Research 07: General learning science for Numberkit (how people learn and remember, beyond games and apps)

Prepared 2026-09-27. Scope: laboratory, classroom, and field research on human learning and memory across ages. Educational games and apps are deliberately excluded as evidence. Each finding carries a note on how it could translate to an app and whether that translation is faithful (the app would do what the studies did) or a stretch (the app would change the conditions that produced the effect).

Verification note. Every reference in the list was checked this session against Crossref, the publisher page, PubMed, or the full text. Effect sizes and sample sizes come from abstracts or full text read this session, unless marked "(not verified here)". Where the evidence in children is thin or missing, the text says so.

1. Summary of key findings

1. Retrieval and spacing are the two most secure findings in the field, at every age tested. Dunlosky et al. (2013) rated only practice testing and distributed practice "high utility". A later meta-analysis of the same ten techniques (Donoghue & Hattie, 2021; 242 studies, 169,179 participants, mean effect 0.56) again ranked these two first. Classroom quizzing gives g = 0.50 (Yang et al., 2021; 222 studies, 48,478 students). Fourth graders benefit too (Karpicke et al., 2016).
2. Retrieval helps transfer only under certain conditions. Pan and Rickard (2018) found d = 0.40 overall (N = 10,396). Transfer was weakest to rearranged stimulus-response items: practising 7 x 8 = 56 does not reliably teach 56 / 8 = 7. After a publication-bias correction, transfer was often absent unless the practice had elaborated retrieval, high initial success, or an answer format that matched the test. Inverse facts have to be practised in their own right.
3. Asking a question before teaching (a prequestion, or pretest) helps with the content that was asked about, and does nothing for the rest. Two meta-analyses agree: g = 0.54 for asked content versus 0.04 for other content (St. Hilaire et al., 2023), and g = 0.66 versus 0.01 (King-Shepard et al., 2025). The benefit holds even when the learner's guesses are wrong, as long as the correct answer follows. It has been replicated in children from kindergarten upward, but not in preschoolers (Carneiro et al., 2018).
4. Successive relearning (retrieve until correct, then come back in later sessions and retrieve until correct again) is among the strongest classroom manipulations. In a college course, scores were 10% higher on the first test and 40% higher about a month later (Rawson et al., 2013). It is close to what a good mastery engine already does. The novel parts are the criterion of several correct recalls within a session and the relearning sessions spread across weeks.
5. Interleaving problem types works in real mathematics classrooms, including with children. Fourth graders scored 77% versus 38% (Taylor & Rohrer, 2010). A randomised trial of 54 seventh-grade classes found d = 0.83 (Rohrer et al., 2020). Learners reliably judge the opposite: in Kornell and Bjork (2008), 78% did better with interleaving, yet 78% rated massing as at least as good. They read effort as poor learning (Kirk-Johnson et al., 2019).
6. Self-explanation (g = 0.55; Bisra et al., 2018), worked examples in mathematics (g = 0.48; Barbieri et al., 2023), comparing solution methods side by side (Rittle-Johnson & Star, 2007), and gesture (Goldin-Meadow et al., 2009; Novack et al., 2014) all have child-level evidence in mathematics. All four sit on concrete representations, and few apps use them.
7. Guidance must fade as expertise grows. Worked examples help novices and can hurt more knowledgeable learners (the expertise reversal effect; Kalyuga et al., 2003). Fading worked steps, backward from the last step, beats an abrupt switch (Atkinson, Renkl & Merrill, 2003). Adaptive fading has been tested inside a tutoring system (Salden et al., 2010).
8. Consolidation happens between sessions, not only inside them. Sleep after learning improves declarative memory in school-age children (Backhaus et al., 2008). Children extract hidden rules during sleep better than adults do (Wilhelm et al., 2013). Quiet rest after learning has a small, variable benefit: d = 0.38 in a meta-analysis of 10 studies (Humiston et al., 2019), with more failures to replicate in healthy young adults. Test scores fall later in the school day, and a 20 to 30 minute break restores part of the loss (Sievertsen et al., 2016; all Danish public school children, 2009 to 2013).
9. Far transfer is rare. It shows up where learning included comparison of structurally similar cases and varied practice (Gick & Holyoak, 1980; Gentner et al., 2003), not where it relied on general "brain training" (Sala & Gobet, 2017).
10. Some popular ideas should be avoided or used with care. Learning styles have no supporting evidence (Pashler et al., 2008). Deliberate practice explains only 4% of the variance in education outcomes (Macnamara et al., 2014). The benefit of handwriting over typing is unsettled (Morehead et al., 2019; Voyer et al., 2022). The contextual interference benefit is much smaller for children than adults in motor learning (Brady, 2004).

