How people learn mathematics: evidence from cognition and classrooms, and what an app can take from it

Literature review prepared 2026-09-27 for Numberkit. Scope: mathematical cognition and mathematics education research (classrooms, tutoring, laboratory and field experiments), from early number through algebra and adult numeracy. Research on educational games and apps is deliberately excluded. Citations were checked against publisher pages, ERIC, PubMed, NCES, government report PDFs, or the Crossref index during preparation. Figures that come from memory and could not be re-checked in this session are marked "(unverified)". Ages are given for each study where the source reports them. Effect sizes are standardised mean differences (d or g) unless stated as correlations (r).

This review overlaps with research 02 (concepts, representations, and explanation) and research 01 (fact fluency and anxiety). It does not repeat their findings on retrieval practice, spacing, and timing; it adds the cognitive and classroom evidence around them.

1. Summary of key findings

  1. Numerical magnitude is the thread that runs through all of school number. Symbolic magnitude comparison correlates with mathematics achievement at r = .30, more than non-symbolic (dot) comparison at r = .24 (Schneider et al. 2017, 284 effects, N = 17,201). Number line estimation correlates at r = .44, and the correlation rises with age because it is strongest for fractions (Schneider et al. 2018, ages 4 to 14, N = 10,576). Siegler's integrated theory (Siegler, Thompson & Schneider 2011) treats whole numbers, fractions, decimals, and negatives as one magnitude system on one line. An app that places every new kind of number on the same number line is building on the best-supported idea in the field.
  2. Training the approximate number system (dots) is not a route to arithmetic. The correlation with achievement is real but small; a systematic review found no conclusive evidence that ANS training improves symbolic mathematics (Szucs & Myers 2017). Spend time on symbolic magnitude, not dot games.
  3. Fraction and division knowledge at age 10 to 12 predicts high-school mathematics five or six years later, after controlling for whole-number arithmetic, IQ, working memory, and family income (Siegler et al. 2012, US and UK national samples). Equal-sign knowledge at age 7 to 8 predicts algebra two years later (Matthews & Fuchs 2020, N = 177). These two are the highest-leverage content areas for a primary-age app that intends to keep going.
  4. Spatial training transfers to mathematics, modestly. Spatial skills are trainable (g = .47, 217 studies; Uttal et al. 2013), and spatial training improves mathematics (g = .28, 29 studies, N = 3,765; Hawes, Gilligan-Lee & Mix 2022).
  5. Comparing is one of the strongest classroom-tested moves. Comparing two solution methods side by side beat studying them one at a time for procedural knowledge and flexibility in 12- to 13-year-olds (Rittle-Johnson & Star 2007). Comparing incorrect with correct examples beat two correct examples for decimal magnitude in 9- to 11-year-olds (Durkin & Rittle-Johnson 2012). The IES algebra practice guide (Star et al. 2015) builds three recommendations on solved problems, structure, and choosing among strategies.
  6. Misconceptions are predictable, and some are manufactured by practice. Whole-number bias in fractions and decimals (Van Hoof et al. 2015), the operational reading of "=" (McNeil & Alibali 2005; Knuth et al. 2006), and letters read as objects (Kuchemann 1978) all follow from the exercises children do. Changing the format of arithmetic practice (operations on the right, "is the same as", problems grouped by equal sums) improved equivalence understanding in 7- to 8-year-olds (McNeil et al. 2015, N = 166). This is cheap for an app to do and rarely done.
  7. Explaining helps, if the explanation is scaffolded. Self-explanation prompts: g = .55 across domains (Bisra et al. 2018); in mathematics, small to moderate immediate gains, stronger with scaffolding, weaker evidence for retention and for classrooms (Rittle-Johnson, Loehr & Durkin 2017). An app can prompt and scaffold self-explanation; it cannot replace a listener who asks follow-up questions.
  8. Word problems yield to schema instruction, not to general heuristics. Teaching problem types and their structure produced large effects in 8- to 9-year-olds (Fuchs et al. 2008, "Hot Math", ES = 1.34 for tutored at-risk pupils) and moderate, durable effects for ratio and proportion in 12- to 13-year-olds (Jitendra et al. 2009, d = .45 post, .56 at 4 months). General Polya-style heuristics without domain knowledge are weak (Schoenfeld 1985).
  9. Cross-cultural "mastery" teaching is partly evidenced and partly enthusiasm. The English trial of Mathematics Mastery found d = .10 in Year 1 and .06 in Year 7 (pooled .07, just significant; Jerrim & Vignoles 2016). The Shanghai teacher exchange found positive KS1 effects in a subsample and no evidence of KS2 effects (Boylan et al. 2019). The ingredients that have their own experimental support (variation, comparison, representations, spacing, interleaving) are more portable than the packages.
  10. Ordering matters within a topic; the developmental sequence helps but is not magic. Learning-trajectory curricula have large effects in preschool (Building Blocks: .47 versus another curriculum, 1.07 versus control; Clements & Sarama 2008), and teaching levels in order beat "teach to the target" in follow-up tests, but children in the alternatives also learned, just less.
  11. Math anxiety operates through working memory in adults as well as children (Ashcraft & Kirk 2001), is reliably associated with lower achievement (Hembree 1990; Namkung, Peng & Lin 2019), and falls when skills rise. Adult numeracy research is thin; its best-supported points are relevance of context, diagnostic starting points, and the anxiety of returners (Alma Economics 2023; Coben et al. 2007).

