. doi:10.1002/acp.3814

Two within-subjects classroom experiments (each student massed one topic and spaced the other). Who: Experiment 1 n = 44, Experiment 2 n = 55; Year 7 (ages 11–12); one large school in the United Kingdom.

Finding. Twelve practice problems in one session or spread over three weekly sessions; test one month later. Spacing raised scores: d = 0.61 (95% CI 0.28–0.92) and d = 0.39 (0.11–0.66). Students' predicted scores were roughly accurate after spaced practice but badly overconfident after massed practice.

Small samples, one school, two non-curriculum topics (Venn diagrams, permutations).

Barzagar Nazari, K., & Ebersbach, M. (2019). Distributing mathematical practice of third and seventh graders: Applicability of the spacing effect in the classroom. Applied Cognitive Psychology, 33(2), 288–298. doi:10.1002/acp.3485

Classroom experiment, massed versus distributed practice, Bayesian analysis. Who: N = 213, grades 3 and 7, school classrooms (authors based in Germany).

Finding. After a curriculum lesson, three practice sets either all in one day or one per day over three days. Tests one and six weeks later: grade 7 did better after distributed practice at both delays; grade 3 only at one week.

Effect sizes not given in the abstract; the benefit for younger children was less durable.

Rohrer, D., & Taylor, K. (2006). The effects of overlearning and distributed practise on the retention of mathematics knowledge. Applied Cognitive Psychology, 20, 1209–1224. doi:10.1002/acp.1266

Two randomised lab experiments. Who: n = 216 college students (across both experiments), USA.

Finding. Students learned a math(s) procedure and were tested one or four weeks later. Splitting ten practice problems across two sessions instead of one roughly doubled scores on the four-week test. Doing extra problems in the same session ("overlearning") had a negligible effect on retention.

Lab study with undergraduates and a single procedure; exact scores not given in the abstract in the record.

Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35(6), 481–498. doi:10.1007/s11251-007-9015-8

Two randomised lab experiments. Who: Experiment 1 n = 66, Experiment 2 n = 18; undergraduates, University of South Florida, USA.

Finding. Experiment 1 (permutations): spaced practice 74% versus massed 49% on a test one week later. Experiment 2 (volumes of four solids): mixed practice 63% versus blocked 20%, d = 1.34, even though the blocked group did better during practice (89% versus 60%). Practice that feels easier produced worse test results.

Experiment 2 had only 18 students; one-week delay; lab setting.

Hopkins, R. F., Lyle, K. B., Hieb, J. L., & Ralston, P. A. S. (2016). Spaced retrieval practice increases college students' short- and long-term retention of mathematics knowledge. Educational Psychology Review, 28(4), 853–873. doi:10.1007/s10648-015-9349-8

Classroom experiment; spaced versus massed quiz questions manipulated both within and between students. Who: Engineering students in a precalculus course (n not given in the abstract), USA; a subset followed into calculus the next semester.

Finding. Content quizzed at spaced intervals through the semester was retained better than content quizzed only soon after the lesson. Students given some spaced retrieval practice did better on the precalculus final and on exams in the following calculus course than those whose practice was all massed.

Effect sizes not given in the abstract. Follow-up (Lyle et al., 2020, same journal, 32(1), 277–295): across-semester retention improved only by spacing, not by extra practice.

Yeo, D. J., & Fazio, L. K. (2019). The optimal learning strategy depends on learning goals and processes: Retrieval practice versus worked examples. Journal of Educational Psychology, 111(1), 73–90. doi:10.1037/edu0000268

Three randomised lab experiments. Who: n not given in the abstract; students learning a novel math(s) procedure, USA.

Finding. For a new math(s) procedure, repeatedly studying worked examples beat repeated testing on an immediate test. After one week, testing matched studying when practice problems varied, and beat it when they were identical. For memorising the text of an example, testing won.

A boundary condition on the testing effect; the advantage of retrieval depends on delay and material. Lab study.

Ebersbach, M., & Barzagar Nazari, K. (2020). No robust effect of distributed practice on the short- and long-term retention of mathematical procedures. Frontiers in Psychology, 11, 811. doi:10.3389/fpsyg.2020.00811

Randomised experiment, 3 (gap between practice sessions: 0, 1 or 11 days) x 2 (test after 1 or 5 weeks), between subjects. Who: n = 235 university students (mean age about 24), Germany.

Finding. Students learned permutation problems. No main effect of spacing and no interaction with retention interval; scores fell between one and five weeks regardless of schedule.

Conflicts with the same authors' school result and with Rohrer and Taylor (2006) on the same topic; authors suggest spacing matters less for procedures than for facts.

Brunmair, M., & Richter, T. (2019). Similarity matters: A meta-analysis of interleaved learning and its moderators. Psychological Bulletin, 145(11), 1029–1052. doi:10.1037/bul0000209

Multilevel meta-analysis of 59 studies, 238 effect sizes, 158 samples. Who: Mixed lab and classroom samples; all ages.

Finding. Overall interleaving effect g = 0.42 versus blocked practice. For mathematical tasks the effect was small, g = 0.34; for paintings g = 0.67; for expository texts no reliable effect; for word lists blocking was better (g = -0.39). Interleaving helped most when categories were similar to each other and material was complex.

Many included studies are lab-based with short delays; the math(s) estimate pools varied tasks.

Yang, C., Luo, L., Vadillo, M. A., Yu, R., & Shanks, D. R. (2021). Testing (quizzing) boosts classroom learning: A systematic and meta-analytic review. Psychological Bulletin, 147(4), 399–435. doi:10.1037/bul0000309

Meta-analysis of 222 independent classroom studies. Who: 48,478 students, elementary school to university, many countries.

Finding. Classroom quizzing raised achievement by g = 0.499 overall. For mathematics/statistics (27 effects) g = 0.433, 95% CI 0.193–0.674. Subject area did not significantly moderate the effect (p = .69). Effects were larger with corrective feedback and repeated quizzes.

Classroom studies vary in rigour; the math(s) estimate has a wide interval; control conditions differ across studies.

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. doi:10.1177/1529100612453266

Narrative evidence review (no quantitative pooling) of ten learning techniques. Who: Learners of all ages, lab and classroom studies.

Finding. Practice testing and distributed practice were rated high utility. Interleaved practice, elaborative interrogation and self-explanation were rated moderate. Summarising, highlighting, keyword mnemonics, imagery for text and rereading were rated low.

Expert ratings, not effect sizes; the interleaving rating predates most of the math(s) classroom RCTs above.

Pashler, H., Bain, P. M., Bottge, B. A., Graesser, A., Koedinger, K., McDaniel, M., & Metcalfe, J. (2007). Organizing instruction and study to improve student learning. IES practice guide. Washington, DC: National Center for Education Research, Institute of Education Sciences, U.S. Department of Education. https://ies.ed.gov/ncee/wwc/PracticeGuide/1

Government evidence synthesis with graded recommendations. Who: K–12 and college learners; not math(s)-specific.

Finding. Evidence ratings: "space learning over time" moderate; "interleave worked example solutions with problem-solving exercises" moderate; "use quizzes to re-expose students to key content" strong; "ask deep explanatory questions" strong; quizzes to identify gaps, and pre-questions, minimal.

Written in 2007, before the classroom interleaving trials; panel judgement, not pooled effect sizes.

Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morphy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79(3), 1202–1242. doi:10.3102/0034654309334431

Meta-analysis of 42 intervention studies (RCTs and high-quality quasi-experiments). Who: School-age students with learning disabilities; mostly US; total n not given in the abstract.

Finding. Most instructional components had significant positive mean effects, ranging from 0.21 to 1.56. Explicit, step-by-step teacher modelling (g = 1.22, 11 studies) and teaching heuristic problem-solving strategies (g = 1.56, 4 studies) were the two components that mattered most in the regression.

Population is students with LD; component-level effect sizes confirmed only from a secondary summary (Bryant, 2009), the abstract gives the 0.21–1.56 range.

Slavin, R. E., & Lake, C. (2008). Effective programs in elementary mathematics: A best-evidence synthesis. Review of Educational Research, 78(3), 427–515. Slavin, R. E., Lake, C., & Groff, C. (2009). Effective programs in middle and high school mathematics: A best-evidence synthesis. Review of Educational Research, 79(2), 839–911. doi:10.3102/0034654308317473 · doi:10.3102/0034654308330968

Best-evidence syntheses; 87 studies (36 randomised) for elementary, 100 studies (26 randomised) for grades 6–12; minimum 12-week duration, independent outcome measures. Who: K–12 students, mainly US.

Finding. Elementary: median ES +0.10 for textbooks/curricula (13 studies), +0.19 for computer-assisted instruction (38), +0.33 for instructional-process programmes such as cooperative learning and classroom management (36). Secondary (report version): weighted mean ES +0.03 curricula, +0.10 CAI, +0.18 instructional process, +0.42 for cooperative learning (9 studies).

Many small or matched (non-random) studies; figures above are from the authors' full report versions, which may differ slightly from the journal text.

Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380–400. doi:10.1037/a0031084

Meta-analysis of 55 studies comparing manipulative-based instruction with abstract-symbol-only instruction. Who: N = 7,237 students, kindergarten to college.

Finding. Aggregated mean effect d = 0.37 (95% CI 0.30–0.44) in favour of manipulatives. Larger for retention (k = 53) than for problem solving, transfer or justification (small effects). Effects were larger with high instructional guidance (d = 0.46 vs 0.29), for fractions (0.69) than arithmetic (0.27) or algebra (0.21), and in shorter interventions.

Heterogeneous; within-subject and published studies gave much larger effects than randomised ones; manipulatives work best when the teacher guides their use, not as free play.

Stockard, J., Wood, T. W., Coughlin, C., & Rasplica Khoury, C. (2018). The effectiveness of Direct Instruction curricula: A meta-analysis of a half century of research. Review of Educational Research, 88(4), 479–507. doi:10.3102/0034654317751919

Meta-analysis (mixed models) of 328 studies, 413 designs, 3,999 effects, 1966–2016. Who: Mostly US school students; math(s) subset 70 studies, 685 effects.

Finding. Overall effect 0.54 (95% CI 0.49–0.59); for mathematics 0.55 (0.46–0.65), against comparison instruction. Effects were larger with more years of exposure and 60+ minutes daily, and declined little after the programme ended.

Covers scripted DI curricula (e.g. Connecting Math Concepts), not "explicit teaching" generally; many older quasi-experimental studies; authors associated with the DI research community.

Fuchs, L. S., 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). Institute of Education Sciences, What Works Clearinghouse. https://ies.ed.gov/ncee/wwc/PracticeGuide/26

Government evidence synthesis (WWC practice guide) with graded recommendations. Who: Elementary (K–6) students needing intervention; US studies.

Finding. Six recommendations, all rated strong evidence: systematic (explicit, sequenced) instruction; precise mathematical language; concrete and semi-concrete representations; number lines; targeted word-problem instruction; regular timed fluency activities.

Aimed at intervention for struggling pupils, not whole-class teaching of advanced children; effect sizes are in the underlying study tables, not in the summary.

Booth, J. L., Oyer, M. H., Paré-Blagoev, E. J., Elliot, A. J., Barbieri, C., Augustine, A., & Koedinger, K. R. (2015). Learning algebra by example in real-world classrooms. Journal of Research on Educational Effectiveness, 8(4), 530–551. See also Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra. Learning and Instruction, 25, 24–34. doi:10.1080/19345747.2015.1055636

Two RCTs (students randomised within classrooms); 2013 paper adds two smaller in-vivo experiments. Who: Exp. 1 n = 56 Algebra I students, 3 classrooms; Exp. 2 n = 425 students, 28 classes, 5 US districts.

Finding. Replacing some practice problems with correct and incorrect worked examples to explain raised posttest scores in Exp. 1 (ηp² = .10, p = .03). In Exp. 2 there was no overall main effect, but a significant condition × prior-knowledge interaction: lower-prior-knowledge students gained more. The 2013 study (n = 134) found higher conceptual scores (63% vs 58%) with fewer practice problems.

Small effects; the larger trial's benefit was confined to weaker students.

Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574. doi:10.1037/0022-0663.99.3.561

Randomised experiment, classroom-based. Who: n = 70 seventh-graders; country not given in the abstract (US authors).

Finding. Students who studied worked examples of two different methods side by side and compared them made greater gains in procedural knowledge and flexibility than students who studied the same examples one at a time; conceptual gains were comparable.

Short intervention, single topic (linear equations); effect sizes not given in the abstract in the record; later studies by the same group found comparison only helps students who already know one method.

Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89. Replicated in class by Carroll, W. M. (1994). Using worked examples as an instructional support in the algebra classroom. Journal of Educational Psychology, 86(3), 360–367. doi:10.1207/s1532690xci0201_3 · doi:10.1037/0022-0663.86.3.360

Five lab experiments (Sweller & Cooper); two classroom experiments (Carroll). Who: Year 9, Year 11 and university math(s) students; 74 US high-school students including a remedial class (Carroll).

Finding. Studying worked examples took far less time than solving equivalent problems, and students then solved similar problems faster with fewer errors. Carroll: worked-example students outperformed controls on posttests after fewer practice problems, with fewer errors and less teacher help.

Gains were specific to problems with the same structure; small samples; effect sizes not given in the abstract.

