Teaching Math Concepts to 9-Year-Olds with Visual Models, Animation and Explanation: Evidence Review

Compiled 2026-09-24. Product context: a concept-teaching math app for a 9-year-old (US grade 3-5 / UK Year 4-5) whose school app, Times Tables Rock Stars (TTRS, UK-origin), is pure timed answer-drilling with no visual models or explanations.

Verification note: every citation below was checked against at least one primary or indexing source (publisher page, ERIC, PubMed, Semantic Scholar DOI record, or the original PDF) during this review unless it is explicitly marked "unverified" or "from recollection". Numbers marked "from recollection" are effect sizes I could not re-read from the primary source because the publisher blocked access; treat them as approximate.

1. Summary of key findings

  • Concrete-to-abstract "concreteness fading" is the best-supported way to sequence representations. It beats concrete-only, abstract-only, and abstract-to-concrete on transfer in children (Fyfe, McNeil & Borjas 2015, grades 2-3) and adults (Goldstone & Son 2005; McNeil & Fyfe 2012). The three-step sequence (physical/animated objects -> schematic picture -> symbols) should be explicit, with each step linked to the last.
  • Manipulatives in general help (Carbonneau, Marley & Selig 2013: 55 studies, N=7,237, small-to-moderate d overall, moderate-to-large for retention), but the effect depends on explicitly linking the manipulative to the symbols and fading it. Both the WWC (2021) and the EEF (2022) say manipulatives are a scaffold to be removed once the child is accurate without them.
  • The number line is the single most evidence-backed representation for this age. The WWC 2021 guide gives it its own "strong evidence" recommendation (14 studies). Number line training causally improves whole-number arithmetic learning (Booth & Siegler 2008), fraction magnitude understanding in grades 2-3 (Hamdan & Gunderson 2017, number line beat the area/pie model), and at-risk fourth graders' fraction learning (Fuchs et al. 2013, n=259, ES 0.29-2.50; five RCTs summarised in Fuchs et al. 2017).
  • Conceptual and procedural knowledge develop iteratively and bidirectionally (Rittle-Johnson, Siegler & Alibali 2001; Rittle-Johnson, Schneider & Star 2015). There is no strong evidence that concepts must always come first, but for 7-9-year-olds, conceptual instruction before problem solving outperformed the reverse (Fyfe, DeCaro & Rittle-Johnson 2014), and spending more time on the concept beat splitting time between concept and procedure (Rittle-Johnson, Fyfe & Loehr 2016).
  • Worked examples work (Barbieri et al. 2023 meta-analysis: g=0.48, 55 studies), and incorrect examples paired with correct ones help 4th-5th graders learn decimals (Durkin & Rittle-Johnson 2012). Self-explanation prompts have a robust effect (Bisra et al. 2018: g=0.55, 69 effects), but for young children the prompts should ask the child to explain a high-quality explanation given to them, not invent one (Rittle-Johnson 2006; Fuchs et al. 2017 "supported self-explaining").
  • Productive failure (problem solving before instruction) has a moderate positive meta-analytic effect overall (Sinha & Kapur 2021: g=0.36), but the effect reverses for grades 2-5, and a Year 5 experiment found explicit-instruction-first clearly better (Ashman, Kalyuga & Sweller 2020). For a 9-year-old, explain-first (with a brief warm-up exploration at most) is the safer default.
  • The WWC 2021 elementary intervention guide rates all six of its recommendations "strong": systematic instruction, mathematical language, concrete/semi-concrete representations, number lines, word-problem instruction, and brief timed fluency activities. Timed activities are recommended only after concepts have been taught, never to introduce content.
  • Multimedia design principles transfer to children with one important twist: for primary-school children, narration should be spoken rather than written when paired with pictures (Herrlinger et al. 2017, grade 4). Seductive details (decorative extras) reliably hurt learning (Sundararajan & Adesope 2020). Animation beats static pictures modestly (Höffler & Leutner 2007: d=0.37), mostly when the animation shows the actual process rather than decorating it.
  • Number Talks have almost no rigorous efficacy evidence (Matney, Lustgarten & Nicholson 2020). The claim that timed tests cause math anxiety (Boaler 2014) rests on correlational data and is contested; the WWC gives brief timed fluency practice a strong rating.
  • Curriculum: US grade 3 = multiplication/division within 100 fluency and unit fractions; grade 4 = multi-digit multiplication, fraction equivalence/addition, decimals; grade 5 = fluent multi-digit multiplication, fraction operations, decimal operations. England Year 4 requires recall of tables to 12 x 12 and a statutory 25-question, 6-seconds-per-question Multiplication Tables Check at age 8-9.