2. Detailed findings

2.1 The canonical syntheses

Dunlosky, Rawson, Marsh, Nathan & Willingham (2013), Psychological Science in the Public Interest, 14(1), 4-58, doi:10.1177/1529100612453266. The review rated ten techniques on how well their benefits generalise across learners, materials, criterion tasks, and learning conditions.

• High utility: practice testing and distributed practice.
• Moderate: elaborative interrogation, self-explanation, and interleaved practice. These were marked down mainly because of limited evidence across ages and materials, not because of negative findings.
• Low: summarisation, highlighting or underlining, the keyword mnemonic, imagery for text, and rereading.

The evidence base runs from young children to adults, but most of it comes from undergraduates. Donoghue & Hattie (2021), Frontiers in Education, doi:10.3389/feduc.2021.581216, meta-analysed the same ten techniques: 242 studies, 1,619 effects, 169,179 participants, overall mean 0.56. Distributed practice and practice testing were again the most effective. They caution that most outcomes were surface or factual, and that effects were larger for lower-ability students.

• App translation: spacing and retrieval are the backbone. Rereading and highlighting have no analogue worth building. Faithful.

Weinstein, Madan & Sumeracki (2018), Cognitive Research: Principles and Implications, 3:2, doi:10.1186/s41235-017-0087-y. This tutorial names six strategies with robust support: spaced practice, interleaving, retrieval practice, elaboration, concrete examples, and dual coding. It is useful because it describes classroom implementations and flags where the applied evidence is thin, notably elaboration and dual coding for younger learners.

Roediger & Pyc (2012), Journal of Applied Research in Memory and Cognition, 1(4), 242-248, doi:10.1016/j.jarmac.2012.09.002. The paper argues that three cheap techniques, distributed practice, retrieval practice, and explanatory questioning, can be added to any curriculum at almost no cost.

Pashler, Bain, Bottge, Graesser, Koedinger, McDaniel & Metcalfe (2007), IES Practice Guide NCER 2007-2004, "Organizing Instruction and Study to Improve Student Learning" (https://files.eric.ed.gov/fulltext/ED498555.pdf). It makes seven recommendations, with levels of evidence read from the guide's Table 2:

• App translation: recommendation 2 (alternate worked examples with problems) and recommendation 6a (delayed judgments of learning) are the two least often seen in commercial apps. Since 2007, two meta-analyses (section 2.2) have strengthened 5a, prequestions. Faithful.

2.2 Retrieval practice beyond flashcards

Magnitude and age range.

• Adesope, Trevisan & Sundararajan (2017), Review of Educational Research, 87(3), doi:10.3102/0034654316689306: practice tests beat restudy and every other comparison condition. Effects were moderated by test format, feedback, and participant characteristics.
• Yang et al. (2021), Psychological Bulletin, 147(4), doi:10.1037/bul0000309: g = 0.499 across 222 classroom studies (48,478 students), from primary school to university.
• Karpicke, Blunt & Smith (2016), Frontiers in Psychology, 7:350, doi:10.3389/fpsyg.2016.00350: fourth graders (mean age 10.0; three experiments with 40, 40, and 42 children) recalled more after retrieval practice than after restudy (55% vs 44%, and 42% vs 28%) and recognised more (77% vs 66%). The authors scaffolded retrieval with category cues and first letters so that initial success stayed high (75 to 87%). They argue retrieval practice without adequate initial success gives little benefit for children.
• App translation: keep first-pass retrieval success high by cueing, then remove the cues. Faithful.

Pretesting and errorful generation.