Fyfe, E. R., McNeil, N. M., & Borjas, S. (2015). Benefits of "concreteness fading" for children's mathematics understanding. Learning and Instruction, 35, 104–120. Background review: Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26(1), 9–25. doi:10.1016/j.learninstruc.2014.10.004

Randomised experiments (2015); narrative systematic review without pooling (2014). Who: Elementary-school children learning mathematical equivalence (e.g. 3 + 4 + 5 = 3 + __); n not given in the abstracts fetched.

Finding. Children taught with a progression from puppets and objects, to pictures, to symbols solved more transfer problems than children given the same materials in reverse order or concrete-only/abstract-only teaching.

Full abstract and effect sizes could not be retrieved (publisher blocked); study details confirmed from the first author's later open-access paper. Short lab-style sessions.

Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenenbaum, H. R. (2011). Does discovery-based instruction enhance learning? Journal of Educational Psychology, 103(1), 1–18. doi:10.1037/a0021017

Two meta-analyses, 164 studies in total. Who: Learners of all ages across subjects (math(s), science, problem solving); total n not given in the abstract.

Finding. Explicit instruction beat unassisted discovery (d = −0.38 for discovery). But "enhanced" discovery with feedback, worked examples, scaffolding or prompts to explain beat other kinds of instruction (d = 0.30).

Not math(s)-specific; wide variety of tasks and outcome measures; "discovery" defined broadly.

Sinha, T., & Kapur, M. (2021). When problem solving followed by instruction works: Evidence for productive failure. Review of Educational Research, 91(5), 761–798. doi:10.3102/00346543211019105

Meta-analysis of 53 studies, 166 comparisons. Who: School and university learners, largely math(s) and science topics; total n not given in the abstract.

Finding. Problem solving before instruction beat instruction before problem solving, g = 0.36 (95% CI 0.20–0.51); g = 0.37–0.58 when implemented faithfully to Productive Failure design. The effect reversed for grades 2–5 and for domain-general skills, where teach-first was better.

Second author is the originator of Productive Failure; the publication-bias-adjusted estimate (g = 0.87) runs the unusual direction; few young-child studies.

Bisra, K., Liu, Q., Nesbit, J. C., Salimi, F., & Winne, P. H. (2018). Inducing self-explanation: A meta-analysis. Educational Psychology Review, 30(3), 703–725. doi:10.1007/s10648-018-9434-x

Meta-analysis of 64 studies, 69 effect sizes. Who: Learners across ages and subjects, including many math(s) tasks; total n not given in the abstract.

Finding. Asking learners to explain material or worked steps to themselves improved learning, random-effects g = 0.55 versus no prompts; the effect held across 20 moderators.

Not math(s)-specific; mostly short lab or computer-based studies.

Kulik, C.-L. C., Kulik, J. A., & Bangert-Drowns, R. L. (1990). Effectiveness of mastery learning programs: A meta-analysis. Review of Educational Research, 60(2), 265–299. doi:10.3102/00346543060002265

Meta-analysis of 108 controlled evaluations. Who: Upper-elementary, high-school and college students, various subjects; total n not given in the abstract.

Finding. Mastery programmes (learn a unit, test, re-teach until a criterion is met) raised exam performance, with larger gains for weaker students; they also improved attitudes but increased time on task, and self-paced versions lowered college completion rates.

Overall mean effect size is not given (often cited as about 0.5 SD, unverified here); Slavin's 1987 best-evidence review found much smaller effects on standardised tests.

Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., Koedinger, K. R., & Ogbuehi, P. (2012). Improving mathematical problem solving in grades 4 through 8. Institute of Education Sciences, What Works Clearinghouse. https://ies.ed.gov/ncee/wwc/PracticeGuide/16

Government evidence synthesis (WWC practice guide) with graded recommendations. Who: Students in grades 4–8; US studies.

Finding. Strong evidence: teach students to monitor and reflect on their problem solving; teach the use of visual representations. Moderate evidence: expose students to multiple strategies; help them articulate concepts and notation. Minimal evidence: preparing problems for whole-class use.

Guide ratings, not pooled effect sizes; revised 2018; publication number not given in the abstract.

Nickow, A., Oreopoulos, P., & Quan, V. (2024). The promise of tutoring for PreK–12 learning: A systematic review and meta-analysis of the experimental evidence. American Educational Research Journal, 61(1), 74–107. doi:10.3102/00028312231208687

Meta-analysis of 89 randomised controlled trials of one-to-one or small-group tutoring. Who: PreK to grade 12, mostly United States.

Finding. Pooled effect 0.288 SD versus no tutoring; math(s) 0.268 SD, literacy 0.290 SD. Largest effects with teacher or paraprofessional tutors, in earlier grades, at three or more sessions a week, during school. The 2020 working paper (NBER w27476, 96 studies) reported 0.37 SD overall, 0.38 SD for math(s); teacher tutors 0.50, paraprofessionals 0.40, volunteers 0.21.

Only about a quarter of studies were math(s); no sign of publication bias; the journal estimate is lower than the working paper's.

Guryan, J., Ludwig, J., Bhatt, M. P., Cook, P. J., Davis, J. M. V., Dodge, K., Farkas, G., Fryer, R. G., Jr., Mayer, S., Pollack, H., Steinberg, L., & Stoddard, G. (2023). Not too late: Improving academic outcomes among adolescents. American Economic Review, 113(3), 738–765. doi:10.1257/aer.20210434

Two large school-based RCTs of Saga Education's daily, in-school, two-to-one math(s) tutoring by paraprofessional tutors. Who: n = 2,633 (first trial) and 2,710 (replication); grades 9–10; Chicago public high schools.

Finding. Math(s) test scores rose 0.18 to 0.40 SD (0.16 SD first trial, 0.37 SD replication) relative to controls; math(s) and other grades also improved, with effects persisting. Cost about $3,500–$4,300 per student per year. The 106-boy pilot (Cook et al., 2014, NBER w19862) found 0.65 SD on math(s) tests.

One provider and city; the two trials gave different point estimates.

Dietrichson, J., Bøg, M., Filges, T., & Klint Jørgensen, A.-M. (2017). Academic interventions for elementary and middle school students with low socioeconomic status: A systematic review and meta-analysis. Review of Educational Research, 87(2), 243–282. doi:10.3102/0034654316687036

Systematic review and meta-analysis of 101 studies (2000–2014), mostly RCTs, with standardised math(s) and reading tests as outcomes. Who: Elementary and middle-school pupils of low socioeconomic status in OECD/EU countries.

Finding. Pooled effects by intervention component: tutoring 0.36 SD, feedback and progress monitoring 0.32 SD, cooperative learning 0.22 SD, each versus business-as-usual controls.

Large unexplained variation in effects within each component; math(s) and reading pooled together in the headline figures.

Education Endowment Foundation (2021, review last updated). One to one tuition. Teaching and Learning Toolkit. https://educationendowmentfoundation.org.uk/education-evidence/teaching-learning-toolkit/one-to-one-tuition

Government-funded evidence synthesis (meta-analysis of 123 studies meeting inclusion criteria). Who: School-age pupils, mainly UK and US studies.

Finding. "+5 months" additional progress over a year; rated "high impact for moderate cost based on moderate evidence" (moderate evidence security padlocks). Primary +6 months, secondary +4. The entry says effects in mathematics appear substantially lower (+2 months) than in literacy (+6 months). Short regular sessions (about 30 minutes, 3–5 times a week, up to ten weeks) linked to lessons are recommended.

Months-progress is a translation of effect sizes; many included studies were not independently evaluated.

Education Endowment Foundation (2021, review last updated July 2021). Small group tuition. Teaching and Learning Toolkit. https://educationendowmentfoundation.org.uk/education-evidence/teaching-learning-toolkit/small-group-tuition

Government-funded evidence synthesis (meta-analysis of 62 studies). Small-group tuition means one adult working with two to five pupils. Who: School-age pupils, mainly UK and US studies.

Finding. "+4 months" additional progress over a year; summary rating "moderate impact for low cost based on moderate evidence". Studies in mathematics show a slightly smaller impact (+3 months) than reading (+4 months). Impact is larger when targeted at diagnosed needs and when staff are trained.

Evidence security rated down because a large share of studies were not independently evaluated; months-progress is a translation of effect sizes.

Banerjee, A. V., Cole, S., Duflo, E., & Linden, L. (2007). Remedying education: Evidence from two randomized experiments in India. Quarterly Journal of Economics, 122(3), 1235–1264. doi:10.1162/qjec.122.3.1235

Two school-level RCTs: a remedial tutor (balsakhi) for lagging pupils, and computer-assisted math(s) practice. Who: 122 primary schools in Vadodara and 77 in Mumbai; grades 2–4; 2001–2003.

Finding. Balsakhi raised average test scores by 0.14 SD (year one) and 0.28 SD (year two), largest in math(s), about 0.6 SD among children actually tutored. Computer-assisted learning raised grade-4 math(s) by 0.35 SD and 0.47 SD. Gains faded but partly persisted a year after the programmes ended.

Low-resource setting; tutors were community women with two weeks' training, so transfer to rich countries is uncertain.

Muralidharan, K., Singh, A., & Ganimian, A. J. (2019). Disrupting education? Experimental evidence on technology-aided instruction in India. American Economic Review, 109(4), 1426–1460. doi:10.1257/aer.20171112

Lottery-based RCT of vouchers for a personalised, computer-adaptive after-school programme. Who: n = 619 students, mostly grades 6–9, Delhi, 2015–16.

Finding. After 4.5 months, lottery winners scored 0.37 SD higher in math(s) and 0.23 SD higher in Hindi than losers. Instrumental-variable estimates imply about 0.6 SD in math(s) for 90 days of attendance. Gains were similar in absolute terms for all pupils, so the weakest gained most relative to their starting point.

Small sample, one city, short duration; the software targeted each child's actual level rather than the grade curriculum, which may be the active ingredient.

Clements, D. H., Sarama, J., Spitler, M. E., Lange, A. A., & Wolfe, C. B. (2011). Mathematics learned by young children in an intervention based on learning trajectories: A large-scale cluster randomized trial. Journal for Research in Mathematics Education, 42(2), 127–166. doi:10.5951/jresematheduc.42.2.0127

Cluster RCT; 42 schools randomly assigned, with teacher professional development built on learning trajectories. Who: n = 1,375 preschoolers in 106 classrooms in low-resource US communities.

Finding. Building Blocks children learned more mathematics than controls, g = 0.72. The earlier 36-classroom RCT (Clements & Sarama, 2008, AERJ, 45(2), 443–494) found effect sizes of 0.47 versus a comparison curriculum and 1.07 versus business as usual.

Outcome measure developed by the same team; What Works Clearinghouse rated the 2011 study as meeting standards without reservations.

Ramani, G. B., & Siegler, R. S. (2008). Promoting broad and stable improvements in low-income children's numerical knowledge through playing number board games. Child Development, 79(2), 375–394. doi:10.1111/j.1467-8624.2007.01131.x

RCT; children randomly assigned to a 1–10 linear number board game or an identical colour game. Who: n = 136 Head Start preschoolers (mean age 5.4), 10 centres, Pittsburgh, USA.

Finding. About one hour of play (four short sessions) improved magnitude comparison, number-line estimation, counting and numeral identification versus the colour game; gains held nine weeks later. Effect sizes not given in the abstract. A follow-up (Siegler & Ramani, 2009, J. Educational Psychology, 101(3), 545ff.; n = 88) found a linear board worked but a circular one did not.

Small lab-style studies, short delays, researcher-designed measures.

Agodini, R., Harris, B., Thomas, M., Murphy, R., & Gallagher, L. (2010). Achievement effects of four early elementary school math curricula: Findings for first and second graders (NCEE 2011-4001). Institute of Education Sciences, US Department of Education. https://ies.ed.gov/ncee/pubs/20114001/

RCT; 110 schools in 12 districts randomly assigned to Investigations, Math Expressions, Saxon Math or Scott Foresman–Addison Wesley (SFAW). Who: 4,716 first-graders (109 schools) and 3,344 second-graders (71 schools), USA.

Finding. Grade 1: Math Expressions beat Investigations and SFAW by 0.11 SD (50th to 54th percentile). Grade 2: Math Expressions and Saxon beat SFAW by 0.12 and 0.17 SD. Other pairwise differences were not significant.

After adjusting for multiple comparisons only the Saxon–SFAW grade-2 difference (0.17 SD) survived; participating schools were volunteers, not a national sample.

Pane, J. F., Griffin, B. A., McCaffrey, D. F., & Karam, R. (2014). Effectiveness of Cognitive Tutor Algebra I at scale. Educational Evaluation and Policy Analysis, 36(2), 127–144. doi:10.3102/0162373713507480

Matched-pair school-level RCT over two years; blended mastery-learning software plus textbook versus existing algebra course. Who: Middle and high schools in seven US states; WWC counts 5,738 high-school students (grades 9–12) in the year-2 analysis.

Finding. No effect in year one. In year two, high schools using the programme scored about 0.2 SD higher on an algebra proficiency exam (median student up roughly eight percentile points); the middle-school estimate was similar but not significant.

The What Works Clearinghouse re-analysis found no statistically significant effects, so the result is fragile; first-year implementation problems were common.