2. Detailed findings

2.1 Mathematical cognition

The approximate number system (ANS). Halberda, Mazzocco and Feigenson (2008, Nature 455, 665-668, doi:10.1038/nature07246) found that 14-year-olds' precision at judging which of two briefly flashed dot arrays was more numerous correlated with their mathematics achievement back to kindergarten, independent of IQ and visuospatial ability (N = 64). This launched a decade of work. The Schneider et al. (2017, Developmental Science 20, e12372, doi:10.1111/desc.12372) meta-analysis (45 articles, 284 effects, N = 17,201, lifespan) found non-symbolic comparison correlated with mathematics at r = .24 and symbolic (digit) comparison at r = .30, significantly higher. Training studies are the causal test. Park and Brannon (2013, Psychological Science, doi:10.1177/0956797613482944) reported that ANS training improved adults' arithmetic, but Szucs and Myers (2017, Trends in Neuroscience and Education 6, 187-203, doi:10.1016/j.tine.2016.11.002) reviewed the training literature and concluded there is no conclusive evidence that ANS training improves symbolic arithmetic, noting that many "ANS" studies trained symbolic skills too and that contested results are cited without their controversies.

How it could translate to an app. Faithful: assess and practise symbolic magnitude (which is bigger, 0.3 or 0.25; place 3/8 on a line). A stretch: dot-comparison games as a path to arithmetic. The evidence says do not build them.

Symbolic number and the number line. Siegler and Booth (2004, Child Development 75, doi:10.1111/j.1467-8624.2004.00684.x) showed that children's placements on a 0-100 line shift from compressed (logarithmic-looking) to linear between kindergarten and second grade, and linearity correlates with arithmetic. Fazio, Bailey, Thompson and Siegler (2014, Journal of Experimental Child Psychology, doi:10.1016/j.jecp.2014.01.013) found in 10- to 11-year-olds that fraction magnitude knowledge correlated with achievement more strongly than whole-number or non-symbolic magnitude. The Schneider et al. (2018, Child Development 89(5), doi:10.1111/cdev.13068) meta-analysis (263 effects, N = 10,576, mean ages 4 to 14) found number line estimation correlated with broader mathematics at r = .44; the correlation grew with age because it was higher for fractions.

The integrated theory. Siegler, Thompson and Schneider (2011, Cognitive Psychology 62(4), 273-296, doi:10.1016/j.cogpsych.2011.03.001) argue that numerical development is a process of learning that all real numbers have magnitudes that can be located on a number line, and of learning which properties of whole numbers do not generalise (a unique successor, "more digits means bigger", multiplication always makes bigger). Siegler and Braithwaite (2017, Annual Review of Psychology 68, doi:10.1146/annurev-psych-010416-044101) extend this to negatives and decimals. The most direct intervention evidence is Fuchs et al. (2013, Journal of Educational Psychology 105(3), 683-700): at-risk 9- to 10-year-olds taught fractions through the measurement (number line) interpretation outperformed controls taught mainly part-whole and procedures, and the effects were mediated by improved magnitude understanding.

How it could translate to an app. Faithful and highly feasible: one persistent number line that extends as the child advances (0-100, then 0-1000, then fractions between 0 and 2, decimals, then negatives), with placement tasks as both teaching and assessment. Numberkit's Line Landing is already this; the evidence supports making it the spine of fractions and decimals rather than a side activity, and explicitly revisiting "which whole-number rule breaks here".

Working memory. Peng, Namkung, Barnes and Sun (2016, Journal of Educational Psychology 108, 455-473, doi:10.1037/edu0000079; 110 studies, 829 effects) found r = .35 between working memory and mathematics, similar for verbal, numerical, and visuospatial working memory, strongest for word problems and whole-number calculation, weakest for geometry. Working-memory training itself does not transfer well to mathematics (that literature is outside this review's scope and is consistent with research 06). The practical implication is load management: fluent facts free capacity (National Mathematics Advisory Panel 2008), and presentation should not add load.