2. Detailed findings

2.1 CRA and concreteness fading

The Concrete-Representational-Abstract (CRA) sequence comes from special-education research (Bruner's enactive-iconic-symbolic stages). "Concreteness fading" is the cognitive-science version: Goldstone and Son (2005, Journal of the Learning Sciences 14(1), 69-110, doi:10.1207/s15327809jls1401_4) had undergraduates learn complex-systems principles from simulations whose elements were concrete throughout, idealized throughout, concrete-then-idealized, or idealized-then-concrete. Transfer was best when concrete elements became idealized.

Fyfe, McNeil, Son and Goldstone (2014, Educational Psychology Review 26, 9-25, doi:10.1007/s10648-014-9249-3) reviewed the math and science evidence and proposed a three-step progression: (1) concrete, enactive representation, (2) an iconic/schematic picture that strips extraneous detail, (3) abstract symbols. Their argument: concrete grounding helps interpret symbols and build memorable images, while fading prevents the well-documented failure of concrete-only learning to transfer (Kaminski, Sloutsky & Heckler 2008, Science 320, 454-455, doi:10.1126/science.1154659, found that undergraduates who learned a group-theory structure via generic symbols transferred better than those taught with concrete measuring-cup instantiations).

The key child study is Fyfe, McNeil and Borjas (2015, Learning and Instruction 35, 104-120, doi:10.1016/j.learninstruc.2014.10.004). Second and third graders learned mathematical equivalence (e.g., 3 + 4 = __ + 2) with puppets and balance scales (concrete), symbols only (abstract), concrete faded to abstract, or abstract then concrete. Concreteness fading produced the best transfer, beat the reverse order, and helped both low and high prior-knowledge children. McNeil and Fyfe (2012, Learning and Instruction 22, 440-448) found the same pattern with undergraduates on modular arithmetic.

Carbonneau, Marley and Selig (2013, Journal of Educational Psychology 105(2), 380-400, doi:10.1037/a0031084) meta-analysed 55 studies (N=7,237, K through college) comparing manipulatives to symbols-only instruction: small-to-moderate positive effect overall, moderate-to-large on retention, small on problem solving, transfer and justification, moderated by instructional features (explicit linking, amount of guidance).

Practice guidance converges. WWC (2021, Recommendation 3) says representations are "thinking tools" that must be explicitly connected to notation, that a few exposures are not enough, and: "Only fade out concrete and semi-concrete representations as students become accurate with doing the work abstractly... Revisit concrete and semi-concrete representations periodically." The EEF (2022, Recommendation 2) says manipulatives "should act as a 'scaffold', which can be removed once independence is achieved", that the removal decision should follow "the pupils' improved knowledge and understanding, not their age", and that children moving away from manipulatives "may find it helpful to draw diagrams or imagine using the manipulatives".

Sequencing in a digital app: the WWC guide explicitly notes that pictures of concrete and semi-concrete representations "are sometimes presented virtually on a computer or tablet screen", i.e., on-screen draggable objects count as the concrete stage. The evidence supports: animated objects the child acts on (stage 1) -> the same quantities redrawn as arrays / bars / number lines with the objects removed (stage 2) -> symbols only, with stage-2 pictures available on demand (stage 3). Fading should be triggered by accuracy at the current stage, not by a fixed timer, and earlier stages should be revisited when a new sub-concept is introduced.

2.2 Visual models: what the evidence supports for each

Number line (whole numbers). Booth and Siegler (2008, Child Development 79(4), 1016-1031, doi:10.1111/j.1467-8624.2008.01173.x): first graders' number-line estimation accuracy predicted arithmetic learning after controlling for prior arithmetic, memory and achievement, and randomly assigned children shown accurate visual magnitude representations of addends and sums learned unfamiliar problems better. Siegler and Ramani (2008, Developmental Science 11, 655-661; Ramani & Siegler 2008, Child Development 79, 375-394): four 15-20 minute sessions of a linear (not circular, not colour) 1-10 board game eliminated the low-income/middle-income gap in preschoolers' magnitude comparison, number-line estimation, counting and numeral identification, with gains at 9-week follow-up. Number-line estimation in kindergarten predicts later achievement (Jordan, Kaplan, Ramineni & Locuniak 2009, Developmental Psychology 45(3), 850-867), and Siegler et al. (2012, Psychological Science 23(7), 691-697) found fraction and division knowledge at age 10-12 predicted high-school algebra and overall math achievement in US and UK cohorts after controlling for IQ, family income and other math.