Roschelle, J., Feng, M., Murphy, R. F., & Mason, C. A. (2016). Online mathematics homework increases student achievement. AERA Open, 2(4), article 2332858416673968. doi:10.1177/2332858416673968

School-level RCT; teachers assigned homework through ASSISTments (hints and immediate feedback for pupils, reports for teachers) plus training, versus usual homework. Who: n = 2,850 seventh-graders in 43 schools in Maine, USA, where every seventh-grader has a laptop.

Finding. Treatment pupils scored higher on the end-of-year standardised math(s) test, g = 0.18 (50th to 58th percentile), with the largest gains for pupils with low prior achievement.

One state with unusual laptop provision; the effect size appears in a companion SRI report rather than the article abstract.

Jaciw, A. P., Hegseth, W. M., Lin, L., Toby, M., Newman, D., Ma, B., & Zacamy, J. (2016). Assessing impacts of Math in Focus, a "Singapore Math" program. Journal of Research on Educational Effectiveness, 9(4), 473–502. doi:10.1080/19345747.2016.1164777

Cluster RCT; grade-level teams randomly assigned to Math in Focus or business-as-usual curriculum. Who: 12 schools, 22 grade-level teams, grades 3–5, USA; student n not given in the abstract.

Finding. Effects on mathematics achievement of 0.11 to 0.15 SD in favour of Math in Focus after one year, with no significant differences by prior achievement, minority status or grade.

Small number of clusters; one school year; modest effects relative to the programme's reputation.

Fryer, R. G., Jr. (2017). The production of human capital in developed countries: Evidence from 196 randomized field experiments. In Handbook of Economic Field Experiments (pp. 95–322). Elsevier. doi:10.1016/bs.hefe.2016.08.006

Review with fixed- and random-effects meta-analysis of 196 RCTs, grouped by intervention type. Who: Children and adolescents in developed countries, mostly USA.

Finding. High-dosage tutoring had the largest school-based meta-coefficient: 0.309 SD (SE 0.106) for math(s) and 0.229 SD for reading; 54% of tutoring estimates were significantly positive and none significantly negative. Low-dosage tutoring, teacher incentives, general professional development and vouchers averaged near zero in math(s) (0.015, 0.022, 0.019 and 0.024 SD).

Only six math(s) tutoring studies behind the headline figure; heterogeneous outcomes pooled across very different programmes.

Berkowitz, T., Schaeffer, M. W., Maloney, E. A., Peterson, L., Gregor, C., Levine, S. C., & Beilock, S. L. (2015). Math at home adds up to achievement in school. Science, 350(6257), 196–198. doi:10.1126/science.aac7427

Randomised field experiment (families assigned to a math(s)-story app or a matched reading app). Who: n = 587 first-graders and their parents, Chicago-area schools, USA; one school year.

Finding. The math(s)-app group gained more on a standardised math(s) test than reading-app controls; children who used the app often were about three months ahead by year end (figure from the university release). Gains were largest for children of math(s)-anxious parents.

Effect size in SD units not given in the abstract; the dose-response comparison is not randomised (use was self-selected); the follow-up below found no overall main effect by grade 3.

Schaeffer, M. W., Rozek, C. S., Berkowitz, T., Levine, S. C., & Beilock, S. L. (2018). Disassociating the relation between parents' math anxiety and children's math achievement: Long-term effects of a math app intervention. Journal of Experimental Psychology: General, 147(12), 1782–1790. doi:10.1037/xge0000490

Longitudinal follow-up of the 2015 RCT. Who: The same cohort (587 first-graders, about 40 Chicago-area classrooms) followed to the end of grade 3, USA.

Finding. No overall main effect of the math(s) app on math(s) achievement by grade 3. Children of highly math(s)-anxious parents in the reading-app group learned less math(s) each year than peers; in the math(s)-app group this gap was absent, even though app use had fallen to roughly monthly.

A moderation result, not an average gain; the authors state there is no main effect, and the paper has been criticised for its framing. Sample retained at follow-up not given in the abstract.

Harackiewicz, J. M., Rozek, C. S., Hulleman, C. S., & Hyde, J. S. (2012). Helping parents to motivate adolescents in mathematics and science: An experimental test of a utility-value intervention. Psychological Science, 23(8), 899–906. doi:10.1177/0956797611435530

RCT (families randomised; two mailed brochures and a website for parents). Who: n = 188 families (181 with transcripts) from the Wisconsin Study of Families and Work; teenagers in grades 10–12, USA.

Finding. Students whose parents received the materials took 8.31 semesters of math(s) and science in the last two years of high school versus 7.50 for controls, about 0.8 of a semester more.

Outcome is course-taking, not test scores; sample mostly well-educated two-parent families (parents averaged 15.4 years of schooling).

Rozek, C. S., Svoboda, R. C., Harackiewicz, J. M., Hulleman, C. S., & Hyde, J. S. (2017). Utility-value intervention with parents increases students' STEM preparation and career pursuit. Proceedings of the National Academy of Sciences, 114(5), 909–914. doi:10.1073/pnas.1607386114

Follow-up of the 2012 RCT. Who: Same Wisconsin families; outcomes measured through the end of high school and five years after the intervention, USA.

Finding. The intervention raised ACT mathematics and science scores by 12 percentile points. Higher STEM preparation in turn predicted more STEM career interest and more college STEM courses five years on.

Downstream career effects are indirect (mediation), not a direct randomised contrast; single moderate-sized sample; number analysed at follow-up not given in the abstract.

Yeager, D. S., Hanselman, P., Walton, G. M., Murray, J. S., Crosnoe, R., Muller, C., ... Dweck, C. S. (2019). A national experiment reveals where a growth mindset improves achievement. Nature, 573(7774), 364–369. doi:10.1038/s41586-019-1466-y

Pre-registered RCT, students randomised within schools; two online sessions under one hour in total. Who: n = 12,490 ninth-graders in 65 US public high schools, a nationally representative sample.

Finding. Among the pre-registered lower-achieving half (n = 6,320), core GPA rose by 0.10 grade points (95% CI 0.04–0.16; standardised effect 0.11). Across all students, enrolment in advanced math(s) (algebra II or higher) in grade 10 rose from 33% to 36% (+3 percentage points). No GPA effect for higher achievers.

Effects small and concentrated where peer norms supported effort; several authors disseminate mindset programmes (disclosed); see the two meta-analyses below.

Foliano, F., Rolfe, H., Buzzeo, J., Runge, J., & Wilkinson, D. (2019). Changing Mindsets: Effectiveness trial. Evaluation report. London: Education Endowment Foundation (evaluated by NIESR). https://niesr.ac.uk/wp-content/uploads/2021/10/Changing-Mindsets_0-4.pdf

Two-arm cluster RCT (schools randomised); eight weekly teacher-led growth-mindset lessons plus teacher training. Who: 101 schools, 5,018 Year 6 pupils (age 10–11), England.

Finding. No effect on Key Stage 2 math(s): effect size −0.01 (95% CI −0.04 to 0.01), 0 months' progress, n = 4,454. Reading and GPS also zero. No effect for pupils eligible for free school meals.

EEF gave the result a high security rating. Most control-school teachers already knew growth-mindset theory, which may dilute the contrast.

Macnamara, B. N., & Burgoyne, A. P. (2023). Do growth mindset interventions impact students' academic achievement? A systematic review and meta-analysis with recommendations for best practices. Psychological Bulletin, 149(3–4), 133–173. doi:10.1037/bul0000352

Systematic review with three meta-analyses, plus study-quality coding. Who: 63 studies, N = 97,672 students, all ages; mostly USA.

Finding. Overall d = 0.05 (95% CI 0.02–0.09), not significant after publication-bias correction. Studies showing the intervention actually changed mindsets: d = 0.04 (CI −0.01 to 0.10). Highest-quality six studies (N = 13,571): d = 0.02 (CI −0.06 to 0.10). Authors with a financial interest reported larger effects.

Outcomes are general achievement, not math(s)-specific; the authors' conclusions are contested by mindset researchers (published replies exist).

Sisk, V. F., Burgoyne, A. P., Sun, J., Butler, J. L., & Macnamara, B. N. (2018). To what extent and under which circumstances are growth mind-sets important to academic achievement? Two meta-analyses. Psychological Science, 29(4), 549–571. doi:10.1177/0956797617739704

Meta-analysis of correlations (k = 273) and of intervention studies (k = 43). Who: N = 365,915 (correlational) and N = 57,155 (interventions); children, adolescents and adults.

Finding. Mindset correlated weakly with achievement, r = .10 (95% CI .08–.13). Interventions: d = 0.08 (95% CI 0.02–0.14). Low-SES students (7 effect sizes) d = 0.34 (CI 0.07–0.62); middle/upper-class students d = 0.03, not significant.

Outcomes not math(s)-specific; subgroup estimates rest on few studies; large heterogeneity.

Ramirez, G., & Beilock, S. L. (2011). Writing about testing worries boosts exam performance in the classroom. Science, 331(6014), 211–213. doi:10.1126/science.1199427

Two lab experiments and two randomised classroom experiments. Who: Lab: college students doing math(s) under pressure (n = 20 and 47). Classroom: 106 ninth-graders sitting a biology final at a Midwestern US high school over two years.

Finding. Test-anxious students who wrote about their worries for 10 minutes averaged B+ on the final versus B− for anxious students who wrote about a biology topic (about 6% higher), matching low-anxiety peers. Low-anxiety students were unaffected.

The classroom outcome was biology, not math(s); math(s) was tested only in the lab with small samples; a 2014 erratum was published.

Park, D., Ramirez, G., & Beilock, S. L. (2014). The role of expressive writing in math anxiety. Journal of Experimental Psychology: Applied, 20(2), 103–111. doi:10.1037/xap0000013

Lab experiment, random assignment to writing or sitting quietly. Who: N = 80 university students varying in math(s) anxiety, USA.

Finding. In the control group, high math(s)-anxious students did significantly worse on math(s) problems than low-anxious students; after seven minutes of writing about their feelings this gap was significantly reduced. Word problems showed no such pattern.

Adults in a lab, single session, immediate test; effect size not given in the abstract.

Flore, P. C., & Wicherts, J. M. (2015). Does stereotype threat influence performance of girls in stereotyped domains? A meta-analysis. Journal of School Psychology, 53(1), 25–44. doi:10.1016/j.jsp.2014.10.002

Meta-analysis of 47 effect sizes from (quasi-)experiments. Who: Girls under 18 taking math(s), science or spatial tests; multiple countries.

Finding. Mean effect of a threat manipulation on girls' scores d = −0.22, significantly below zero. No moderator (test difficulty, presence of boys, national gender equality, control type) was significant.

Several indicators of publication bias; the authors warn the literature may be seriously distorted and call for a large preregistered test (next entry).

Flore, P. C., Mulder, J., & Wicherts, J. M. (2018). The influence of gender stereotype threat on mathematics test scores of Dutch high school students: A registered report. Comprehensive Results in Social Psychology, 3(2), 140–174. doi:10.1080/23743603.2018.1559647

Preregistered registered report; students randomised within class to a threat or nullified-stereotype instruction before a math(s) test. Who: N = 2,064 second-year secondary students (age 13–14, about US grade 8), average-to-high tracks, Netherlands.

Finding. No stereotype threat effect: threat coefficient −0.06 points (SE 0.16) on a 20-item test; threat × gender interaction 0.16 (SE 0.31), not significant. Bayes factor 28 in favour of no effect over the threat hypothesis. Boys outscored girls regardless of condition.

One country and one manipulation; Dutch students may not hold the stereotype.

Barroso, C., Ganley, C. M., McGraw, A. L., Geer, E. A., Hart, S. A., & Daucourt, M. C. (2021). A meta-analysis of the relation between math anxiety and math achievement. Psychological Bulletin, 147(2), 134–168. doi:10.1037/bul0000307

Meta-analysis of 747 effect sizes (1992–2018). Who: Children through adults, many countries.

Finding. Correlation r = −.28 between math(s) anxiety and math(s) achievement. Present from childhood to adulthood; smaller in grades 3–5 and at university, smaller for exam grades, larger for certain anxiety scales.

Correlational only; direction of causation unresolved. Published studies report stronger correlations than unpublished ones.

Maloney, E. A., Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2015). Intergenerational effects of parents' math anxiety on children's math achievement and anxiety. Psychological Science, 26(9), 1480–1488. doi:10.1177/0956797615592630

Longitudinal field study across one school year; not an experiment. Who: n = 438 first- and second-graders (90 classrooms, 29 schools, three Midwestern US states) and their primary caregivers.

Finding. Parents' math(s) anxiety predicted lower math(s) growth and higher child math(s) anxiety only when parents reported helping often with math(s) homework (interaction p = .033). No such relation for reading.

Correlational; help frequency self-reported; families who returned surveys were higher-SES than the full sample; effect in grade-equivalent units only shown graphically.

Krueger, A. B. (1999). Experimental estimates of education production functions. The Quarterly Journal of Economics, 114(2), 497–532. doi:10.1162/003355399556052

Randomised controlled trial (students and teachers randomly assigned to small, regular, or regular-with-aide classes), with adjustment for attrition and class switching. Who: n = about 11,600 students, kindergarten to grade 3, Tennessee, USA, from 1985–86.

Finding. Standardised test performance (Stanford Achievement Test, reading and math(s)) rose by about 4 percentile points in the first year in a small class (13–17 pupils) versus a regular class (22–25), and the gap grew by about 1 percentile point per year after that. Teacher aides had little effect. Gains were larger for minority and free-lunch pupils.