How it could translate to an app. Faithful: keep extraneous load low in word problems (one representation at a time, text read aloud, diagrams that carry the structure), and treat fact fluency as a prerequisite for multi-step work, not a goal in itself.

Spatial skills. Mix and Cheng (2012, Advances in Child Development and Behavior 42, 197-243) review a well-established correlation between spatial ability and mathematics and call for mechanistic accounts. Uttal et al. (2013, Psychological Bulletin 139(2), 352-402, doi:10.1037/a0028446; 217 studies) found spatial training effective (g = .47), durable, and transferable to untrained spatial tasks. Hawes, Gilligan-Lee and Mix (2022, Developmental Psychology, 29 studies, N = 3,765, k = 89) found spatial training improved spatial skill (g = .49) and mathematics (g = .28). Mental rotation, visualising transformations, and number line tasks are the plausible bridges.

How it could translate to an app. Moderately faithful: spatial tasks tied to mathematics (composing and decomposing rectangles for the area model, folding and partitioning for fractions, mental rotation of arrays for commutativity) are both spatial practice and mathematics. Stand-alone "brain training" spatial puzzles would be a stretch and would break the project rule that the mechanic is the math.

2.2 Learning trajectories and Cognitively Guided Instruction

Learning trajectories. Clements and Sarama (2004, Mathematical Thinking and Learning 6(2), doi:10.1207/s15327833mtl0602_1) define a trajectory as a goal, a developmental progression of levels of thinking, and instructional tasks matched to each level. Their Building Blocks preschool curriculum, evaluated in a cluster RCT with 36 classrooms and 26 weeks of instruction (Clements & Sarama 2008, American Educational Research Journal 45(2), 443-494, doi:10.3102/0002831207312908), produced effects of .47 against a comparison curriculum and 1.07 against business as usual (ages 3 to 5). Later tests of the trajectory assumption itself (for example Clements et al. 2019 on shape composition; Sarama et al. 2021 on length measurement, AERA Open) compared teaching each level in order with teaching directly at the target level: the ordered sequence was more efficacious, but children in the counterfactual also learned. A 2020 three-arm trial (JRME 51(3)) found Building Blocks effects mostly positive but only some significant. Confrey and colleagues (Confrey, Maloney, Nguyen & Rupp 2014, in Learning over Time, IAP) developed an equipartitioning trajectory (fair sharing, from halving to arbitrary splits) as the foundation for rational number, starting with the child's own sharing actions.

How it could translate to an app. Faithful: a prerequisite graph whose nodes are levels of thinking, not just topics, with diagnostic items per level. Numberkit's KC graph is structurally this. For fractions, the equipartitioning trajectory (Fair Shares) is a well-reasoned entry point, though its evidence is design research and clinical interviews rather than RCTs.

Cognitively Guided Instruction (CGI). Carpenter, Fennema, Peterson, Chiang and Loef (1989, American Educational Research Journal 26(4), 499-531, doi:10.3102/00028312026004499) randomly assigned 40 first-grade teachers (children aged 6 to 7) to a month-long workshop on the research-based taxonomy of addition and subtraction problem types (join, separate, part-part-whole, compare) and children's strategies (direct modelling, counting, derived facts), or to control. Instructional practices were not prescribed. CGI teachers taught more problem solving and fewer facts, listened more to children's strategies, and their pupils did better on problem solving and, as commonly reported, no worse on number facts (the fact-recall comparison is recalled from secondary summaries, unverified here). The mechanism is teacher knowledge of how children's strategies develop, which the teacher uses to pose the next problem.

How it could translate to an app. Partly faithful. The problem-type taxonomy and the strategy progression (model, count, derive, recall) can be encoded: generate word problems across all types (including the hard ones: compare, start-unknown), and infer strategy from response time and errors. The core of CGI, a teacher listening to a child describe a strategy and responding, depends on a live teacher. An app can ask the child to pick which of several strategy pictures matches what they did, which is an approximation with unknown validity.

2.3 International and cross-cultural evidence

TIMSS video studies. Stigler and Hiebert (1999, The Teaching Gap, Free Press) and Hiebert et al. (2003, Teaching Mathematics in Seven Countries, NCES 2003-013) compared videotaped eighth-grade lessons (ages 13 to 14). The key finding is not that high-achieving countries use a single method; it is that they keep the cognitive demand of "making connections" problems during the lesson. Of problems stated as making connections, less than 1 percent in the US and 8 percent in Australia were discussed in a way that made the connections, against 37 to 52 percent in the other countries. Japanese lessons in 1995 typically centred on one problem solved several ways and then compared. This is correlational, based on a sample of lessons, and cannot show that the practice causes achievement.