Number line (fractions). Siegler, Thompson and Schneider (2011, Cognitive Psychology 62, 273-296) argue fraction learning is fundamentally about magnitude on a line. Hamdan and Gunderson (2017, Developmental Psychology 53(3), 587-596) randomised second and third graders to number-line training, area-model (circle) training, or control; only number-line training transferred to a symbol-only fraction comparison task. Gunderson et al. (2019, Journal of Experimental Child Psychology) followed up showing the one-dimensionality of the number line is the critical feature. Fuchs et al. (2013, Journal of Educational Psychology 105(2), doi:10.1037/a0032446): 259 at-risk fourth graders, 12 weeks, 3 x 30 min/week; a measurement-interpretation (number-line-centred) intervention beat a part-whole/procedures control on every conceptual and procedural outcome (ES 0.29 to 2.50), and gains in number-line accuracy mediated the effect. Fuchs, Malone, Schumacher, Namkung and Wang (2017, Journal of Learning Disabilities 50(6), 631-639, doi:10.1177/0022219416677249) summarise five RCTs: mean Hedges g of 1.72 on fraction calculation and 0.58 on released NAEP items, with the achievement gap to non-at-risk peers narrowing by about 1 SD, "even though the control group allocated more instructional time to computational procedures".

Arrays and area model (multiplication). Barmby, Harries, Higgins and Suggate (2009, Educational Studies in Mathematics 70, 217-241, doi:10.1007/s10649-008-9145-1) studied primary children using an interactive array on laptops; the array supported reasoning about commutativity and distributivity, but this is a qualitative/descriptive study, not an RCT. The WWC 2021 guide lists arrays among recommended semi-concrete representations. Direct causal evidence that the area model beats other multiplication representations is thin; the case rests on the WWC "strong" representation rating and on the Common Core progressions, which build multi-digit multiplication and the standard algorithm on area/array decompositions.

Bar models / Singapore tape diagrams. WWC "Improving Mathematical Problem Solving in Grades 4 Through 8" (2012, rev. 2018) rates "teach students how to use visual representations" as strong evidence, based on six studies meeting standards, mostly schema-based instruction in which students learn to match problem type to a diagram (Jitendra and colleagues). The Singapore-specific evidence is weaker: the WWC intervention report on Singapore Math (December 2015) found no studies meeting standards; Jaciw et al. (2016, Journal of Research on Educational Effectiveness 9(4), doi:10.1080/19345747.2016.1164777) ran an RCT of Math in Focus with mixed/small effects (details behind paywall; unverified beyond citation). Bar-model studies at elementary level are mostly single-case or quasi-experimental (e.g., Mahoney 2012 dissertation, single-subject). Conclusion: schema diagrams for word problems are well supported; the bar model specifically is a plausible instance, not separately proven.

Base-ten blocks and place value. Fuson and Briars (1990, JRME 21(3), 180-206) taught first and second graders (N=169 and 75) multidigit addition/subtraction with base-ten blocks, immediately recording each block action in numerals; most classes demonstrated meaningful place-value and four-digit addition, but this is a teaching experiment without a randomised control. Carpenter, Franke, Jacobs, Fennema and Empson (1998, JRME 29(1), 3-20) followed 82 children grades 1-3: those who used invented strategies before learning the standard algorithm showed better base-ten knowledge and better transfer. Both support blocks plus explicit linking to written notation.

Ratio tables. I found no controlled evaluations of ratio tables specifically at grades 3-5; they are a Realistic Mathematics Education device supported mainly by design research. Treat as plausible, unproven.

2.3 Procedural vs conceptual knowledge

Rittle-Johnson, Siegler and Alibali (2001, Journal of Educational Psychology 93(2), 346-362) proposed the iterative model from decimal-fraction studies: conceptual gains improve procedures, procedural gains improve concepts, mediated by better problem representation. Rittle-Johnson, Schneider and Star (2015, Educational Psychology Review 27, 587-597, doi:10.1007/s10648-015-9302-x) reviewed the evidence and concluded the relations are bidirectional, but that "alternative orderings of instruction on concepts and procedures have rarely been compared, with limited empirical support for one ordering of instruction over another".