The abstract pools reading and math(s); math(s)-only estimates were not verified in a fetched record. Single US state, 1980s.

Angrist, J. D., & Lavy, V. (1999). Using Maimonides' rule to estimate the effect of class size on scholastic achievement. The Quarterly Journal of Economics, 114(2), 533–575. doi:10.1162/003355399556061

Regression discontinuity / instrumental variables, using the rule that splits a grade into two classes once enrolment passes 40. Who: Population of Israeli public-school classes; grade 4 and 5 pupils tested in 1991, grade 3 in 1992. Unit of analysis is the class; class counts not given in the abstract.

Finding. Smaller classes raised grade 5 reading and math(s) scores. With enrolment controls, the IV estimate for grade 5 math(s) was about −0.23 points per extra pupil (reading −0.26 to −0.28), so a class 10 pupils smaller scores roughly 2–3 points higher. Smaller effect for grade 4 reading, none for grade 3.

Grade 3 null may reflect different testing that year. Later re-analyses (not fetched here) have debated score manipulation near the cutoff.

Jacob, B. A., & Lefgren, L. (2004). Remedial education and student achievement: A regression-discontinuity analysis. The Review of Economics and Statistics, 86(1), 226–244. doi:10.1162/003465304323023778

Regression discontinuity around promotion test cut-offs under Chicago's 1996 accountability policy. Who: Chicago Public Schools grade 3 and grade 6 cohorts 1997–1999 (base sample of about 347,000 students across pre- and post-policy cohorts); over 30,000 third graders and 21,000 sixth graders per year attended summer school.

Finding. For grade 3, summer school plus retention raised math(s) and reading achievement by roughly 20% of a year's learning; effects faded by 25–40% by the second year but stayed significant. For grade 6, no net benefit, and retention had little effect on math(s).

Low-income urban district; effects are the combined programme, not summer school alone; test-based selection means the estimates apply to pupils near the cut-off.

Steenbergen-Hu, S., Makel, M. C., & Olszewski-Kubilius, P. (2016). What one hundred years of research says about the effects of ability grouping and acceleration on K–12 students' academic achievement: Findings of two second-order meta-analyses. Review of Educational Research, 86(4), 849–899. doi:10.3102/0034654316675417

Two second-order meta-analyses (13 ability-grouping meta-analyses; several acceleration meta-analyses). Who: K–12 students, mostly US studies spanning about a century.

Finding. Within-class grouping g = 0.19–0.30, cross-grade subject grouping g = 0.26, special grouping for the gifted g = 0.37; between-class grouping (streaming/tracking) g = 0.04–0.06, with no difference for high-, medium- or low-ability students. Accelerated students outperformed same-age peers (g = 0.70) and matched older non-accelerated peers (g = 0.09); overall acceleration g = 0.42.

Second-order synthesis inherits the mostly non-randomised designs of the underlying studies; not math(s)-specific.

Lavy, V. (2015). Do differences in schools' instruction time explain international achievement gaps? Evidence from developed and developing countries. The Economic Journal, 125(588), F397–F424. doi:10.1111/ecoj.12233

Quasi-experimental; within-student comparison across subjects (math(s), science, language) so pupil fixed effects absorb ability and family background. Who: PISA 2006, over 400,000 15-year-olds in more than 50 countries.

Finding. One extra hour per week of instruction in a subject raised that subject's score by about 0.15 SD of the within-student test-score distribution in OECD and Eastern European samples, but only about 0.025 SD in developing countries. Returns were higher where schools faced accountability or had more autonomy.

Effect sizes quoted from the NBER working paper version; instruction time is self-reported by pupils.

Fan, H., Xu, J., Cai, Z., He, J., & Fan, X. (2017). Homework and students' achievement in math and science: A 30-year meta-analysis, 1986–2015. Educational Research Review, 20, 35–54. doi:10.1016/j.edurev.2016.11.003

Meta-analysis of 28 studies (41 independent correlations), correlational designs including PISA/TIMSS samples. Who: About 312,800 students, mainly primary to upper secondary, mostly US (31 of 41 correlations).

Finding. Small positive association overall, r = 0.23; math(s) r = 0.20, science r = 0.26. Stronger in elementary (r = 0.32) and high school (r = 0.30) than middle school (r = 0.15). US r = 0.27 versus r = 0.07 in Europe and Asia. Time spent on homework correlated only r = 0.19.

Correlational, so no causal claim; effects heterogeneous and several moderator estimates rest on few studies.

Codding, R. S., Burns, M. K., & Lukito, G. (2011). Meta-analysis of mathematic basic-fact fluency interventions: A component analysis. Learning Disabilities Research & Practice, 26(1), 36–47. doi:10.1111/j.1540-5826.2010.00323.x

Meta-analysis of 17 single-case design studies, with component analysis. Who: 55 participants, school-age students (many with math(s) difficulties); setting details not given in the abstract.

Finding. Drill-and-practice combined with modelling produced the largest effect sizes for basic-fact fluency. Interventions with more than three components beat those with fewer, and using more than one treatment agent (e.g. peer plus teacher) was more effective than one.

Single-case designs and a very small pooled sample; numeric effect sizes not given in the abstract.

Ellington, A. J. (2003). A meta-analysis of the effects of calculators on students' achievement and attitude levels in precollege mathematics classes. Journal for Research in Mathematics Education, 34(5), 433–463. doi:10.2307/30034795

Meta-analysis of calculator-use studies in K–12 classrooms. Who: K–12 students, mostly US; number of studies not given in the abstract (the author's 2000 dissertation on which it builds pooled 53 studies, 307 effect sizes).

Finding. Calculator use during instruction did not harm students' operational (paper-and-pencil) skills, and problem-solving skills and attitudes to math(s) improved; benefits were largest when calculators were allowed in both instruction and testing.

Mostly short studies from the 1980s–90s; findings for the published article taken from the record abstract and the underlying dissertation, not the full text.

Kluger, A. N., & DeNisi, A. (1996). The effects of feedback interventions on performance: A historical review, a meta-analysis, and a preliminary feedback intervention theory. Psychological Bulletin, 119(2), 254–284. doi:10.1037/0033-2909.119.2.254

Meta-analysis of 607 effect sizes. Who: 23,663 observations across lab and field studies of task performance (not math(s)-specific).

Finding. Feedback improved performance on average, d = 0.41, but more than one third of effects were negative. Effectiveness fell as feedback drew attention away from the task and toward the self (e.g. praise or person-focused comments).

Broad performance literature, not classroom math(s); the theory part is labelled preliminary by the authors.

Wisniewski, B., Zierer, K., & Hattie, J. (2020). The power of feedback revisited: A meta-analysis of educational feedback research. Frontiers in Psychology, 10, 3087. doi:10.3389/fpsyg.2019.03087

Meta-analysis of 435 studies (k = 994 effect sizes), random-effects model. Who: N > 61,000 students; median publication year 1985.

Finding. Overall d = 0.48 on learning (0.55 before removing outliers), with 17% of effects negative and high heterogeneity (I² = 83%). Larger for cognitive outcomes (d = 0.51) than motivational (d = 0.33). High-information feedback (task, process and self-regulation combined) d = 0.99; corrective feedback d = 0.46; reinforcement/punishment d = 0.24.

Egger's test indicated publication bias in journal articles but not dissertations; not math(s)-specific.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406. doi:10.3102/00028312042002371

Correlational; linear mixed models of one-year gain scores, students nested in teachers in schools. Who: 1,190 first graders (334 teachers) and 1,773 third graders (365 teachers) in 115 mostly low-income US schools.

Finding. Teachers' score on a 30-item "mathematical knowledge for teaching" test predicted pupils' math(s) gains in both grades after controlling for student and teacher covariates; press coverage of the study put it at roughly one third to one half of a month's extra growth per SD of teacher knowledge.

Observational, so teacher knowledge may proxy for other teacher qualities; the month-equivalent figure comes from an Education Week report, not the article's abstract.

Cooper, H., Nye, B., Charlton, K., Lindsay, J., & Greathouse, S. (1996). The effects of summer vacation on achievement test scores: A narrative and meta-analytic review. Review of Educational Research, 66(3), 227–268. doi:10.3102/00346543066003227

Narrative review of 39 studies with meta-analysis of the 13 most recent. Who: US school pupils across grades; pooled sample size not given in the abstract.

Finding. Scores fell over summer by about one month on a grade-equivalent scale, or one tenth of a standard deviation relative to spring scores. The loss was larger for math(s) than reading, and largest for math(s) computation and spelling. Losses grew with grade level; no moderation by gender or race.

Older studies on grade-equivalent scales; later work (Atteberry & McEachin below; von Hippel) questions how much of the classic pattern is a measurement artefact.

Atteberry, A., & McEachin, A. (2021). School's out: The role of summers in understanding achievement disparities. American Educational Research Journal, 58(2), 239–282. doi:10.3102/0002831220937285

Descriptive longitudinal growth modelling of a large test-score panel (not causal). Who: About 18 million US students, grades 1–8, roughly 200 million math(s) and reading MAP scores, 7,500 districts, all 50 states, 2008–2016.

Finding. On average pupils lost 25–34% of each school-year math(s) gain over the following summer (17–28% in ELA). Variation was large: some pupils kept learning, others lost almost all of the year's gain; 52% lost ground in ELA in five consecutive summers. Race and income explained only about 4% of the variance in summer loss.

Published online 2020; test taken on different dates, so summer is partly imputed; no intervention tested.

Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., Pagani, L. S., Feinstein, L., Engel, M., Brooks-Gunn, J., Sexton, H., Duckworth, K., & Japel, C. (2007). School readiness and later achievement. Developmental Psychology, 43(6), 1428–1446. doi:10.1037/0012-1649.43.6.1428

Correlational; coordinated regressions in six longitudinal datasets (US, UK, Canada) pooled by meta-analysis of 228 standardised coefficients. Who: Children followed from school entry (about age 5) into later primary and secondary school; pooled n not given in the abstract.

Finding. School-entry math(s) was the strongest predictor of later reading and math(s) achievement (average standardised coefficient 0.33), ahead of early reading (0.13) and attention (0.07), after controlling for prior cognitive skills and family background. Socio-emotional measures did not predict achievement.

Prediction, not causation; Watts et al. (2014, Educational Researcher) replicate with β = 0.24 for 54-month math(s) to age-15 math(s) in the NICHD SECCYD (n = 828).

McEachin, A., Domina, T., & Penner, A. (2020). Heterogeneous effects of early algebra across California middle schools. Journal of Policy Analysis and Management, 39(3), 772–800. doi:10.1002/pam.22202

Regression discontinuity at schools' 7th-grade test-score placement cutoffs. Who: Several hundred California middle schools; student n not given in the abstract.

Finding. Placement into 8th-grade algebra raised advanced-math(s) enrolment by 30 percentage points in grade 9 and 16 in grade 11. Grade-10 math(s) scores rose 0.05 SD. Benefits were substantially larger where schools set the cutoff higher.

Estimates apply to pupils near each school's cutoff; the achievement gain is small.

Clotfelter, C. T., Ladd, H. F., & Vigdor, J. L. (2015). The aftermath of accelerating algebra: Evidence from district policy initiatives. Journal of Human Resources, 50(1), 159–188. doi:10.3368/jhr.50.1.159

Quasi-experiment using sharp district policy shifts in the timing of algebra. Who: Two large North Carolina districts; n not given in the abstract.

Finding. No evidence of a positive average effect of acceleration in any specification; significant negative effects on Algebra I and Geometry performance. Effect sizes not given in the abstract. "Benign or beneficial for higher-performing students but unambiguously harmful to the lowest performers."

A policy of broad acceleration, not a selective programme; weaker teacher assignment explains only a small part.

Dougherty, S. M., Goodman, J. S., Hill, D. V., Litke, E. G., & Page, L. C. (2017). Objective course placement and college readiness: Evidence from targeted middle school math acceleration. Economics of Education Review, 58, 141–161. doi:10.1016/j.econedurev.2017.04.002

Regression discontinuity on a prior-test-score placement rule (design named in the companion paper and NBER version; the abstract says placement "uses prior test scores"). Who: Middle-school pupils in one large US district; n not given in the abstract.

Finding. Accelerated placement of pupils near the threshold raised later Precalculus enrolment by one-seventh, and by over one-third for girls and non-low-income pupils. It also raised college readiness and intentions to pursue a bachelor's degree.

Outcomes are course-taking and readiness, not test-score gains; one district.

Card, D., & Giuliano, L. (2016). Can tracking raise the test scores of high-ability minority students? American Economic Review, 106(10), 2783–2816. doi:10.1257/aer.20150484

Rank-based regression discontinuity plus a between-school/cohort analysis. Who: Fourth graders in a large urban US district; n not given in the abstract.

Finding. Non-gifted high achievers placed in a separate gifted/high-achiever class gained, with effects concentrated among Black and Hispanic pupils: 0.5 SD in grade-4 reading and math(s), persisting to grade 6. No spillovers, positive or negative, on non-participants.

One district. The earlier working paper (NBER w20453) found no reading or math(s) effect for pupils admitted by IQ threshold.