• Kornell, Hays & Bjork (2009), JEP: Learning, Memory, and Cognition, 35(4), 989-998, doi:10.1037/a0015729: failed retrieval attempts on fictional trivia and weak associates, followed by the answer, beat simply studying for the same time (adults).
• Richland, Kornell & Kao (2009), JEP: Applied, 15(3), 243-257, doi:10.1037/a0016496: pretesting on an essay about vision improved later learning even though most pretest answers were wrong (undergraduates).
• Pan & Carpenter (2023), Educational Psychology Review, 35:97, doi:10.1007/s10648-023-09814-5, a narrative review: pretesting works "if there is an opportunity to study the correct answers afterwards". It works with text, video, and lectures. For paired associates it needs a semantic link between cue and answer, and without that link cued-recall benefits often vanish. The benefit was replicated in kindergarten and early-elementary children but not preschoolers (Carneiro, Lapa & Finn, 2018, Journal of Experimental Child Psychology, 166, 400-420, doi:10.1016/j.jecp.2017.09.010). In fourth and fifth graders, retrieval practice after reading outperformed prequestions at one week (de Lima & Jaeger, 2020, JARMAC, as cited by Pan & Carpenter).
• Two meta-analyses:
  • St. Hilaire, Chan & Ahn (2023), Psychonomic Bulletin & Review, doi:10.3758/s13423-023-02353-8, preregistered: specific effect g = 0.54 (k = 97), general effect g = 0.04 (k = 91).
  • King-Shepard, Walker, Nokes-Malach, Carpenter & Fraundorf (2025), Educational Psychology Review, doi:10.1007/s10648-025-10075-7: g = 0.66 on prequestioned content, g = 0.01 on other content, robust across samples and materials.
• Productive failure is related: problem solving before instruction versus instruction first. Sinha & Kapur (2021), Review of Educational Research, 91(5), doi:10.3102/00346543211019105: g = 0.36 overall (53 studies, 166 comparisons), up to d = 0.58 with high fidelity to the design, and benefits mainly in conceptual understanding and transfer. For second to fifth graders the direction reversed and favoured instruction first.
• App translation: before a Learn segment on, say, the area model, ask one or two prediction questions whose answers the segment then shows, and make sure the segment reveals exactly those answers. Faithful for targeted items. Do not expect a general benefit for the rest of the lesson, and do not use open-ended "invent a method" productive failure with 9-year-olds, where the evidence runs the other way. The transfer to an app's short Learn segments is a modest stretch: most studies used text passages and lectures, not interactive representations.

Generation effect.

• Slamecka & Graf (1978), JEP: Human Learning and Memory, 4(6), 592-604, doi:10.1037/0278-7393.4.6.592.
• Bertsch, Pesta, Wiscott & McDaniel (2007), Memory & Cognition, 35(2), 201-210, doi:10.3758/BF03193441: 86 studies, 445 effect sizes, mean d = 0.40 for generating over reading, with large moderator variability. Mostly adults and word-level materials.
• App translation: the child produces the answer or the next step instead of confirming a shown one. Multiple-choice recognition gives up part of this effect. Faithful.

Successive relearning.

• Rawson, Dunlosky & Sciartelli (2013), Educational Psychology Review, 25(4), 523-548, doi:10.1007/s10648-013-9240-4: college students in an introductory psychology course practised concepts to a criterion of three correct recalls, in sessions spaced across the semester. Relative to material they studied on their own, exam scores were 10% higher, and about 40% higher at a delay of nearly a month.
• Rawson & Dunlosky (2022), Current Directions in Psychological Science, doi:10.1177/09637214221100484: they argue time on task should be tailored and treated as an outcome, not fixed.
• The evidence base is overwhelmingly college students. Child studies were not found this session.
• App translation: in each session, re-queue a fact until the child retrieves it correctly more than once (not only once), and revisit it in later sessions to the same criterion. Faithful in mechanism. For children the dose (number of correct recalls) is untested.

Test-potentiated learning.

• Arnold & McDermott (2013), JEP: LMC, 39(3), 940-945, doi:10.1037/a0029199: an unsuccessful test makes the restudy that follows more effective, separately from the direct effect of retrieval (adults).
• App translation: after a miss, show the worked representation right away. The miss has primed the child to encode it. Faithful.

Feedback timing.

• Mullet, Butler, Verdin, von Borries & Marsh (2014), JARMAC, 3(3), 222-229, doi:10.1016/j.jarmac.2014.05.001: in a college engineering course, homework feedback delayed one week produced better transfer on exams than immediate feedback, even though students preferred immediate feedback.
• Hypercorrection: errors made with high confidence are more likely to be corrected after feedback than low-confidence errors (Butterfield & Metcalfe, 2001, JEP: LMC, 27(6), doi:10.1037/0278-7393.27.6.1491). This holds in grade 3 to 6 children, who also claimed they "knew it all along" (Metcalfe & Finn, 2012, Learning and Instruction, doi:10.1016/j.learninstruc.2011.10.004).
• App translation: for fact fluency, keep immediate corrective feedback, because an uncorrected error can be learned. The delay-benefit evidence is for adults and conceptual transfer, so delaying feedback for children is a stretch. A faithful low-cost addition is a delayed re-display: re-present missed items and their explanations in the next session, not only instantly. Asking for a quick confidence rating before feedback would exploit hypercorrection, but the rating costs time and the evidence for doing it in children's apps is indirect.