Child experiments since then:

  • Fyfe, DeCaro and Rittle-Johnson (2014, British Journal of Educational Psychology 84, 502-519, doi:10.1111/bjep.12035): 122 second and third graders; conceptual instruction before problem solving produced greater conceptual and procedural knowledge of equation structures than instruction after, and raised the quality of explanations and attempted procedures.
  • DeCaro and Rittle-Johnson (2012, Journal of Experimental Child Psychology 113(4), 552-568): 159 second to fourth graders; exploring unfamiliar problems before instruction improved understanding versus instruct-then-practice, because exploration led children to try more strategies and attend to key features. (The two Vanderbilt studies differ in what the "instruction" contained; the 2014 paper reconciles them by arguing telling-first works when the instruction is conceptual and the problems are novel enough to be confusing without it.)
  • Rittle-Johnson, Fyfe and Loehr (2016, British Journal of Educational Psychology 86, 576-591, doi:10.1111/bjep.12124): 180 second graders; two doses of conceptual instruction produced better retention of both conceptual and procedural knowledge than one dose of conceptual plus one of procedural instruction, and order relative to problem solving did not matter. The authors conclude "spending more time on conceptual instruction may be more beneficial than time spent teaching a procedure".
  • Fuchs et al. (2013) delayed procedural fraction instruction until lesson 22 of 36, after the magnitude concept was established, and still beat a procedure-heavy control on procedures.

Bottom line: teach the concept and the procedure in the same unit, concept-heavy, with the procedure derived from the representation rather than presented as a separate rule.

2.4 Worked examples, self-explanation, comparison, productive failure, expertise reversal

Worked examples. Sweller and Cooper (1985, Cognition and Instruction 2(1), 59-89) showed studying worked algebra examples beat solving equivalent problems. Renkl (2014, Cognitive Science 38(1), 1-37, doi:10.1111/cogs.12086) integrates the literature: examples work for novices, should be accompanied by self-explanation prompts, and should fade into completion problems and then full problems as competence grows. Barbieri, Miller-Cotto, Clerjuste and Chawla (2023, Educational Psychology Review 35, article 11, doi:10.1007/s10648-023-09745-1): 55 studies, 181 effects, elementary through adult; mean g=0.48 on math performance (about 18 percentile points); moderators included correct vs incorrect examples, pairing with self-explanation, and timing.

Incorrect examples. Durkin and Rittle-Johnson (2012, Learning and Instruction 22, 206-214): 74 fourth and fifth graders learning decimal magnitude; comparing correct and incorrect examples improved procedures and concepts and reduced misconceptions more than correct examples alone. This is directly the target age.

Self-explanation. Chi, Bassok, Lewis, Reimann and Glaser (1989, Cognitive Science 13, 145-182) originated the effect. Bisra, Liu, Nesbit, Salimi and Winne (2018, Educational Psychology Review 30, 703-725, doi:10.1007/s10648-018-9434-x): 69 effects from 64 reports, random-effects g=0.55. With children the picture is subtler: Rittle-Johnson (2006, Child Development 77(1), 1-15): 85 third to fifth graders learning equivalence; self-explanation promoted transfer of a correct procedure regardless of whether the procedure was taught or invented, but did not improve an independent conceptual measure, because children's own explanations rarely included conceptual content. Fuchs et al. (2017) therefore used "supported self-explaining", in which at-risk fourth graders practise reproducing a high-quality explanation of fraction comparisons; this improved magnitude understanding in the Year 4 RCT.

Comparison of solution methods. Rittle-Johnson and Star (2007, Journal of Educational Psychology 99(3), 561-574): 70 seventh graders; comparing two methods side by side beat studying them sequentially on procedural knowledge and flexibility (WWC: meets standards without reservations). Star and Rittle-Johnson (2009, Journal of Experimental Child Psychology 102, 408-426): 157 fifth and sixth graders learning computational estimation; comparison improved flexibility at posttest and retention, and improved conceptual knowledge mainly for those with some prior knowledge. Rittle-Johnson, Star and Durkin (2009, Journal of Educational Psychology 101(4), 836-852): novices who did not already know one method benefited from comparison only when the pace was slowed; comparison is most effective once the child owns at least one method.

Productive failure / invention. Schwartz and Martin (2004, Cognition and Instruction 22(2), 129-184) and Schwartz, Chase, Oppezzo and Chin (2011, Journal of Educational Psychology 103(4), 759-775) showed with high-schoolers that inventing with contrasting cases before a lecture improves transfer. Sinha and Kapur (2021, Review of Educational Research 91(5), doi:10.3102/00346543211019105): 53 studies, 166 comparisons; problem-solving-then-instruction beat instruction-then-problem-solving overall (g=0.36, 95% CI 0.20-0.51; 0.37-0.58 with high fidelity), but the abstract explicitly reports that the advantage reversed for younger learners in grades 2-5. Loibl, Roll and Rummel (2017, Educational Psychology Review 29, 693-715) show the effect depends on contrasting cases and on the instruction building on students' attempts. Ashman, Kalyuga and Sweller (2020, Educational Psychology Review 32, 229-247): two randomised experiments with Year 5 pupils (N=64 and 71); explicit instruction first beat problem-solving first on similar problems, and also on transfer when element interactivity was high.