Bui, S. A., Craig, S. G., & Imberman, S. A. (2014). Is gifted education a bright idea? Assessing the impact of gifted and talented programs on students. American Economic Journal: Economic Policy, 6(3), 30–62. doi:10.1257/pol.6.3.30

Regression discontinuity at the eligibility cutoff, plus a randomised lottery for elite magnet GT schools. Who: Pupils in a large US district; n not given in the abstract.

Finding. Marginal pupils got higher-achieving peers and a more advanced curriculum, but achievement "neither improves nor worsens" in the short run. Lottery winners did better only in science. Effect sizes not given in the abstract.

Applies to pupils near the cutoff, not to the most able; short-run outcomes; bounds cannot rule out zero.

Abdulkadiroğlu, A., Angrist, J., & Pathak, P. (2014). The elite illusion: Achievement effects at Boston and New York exam schools. Econometrica, 82(1), 137–196. doi:10.3982/ECTA10266

Fuzzy regression discontinuity at admissions cutoffs. Who: Applicants to Boston and New York City exam schools; n not given in the abstract.

Finding. Admission brings much higher-achieving classmates, yet these changes "have little causal effect on test scores or college quality". Effect sizes not given in the abstract.

Applies to applicants near the cutoff. The working paper notes modest gains in high-school English for minority applicants in Boston.

Dobbie, W., & Fryer, R. G. (2014). The impact of attending a school with high-achieving peers: Evidence from the New York City exam schools. American Economic Journal: Applied Economics, 6(3), 58–75. doi:10.1257/app.6.3.58

Regression discontinuity at admissions cutoffs. Who: Applicants to three New York City exam high schools; n not given in the abstract.

Finding. Pupils just eligible had peers scoring 0.17 to 0.36 SD higher on grade-8 state tests, but this had "little impact on college enrollment, college graduation, or college quality".

Outcomes are college outcomes, not math(s) scores; applies to pupils near the cutoff.

Cohodes, S. R. (2020). The long-run impacts of specialized programming for high-achieving students. American Economic Journal: Economic Policy, 12(1), 127–166. doi:10.1257/pol.20180315

Fuzzy regression discontinuity. Who: Boston Public Schools pupils, grades 4–6; n not given in the abstract.

Finding. The accelerated curriculum in dedicated classrooms had "positive yet imprecise impacts on test scores" and raised high-school graduation and college enrolment, driven by Black and Latino pupils. Effect sizes not given in the abstract. Little evidence that peers explain the gains.

General achievement and attainment, not math(s) specifically; test-score effects are imprecise.

Card, D., & Giuliano, L. (2016). Universal screening increases the representation of low-income and minority students in gifted education. Proceedings of the National Academy of Sciences, 113(48), 13678–13683. doi:10.1073/pnas.1605043113

Natural experiment: a district introduced universal screening of second graders with unchanged eligibility standards. Who: Second graders in a large urban US district; n not given in the abstract.

Finding. Screening every child led to "large increases" in the share of economically disadvantaged and minority pupils placed in gifted programmes. Black and Hispanic pupils, low-income pupils, English learners and girls had been under-referred by parents and teachers. Sizes not given in the abstract.

Outcome is identification, not learning; one district.

Gavin, M. K., Casa, T. M., Adelson, J. L., Carroll, S. R., & Sheffield, L. J. (2009). The impact of advanced curriculum on the achievement of mathematically promising elementary students. Gifted Child Quarterly, 53(3), 188–202. doi:10.1177/0016986209334964

Field test with experimental and comparison groups; the abstract does not say assignment was random. Who: Grades 3–5 in 11 urban and suburban US schools; almost 50% economically disadvantaged; n not given in the abstract.

Finding. Pupils taught the advanced units outscored the comparison group at every grade: effect sizes 0.29 to 0.59 on ITBS Concepts and Estimation, and 0.69 to 0.97 on open-response assessments.

Evaluated by the curriculum's developers; open-response tests were project-made; replicated with a second cohort.

Steenbergen-Hu, S., & Moon, S. M. (2011). The effects of acceleration on high-ability learners: A meta-analysis. Gifted Child Quarterly, 55(1), 39–53. doi:10.1177/0016986210383155

Meta-analysis of 38 primary studies (1984–2008). Who: High-ability learners, P–12 and postsecondary; total n not given in the abstract.

Finding. Academic achievement g = 0.180, 95% CI [−0.072, 0.431]; social–emotional development g = 0.076, 95% CI [−0.025, 0.176] (random effects). No strong moderators.

Both intervals include zero, although the authors describe the effect as positive. Primary studies are mostly non-randomised. General achievement, not math(s) only.

Park, G., Lubinski, D., & Benbow, C. P. (2013). When less is more: Effects of grade skipping on adult STEM productivity among mathematically precocious adolescents. Journal of Educational Psychology, 105(1), 176–198. doi:10.1037/a0029481

Longitudinal (40 years), exact plus propensity-score matching. Correlational. Who: 363 grade skippers and 657 matched controls, from 3,467 top-1% math(s) students, United States.

Finding. Grade skippers were more likely to earn advanced STEM degrees and publish in STEM, did so earlier, and had more citations by age 50. Effect sizes not given in the abstract. Pattern less consistent for women.

Not randomised; matching cannot remove unmeasured differences such as motivation. Outcomes are careers, not math(s) learning.

Clements, D. H., Sarama, J., Wolfe, C. B., & Spitler, M. E. (2013). Longitudinal evaluation of a scale-up model for teaching mathematics with trajectories and technologies: Persistence of effects in the third year. American Educational Research Journal, 50(4), 812–850. doi:10.3102/0002831212469270

Cluster RCT; 42 schools randomly assigned to three conditions (pre-K intervention with follow-through in kindergarten and grade 1, pre-K intervention only, control). Who: 42 schools in two US city districts serving low-resource communities; number of children not given in the abstract.

Finding. At the end of grade 1, intervention children still scored higher than controls: g = 0.51 with follow-through, g = 0.28 without. Follow-through beat no follow-through by g = 0.24.

The pre-K-only effect is well below the g = 0.72 seen at the end of preschool; later analyses (below) show further fading.

Watts, T. W., Duncan, G. J., Clements, D. H., & Sarama, J. (2018). What is the long-run impact of learning mathematics during preschool? Child Development, 89(2), 539–555. doi:10.1111/cdev.12713

Instrumental-variables analysis of a cluster RCT (random assignment to the preschool math(s) intervention used as the instrument). Who: n = 410 children from the Building Blocks/TRIAD trial, followed from age 4 to late elementary school, USA.

Finding. One standard deviation of intervention-produced gain at age 4 led to a 0.24 SD gain in late-elementary math(s). That is about half the size implied by correlational models, and about 35% smaller than heavily controlled regression estimates from national data sets.

One intervention and one sample; the IV estimate assumes the programme affected later math(s) only through preschool math(s) learning.

Bailey, D. H., Nguyen, T., Jenkins, J. M., Domina, T., Clements, D. H., & Sarama, J. S. (2016). Fadeout in an early mathematics intervention: Constraining content or preexisting differences? Developmental Psychology, 52(9), 1457–1469. doi:10.1037/dev0000188

Secondary analysis of a cluster RCT; intervention children compared with a matched subset of controls who had the same scores at the end of treatment. Who: Children from the Building Blocks/TRIAD preschool trial, USA; sample size not given in the abstract.

Finding. About 72% of the fade-out was attributable to pre-existing differences between intervention and control children who had the same post-test score, not to later teaching that held intervention children back.

The matched comparison is not itself randomised; one intervention.

Hart, E. R., Bailey, D. H., Luo, S., Sengupta, P., & Watts, T. W. (2024). Fadeout and persistence of intervention impacts on social–emotional and cognitive skills in children and adolescents: A meta-analytic review of randomized controlled trials. Psychological Bulletin, 150(10), 1207–1236. doi:10.1037/bul0000450

Meta-analysis (meta-regression) of 86 RCTs with a post-test and at least one follow-up. Who: 56,662 participants, infancy to adolescence; 450 outcomes.

Finding. Impacts on both cognitive and social–emotional skills faded, most for interventions with larger initial effects. Persistence was lower at 1–2 years than at 6–12 months. A small positive follow-up effect remained beyond what the post-test impact predicted. No single pooled effect size is given in the abstract.

Not math(s)-specific; outcomes are cognitive, achievement and social–emotional measures in general.

Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Hamlett, C. L., Cirino, P. T., Jordan, N. C., Siegler, R., Gersten, R., & Changas, P. (2013). Improving at-risk learners' understanding of fractions. Journal of Educational Psychology, 105(3), 683–700. doi:10.1037/a0032446

RCT; pupils randomly assigned to small-group intervention or control. Who: n = 259 at-risk fourth graders, USA; 12 weeks, three 30-minute sessions a week.

Finding. The intervention stressed the measurement meaning of fractions (magnitude, number line); the control stressed part–whole pictures and procedures. Every conceptual and procedural fraction outcome favoured the intervention (effect sizes 0.29 to 2.50), and the gap to low-risk classmates narrowed.

The largest effects are on measures close to what was taught; immediate post-test only.

Barbieri, C. A., Rodrigues, J., Dyson, N., & Jordan, N. C. (2020). Improving fraction understanding in sixth graders with mathematics difficulties: Effects of a number line approach combined with cognitive learning strategies. Journal of Educational Psychology, 112(3), 628–648. doi:10.1037/edu0000384

RCT, randomised at pupil level; control received the school's usual intervention. Who: N = 51 sixth graders struggling with fractions (28 intervention, 23 control), USA; 27 lessons over 6 weeks.

Finding. The intervention group learned more: fraction concepts g = 1.09, number line estimation error g = −0.85, magnitude comparison g = 0.82. Gains held at a delayed post-test 7 weeks later.

Small sample. The abstract lists fraction arithmetic as measured but gives no effect size for it.

Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games—but not circular ones—improves low-income preschoolers' numerical understanding. Journal of Educational Psychology, 101(3), 545–560. doi:10.1037/a0014239

Experiment comparing a linear board game, a circular board game and other number activities. Who: Low-income preschoolers, Pittsburgh, USA; the authors' conference abstract of the study gives 88 children. About 1 hour of play.

Finding. The linear game improved magnitude comparison and number line estimation more than the circular game or other number activities. Linear-game children also learned more from later arithmetic training. Effect sizes not given in the abstract.

Small study by the game's developers; brief; outcomes close to the game.

Jordan, N. C., Glutting, J., Dyson, N., Hassinger-Das, B., & Irwin, C. (2012). Building kindergartners' number sense: A randomized controlled study. Journal of Educational Psychology, 104(3), 647–660. doi:10.1037/a0029018

RCT with three arms: number sense intervention, language intervention (active control), business as usual. Who: n = 44 per group (132 in all); high-risk kindergartners from low-income communities, USA.

Finding. Number sense children did better than controls at immediate post-test on number competencies and general math(s) achievement. Many effects held 8 weeks later. The language group did not differ from controls on any math(s) measure. Effect sizes not given in the abstract.

Follow-up was only 8 weeks; modest sample.

Clarke, B., Doabler, C., Smolkowski, K., Kurtz Nelson, E., Fien, H., Baker, S. K., & Kosty, D. (2016). Testing the immediate and long-term efficacy of a Tier 2 kindergarten mathematics intervention. Journal of Research on Educational Effectiveness, 9(4), 607–634. doi:10.1080/19345747.2015.1116034

Randomised block design; at-risk pupils within classrooms randomly assigned to intervention or control. Who: Kindergartners at risk in math(s), USA; sample size not given in the abstract.

Finding. At the end of kindergarten the intervention group was ahead on four of six math(s) measures and had narrowed the gap with not-at-risk classmates. By the winter of first grade there was no effect on achievement. Effect sizes not given in the abstract.

A clear example of fade-out within one year.

Nelson, G., & McMaster, K. L. (2019). The effects of early numeracy interventions for students in preschool and early elementary: A meta-analysis. Journal of Educational Psychology, 111(6), 1001–1022. doi:10.1037/edu0000334

Meta-analysis of 34 studies (52 treatment groups) with meta-regression. Who: Preschool, kindergarten and first-grade pupils, including those with disabilities or at risk for math(s) difficulty.

Finding. Average weighted effect g = 0.64, 95% CI [0.52, 0.76], with two outliers removed. Effects were larger for interventions that included counting with one-to-one correspondence and for those lasting 8 weeks or less.

Mostly immediate post-tests; a companion paper by the same authors reports smaller effects on distal measures (g = 0.35).

Schild, U., Bauch, A., & Nuerk, H.-C. (2020). A finger-based numerical training failed to improve arithmetic skills in kindergarten children beyond effects of an active non-numerical control training. Frontiers in Psychology, 11, 529. doi:10.3389/fpsyg.2020.00529

Controlled intervention study with an active control group (phonology training of similar length); the abstract does not say whether assignment was random. Who: 102 kindergarten children, Germany; 50 daily sessions over 10 weeks.

Finding. All children improved in arithmetic. The finger-training group did not improve more than the active control group. Effect size not given in the abstract.

One study; some analyses used a reduced sample of 63.

Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. doi:10.1177/0956797612440101

Longitudinal, correlational; two nationally representative data sets. Who: Elementary pupils in the USA and the UK followed for 5 or 6 years; sample sizes not given in the abstract.

Finding. Knowledge of fractions and of division uniquely predicted algebra knowledge and overall math(s) achievement in high school, after controlling for other math(s) knowledge, general intellectual ability, working memory, and family income and education. Effect sizes not given in the abstract.