2.3 Elaboration, self-explanation, examples, and dual coding

Elaborative interrogation (asking "why is this true?"). Dunlosky et al. rated it moderate: evidence is mostly fact lists and text with older learners, and it depends on prior knowledge. Child-specific studies exist in the older literature but were not verified this session.

• App translation: short "why does this work?" prompts attached to a representation the child already knows, for example "why is 6 x 7 the same as 7 x 6?" answered by rotating the array. It is a stretch if it becomes free-text typing for a 9-year-old. It is faithful if the child answers by choosing or manipulating on the representation.

Self-explanation.

• Chi, de Leeuw, Chiu & LaVancher (1994), Cognitive Science, 18(3), doi:10.1207/s15516709cog1803_3: eighth graders prompted to self-explain a biology text understood it better.
• Bisra, Liu, Nesbit, Salimi & Winne (2018), Educational Psychology Review, 30, doi:10.1007/s10648-018-9434-x: g = .55 across 69 effect sizes from 64 reports, across levels of education.
• In mathematics with children: Rittle-Johnson (2006), Child Development, 77(1), doi:10.1111/j.1467-8624.2006.00852.x. Grade 3 to 5 children (ages 8 to 11, n = 85) learning mathematical equivalence transferred better when prompted to self-explain, with or without direct instruction. Self-explanation did not improve an independent conceptual measure.
• App translation: after a worked example, ask "which step made the answer?" or "why did we carry here?" as a menu-based choice between explanations. Menu-based prompts are a documented implementation; the free-form versions are richer. Faithful when the prompts target the principle.

Concrete examples and concreteness fading.

• Rawson, Thomas & Jacoby (2015 in print; online 2014), Educational Psychology Review, doi:10.1007/s10648-014-9273-3: studying several illustrative examples improves learning of declarative concepts (college students).
• Fyfe, McNeil, Son & Goldstone (2014), Educational Psychology Review, 26, doi:10.1007/s10648-014-9249-3: a systematic review supporting a progression from concrete to idealised to symbolic ("concreteness fading") in mathematics and science. The review includes children.
• App translation: this is already the plan's rule 2. Faithful.

Dual coding.

• Clark & Paivio (1991), Educational Psychology Review, 3, doi:10.1007/BF01320076: verbal and imagistic codes are separate and additive.
• IES recommendation 3 (moderate): combine graphics with verbal description.
• The strong form ("two channels always double memory") overreaches. The defensible form is to pair words with a picture that shows the same structure, and cut decorative pictures.
• App translation: narration synchronised with the representation it describes. Faithful.

Testing for complex material and transfer.

• Butler (2010), JEP: LMC, 36(5), 1118-1133, doi:10.1037/a0019902: repeated testing beat repeated study at one week on the same questions, on new inferential questions in the same domain, and on questions from a different domain (undergraduates).
• Pan & Rickard (2018), Psychological Bulletin, 144(7), 710-756, doi:10.1037/bul0000151: d = 0.40 for transfer (192 effect sizes, 122 experiments, N = 10,396).
  • Transfer is greatest across test formats and to application and inference questions.
  • It is weakest to rearranged stimulus-response items, to untested material seen during study, and to problems involving worked examples.
  • Response congruency, elaborated retrieval, and initial test performance strongly moderate transfer. With none of these present, bias-corrected estimates often showed no transfer.
• App translation: (a) practise each direction of a fact family explicitly: multiplication, both divisions, and missing-factor forms. (b) Include application items (word problems, area problems) as retrieval targets in their own right. (c) Ask for elaboration ("show it on the array") on some retrievals. Faithful.

2.4 Metacognition and illusions of learning

Fluency illusions and strategy preference.

• Kornell & Bjork (2008), Psychological Science, 19(6), 585-592, doi:10.1111/j.1467-9280.2008.02127.x: interleaving paintings by artist beat massing (Experiment 1a: .61 vs .35, d = 0.99, N = 120). Seventy-eight percent of participants did better with spacing, yet 78% said massing was as good or better. Across Experiments 1a and 2, 85% did at least as well spaced, and 83% rated massed as equal or better. Undergraduates.
• Kirk-Johnson, Galla & Fraundorf (2019), Cognitive Psychology, 115, 101237, doi:10.1016/j.cogpsych.2019.101237: across three studies (interleaving vs blocking; retrieval vs restudy), the strategy that felt more effortful was rated less effective and chosen less often. This is the "misinterpreted-effort hypothesis".
• App translation: never let the child, or the parent, pick the practice schedule on the basis of how it feels. A parent report can show the mismatch between how a session felt and what it achieved, for example a delayed-probe result next to the session's accuracy. Faithful.