Lesson study. A US cluster RCT of lesson study supported by a fractions resource kit (39 teams, mostly elementary teachers, 3 months) improved teachers' and students' fractions knowledge relative to two controls (Lewis & Perry 2017, JRME 48(3), 261-299). The EEF trial of a different lesson study version in English primary schools found no evidence of impact on KS2 attainment (EEF 2017, Murphy et al.). The US positive result involved research-based fractions materials (including linear measurement models), so it may be the materials as much as the process.

Mastery in England. Mathematics Mastery, one year, two RCTs (Year 1: 83 schools, 4,176 pupils; Year 7: 44 schools, 5,938 pupils): d = .10 and .06, neither individually significant, pooled .073 just significant (Jerrim & Vignoles 2016, Economics of Education Review; EEF 2015 reports, ERIC ED581180). The Shanghai Mathematics Teacher Exchange evaluation (Boylan et al. 2019, Sheffield Hallam for DfE) found positive effects on KS1 attainment in the subsample of schools implementing most fully, similar in size to Mathematics Mastery's KS1 effect, and no evidence of effect at KS2; the design was quasi-experimental and the KS1 measure is teacher-assessed. The Inspire Maths (Singapore textbook) Year 1 trial (576 pupils) reported more progress with longer use, and teachers reported lower attainers struggling with the language load (Hall et al., Exeter, 2016; effect sizes not re-checked, unverified). Commentators note that over 70 million pounds of public investment in Teaching for Mastery has produced inconclusive outcome evidence (Boylan and colleagues; Oxford Review of Education 2023 article on "hyperreal" policy borrowing).

Variation theory and bianshi. Marton's variation theory (Marton & Pang 2006, Journal of the Learning Sciences 15(2), doi:10.1207/s15327809jls1502_2) holds that learners discern a feature only when it varies against a background that stays invariant. Gu, Huang and Marton (2004, in Fan et al., How Chinese Learn Mathematics, 309-347) describe Chinese "conceptual variation" (varying non-examples and representations to isolate the concept) and "procedural variation" (a problem changed step by step so each new problem builds on the last). The theory is coherent and widely used in England's mastery materials, but direct experimental tests of bianshi sequences against well-designed alternatives are few; much of the evidence is learning-study design research. The closest rigorous relatives are the comparison studies below (Rittle-Johnson & Star 2007; Alfieri, Nokes-Malach & Schunn 2013, Educational Psychologist, doi:10.1080/00461520.2013.775712, a meta-analysis of case comparisons).

Realistic Mathematics Education (RME). Freudenthal Institute (Netherlands, from 1971): problems in contexts the child can imagine, progressive formalisation through models (the empty number line, the bar/strip, the ratio table). Its models are influential and several have independent support (the number line above). Outcome evidence for RME as a package is mostly quasi-experimental, often small, and concentrated in a few countries; a recent meta-analysis (American Journal of STEM Education) reports positive effects but on this weak base.

How it could translate to an app. Faithful: variation-designed item sets (one feature changes, the rest held), "one problem, several methods, then compare", the bar model and number line as persistent models. A stretch: claiming "mastery" or "Shanghai" as a validated package; claiming whole-class discussion benefits. The TIMSS finding (keeping connection problems demanding) translates as a design rule: do not let hints collapse a connection task into a procedure.

2.4 Big syntheses

US National Mathematics Advisory Panel (2008, Foundations for Success, ERIC ED500486). Among 45 findings: conceptual understanding, procedural fluency, and automatic recall of facts are mutually reinforcing, and debates about their relative importance are misguided; children should have immediate recall of facts to free working memory; proficiency with fractions (including decimals and percent) is a major goal and is severely underdeveloped; explicit instruction benefits students with learning difficulties; neither wholly teacher-directed nor wholly student-centred instruction is supported by research. The panel's evidence bar was strict and it found few high-quality studies for many questions.

IES practice guide on algebra (Star et al. 2015, NCEE 2015-4010). Three recommendations for ages 11 to 18: (1) use solved problems to engage students in analysing algebraic reasoning and strategies (minimal evidence rating, as recalled; the guide's ratings are unverified here); (2) teach students to use the structure of algebraic representations (for example seeing 3(x+2) = 12 as "something times 3 is 12"); (3) teach students to intentionally choose from alternative strategies, including comparing them. Key studies behind it include Rittle-Johnson and Star (2007, Journal of Educational Psychology 99(3), 561-574, doi:10.1037/0022-0663.99.3.561; 70 seventh-graders randomly assigned; comparing methods side by side improved procedural knowledge and flexibility with comparable conceptual gains).