Correlational. Experimental estimates of how early skills carry forward are much smaller than such correlations suggest (Watts et al., 2018, above).

Jitendra, A. K., Harwell, M. R., Dupuis, D. N., Karl, S. R., Lein, A. E., Simonson, G., & Slater, S. C. (2015). Effects of a research-based intervention to improve seventh-grade students' proportional problem solving: A cluster randomized trial. Journal of Educational Psychology, 107(4), 1019–1034. doi:10.1037/edu0000039

Cluster RCT; teachers/classrooms randomised to schema-based instruction (SBI: sort problems by underlying structure, draw a schematic diagram, then solve) or business as usual. Who: 82 teachers/classrooms, n = 1,999 grade-7 students, 50 US school districts (Upper Midwest).

Finding. SBI classes scored higher on a proportional problem-solving test: standardised effect 0.46 SD at posttest and 0.32 SD nine weeks later, versus control.

Researcher-designed outcome; no effect on the standardised GMADE Process and Applications subtest.

Jitendra, A. K., Harwell, M. R., Im, S.-h., Karl, S. R., & Slater, S. C. (2019). Improving student learning of ratio, proportion, and percent: A replication study of schema-based instruction. Journal of Educational Psychology, 111(6), 1045–1062. doi:10.1037/edu0000335

Cluster RCT replicating the 2015 trial in different US regions. Who: 59 math(s) teachers and n = 1,492 grade-7 students, 36 schools, 5 districts.

Finding. Statistically significant differences favouring SBI on both the proximal problem-solving test and a distal (standardised) measure, "with effect sizes similar to those reported in Jitendra et al. (2015)". Exact effect sizes are not given in the abstract.

Same research team as the original. A 2017 conference abstract of this study (52 classrooms, N = 1,379) reported g = 0.59 posttest, 0.35 delayed, 0.28 GMADE; the journal sample differs, so treat those as preliminary.

Lein, A. E., Jitendra, A. K., & Harwell, M. R. (2020). Effectiveness of mathematical word problem solving interventions for students with learning disabilities and/or mathematics difficulties: A meta-analysis. Journal of Educational Psychology, 112(7), 1388–1408. doi:10.1037/edu0000453

Meta-analysis of 31 experimental or quasi-experimental studies (34 effect sizes) after removing 2 influential data points from 33 eligible studies. Who: K–12 students with learning disabilities or math(s) difficulties.

Finding. Mean effect g = 0.56 (95% CI 0.40 to 0.72). Schema-based transfer instruction gave larger effects than plain schema-based instruction, strategy instruction or other approaches.

Effects were larger on researcher-made than norm-referenced tests, with researcher than teacher delivery, and in elementary than secondary grades. Two authors developed SBI.

Barbieri, C. A., Miller-Cotto, D., Clerjuste, S. N., & Chawla, K. (2023). A meta-analysis of the worked examples effect on mathematics performance. Educational Psychology Review, 35(1), Article 35 (as indexed by ERIC; Crossref lists no article number). doi:10.1007/s10648-023-09745-1

Meta-analysis of 55 experimental and quasi-experimental studies (43 articles, 181 effect sizes), robust variance estimation. Who: Elementary grades to postsecondary.

Finding. Average effect of worked examples on math(s) performance g = 0.48 (p = 0.01). Correct examples alone gave larger effects than incorrect examples alone or mixed sets. Adding self-explanation prompts gave a smaller effect than worked examples without them.

The moderator findings run against single studies that favour erroneous examples and self-explanation; moderator comparisons are between studies, not randomised.

Rittle-Johnson, B., Loehr, A. M., & Durkin, K. (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM Mathematics Education, 49(4), 599–611. doi:10.1007/s11858-017-0834-z

Meta-analysis of experiments on prompting self-explanation in math(s); number of studies not given in the abstract. Who: Not stated in the abstract.

Finding. Prompted self-explanation gave "a small to moderate improvement" in procedural knowledge, conceptual knowledge and procedural transfer when tested immediately. Evidence for benefits in classrooms or after a delay was "much more limited". Effects were stronger when explanations were scaffolded. Effect sizes are not given in the abstract.

Mostly immediate tests in lab-like settings; the authors themselves flag the thin classroom and retention evidence.

Star, J. R., Pollack, C., Durkin, K., Rittle-Johnson, B., Lynch, K., Newton, K., & Gogolen, C. (2015). Learning from comparison in algebra. Contemporary Educational Psychology, 40, 41–54. doi:10.1016/j.cedpsych.2014.05.005

Year-long RCT; teachers randomised to a supplemental comparison curriculum (side-by-side worked examples) or business as usual. Who: 141 Algebra I teachers and their students, Massachusetts; student n not given in the abstract.

Finding. "No main effect of condition on student achievement." Teachers used the materials much less often than asked; greater use was associated with greater procedural knowledge. Effect sizes are not given in the abstract.

The dosage link is correlational. An earlier conference report on the first cohort (77 teachers, 1,661 students) recorded about 4.5 hours of use over the whole year.

Schoenherr, J., Strohmaier, A. R., & Schukajlow, S. (2024). Learning with visualizations helps: A meta-analysis of visualization interventions in mathematics education. Educational Research Review, 45, 100639. doi:10.1016/j.edurev.2024.100639

Random-effects meta-analysis of 41 intervention studies. Who: N = 10,562 learners across age groups and math(s) topics.

Finding. Interventions built around external visualisations (drawings, diagrams, graphs) improved math(s) outcomes by g = 0.504 (95% CI 0.379 to 0.630). Technology-based visualisation was no better than paper.

Significant heterogeneity; effects were larger in quasi-experiments and against business-as-usual controls, so the RCT-only figure is likely lower than the headline.

Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34. doi:10.1016/j.learninstruc.2012.11.002

Two in-class randomised experiments inside Cognitive Tutor Algebra software. Who: Experiment 1: 134 high-school students randomised, 116 analysed; Experiment 2: 64 grade-8 students; United States.

Finding. Exp. 1: explaining examples beat extra practice alone on conceptual knowledge (63% vs 58% correct; partial eta-squared = .05), with equal procedural scores despite 8 fewer practice problems. Exp. 2: mixed correct-and-incorrect examples beat correct-only on conceptual features (72% vs 53%).

Small samples, especially Exp. 2 (about 21 per group); a later meta-analysis (Barbieri et al., 2023, above) did not find incorrect examples superior.

Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008). The advantage of abstract examples in learning math. Science, 320(5875), 454–455. doi:10.1126/science.1154659

Randomised lab experiment; one abstract symbolic version of a simple group-theory rule versus one, two or three concrete versions (measuring cups, pizza slices). Who: 80 undergraduates, Ohio State University.

Finding. Students taught with abstract symbols transferred the rule to a new game better than those taught with concrete examples, who scored about at chance. A later replication paper tabulates 76% vs 44% on transfer (n = 18 vs 20) and computes Cohen's d > 1.6.

Adults, one session, artificial task. WWC rated it consistent with its standards but noted 10% of the sample was dropped without explanation. See the two replications below.

Trninic, D., Kapur, M., & Sinha, T. (2020). The disappearing "advantage of abstract examples in learning math". Cognitive Science, 44(7), e12851. doi:10.1111/cogs.12851

Randomised replication and extension of Kaminski et al. (2008) with a redesigned concrete condition. Who: 114 first- and second-year non-math(s) university students in Zurich, three conditions; main comparison n = 38 vs 38.

Finding. Improved concrete examples gave much better initial learning (95% vs 70%; d = 1.276), no difference on abstract transfer (73% vs 78%; p = .835), and better concrete transfer (d = 0.416, p = .033).

Adults, single session, same artificial task. An earlier Belgian replication (De Bock et al., 2011, JRME 42(2), 109–126) reproduced the original result for abstract transfer but found concrete learners transferred better to a concrete context.

McNeil, N. M., Uttal, D. H., Jarvin, L., & Sternberg, R. J. (2009). Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. Learning and Instruction, 19(2), 171–184. doi:10.1016/j.learninstruc.2008.03.005

Two experiments; children solved money word problems with or without play money. Assignment method not given in the abstract. Who: Exp. 1: n = 229, grades 4 and 6; Exp. 2: n = 79, grade 5; United States.

Finding. Children given bills and coins solved fewer problems correctly than children given none. In Exp. 2, realistic money produced the most errors compared with bland money or none, though those errors were least likely to be conceptual. Effect sizes are not given in the abstract.

Measures performance during one session, not lasting learning; money problems only.

Cook, S. W., Mitchell, Z., & Goldin-Meadow, S. (2008). Gesturing makes learning last. Cognition, 106(2), 1047–1058. doi:10.1016/j.cognition.2007.04.010

Randomised experiment; during a lesson on problems like 4 + 3 + 6 = __ + 6, children were told to repeat a strategy in speech, in gesture, or both. Who: n = 84 third- and fourth-graders who failed the pretest; United States.

Finding. All groups improved equally right after the lesson. Four weeks later, children told to gesture kept 85% of their posttest gains on average, versus 33% for children who only spoke (group difference in regression slopes, F(2,78) = 5.79, p = .0045).

Small sample, one lesson on one problem type, one lab; no standardised effect size reported in the passages checked.

Garet, M. S., Heppen, J. B., Walters, K., Parkinson, J., Smith, T. M., Song, M., Garrett, R., Yang, R., & Borman, G. D. (2016). Focusing on Mathematical Knowledge: The Impact of Content-Intensive Teacher Professional Development (NCEE 2016-4010). National Center for Education Evaluation and Regional Assistance, IES. https://eric.ed.gov/?id=ED569154

RCT; teachers randomly assigned within schools. Who: Grade 4 teachers in 94 schools, six districts, five US states (teacher and pupil counts not given in the abstract). 93 hours of training.

Finding. The training raised teachers' math(s) knowledge and some aspects of their teaching, but "did not have a positive impact on student achievement". Effect sizes not given in the abstract.

One year of follow-up. The authors note the knowledge and practices targeted were generally not correlated with pupil math(s) achievement in their data.

Garet, M. S., Wayne, A. J., Stancavage, F., Taylor, J., Eaton, M., Walters, K., Song, M., Brown, S., Hurlburt, S., Zhu, P., Sepanik, S., & Doolittle, F. (2011). Middle School Mathematics Professional Development Impact Study: Findings After the Second Year of Implementation (NCEE 2011-4024). NCEE, IES. https://eric.ed.gov/?id=ED519922

RCT; schools randomly assigned within districts. Who: 12 US districts in year one; second-year sample 39 schools, 92 teachers, 2,132 grade 7 pupils in 6 districts.

Finding. No significant impact on pupils' rational-number achievement: effect size 0.04 (p = 0.37) after year one, –0.01 (p = 0.94) after year two. No significant impact on teacher knowledge either (0.05, p = 0.79 in year two).

Teacher turnover cut the dose received; the second year ran in only half the districts.

Jacob, R., Hill, H., & Corey, D. (2017). The impact of a professional development program on teachers' mathematical knowledge for teaching, instruction, and student achievement. Journal of Research on Educational Effectiveness, 10(2), 379–407. doi:10.1080/19345747.2016.1273411

RCT; teachers randomly assigned within schools; three-year evaluation. Who: 105 fourth- and fifth-grade teachers in 19 low-income schools in one US district.

Finding. "Some limited evidence of positive impacts on teachers' mathematical knowledge for teaching, but no effects on instructional practice or student outcomes." Effect sizes not given in the abstract.

One district; within-school randomisation allows spillover between teachers. The programme is named in the paper, not in the abstract; check before naming it on the page.

Lynch, K., Hill, H. C., Gonzalez, K. E., & Pollard, C. (2019). Strengthening the research base that informs STEM instructional improvement efforts: A meta-analysis. Educational Evaluation and Policy Analysis, 41(3), 260–293. doi:10.3102/0162373719849044

Meta-analysis of 95 experimental and quasi-experimental studies. Who: Pre-K to grade 12 science, technology, engineering and math(s) programmes (pupil numbers not given in the abstract).

Finding. Average weighted impact of +0.21 SD on pupil outcomes. Effects were larger when programmes helped teachers use curriculum materials, built content knowledge or understanding of how pupils learn, included summer workshops, and held meetings to troubleshoot classroom use.

Pools math(s) with science; the comparisons of programme features are correlational across studies. Sits awkwardly beside the three null IES-style trials above.

Kraft, M. A., Blazar, D., & Hogan, D. (2018). The effect of teacher coaching on instruction and achievement: A meta-analysis of the causal evidence. Review of Educational Research, 88(4), 547–588. doi:10.3102/0034654318759268

Meta-analysis of 60 studies with causal designs. Who: Mostly US pre-kindergarten and elementary teachers (numbers not given in the abstract).

Finding. Pooled effects of 0.49 SD on teaching practice and 0.18 SD on pupil achievement. Effects in large effectiveness trials were "only a fraction" of those in small efficacy trials.

Not math(s)-specific: the abstract says much of the evidence is from literacy coaching. No separate math(s) estimate is given in the abstract.

Chetty, R., Friedman, J. N., & Rockoff, J. E. (2014). Measuring the impacts of teachers II: Teacher value-added and student outcomes in adulthood. American Economic Review, 104(9), 2633–2679. doi:10.1257/aer.104.9.2633

Quasi-experiment using teacher turnover between school years; district records linked to tax records. Who: More than one million children, grades 3–8, one large US urban district.