Judgments of learning.

• Nelson & Dunlosky (1991), Psychological Science, 2(4), doi:10.1111/j.1467-9280.1991.tb00147.x: judgments of learning made after a delay, and cued by the stimulus alone, predict later recall far better than immediate ones.
• Rhodes & Tauber (2011), Psychological Bulletin, 137(1), 131-148, doi:10.1037/a0021705: meta-analysis confirming the delayed-JOL effect. Effect size not extracted this session.
• IES 6a (low evidence for the classroom procedure).
• Most of this research is on adults. Children's delayed-JOL accuracy was not verified this session.
• App translation: a "do I know this?" check shown before the answer, a session or more after learning, followed by the actual retrieval. This teaches calibration through the gap between prediction and result. It is a stretch for a 9-year-old unless it is a single tap. The engine should never use the child's judgment to schedule, because judgments are biased.

Self-regulated learning.

• Dent & Koenka (2016), Educational Psychology Review, 28, 425-474, doi:10.1007/s10648-015-9320-8: in elementary and secondary students, achievement correlates r = .20 with metacognitive processes and r = .11 with cognitive strategies. The correlations vary by process, subject, grade, and measure. They are correlational only.
• Teaching strategy choice: McDaniel & Einstein (2020), Perspectives on Psychological Science, 15(6), doi:10.1177/1745691620920723, propose that training must build knowledge of the strategy, belief that it works (ideally from the learner's own experience), commitment, and planning. Each alone is insufficient. It is a framework, not a trial.
• App translation: for older learners on the no-ceiling path, a short in-app experience is a faithful use of the "belief" component. For example, practise half a set blocked and half interleaved, then show the delayed result. For a 9-year-old, keep strategy choice with the engine and explain the benefit to the parent.

2.5 Consolidation: sleep, time of day, rest, and exercise

Sleep.

• Diekelmann & Born (2010), Nature Reviews Neuroscience, 11, 114-126, doi:10.1038/nrn2762: sleep, especially slow-wave sleep, supports consolidation of both declarative and procedural memory, and transforms memories qualitatively (extracting gist and rules).
• Children:
  • Backhaus et al. (2008), Neurobiology of Learning and Memory, 89(1), 76-80, doi:10.1016/j.nlm.2007.08.010: sleep after learning, immediate or delayed, but not wakefulness, enhanced declarative consolidation in children.
  • Wilhelm et al. (2013), Nature Neuroscience, 16(4), 391-393, doi:10.1038/nn.3343: after implicit motor-sequence training, children gained more explicit knowledge of the hidden sequence across sleep than adults did, linked to slow-wave activity.
  • A summary of reviews (Portland Press, Emerging Topics in Life Sciences, 2023, not individually verified) reports reliable sleep benefits for declarative memory in children and weaker, inconsistent ones for procedural memory.
• App translation: an app cannot make a child sleep. It can (a) schedule the first review of new material in the next day's session rather than later the same day, which is faithful to "consolidate, then retrieve". (b) Suggest to parents that a short session before bed suits new content better than a morning session that is followed by a full school day. (b) is a stretch: plausible from the Backhaus design, but not tested as an app intervention.

Time of day and fatigue.

• Sievertsen, Gino & Piovesan (2016), PNAS, 113(10), 2621-2624, doi:10.1073/pnas.1516947113: across all Danish public school children (2009/10 to 2012/13), each hour later in the day lowered test scores by 0.9% of a standard deviation. A 20 to 30 minute break raised them by 1.7% of a standard deviation.
• App translation: treat fatigue as noise in fluency measures. Do not treat a slow late-evening session as evidence of lost mastery. Log the time of day with every response time. Faithful, and cheap.

Wakeful rest.

• Dewar, Alber, Butler, Cowan & Della Sala (2012), Psychological Science, 23(9), doi:10.1177/0956797612441220: 10 minutes of quiet rest after hearing a story, instead of a spot-the-difference game, improved recall at 15 to 30 minutes and at 7 days (adults).
• In 13 to 14 year olds (N = 102), rest beat problem solving at 7 days, but only for children with low immediate recall (Martini et al., 2018/2019, British Journal of Developmental Psychology, doi:10.1111/bjdp.12267).
• The same lab that reported Brokaw et al. (2016) failed to replicate it in a preregistered study (Humiston et al., 2019, Scientific Reports, doi:10.1038/s41598-019-56033-6). Its 10-study meta-analysis still found d = 0.38.
• Weng et al. (2025), Psychonomic Bulletin & Review, doi:10.3758/s13423-025-02665-x: g = 0.45 (37 studies), g = 0.27 at 7 days.
• Parra, Zhang & Radvansky (2026), Psychonomic Bulletin & Review, doi:10.3758/s13423-025-02778-3: 142 effect sizes from 51 studies. Effects are larger in patients and older adults and weaker in young healthy adults.
• App translation: end a Learn segment with a short, quiet screen-off pause, or at least do not start a demanding unrelated game straight after new learning. This is a stretch: the evidence is mixed, strongest in populations unlike a typical child, and never tested inside a learning app. It costs little and does no harm if the pause is short and optional.