Expertise reversal. Kalyuga, Ayres, Chandler and Sweller (2003, Educational Psychologist 38(1), 23-31) and Kalyuga (2007, Educational Psychology Review 19, 509-539): guidance that helps novices (worked examples, integrated explanations) becomes redundant or harmful as knowledge grows; instruction should be adapted to measured knowledge.

Explain first or try first? For a 9-year-old on a new concept: explain first with a worked example on the visual model, then practise with faded support. A short "what do you think happens?" prediction before the explanation is defensible (DeCaro & Rittle-Johnson 2012) as long as it is brief, feedback follows immediately, and the explanation then references the child's attempt. Once the child has one working method, switch to comparison and completion problems, and reduce explanation length (expertise reversal).

2.5 IES / What Works Clearinghouse practice guides

Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades (WWC 2021006, March 2021, grades K-6). All six recommendations rated strong: (1) systematic instruction, 43 studies; (2) mathematical language, 16 studies; (3) well-chosen concrete and semi-concrete representations, 28 studies (19 without reservations); (4) number lines, 14 studies; (5) deliberate word-problem instruction, 18 studies; (6) timed activities, 27 studies (21 without reservations). On timed activities the panel writes that they should last 1-5 minutes, be added "once students have been working on a concept over many lessons", and "Do not use timed activities to introduce and teach mathematics concepts and operations"; it "does not recommend merely giving students timed worksheets or putting students on a computer-based program without supporting their learning". Timed activity should include real-time feedback, improvement goals and gradually harder items.

Developing Effective Fractions Instruction for K-8 (NCEE 2010-4039). Rec 1 (build on sharing/proportionality intuitions): minimal. Rec 2 (fractions are numbers; use number lines): moderate. Rec 3 (understand why computation procedures make sense): moderate. Rec 4 (proportional reasoning before cross-multiplication): minimal. Rec 5 (teacher PD): minimal.

Improving Mathematical Problem Solving in Grades 4 Through 8 (NCEE 2012-4055, rev. 2018). Rec 1 (prepare problems for whole-class use): minimal. Rec 2 (monitor and reflect on the problem-solving process): strong. Rec 3 (teach visual representations): strong. Rec 4 (expose students to multiple strategies): moderate. Rec 5 (articulate concepts and notation): moderate.

The UK EEF guidance Improving Mathematics in Key Stages 2 and 3 (2017, updated 2022) has eight recommendations: assessment, manipulatives and representations, problem-solving strategies, a rich network of mathematical knowledge, independence and motivation, tasks that challenge, structured interventions, and transition support. It rates the evidence descriptively rather than with tiers.

2.6 Multimedia principles for children

Mayer's cognitive theory of multimedia learning (Mayer 2009/2014, Multimedia Learning, Cambridge University Press) yields the familiar principles. Cromley and Chen (2025, Educational Research Review 49, 100730) meta-analysed 92 Mayer-authored articles (181 studies, 591 effects): removing seductive details g=1.00, modality g=0.82, personalization g=0.70, multimedia g=0.68, sentence-level coherence g=0.63, cueing/signaling g=0.24. Independent meta-analyses: signaling (Schneider, Beege, Nebel & Rey 2018, Educational Research Review 23, 1-24; small-to-medium effects on retention and transfer across school and university samples), segmenting (Rey et al. 2019, Educational Psychology Review 31, 389-419), conversational style/personalization (Ginns, Martin & Marsh 2013, Educational Psychology Review 25, 445-472).

Seductive details. Sundararajan and Adesope (2020, Educational Psychology Review 32, 707-734, doi:10.1007/s10648-020-09522-4) meta-analysed 35 years of research: interesting-but-irrelevant text, images, sounds and animations reliably reduce learning, small-to-moderate overall, larger with dynamic media and system-paced delivery. Decorative animation and reward-style visual noise inside an explanation are the concrete risk for a children's app.