Worked examples. Barbieri, Miller-Cotto, Clerjuste and Chawla (2023, Educational Psychology Review 35, doi:10.1007/s10648-023-09745-1; 55 studies, 181 effects, elementary to adult) found g = .48 for worked examples in mathematics; correct examples alone produced larger average gains than incorrect ones. The expertise reversal effect (Kalyuga et al. 2003, Educational Psychologist, doi:10.1207/s15326985ep3801_4) predicts that examples help novices and should fade as skill grows.

Productive failure. Sinha and Kapur (2021, Review of Educational Research 91(5), 761-798, doi:10.3102/00346543211019105; 53 studies, 166 comparisons, N > 12,000) found problem solving before instruction improved conceptual knowledge and transfer (d = .36; up to .58 with high fidelity) without harming procedures, but effects favoured instruction first for younger learners (grades 2 to 5).

EEF. The EEF guidance "Improving Mathematics in Key Stages 2 and 3" (2017, updated 2022) makes eight recommendations: use assessment to build on existing knowledge; use manipulatives and representations; teach strategies for solving problems; build a rich network of mathematical knowledge; develop independence and motivation (metacognition); use tasks and resources to challenge and support; use structured interventions for additional support; support primary-to-secondary transition. The accompanying evidence review (Hodgen et al. 2018, Leicester/Nottingham) grades evidence behind each.

Hattie and meta-meta-analysis. Hattie's Visible Learning (2009; sequel 2023) ranks influences by average effect size. Simpson (2017, Journal of Education Policy 32(4), 450-466, doi:10.1080/02680939.2017.1280183) shows that standardised effect sizes depend on comparison group, range restriction, and test design, so league tables of averaged effects may rank research designs rather than teaching approaches. Use Hattie's list to find literature, not to choose features.

How it could translate to an app. Faithful: worked examples that fade; structure-reading items; strategy-choice items; diagnostic starting points (EEF 1). A stretch: productive failure for 9-year-olds, which the meta-analysis itself flags as reversing at primary age.

2.5 Conceptual change and misconceptions

Whole-number bias. Children apply natural-number properties to rational numbers: longer decimal is bigger (0.25 > 0.3), bigger denominator is bigger fraction, multiplying always makes bigger, there is no number between 0.3 and 0.4. Resnick et al. (1989, JRME 20(1), doi:10.5951/jresematheduc.20.1.0008) described decimal rules children use. Van Hoof, Verschaffel and Van Dooren (2015, Educational Studies in Mathematics, doi:10.1007/s10649-015-9613-3) tracked the bias from primary through secondary: weakest for size, stronger for operations, strongest for density, and still present in secondary pupils. Lortie-Forgues, Tian and Siegler (2015, Developmental Review) explain why fraction and decimal arithmetic is hard and note little improvement in proficiency over three decades.

The equals sign. Many children read "=" as "put the answer here". McNeil and Alibali (2005, Child Development, doi:10.1111/j.1467-8624.2005.00884.x) showed that knowledge of the operational pattern (operations on the left, answer on the right) hinders learning equations. Knuth, Stephens, McNeil and Alibali (2006, JRME 37(4), 297-312) found that middle-school students (ages 11 to 14) with a relational view solved equations better. Matthews and Fuchs (2020, Child Development 91(1), e14-e28, doi:10.1111/cdev.13144; N = 177, mean age 7.6) found second-grade equal-sign knowledge predicted fourth-grade algebra after controls. The repair is cheap: McNeil, Fyfe and Dunwiddie (2015, Journal of Educational Psychology 107(2), 423-436; 166 second-graders) varied the format of ordinary fact practice (__ = 4 + 3; "is the same as"; problems grouped by equal sums such as 4 + 3, 5 + 2, 6 + 1) and improved equivalence understanding relative to traditional workbooks, with the same amount of practice.

Negative numbers. Bofferding (2014, JRME 45(2), 194-245; 61 first-graders) found children first apply whole-number principles to negatives (for example -8 > -2 because 8 > 2) and some later build formal mental models, supporting instruction that contrasts order and magnitude.

Letters as objects. Kuchemann (1978, Mathematics in School; CSMS test of about 3,000 English pupils aged 12 to 15) classified letter interpretations from "letter evaluated" to "letter as variable", and identified "letter as object" (a for apples) as common. A 2025 latent transition analysis in Educational Studies in Mathematics calls the misconception "omnipresent". Fruit-salad algebra ("3a + 2b is 3 apples and 2 bananas") manufactures it.