Exercise.

• Roig, Nordbrandt, Geertsen & Nielsen (2013), Neuroscience & Biobehavioral Reviews, 37(8), doi:10.1016/j.neubiorev.2013.06.012: acute and long-term cardiovascular exercise both benefit memory in meta-analysis. Reviews since then report larger benefits when exercise follows encoding than when it precedes it, and harm when it happens during encoding (not individually verified here).
• School-based physical activity meta-analyses report chronic programmes improving academic achievement, especially mathematics, with little evidence for single acute bouts (Frontiers in Public Health, 2025, not individually verified).
• App translation: outside the app's remit. At most, suggest active play after a session and not during it. Stretch.

2.6 Transfer and analogical reasoning

• Gick & Holyoak (1980), Cognitive Psychology, 12(3), 306-355, doi:10.1016/0010-0285(80)90013-4: adults who read an analogous military story rarely used it on Duncker's radiation problem until given a hint to use it, after which most did. Gick & Holyoak (1983) reported that comparing two analogues, which induces a schema, raised spontaneous transfer (about 52% vs about 30% for one story, as reported in secondary sources).
• Gentner, Loewenstein & Thompson (2003), Journal of Educational Psychology, 95(2), 393-408, doi:10.1037/0022-0663.95.2.393: comparing two cases side by side ("analogical encoding") produced much more transfer than studying the same two cases one after the other (adult learners of negotiation).
• Alfieri, Nokes-Malach & Schunn (2013), Educational Psychologist, 48(2), doi:10.1080/00461520.2013.775712: a meta-analysis of case comparisons found benefits across laboratory and classroom settings. Effect size not extracted this session.
• Mathematics and children: Rittle-Johnson & Star (2007), Journal of Educational Psychology, 99(3), 561-574, doi:10.1037/0022-0663.99.3.561. Seventy seventh graders who compared two solution methods side by side gained more procedural knowledge and flexibility than those who studied the same methods one at a time.
• Barnett & Ceci (2002), Psychological Bulletin, 128(4), 612-637, doi:10.1037/0033-2909.128.4.612: a nine-dimension taxonomy of transfer distance (content: skill, performance change, memory demands; context: knowledge domain, physical, temporal, functional, social, modality). Evidence for transfer is substantial under some conditions; many critical conditions are untested.
• Sala & Gobet (2017), Current Directions in Psychological Science, 26(6), doi:10.1177/0963721417712760: chess, music, and working-memory training show little far transfer to children's academic skills.
• Varied practice: contextual interference (random rather than blocked practice) was first shown for motor skills (Shea & Morgan, 1979). Brady (2004), Perceptual and Motor Skills, 99(1), doi:10.2466/pms.99.1.116-126: mean effect .38 across 61 studies, but .57 in basic research versus .19 applied, and .50 for adults versus .10 for younger learners. Schmidt & Bjork (1992), Psychological Science, 3(4), doi:10.1111/j.1467-9280.1992.tb00029.x, argued the same pattern holds across motor and verbal learning: manipulations that slow acquisition can improve retention and transfer. In mathematics, interleaving has strong child evidence (section 2.4 and "Strategies apps rarely use").
• App translation: (a) show two strategies or two representations of the same problem side by side and ask what is the same. This is faithful to Rittle-Johnson and Star, and it fits the trick framework (for example, two ways to get 9 x 7). (b) Present each new procedure through at least two surface contexts before expecting transfer. (c) Do not promise "general thinking skills". Faithful.