Animation vs static. Höffler and Leutner (2007, Learning and Instruction 17(6), 722-738): 26 studies, 76 comparisons, d=0.37 for animation over static pictures; d=0.40 when the animation is representational (it depicts the process being learned) rather than decorative, d=1.06 for procedural-motor knowledge. Berney and Bétrancourt (2016, Computers and Education 101, 150-167, doi:10.1016/j.compedu.2016.06.005) updated this with a smaller but still positive overall effect (about g=0.2, from recollection; unverified against the paper). Animation earns its place when it shows a quantity being regrouped, an array being split, or a point sliding on a number line, not when it is a mascot.

Children specifically. Most multimedia studies use adults. Herrlinger, Höffler, Opfermann and Leutner (2017, Research in Science Education 47, 685-704) tested fourth graders learning from a biology text: pictures helped, but only when the text was spoken; written text plus pictures overloaded children's visual channel. Ashman et al. (2020) also found high element-interactivity content requires explicit guidance for Year 5 pupils. Practical reading: for a 9-year-old, narrate explanations aloud over the visual, keep on-screen text minimal and synchronised, segment into short learner-paced steps, and signal the relevant part of the picture as it is mentioned.

2.7 Number sense, estimation, subitizing, number talks, flexible strategies

Number sense and estimation. Kindergarten number competence (counting, magnitude comparison, number-line estimation, simple arithmetic) predicts growth in achievement through grade 3 (Jordan et al. 2009). Magnitude representations are causally related to arithmetic learning (Booth & Siegler 2008), and linear number-line games improve them (Siegler & Ramani 2008). Fraction magnitude is the analogous foundation at grades 3-5 (Siegler et al. 2011; Fuchs et al. 2013). Subitizing (instant recognition of small quantities, and "conceptual subitizing" of dot patterns as composed groups) is advocated by Clements (1999, Teaching Children Mathematics 5(7), 400-405), a practitioner article; the causal evidence at age 9 is thin and mostly embedded in early-numeracy programmes.

Number Talks. Parrish (2010, Number Talks; 2011, Teaching Children Mathematics 18(3), 198-206) and Boaler (2015, Mathematical Mindsets; youcubed "Fluency Without Fear") promote short mental-computation discussions. Matney, Lustgarten and Nicholson (2020, Investigations in Mathematics Learning, "Black holes of research on instructional practice: the case of Number Talks") searched the peer-reviewed literature and found "a shallow depth of articles" from which efficacy could be judged; the evidence base is dissertations and practitioner pieces. The mechanism (comparing strategies aloud) is supported indirectly by the comparison studies in 2.4, but Number Talks as a package are unproven.

Flexible strategies vs standard algorithms. Carpenter et al. (1998) found children who invented strategies before the algorithm had better base-ten understanding and transfer. Blöte, Van der Burg and Klein (2001, Journal of Educational Psychology 93, 627-638; unverified in this review) reported Dutch second graders taught flexible strategies before a single procedure were more flexible and accurate. But Torbeyns and Verschaffel (2016, European Journal of Psychology of Education 31, 99-116) showed Flemish fourth graders default to the written algorithm even on items designed to invite mental strategies, and choose strategies by personal mastery rather than by number features; adaptive choice develops slowly. For multiplication facts, Woodward (2006, Learning Disability Quarterly 29(4), 269-289): 58 fourth graders (15 with IEPs); strategy instruction integrated with timed practice matched drill-only on automaticity and beat it on application and maintenance measures. Codding, Burns and Lukito (2011, Learning Disabilities Research and Practice 26, 36-47) found in single-case studies that drill with modelling and multi-component packages produced the largest fluency gains. Net: teach derived-fact strategies (doubling, 5s-plus-one-group, 9s as 10s-minus-one, commutativity, distributivity on an array) and then drill, rather than drill alone.

2.8 Curriculum scope

US Common Core (standards verified from the CCSS 3-5 domain progressions; narrative progressions at achievethecore.org/page/254 and ime.math.arizona.edu/progressions):

Grade 3. OA: interpret products and quotients (equal groups, arrays, area), properties of operations (commutative, associative, distributive) as strategies, division as unknown factor, 3.OA.7 "fluently multiply and divide within 100" with all products of two one-digit numbers known from memory by end of grade, two-step word problems, arithmetic patterns. NBT: round to 10/100, 3.NBT.2 fluently add and subtract within 1000, multiply one-digit numbers by multiples of 10. NF: unit fractions 1/b as one part of a whole partitioned into b equal parts, fractions as numbers on the number line (3.NF.2), equivalence and comparison with same numerator or denominator; denominators limited to 2, 3, 4, 6, 8. Measurement: area as tiling, connecting area to multiplication and to the distributive property (3.MD.7), which is the seed of the area model.