Repairs. Contrasting correct and incorrect examples: Durkin and Rittle-Johnson (2012, Learning and Instruction 22; N = 74, ages 9 to 11) found comparing incorrect with correct decimal examples beat comparing two correct ones for procedures and concepts and reduced misconceptions. Refutation texts (explicitly naming and refuting a misconception) have solid support in science education; their evidence in mathematics specifically is thinner (not verified in this session). Cognitive conflict alone (showing the child a counterexample) can entrench a misconception if the child lacks a replacement idea; the conceptual-change literature (Vosniadou's framework theory, applied to rational number by Vamvakoussi and Vosniadou) treats the change as gradual, with synthetic models along the way.

How it could translate to an app. Highly faithful and rarely done: (a) write fact practice in equivalence formats from the start; (b) diagnose misconceptions from answer patterns (Numberkit's bug rules already do this) and, when one matches, show the child an incorrect worked example to critique beside a correct one; (c) place fractions, decimals, and negatives on the same line and ask "which whole-number rule broke?"; (d) never use letters as object labels.

2.6 Mathematical discourse and explanation

Explaining to others. Webb (1991, JRME 22(5), 366-389) found in small-group mathematics that giving elaborated explanations is associated with higher achievement, giving only answers is not, and receiving less help than requested is negatively associated. Fiorella and Mayer (2013, Contemporary Educational Psychology 38(4), 281-288) found that actually teaching (explaining on video) produced better delayed learning than only expecting to teach. Chi and Wylie's ICAP framework (2014, Educational Psychologist, doi:10.1080/00461520.2014.965823) orders engagement as interactive > constructive > active > passive.

Self-explanation. Chi et al. (1994, Cognitive Science, doi:10.1207/s15516709cog1803_3) showed prompting self-explanations improved understanding. Bisra et al. (2018, Educational Psychology Review; 64 reports, 69 effects) found g = .55. In mathematics, Rittle-Johnson, Loehr and Durkin (2017, ZDM 49(4), 599-611) found small to moderate immediate effects on procedural and conceptual knowledge and transfer, stronger when explanations were scaffolded, with limited evidence for classrooms and delayed retention. Fuchs et al. (2016, Journal of Educational Psychology, doi:10.1037/edu0000073) added supported self-explaining (children taught to explain with a model explanation and structure) to a fraction intervention for 9- to 10-year-olds; the supported-explaining arm improved explanation quality (details of other outcomes not re-checked, unverified).

Analogies. Richland, Zur and Holyoak (2007, Science 316, doi:10.1126/science.1142103) found that teachers in Hong Kong and Japan provided more cognitive supports for analogies (visual side-by-side representations, gestures between them) than US teachers.

How it could translate to an app. Partly faithful. Feasible: menu-based or fill-in self-explanations ("Why is 0.3 bigger? Because the 3 is in the tenths / because 3 > 25"), explanation of a worked example step, choosing which of two explanations is right, and "tell a grown-up how" prompts routed through the parent screen. Research 02 and Numberkit's A.5 task already do some of this. Honest limits: the benefits of discourse come from a responsive listener who probes, disagrees, and asks for justification; free-text or spoken explanations cannot be deterministically evaluated, and the project rule forbids live AI evaluation of the child. The parent script (Task 3.11) is the most faithful route to real explanation to another person.

2.7 Problem solving and word problems

Schema-based instruction (SBI). Children are taught to recognise problem types (change, combine or group, compare, equal groups, ratio) by their underlying structure, represent them with a schematic diagram, and then solve. Fuchs et al. (2008, Exceptional Children; also the classroom trial in Journal of Educational Psychology) randomised 119 third-grade classes (ages 8 to 9) and at-risk pupils within them; tutored pupils in Hot Math classrooms outperformed tutored pupils in conventional classrooms, ES = 1.34. Hot Math adds "schema broadening": explicitly teaching that superficial features (irrelevant information, different vocabulary, charts) do not change the type, which supports transfer. Jitendra et al. (2009, Contemporary Educational Psychology 34, 250-264; seventh-graders, classroom randomisation, 10 sessions) found d = .45 on proportional problem solving, maintained at 4 months (d = .56). A PMC review (Powell 2011, "Solving word problems using schemas") summarises the program of work.

General heuristics. Polya's four phases (understand, plan, carry out, look back) are widely taught. Schoenfeld (1985, Mathematical Problem Solving) found that general heuristics are too broad to use without domain-specific tactics, metacognitive control, and knowledge; heuristics instruction worked in his university course when heavily elaborated. Evidence for teaching generic heuristics to children is weak; the EEF's "teach strategies for solving problems" recommendation leans on worked examples, representations, and metacognition rather than generic steps.