2.7 Cognitive load theory: newer findings, and learning styles

• Element interactivity. Sweller (2010), Educational Psychology Review, 22, 123-138, doi:10.1007/s10648-010-9128-5: intrinsic load depends on how many elements must be processed together, relative to the learner's expertise. Most cognitive-load effects appear only with high-interactivity material. Times tables facts are low-interactivity (each fact stands alone). Long division and the area model are high-interactivity.
• Expertise reversal. Kalyuga, Ayres, Chandler & Sweller (2003), Educational Psychologist, 38(1), 23-31, doi:10.1207/S15326985EP3801_4: guidance that helps novices becomes redundant, and can hurt, for learners with more knowledge.
• Worked examples and fading.
  • Barbieri, Miller-Cotto, Clerjuste & Chawla (2023), Educational Psychology Review, 35:11, doi:10.1007/s10648-023-09745-1: worked examples in mathematics, g = 0.48 (55 studies, 181 effect sizes), elementary school to adult.
  • Renkl & Atkinson (2003), Educational Psychologist, 38(1), doi:10.1207/S15326985EP3801_3; Atkinson, Renkl & Merrill (2003), Journal of Educational Psychology, 95(4), 774-783, doi:10.1037/0022-0663.95.4.774: fading worked steps backward from the last step, combined with self-explanation prompts, improved transfer (high school and college learners, probability problems).
  • Salden, Aleven, Schwonke & Renkl (2010), Instructional Science, 38, doi:10.1007/s11251-009-9107-8: fading steps adaptively inside a Cognitive Tutor, once the individual learner had mastered each step, outperformed fixed fading in the laboratory study (reported; details not re-read here).
• Guidance fading effect is the general form: move from worked examples to completion problems to full problems as expertise grows.
• Collective working memory. Kirschner, Sweller, Kirschner & Zambrano (2018), International Journal of Computer-Supported Collaborative Learning, 13, 213-233, doi:10.1007/s11412-018-9277-y: collaboration helps when the task exceeds one person's working memory. For simple tasks the cost of coordinating outweighs the benefit. For a single-child app, the only relevance is parent-child joint sessions on high-interactivity topics, which would be a stretch.
• Learning styles. Pashler, McDaniel, Rohrer & Bjork (2008), Psychological Science in the Public Interest, 9(3), 105-119, doi:10.1111/j.1539-6053.2009.01038.x: validating "meshing" needs a crossover interaction (learners with style A do better with method A, those with style B with method B). Almost no studies used that design, and those that did contradicted meshing.
• App translation: fade by step and by learner, with each step's fade triggered by that learner's accuracy on it. Expect fading to matter for multi-step procedures and not for single facts. Never offer a "visual learner mode". Faithful.

2.8 Deliberate practice and expertise

• Ericsson, Krampe & Tesch-Römer (1993), Psychological Review, 100(3), 363-406, doi:10.1037/0033-295X.100.3.363: they defined deliberate practice as effortful, goal-directed practice at the edge of current ability, with immediate informative feedback and repetition, and reported strong links between accumulated practice and violin skill.
• Macnamara, Hambrick & Oswald (2014), Psychological Science, 25(8), 1608-1618, doi:10.1177/0956797614535810: deliberate practice explained 26% of performance variance in games, 21% in music, 18% in sports, 4% in education, and under 1% in professions.
• Macnamara, Moreau & Hambrick (2016), Perspectives on Psychological Science, 11(3), doi:10.1177/1745691616635591: 18% in sports overall, 1% among elite performers.
• Ericsson disputed their broad coding of "deliberate practice", noting that they included lectures and group activities that do not meet his definition. That debate is unresolved.
• App translation: keep the design features that have independent support: tasks at the edge of competence, immediate informative feedback, targeted repetition of errors. Do not claim practice volume guarantees expertise, and do not reward volume (already rule 4). Faithful to the design principles; the 10,000-hours message should be avoided.