Grade 4. OA: multiplicative comparison, multi-step problems with remainders, factors and multiples, primes, patterns. NBT: place value to 1,000,000 with "a digit in one place represents ten times what it represents in the place to its right", 4.NBT.4 fluently add and subtract multi-digit numbers with the standard algorithm, multiply up to four-digit by one-digit and two-digit by two-digit "using strategies based on place value and the properties of operations" illustrated by "equations, rectangular arrays, and/or area models", divide up to four-digit dividends by one-digit divisors with the same representations. NF: equivalence via (n x a)/(n x b) with visual models, comparison by common denominators or benchmarks, addition/subtraction with like denominators including mixed numbers, multiplying a fraction by a whole number, decimals for tenths and hundredths and comparison; denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, 100.

Grade 5. OA: order of operations with grouping symbols, numerical patterns. NBT: place value to thousandths, powers of ten, 5.NBT.5 fluently multiply multi-digit whole numbers with the standard algorithm, divide by two-digit divisors, add/subtract/multiply/divide decimals to hundredths with models. NF: add and subtract with unlike denominators, fraction as division, multiply fractions by fractions with area models, scaling, divide unit fractions by whole numbers and vice versa.

England National Curriculum (statutory programmes of study, DfE 2013; text verified on gov.uk). Year 4 (age 8-9): count in multiples of 6, 7, 9, 25 and 1,000; place value in four-digit numbers; add and subtract to four digits with columnar methods; "recall multiplication and division facts for multiplication tables up to 12 x 12"; multiply two- and three-digit by one-digit using formal written layout; decimal equivalents of tenths and hundredths; add and subtract fractions with the same denominator. Year 5 (age 9-10): numbers to at least 1,000,000; columnar methods beyond four digits; long multiplication of up to four digits by two digits; short division; add and subtract fractions whose denominators are multiples of the same number; multiply proper fractions and mixed numbers by whole numbers; percentages; decimals to three places.

Multiplication Tables Check (MTC). Statutory in England from the 2021/22 academic year for all Year 4 pupils in state schools, taken in June when most pupils are 8 or 9. 25 questions, 6 seconds each with a 3-second gap, drawn from the 2 to 12 tables (DfE, "Multiplication tables check attainment" statistics and methodology; STA assessment framework 2018, updated 2022). The framework caps how often each table appears, and practice sites such as Mathsframe reproduce the distribution showing the 6, 7, 8, 9 and 12 tables carrying higher maximum counts than 2, 3, 4, 5, 10 and 11; I could not re-read the framework PDF itself in this review, so treat the exact weighting as unverified. TTRS is a commercial UK product built to train exactly this format; it measures recall speed and does not teach the meaning of multiplication, which is the gap this project targets.