Finding. Pupils assigned to high value-added teachers were more likely to attend college, earned more, and were less likely to have children as teenagers. Replacing a bottom-5% teacher with an average one would raise pupils' lifetime income by about $250,000 per classroom (present value).

Value-added is measured on math(s) and English tests together, not math(s) alone. Not randomised; a What Works Clearinghouse quick review said baseline equivalence was not clearly established. The companion paper (part I, https://doi.org/10.1257/aer.104.9.2593 tests for bias.

Angrist, J. D., Lavy, V., Leder-Luis, J., & Shany, A. (2019). Maimonides' rule redux. American Economic Review: Insights, 1(3), 309–324. doi:10.1257/aeri.20180120

Regression discontinuity / instrumental variables using the 40-pupil class cap; a re-run of Angrist & Lavy (1999) by its own authors on newer data. Who: Large Israeli samples, 2002–2011 (exact n not given in the abstract).

Finding. "Newer estimates show no evidence of class size effects." Schools were found to manipulate enrolment near the cut-offs; a modified rule using birthdays avoids this and still gives "precisely estimated zeros".

The abstract does not name the test subjects. The authors discuss, but do not settle, why the 1990s effect vanished.

Hoxby, C. M. (2000). The effects of class size on student achievement: New evidence from population variation. Quarterly Journal of Economics, 115(4), 1239–1285. doi:10.1162/003355300555060

Natural experiment: year-to-year variation in cohort size, plus discontinuities at maximum class-size rules. Who: 649 elementary schools, USA (state not named in the abstract).

Finding. Class size "does not have a statistically significant effect on student achievement". Rules out effects as small as 2 to 4 percent of a standard deviation for a 10 percent cut in class size.

Conflicts with the Tennessee STAR experiment already on the page. The working-paper abstract lists math(s), reading and writing scores.

Banerjee, A., Banerji, R., Berry, J., Duflo, E., Kannan, H., Mukerji, S., Shotland, M., & Walton, M. (2017). From proof of concept to scalable policies: Challenges and solutions, with an application. Journal of Economic Perspectives, 31(4), 73–102. doi:10.1257/jep.31.4.73

Series of cluster RCTs in four Indian states. Who: Primary pupils, grades 3–5 in the later trials; e.g. 400 schools in Haryana, about 120 schools per arm in Uttar Pradesh.

Finding. Giving teachers materials and training alone did nothing (Bihar, Uttarakhand). In Uttar Pradesh, in-school "learning camps" grouping children by level raised math(s) by 0.73–0.81 levels on a 0–4 scale (control mean 1.7).

Effects are in test "levels", not SD. The Haryana version did not target math(s) and showed no math(s) gain (–0.007). Effect sizes are from Table 3 of the working paper, not the abstract.

Duflo, E., Dupas, P., & Kremer, M. (2011). Peer effects, teacher incentives, and the impact of tracking: Evidence from a randomized evaluation in Kenya. American Economic Review, 101(5), 1739–1774. doi:10.1257/aer.101.5.1739

Cluster RCT: 121 primary schools; 60 split grade 1 by initial achievement, 61 assigned pupils to classes at random. Who: Grade 1 pupils in Kenya (pupil count not given in the abstract).

Finding. Pupils in tracking schools scored 0.14 SD higher after 18 months, and the effect persisted a year after the programme ended. Pupils at all levels gained, including the lower half.

The 0.14 SD figure is from the NBER working-paper abstract (w14475), not the journal abstract. The abstract does not separate math(s) from language. Very large classes and contract teachers; may not transfer to rich-country schools.

Taylor, E. (2014). Spending more of the school day in math class: Evidence from a regression discontinuity in middle school. Journal of Public Economics, 117, 162–181. doi:10.1016/j.jpubeco.2014.06.002

Regression discontinuity on a prior-test-score cut-off for assignment to a second, remedial math(s) class. Who: Middle-school pupils with below-cut-off math(s) scores (district and n not given in the abstract).

Finding. Math(s) achievement grew 0.16–0.18 student SD faster under the two-class requirement. A year after returning to one class the gain had decayed by as much as half; two years later one-third remained.

Applies to pupils near the cut-off. The extra math(s) class crowds out teaching in other subjects.

Jerrim, J., Macmillan, L., Micklewright, J., Sawtell, M., & Wiggins, M. (2018). Does teaching children how to play cognitively demanding games improve their educational attainment? Journal of Human Resources, 53(4), 993–1021. doi:10.3368/jhr.53.4.0516.7952R

School-randomised controlled trial (EEF effectiveness trial). Who: 4,009 Year 5 pupils in 100 schools across 11 English local authorities; 30 hours of chess lessons from a chess tutor, replacing lessons such as music or PE.

Finding. No evidence of an effect on Key Stage 2 math(s) scores one year after the lessons ended, nor on reading or science, nor for pupils on free school meals. Effect size not given in the abstract.

Outcome measured a year after the programme; compared with normal lessons, not another game. School and pupil counts come from the EEF report of the same trial.

Roberts, G., Quach, J., Spencer-Smith, M., Anderson, P. J., Gathercole, S., Gold, L., Sia, K.-L., Mensah, F., Rickards, F., Ainley, J., & Wake, M. (2016). Academic outcomes 2 years after working memory training for children with low working memory: A randomized clinical trial. JAMA Pediatrics, 170(5), e154568. doi:10.1001/jamapediatrics.2015.4568

Population-based RCT, intention-to-treat, registered. Who: 452 children aged 6–7 with low working memory (from 1,723 screened), 44 schools, Melbourne, Australia; 88% retained at 2 years.

Finding. 20–25 sessions of Cogmed gave a temporary gain in visuospatial short-term memory (effect size 0.43 at 6 months, 0.49 at 12 months, gone at 24). No benefit to reading, spelling or math(s); math(s) was worse at 2 years (mean difference −3.0, 95% CI −5.4 to −0.7, P = .01) versus usual teaching.

Children were selected for low working memory; training replaced classroom time. Cost A$1,035 per child.

Melby-Lervåg, M., & Hulme, C. (2013). Is working memory training effective? A meta-analytic review. Developmental Psychology, 49(2), 270–291. doi:10.1037/a0028228

Meta-analysis of 23 studies (30 group comparisons), randomised and non-randomised controlled designs. Who: Children and adults, clinical and typically developing samples.

Finding. Training gave reliable short-term gains on working-memory tasks. There was "no convincing evidence" that it generalised to other skills, including nonverbal and verbal ability, attention, word decoding and arithmetic. Effect sizes not given in the abstract.

Mixed ages and clinical conditions; few studies measured arithmetic. Later, larger meta-analyses (already on the page) reached the same conclusion.

Dunning, D. L., Holmes, J., & Gathercole, S. E. (2013). Does working memory training lead to generalized improvements in children with low working memory? A randomized controlled trial. Developmental Science, 16(6), 915–925. doi:10.1111/desc.12068

RCT with three arms: adaptive training, non-adaptive low-load training (active control), no training. Who: Children aged 7–9 with low working memory; country and n not given in the abstract.

Finding. Adaptive training improved several untrained working-memory tests, and verbal working-memory gains lasted a year. There was no evidence of change in classroom-style tasks that load working memory, or in any other cognitive or academic assessment. Effect sizes not given in the abstract.

Academic outcomes are described only generally in the abstract; math(s) is not named separately. Selected low-working-memory sample.

Sala, G., & Gobet, F. (2017). Does chess instruction improve mathematical problem-solving ability? Two experimental studies with an active control group. Learning & Behavior, 45(4), 414–421. doi:10.3758/s13420-017-0280-3

Two three-group experiments (chess, active control game, passive control). Who: Experiment 1: n = 233 third and fourth graders, 25 hours of chess versus checkers versus nothing. Experiment 2: n = 52, with Go as the active control.

Finding. No statistically significant differences between groups in mathematical problem solving or metacognition in either experiment. Effect sizes not given in the abstract.

Experiment 2 is small; allocation method not given in the abstract. Authors conclude any effect is modest and chess should not replace math(s) lessons.

Sala, G., Aksayli, N. D., Tatlidil, K. S., Tatsumi, T., Gondo, Y., & Gobet, F. (2019). Near and far transfer in cognitive training: A second-order meta-analysis. Collabra: Psychology, 5(1). Page or article number not stated in the Crossref record. doi:10.1525/collabra.203

Second-order meta-analysis (models with k = 99, 119 and 233 effect sizes). Who: Children, adults and older adults; working-memory training, video games, music, chess and exergames.

Finding. Working-memory training improves similar memory tasks. Far-transfer effects were small or null, and "when placebo effects and publication bias were controlled for, the overall effect size and true variance equaled zero", whatever the population or programme.

Far transfer here means general cognitive and academic outcomes, not math(s) alone. Same research group as several of the pooled meta-analyses.

Hawes, Z. C. K., Gilligan-Lee, K. A., & Mix, K. S. (2022). Effects of spatial training on mathematics performance: A meta-analysis. Developmental Psychology, 58(1), 112–137. doi:10.1037/dev0001281

Meta-analysis of 29 controlled pre-post studies (89 effect sizes). Who: N = 3,765, ages 3 to 20.

Finding. Spatial training improved math(s) by g = 0.28 (SE = 0.07) relative to controls, and spatial skills by g = 0.49. Effects were larger for older participants, for training with physical manipulatives rather than computers, and for math(s) outcomes close to what was trained.

Mostly small studies. Type of control group and publication status did not moderate, and bias tests were nonsignificant, but the authors say the transfer mechanism is poorly understood.

Judd, N., & Klingberg, T. (2021). Training spatial cognition enhances mathematical learning in a randomized study of 17,000 children. Nature Human Behaviour, 5(11), 1548–1554. doi:10.1038/s41562-021-01118-4

Randomised comparison of training mixes inside a math(s) app. Who: 17,648 children aged 6–8; seven weeks of math(s) training plus randomly assigned spatial tasks (mental rotation or tangram, visuospatial working memory, or non-verbal reasoning).

Finding. The type of cognitive training affected math(s) learning; visuospatial working-memory and reasoning training helped most. Effect sizes not given in the abstract.

Every group did math(s) training and some cognitive training, so there is no untrained control. Math(s) was measured inside the same app, not by a school test.

Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N. S. (2013). The malleability of spatial skills: A meta-analysis of training studies. Psychological Bulletin, 139(2), 352–402. doi:10.1037/a0028446

Meta-analysis of 217 training studies. Who: All ages; n not given in the abstract.

Finding. Training improved spatial skills by g = 0.47 (SE = 0.04) relative to controls. Gains lasted across delays and transferred to untrained spatial tasks.

The outcome is spatial skill, not math(s). The suggestion that this would raise STEM participation is the authors' inference from correlational studies, not a tested result.

Scherer, R., Siddiq, F., & Sánchez Viveros, B. (2019). The cognitive benefits of learning computer programming: A meta-analysis of transfer effects. Journal of Educational Psychology, 111(5), 764–792. doi:10.1037/edu0000314

Three-level meta-analysis of 105 studies (539 effect sizes). Who: Kindergarten to university; n not given in the abstract.

Finding. Overall transfer g = 0.49 (95% CI 0.37–0.61); near transfer g = 0.75, far transfer g = 0.47. Mathematical skills were among the outcomes that benefited; school achievement and literacy benefited least. The math(s)-specific effect size is not given in the abstract.

Effects were significantly larger with untreated than with active control groups, and larger in published than in unpublished studies. Many studies are old Logo-era experiments.

Owen, A. M., Hampshire, A., Grahn, J. A., Stenton, R., Dajani, S., Burns, A. S., Howard, R. J., & Ballard, C. G. (2010). Putting brain training to the test. Nature, 465(7299), 775–778. doi:10.1038/nature09042

Six-week online randomised study. Who: 11,430 participants (country not given in the abstract), training several times a week on reasoning, memory, planning, visuospatial and attention tasks.

Finding. Participants improved on every trained task, but "no evidence was found for transfer effects to untrained tasks, even when those tasks were cognitively closely related". Effect sizes not given in the abstract.

A general-public sample, not schoolchildren; outcomes were cognitive tests, not math(s). Training dose was modest and self-administered.

Mehr, S. A., Schachner, A., Katz, R. C., & Spelke, E. S. (2013). Two randomized trials provide no consistent evidence for nonmusical cognitive benefits of brief preschool music enrichment. PLoS ONE, 8(12), e82007. doi:10.1371/journal.pone.0082007

Two RCTs; music classes versus visual-arts classes (Exp. 1) or no classes (Exp. 2). Who: Preschool children attending with a parent; six weeks of classes; n not given in the abstract.

Finding. An apparent spatial advantage for the music group in Experiment 1 did not replicate. Combined, children given music classes did no better on spatial reasoning, visual form analysis, numerical discrimination or vocabulary.

Small samples and only six weeks, so it cannot rule out effects of years of lessons. The only number-related outcome is numerical discrimination. Open access.

Lynch, K., An, L., & Mancenido, Z. (2023). The impact of summer programs on student mathematics achievement: A meta-analysis. Review of Educational Research, 93(2), 275–315. doi:10.3102/00346543221105543

Meta-analysis of 37 experimental and quasi-experimental studies. Who: Children in pre-K to grade 12; total n not given in the abstract.