Estimation and checking. Number line and magnitude knowledge predict estimation and achievement (above). A "does this make sense?" check (estimate first, then compare) is a natural corollary but the specific effect of an estimation-checking habit on error rates in children has limited direct experimental evidence (not verified here).

Fluency and working memory. Price, Mazzocco and Ansari (2013, Journal of Neuroscience, doi:10.1523/JNEUROSCI.2936-12.2013) found that the degree to which adolescents used fact-retrieval brain networks during single-digit arithmetic predicted their PSAT maths scores. With Peng et al. (2016) showing word problems are the most working-memory-dependent skill, the case that fluency frees resources for problem solving is plausible and endorsed by the National Mathematics Advisory Panel, though mostly correlational.

How it could translate to an app. Faithful: word problems generated by schema type (theme slots fill the nouns, which fits Numberkit's typed theme slots), a schema diagram (bar model) the child fills before calculating, schema broadening (irrelevant numbers, varied wording), and "estimate first" before multi-digit calculation. A stretch: teaching Polya steps as a poster.

2.8 Algebra readiness and later mathematics

Siegler et al. (2012, Psychological Science 23(7), 691-697, doi:10.1177/0956797612440101) analysed US and UK national longitudinal data: fraction and division knowledge at ages 10 to 12 uniquely predicted algebra and overall achievement at 16 to 17, controlling for whole-number arithmetic, IQ, working memory, and family income and education. Booth and Newton (2012, Contemporary Educational Psychology; middle school) found fraction magnitude knowledge, especially of unit fractions, related to algebra readiness more than whole-number magnitude. Matthews and Fuchs (2020) link early equals-sign knowledge to algebra. Lee, Ng, Bull and Pe (2011, Journal of Educational Psychology 103, doi:10.1037/a0023068) linked pattern proficiency to algebraic word problems in Singaporean children. Jordan et al. (2009, Developmental Psychology, doi:10.1037/a0014939) found kindergarten number competence predicted mathematics through third grade.

How it could translate to an app. Faithful: treat fractions-as-numbers, division (including remainders and division as the inverse of multiplication), and relational equivalence as priority strands, not waypoints. Beyond primary, the IES algebra guide gives a design brief: analyse solved problems, read structure, choose strategies. An app with no age ceiling should add structure-reading items early (for example, 37 + 48 = 40 + __ ) so algebra is not a new language at 11.

2.9 Adolescent and adult learning, and math anxiety

Anxiety and working memory. Ashcraft and Kirk (2001, Journal of Experimental Psychology: General 130, 224-237) showed in adults that high math anxiety reduces working memory span during arithmetic and that the performance deficit is largest when problems load working memory (carrying). Hembree (1990, JRME 21(1), 33-46; 151 studies) found anxiety related to poorer performance and avoidance, and that one-to-one treatments (systematic desensitisation, cognitive restructuring) reduce anxiety and raise performance. Namkung, Peng and Lin (2019, Review of Educational Research, doi:10.3102/0034654319843494) found a moderate negative correlation among school-age students (about r = -.34, as reported in research 01). Ramirez, Shaw and Maloney (2018, Educational Psychologist, doi:10.1080/00461520.2018.1447384) frame anxiety as reciprocal with avoidance and skill. Park, Ramirez and Beilock (2014, Journal of Experimental Psychology: Applied 20(2), 103-111; N = 80 undergraduates) found brief expressive writing before a test reduced the gap between high- and low-anxious students. Sammallahti et al. (2023, meta-analysis of 50 studies) report math-anxiety interventions reduce anxiety (g = -.47) and raise performance (g = .50), with cognitive-support and emotion-regulation approaches effective and older students (over 12) showing the largest anxiety decrease (journal and DOI not re-checked, unverified).

Adult numeracy. The DfE-commissioned systematic review (Alma Economics 2023, 209 studies) found the evidence mostly small case studies; its strongest finding is that adults engage more with content situated in a context relevant to them; lecturing is unlikely to work; diagnostic assessment helps but can alienate returners, so should be welcoming; teacher subject expertise predicts outcomes. The NRDC effective-practice study (Coben et al. 2007) found adult learners diverse, many (mostly women) anxious about returning, and confidence rising with achievement. The Multiply programme (England, 2022-2025, 270 million pounds, about 210,000 learners) engaged many anxious adults, but its evaluation could not show improved numeracy; six adult numeracy RCTs (family numeracy, adapted mastery, contextualised approach, and others) were published on GOV.UK in 2026 (results not read in this session).