2.9 Other strong findings rarely implemented

• Drawing to learn. Fernandes, Wammes & Meade (2018), Current Directions in Psychological Science, 27(5), doi:10.1177/0963721418755385: drawing a to-be-remembered word or definition beats writing it, visualising it, or tracing it, in younger and older adults. Fiorella & Zhang (2018), Educational Psychology Review, 30, doi:10.1007/s10648-018-9444-8, review boundary conditions for learner-generated drawing in science text learning: it helps when drawing is supported, for example with partial drawings or trained, and when the drawing captures the relevant structure. Children's evidence exists in the science-text literature but was not verified here.
  • App translation: ask the child to draw the array or the area model for a new fact. The app cannot easily grade free drawing, so a faithful version uses a constrained canvas: shade the rows, draw the cut. Stretch if the drawing goes ungraded and unexamined.
• Gesture and enactment.
  • Goldin-Meadow, Cook & Mitchell (2009), Psychological Science, 20(3), 267-272, doi:10.1111/j.1467-9280.2009.02297.x: third graders told to produce a correct grouping gesture while solving equivalence problems learned more than children given a partially correct gesture or no gesture. The effect was mediated by children putting into words the information their gestures carried.
  • Novack, Congdon, Hemani-Lopez & Goldin-Meadow (2014), Psychological Science, 25(4), 903-910, doi:10.1177/0956797613518351: action on objects, concrete gesture, and abstract gesture all taught third graders the trained problems, but only gesture generalised.
  • App translation: a physical-gesture instruction ("trace the rows with your finger as you count") before touch input is a faithful, cheap use. Replacing gesture with drag-and-drop is closer to "action on objects", which did not generalise in Novack et al. That substitution is a stretch.
• Handwriting versus typing. The widely cited Mueller & Oppenheimer (2014) laptop result did not replicate robustly (Morehead, Dunlosky & Rawson, 2019, Educational Psychology Review, 31, 753-780, doi:10.1007/s10648-019-09468-2: small, non-significant effects favouring longhand). A meta-analysis of 39 experiments found medium has little influence on achievement in controlled settings (Voyer, Ronis & Byers, 2022, Contemporary Educational Psychology, 68, 102025, doi:10.1016/j.cedpsych.2021.102025). For mathematics specifically, the evidence is thin and comes from human-computer interaction. Anthony, Yang & Koedinger reported that handwritten input in an algebra tutor gave similar learning in less time than typing, with middle and high school students (2007 workshop paper, doi:10.1145/1290144.1290153; later journal versions not re-read). Oviatt's work reports that pen input elicits more diagrams than keyboards (not verified here). Early letter learning favours handwriting in young children, but that is literacy, not mathematics. Flag: mixed and not specific to arithmetic.
• Interleaved review inside new lessons, and cumulative tests. Rohrer et al. (2020) interleaved earlier problem types into later assignments across four months, with a delayed test (section 3, rank 1). The IES guide recommends revisiting content after weeks to months. Cumulative review is the classroom analogue of an app mixing old strands into current practice.
• Spaced re-exposure to worked examples. Distributed practice research (Cepeda, Pashler, Vul, Wixted & Rohrer, 2006, Psychological Bulletin, 132(3), doi:10.1037/0033-2909.132.3.354) applies to studying, not only testing. Re-presenting a worked example briefly at a delay before the problem set that uses it is a direct extension. No app-level trial was found. This is a faithful extension of the mechanism but not a tested package.

3. Strategies apps rarely use (ranked)

Ranked first by strength of evidence for this age and domain, then by how easy an app can implement it. "Evidence" reflects child and mathematics evidence where it exists.

4. Contested or weak evidence

• Learning styles and meshing: no supporting crossover evidence (Pashler et al., 2008). Avoid.
• Deliberate practice as the main driver of expertise: 4% of variance in education (Macnamara et al., 2014). The design principles stand; the causal claim is disputed.
• Delayed feedback for children: the benefit is shown for adults on transfer. For fact learning in children, immediate correction is safer. Unresolved.
• Wakeful rest: pooled effects are positive (d = 0.38 to g = 0.45), but many preregistered failures to replicate and weaker effects in healthy young adults (Humiston et al., 2019; Parra et al., 2026).
• Productive failure for young children: favours instruction first for grades 2 to 5 (Sinha & Kapur, 2021). Do not use invention-first designs with the core audience.
• Contextual interference for children: in motor learning the effect is small for younger learners (.10; Brady, 2004). The mathematics-interleaving findings are stronger and cover different mechanisms. Do not generalise from motor studies.
• Handwriting versus typing: the laptop note-taking effect did not replicate robustly. Mathematics-specific evidence is thin, from human-computer interaction papers, and mostly with older students.
• Prequestions for non-asked content: essentially zero (g = 0.01 to 0.04).
• Dual coding "strong form" and "brain training" far transfer: both overclaimed.
• Acute exercise before learning: mixed, and outside an app's remit.
• Correlational self-regulation findings (Dent & Koenka, 2016): small correlations that do not establish that teaching self-regulation raises achievement.

5. References

All DOIs checked against Crossref or publisher pages this session unless noted.

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Items cited from secondary sources only, and not verified here: de Lima & Jaeger (2020, JARMAC), cited via Pan & Carpenter (2023); the Gick & Holyoak (1983) percentages; the 2023 Emerging Topics in Life Sciences summary of sleep reviews; the 2025 school physical activity meta-analysis; Oviatt's pen-interface figures; and the Shea & Morgan (1979) DOI. Volume and page numbers for the other entries were checked against Crossref metadata.