Finding. Children in summer programmes that included math(s) did better in math(s) than controls: average weighted effect +0.10 SD. Effects were similar in higher- and lower-poverty settings.

Pools randomised and quasi-experimental designs; programmes differ widely in length and content.

Augustine, C. H., McCombs, J. S., Pane, J. F., Schwartz, H. L., Schweig, J., McEachin, A., & Siler-Evans, K. (2016). Learning from Summer: Effects of Voluntary Summer Learning Programs on Low-Income Urban Youth. RAND Corporation, RR-1557-WF. doi:10.7249/RR1557

RCT (lottery among applicants), followed over two summers. Who: n = 5,637 third-graders in five US urban districts (Boston, Dallas, Duval County, Pittsburgh, Rochester).

Finding. Being offered one summer produced a math(s) gain of 0.08 SD in the autumn, which had gone by the next assessments. No causal effect on language arts, and none from offering two summers.

Larger, lasting gains for high attenders (0.13 SD near-term) are correlational, not randomised.

von Hippel, P. T., Workman, J., & Downey, D. B. (2018). Inequality in reading and math skills forms mainly before kindergarten: A replication, and partial correction, of "Are Schools the Great Equalizer?" Sociology of Education, 91(4), 323–357. doi:10.1177/0038040718801760

Longitudinal replication in two national US cohorts (correlational), using better-scaled test scores. Who: Children from kindergarten to about grade 2; n not given in the abstract.

Finding. The earlier result that inequality grows mainly in summer rested partly on a measurement artifact. With better scores, most inequality is present at the start of kindergarten; changes afterwards are small and do not replicate consistently. No effect size given in the abstract.

Correlational. About gaps between children, not average summer loss.

Fryer, R. G. (2011). Financial incentives and student achievement: Evidence from randomized trials. Quarterly Journal of Economics, 126(4), 1755–1798. doi:10.1093/qje/qjr045

School-level RCTs. Who: Over 200 urban schools in Dallas, New York and Chicago (the WWC review of the working paper counts about 38,000 students).

Finding. Paying students to read books, for interim test results, or for grades had an effect on achievement that was "statistically 0" in each city. The only significant effect was for English-speaking students in Dallas.

The author says the trials could not rule out effects small enough to still be worth the cost.

Bettinger, E. P. (2012). Paying to learn: The effect of financial incentives on elementary school test scores. Review of Economics and Statistics, 94(3), 686–698. doi:10.1162/REST_a_00217

RCT (eligibility for payments set by randomisation). Who: Grades 3–6, Coshocton, Ohio; n not given in the abstract.

Finding. Cash payments for passing standardised tests raised math(s) scores by about 0.15 SD. Reading, social science and science scores did not improve.

One small district. Contrasts with Fryer's nulls in large cities.

Cristia, J., Ibarrarán, P., Cueto, S., Santiago, A., & Severín, E. (2017). Technology and child development: Evidence from the One Laptop per Child program. American Economic Journal: Applied Economics, 9(3), 295–320. doi:10.1257/app.20150385

School-level RCT, measured after 15 months. Who: 318 rural primary schools in Peru; number of pupils not given in the abstract.

Finding. Computers per pupil rose from 0.12 to 1.18 and use rose sharply at school and at home, but there was no evidence of effects on math(s) or language test scores.

Some inconclusive evidence of gains in general cognitive skills. Laptops came without math(s)-specific software or teaching changes.

Fairlie, R. W., & Robinson, J. (2013). Experimental evidence on the effects of home computers on academic achievement among schoolchildren. American Economic Journal: Applied Economics, 5(3), 211–240. doi:10.1257/app.5.3.211

RCT. Who: n = 1,123 pupils in grades 6–10 at 15 California schools, without a home computer.

Finding. Computer ownership and use rose a lot, but there were no effects on grades, test scores, credits, attendance or discipline. The authors say estimates are precise enough to rule out even modest effects either way.

General achievement, not math(s) alone. Homework time did not change.

Beland, L.-P., & Murphy, R. (2016). Ill communication: Technology, distraction & student performance. Labour Economics, 41, 61–76. doi:10.1016/j.labeco.2016.04.004

Difference-in-differences around schools' own decisions to ban phones. Who: 91 secondary schools in Birmingham, London, Leicester and Manchester; age-16 exam results 2001–2011.

Finding. After a ban, exam scores rose by 6.41% of an SD. The lowest quarter of prior achievers gained 14.23% of an SD; the top quarter were unaffected.

Not randomised; only 21% of surveyed schools replied. Outcome is total GCSE points, not math(s). Figures are from the 2015 working-paper version.

Kessel, D., Hardardottir, H. L., & Tyrefors, B. (2020). The impact of banning mobile phones in Swedish secondary schools. Economics of Education Review, 77, 102009. doi:10.1016/j.econedurev.2020.102009

Difference-in-differences; partial replication of Beland and Murphy. Who: 1,086 of 1,423 Swedish lower-secondary schools (about 75% survey response).

Finding. No impact of phone bans on student performance; the authors "can reject even small-sized gains". A second outcome, the share failing the national math(s) test, also showed very little effect, if any.

Not randomised; the survey collected less detail on how strictly bans were enforced.

Heissel, J. A., & Norris, S. (2018). Rise and shine. Journal of Human Resources, 53(4), 957–992. doi:10.3368/jhr.53.4.0815-7346R1

Natural experiment (instrumental variable using the time-zone boundary in Florida). Who: Florida public-school pupils aged about 8–15; n not given in the abstract.

Finding. Starting one hour later relative to sunrise raised adolescents' scores by 0.08 SD in math(s) and 0.06 SD in reading. The math(s) effect was larger for older children.

Identifies daylight before school, not a clock-time policy change; about 0.01 SD for younger children.

Edwards, F. (2012). Early to rise? The effect of daily start times on academic performance. Economics of Education Review, 31(6), 970–983. doi:10.1016/j.econedurev.2012.07.006

Quasi-experiment using changes in start times within schools over time. Who: All middle-school students in Wake County, North Carolina, 1999–2006; n not given in the abstract.

Finding. A later start gave a two percentile point gain in math(s) scores, with similar results for reading. The effect was stronger for lower-scoring students.

Start times were set for bus scheduling, not randomly; one district. The companion article describes the change as one hour later.

Backes, B., & Cowan, J. (2019). Is the pen mightier than the keyboard? The effect of online testing on measured student achievement. Economics of Education Review, 68, 89–103. doi:10.1016/j.econedurev.2018.12.007

Quasi-experiment: Massachusetts gave the same exam (PARCC) online and on paper during a two-year rollout. Who: Massachusetts public-school pupils; n not given in the abstract.

Finding. Pupils tested online scored about 0.10 SD lower in math(s) and 0.25 SD lower in English. In the second year the math(s) penalty was about one third as large.

Measures test format, not learning; schools chose their mode.

Sala, G., & Gobet, F. (2016). Do the benefits of chess instruction transfer to academic and cognitive skills? A meta-analysis. Educational Research Review, 18, 46–57. doi:10.1016/j.edurev.2016.02.002

Meta-analysis of 24 studies (40 effect sizes). Who: 2,788 young people in chess conditions, 2,433 controls; countries not given in the abstract.

Finding. Moderate overall effect (g = 0.338), larger for mathematics (g = 0.382) than reading (g = 0.248), rising with hours of instruction. But no study used an "ideal design" with pre/post tests, full randomisation and an active control group, so placebo effects cannot be ruled out.

The positive pooled effect rests on weak designs; treat as unproven, not as support.

Melby-Lervåg, M., Redick, T. S., & Hulme, C. (2016). Working memory training does not improve performance on measures of intelligence or other measures of "far transfer": Evidence from a meta-analytic review. Perspectives on Psychological Science, 11(4), 512–534. doi:10.1177/1745691616635612

Meta-analysis of 87 publications, 145 experimental comparisons (pretest–posttest with control group). Who: Ages and pooled n not given in the abstract.

Finding. Training improved the trained tasks, but for far transfer (arithmetic, reading, verbal and nonverbal ability) there was "no convincing evidence of any reliable improvements" against treated controls, and working-memory gains did not predict far-transfer gains.

Sala & Gobet (2017, https://doi.org/10.1037/dev0000265): far transfer g = 0.12 in children aged 3–16, shrinking with design quality. Simons et al. (2016, https://doi.org/10.1177/1529100616661983): same verdict on commercial brain-training.

Sala, G., & Gobet, F. (2020). Cognitive and academic benefits of music training with children: A multilevel meta-analysis. Memory & Cognition, 48(8), 1429–1441. doi:10.3758/s13421-020-01060-2

Multilevel meta-analysis (254 effect sizes). Who: 6,984 children; international.

Finding. Once study-design quality is controlled for, "the overall effect of music training programs is null." Significant effects appeared only in studies without random allocation or with non-active control groups, and did not vary by outcome, age or training length.

The authors' 2017 meta-analysis (https://doi.org/10.1016/j.edurev.2016.11.005 found d = 0.16, shrinking with design quality. Transfer only; musical skill itself is not in question.

Pietschnig, J., Voracek, M., & Formann, A. K. (2010). Mozart effect–Shmozart effect: A meta-analysis. Intelligence, 38(3), 314–323. doi:10.1016/j.intell.2010.03.001

Meta-analysis of nearly 40 studies. Who: Over 3,000 participants, published and unpublished studies.

Finding. Mozart's K.448 versus silence gave a small effect on spatial tasks (d = 0.37); any other music gave the same (d = 0.38); Mozart versus other music was negligible (d = 0.15). Effects were larger in the originating lab and shrank with publication-bias correction. "Little evidence left for a specific, performance-enhancing Mozart effect."

Short-term spatial-task performance, not math(s) learning.

Macnamara, B. N., Hambrick, D. Z., & Oswald, F. L. (2014). Deliberate practice and performance in music, games, sports, education, and professions: A meta-analysis. Psychological Science, 25(8), 1608–1618. doi:10.1177/0956797614535810

Meta-analysis across all major domains in which deliberate practice has been measured; study count not given in the abstract. Who: Pooled n not given in the abstract.

Finding. Deliberate practice explained 26% of the variance in performance for games, 21% for music, 18% for sports, 4% for education and under 1% for professions. "Deliberate practice is important, but not as important as has been argued."

Correlational data on accumulated practice; the education figure is for academic attainment, not math(s) specifically.

Rogowsky, B. A., Calhoun, B. M., & Tallal, P. (2020). Providing instruction based on students' learning style preferences does not improve learning. Frontiers in Psychology, 11, 164. doi:10.3389/fpsyg.2020.00164

Randomised experiment using the Pashler et al. (2008) test: assess style, randomise to modality, same test for all. Who: n = 125 fifth graders (118 with complete data), rural Pennsylvania, USA.

Finding. No relationship between learning-style preference, instructional mode (listening vs reading) and comprehension, either immediately or at two-week retention. Children classed as visual learners did better in both modes.

Comprehension, not math(s); one school. The same team's adult study (2015, J. Educational Psychology, 107(1), 64–78, https://doi.org/10.1037/a0037478 also "failed to statistically support the meshing hypothesis".

Pashler, H., McDaniel, M., Rohrer, D., & Bjork, R. (2008). Learning styles: Concepts and evidence. Psychological Science in the Public Interest, 9(3), 105–119. doi:10.1111/j.1539-6053.2009.01038.x

Commissioned evidence review setting out the required test (an aptitude-by-treatment crossover) and auditing the literature against it. Who: Review; no new participants.

Finding. People do report preferences, but "we found virtually no evidence for the interaction pattern" needed to justify matching instruction to style; the few properly designed studies contradicted it. Conclusion: no justification for learning-styles assessment in general practice.

Review rather than meta-analysis; subsequent RCTs (Rogowsky et al. 2015, 2020) tested and rejected the pattern.

Morehead, K., Dunlosky, J., & Rawson, K. A. (2019). How much mightier is the pen than the keyboard for note-taking? A replication and extension of Mueller and Oppenheimer (2014). Educational Psychology Review, 31(3), 753–780. doi:10.1007/s10648-019-09468-2 · doi:10.1177/0956797614524581

Two randomised experiments: direct replication plus eWriter and no-notes conditions; small internal meta-analysis. Who: Students; n and setting not given in the abstract.

Finding. "Performance did not consistently differ between any groups", even against students who took no notes; pooled effects favouring longhand were small and non-significant. The original Mueller & Oppenheimer (2014, Psychological Science, 25(6), 1159–1168) found laptop note-takers did worse on conceptual questions in three studies.

Lecture comprehension, not math(s); the popular finding did not replicate cleanly.

Agarwal, P. K., D'Antonio, L., Roediger, H. L., McDermott, K. B., & McDaniel, M. A. (2014). Classroom-based programs of retrieval practice reduce middle school and high school students' test anxiety. Journal of Applied Research in Memory and Cognition, 3(3), 131–139. doi:10.1016/j.jarmac.2014.07.002

Student survey following classroom retrieval-practice programmes (correlational, self-report). Who: n = 1,408 middle- and high-school students, USA.

Finding. In classes using regular low-stakes retrieval practice, 92% of students said it helped them learn and 72% said it made them less nervous for unit tests and exams.

Not an RCT and no objective anxiety measure; include only as evidence against the "quizzing causes anxiety" claim, not as proof of a reduction.