Which principles hold across ages. Worked examples (g = .48, elementary to adult), self-explanation (lifespan), magnitude processing (lifespan association), comparison (children to undergraduates), interleaving (Rohrer et al. 2020, 54 seventh-grade classes, d = .83 at one month), and the anxiety-working-memory mechanism all have evidence in both children and adults. What differs for adults: stronger need for relevance, prior negative school experiences, and the risk that child-like presentation alienates.

How it could translate to an app. Faithful: the core engine rules (worked examples fading, comparison, spacing, interleaving, low-stakes timing) do not need an age switch. For older learners, a non-childish skin and contexts drawn from their life (money, measurement, work) matter, and themes as data files make this possible without changing mechanics. Diagnostic placement should feel like a welcome, not a test.

3. Principles apps rarely apply (ranked by evidence and feasibility)

  1. Equivalence-format fact practice. Write some fact items as __ = 4 x 3, 12 = 3 x __, and 3 x 4 = 2 x __, and group items by equal results. RCT evidence in 7- to 8-year-olds (McNeil et al. 2015), longitudinal link to algebra (Matthews & Fuchs 2020), zero extra time. Feasibility: very high.
  2. One number line for every number. Placement tasks as the spine of whole numbers, fractions, decimals, and negatives, with explicit "which rule broke" moments (Siegler et al. 2011; Schneider et al. 2018; Fuchs et al. 2013). Feasibility: high; Numberkit has Line Landing.
  3. Correct and incorrect examples side by side, triggered by a diagnosed misconception. (Durkin & Rittle-Johnson 2012; Barbieri et al. 2023; Rittle-Johnson & Star 2007.) Feasibility: high, since bug rules already detect the misconception.
  4. Compare two correct methods. "Priya did it this way, Sam did it that way. Which is quicker here? When would you use each?" (Rittle-Johnson & Star 2007; IES algebra guide Rec. 3). Feasibility: high for multi-digit arithmetic, tricks, and later algebra.
  5. Schema-typed word problems with a diagram step and schema broadening. (Fuchs et al. 2008; Jitendra et al. 2009.) Feasibility: high with typed theme slots.
  6. Variation-designed item sequences. Change one feature at a time; include near-miss non-examples (Gu, Huang & Marton 2004; Marton & Pang 2006; comparison meta-analysis Alfieri et al. 2013). Evidence: moderate (theory strong, direct tests fewer). Feasibility: high in generators.
  7. Scaffolded self-explanation prompts (menu or fill-in, not free text), fading with skill (Rittle-Johnson et al. 2017; Bisra et al. 2018). Feasibility: medium-high.
  8. Structure-reading items early (compensation, 37 + 48 = 40 + __, seeing 3(x+2) as a chunk later; IES algebra Rec. 2). Feasibility: high.
  9. Spatial tasks that are mathematics (partitioning, composing areas, rotating arrays; Hawes et al. 2022). Evidence moderate for transfer; feasibility medium.
  10. Explain-to-a-person prompts via the parent (Webb 1991; Fiorella & Mayer 2013). Evidence moderate; depends on a live adult; the app's role is to supply the question and what a good answer contains.

4. Contested or weak evidence

  • ANS training as a route to arithmetic: not supported (Szucs & Myers 2017). The ANS-achievement correlation is real but small and weaker than symbolic.
  • "Mastery" and "Shanghai maths" as packages: small effects in RCTs (d about .07 pooled), KS1-only quasi-experimental positives, inconclusive national outcomes. Components are better evidenced than the brand.
  • Bianshi / variation theory: coherent, widely adopted, few controlled experiments.
  • Realistic Mathematics Education as a package: mostly quasi-experimental evidence of low quality; specific RME models (number line, bar) have better independent support.
  • Lesson study: one positive US RCT with research materials, one null English RCT.
  • Productive failure for primary pupils: the meta-analysis shows effects reverse for grades 2 to 5.
  • Generic problem-solving heuristics (Polya steps): weak without domain knowledge.
  • Hattie-style effect-size rankings: methodologically compromised for choosing interventions (Simpson 2017).
  • Self-explanation retention and classroom effects: immediate lab effects are solid; delayed and classroom effects less so.
  • Learning-trajectory ordering: ordered sequences beat teach-to-target, but the advantage is moderate and children learn under alternatives too.
  • Adult numeracy programmes: evidence base mostly case studies; Multiply could not show skill gains.
  • CGI outcome details: the 1989 RCT is strong on teacher behaviour; the pupil-outcome specifics here are from secondary summaries.
  • Refutation texts in mathematics and estimation-as-checking habits: plausible, with less direct mathematical evidence than their popularity suggests (not verified in this session).

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