3. Design implications for a concept-teaching app

  1. Every new concept starts on a representation the child can act on, then fades. Sequence: (a) draggable objects or an animated scene (equal groups, a length being measured), (b) the same quantities as an array/area model, bar, or number line with the objects gone, (c) symbols with the picture one tap away. Advance a stage only when the child is accurate at the current one; drop back one stage on repeated errors (WWC 2021 Rec 3; Fyfe et al. 2015; EEF 2022).
  2. Make the number line the backbone for whole-number magnitude, rounding, fractions and decimals. Every fraction is placed on a 0-1 (or 0-2) line before any part-whole picture; equivalent fractions land on the same point; comparison is "which is further right" (WWC 2021 Rec 4; Hamdan & Gunderson 2017; Fuchs et al. 2013).
  3. Use the area/array model as the single multiplication representation from 3 x 4 through 23 x 47, splitting the rectangle to show distributivity, then mapping each partial product to a line of the written method. Do not introduce the algorithm as a separate rule (CCSS 4.NBT.5 progression; Barmby et al. 2009).
  4. Explain first, then practise, for a 9-year-old on new material. Optionally open with a single 20-second prediction or attempt, then explain on the visual model and reference the child's attempt. Do not run extended discovery phases (Sinha & Kapur 2021 reversal for grades 2-5; Ashman et al. 2020; Fyfe et al. 2014).
  5. Budget more instruction time to the concept than to the procedure; derive the procedure from the model (Rittle-Johnson et al. 2016; Fuchs et al. 2013).
  6. Start each skill with fully worked examples on the model; fade to completion problems (child fills one missing step) and then full problems as accuracy rises. Track accuracy and cut explanation length for a child who is already accurate (Renkl 2014; Barbieri et al. 2023; Kalyuga 2007).
  7. Include incorrect worked examples ("Sam says 0.35 > 0.5 because 35 > 5. Where did Sam go wrong?") once the child has a correct method, especially for decimals and fraction comparison (Durkin & Rittle-Johnson 2012).
  8. Self-explanation should be supported, not open-ended: present a short correct explanation and ask the child to complete or re-say it (choose the reason, drag the justification), rather than "explain why". Open "why" prompts for 8-10-year-olds produce procedural, not conceptual, explanations (Rittle-Johnson 2006; Fuchs et al. 2017).
  9. After the child owns one strategy, show two strategies side by side on the same problem and ask which is easier and why (Rittle-Johnson & Star 2007; Star & Rittle-Johnson 2009). Never compare before the child has one method (Rittle-Johnson, Star & Durkin 2009).
  10. Narrate explanations with spoken audio synchronised to the animation; keep on-screen words to labels and the equation. Written paragraphs beside a picture overload a 9-year-old's visual channel (Herrlinger et al. 2017; modality principle).
  11. Animation must depict the mathematics (groups merging, a rectangle splitting, a point sliding). No mascot reactions, confetti, sound effects or background motion during an explanation; keep rewards to the end of a session (Sundararajan & Adesope 2020; Höffler & Leutner 2007 decorative vs representational).
  12. Segment explanations into learner-paced steps of a few seconds each with a "next" tap, and highlight the part of the model being talked about at that moment (segmenting and signaling).
  13. Use conversational second person ("you have 4 rows...") and the child's name where natural (personalization).
  14. Keep timed fluency practice, but only for facts and sub-skills already taught conceptually, in 1-5 minute bursts, with a visible personal-best chart and gradually harder sets; never use a timer during a concept lesson (WWC 2021 Rec 6; Woodward 2006).
  15. Teach derived-fact strategies explicitly for tables (x2 double, x4 double-double, x5 half of x10, x9 as x10 minus one group, x6 as x5 plus one group, x12 as x10 plus x2) on the array before drilling them, and align drill to the MTC format (2-12, 6-second items) so the child sees the app as helping with the school check.
  16. Interleave problem types in practice sets after initial acquisition (e.g., mix multiplication, division-as-unknown-factor and comparison problems) rather than blocking; a preregistered cluster RCT with seventh graders found d=0.83 for interleaved over blocked practice on a delayed test (Rohrer, Dedrick, Hartwig & Cheung 2020, Journal of Educational Psychology 112(1), 40-52). The age is older than 9, so treat as a strong default rather than a proven effect at grade 4.
  17. Word problems: teach the child to draw the diagram for a problem type (equal groups, comparison, part-part-whole) before writing an equation, and mark why keyword rules fail (WWC 2012 Rec 3; WWC 2021 Rec 5).
  18. Use precise mathematical language consistently (factor, product, numerator, denominator, equal groups) with a tap-to-hear glossary (WWC 2021 Rec 2).
  19. Revisit earlier representations at the start of each new sub-topic (e.g., reopen the array when moving from 6 x 7 to 60 x 7) rather than assuming the abstraction is permanent (WWC 2021 Rec 3 panel advice).
  20. Sequence content to the CCSS 3-5 / England Y4-5 progression above: whole-number magnitude and place value -> multiplication/division meaning and facts -> area model to multi-digit -> fractions as numbers on the line -> equivalence and comparison -> like-denominator operations -> decimals as fractions -> unlike denominators.

4. Contested or weak evidence

  • Productive failure with young children. The headline meta-analytic effect (Sinha & Kapur 2021) is positive, but its own moderator analysis reverses for grades 2-5, and the only randomised primary-school tests I found (Ashman et al. 2020) favour explicit-first. Vanderbilt's own studies split (DeCaro & Rittle-Johnson 2012 vs Fyfe et al. 2014). Do not build the app around invention-first.
  • Timed tests and math anxiety. Boaler (2014, Teaching Children Mathematics 20(8), 469-474) claims timed tests cause anxiety, citing Ramirez, Gunderson, Levine and Beilock (2013, Journal of Cognition and Development 14(2), 187-202), which is a correlational study of 154 first and second graders showing anxiety relates to achievement in high-working-memory children; it did not manipulate timing. No causal study of timed tests producing anxiety was found; the WWC 2021 panel rates brief timed activities strong. The defensible position is: timed practice is fine after concepts are learned, harmful as the only mode of instruction.
  • Number Talks (Parrish, Boaler). Practitioner literature only; Matney et al. (2020) found no adequate efficacy studies. The comparison-of-strategies mechanism is supported separately.
  • Bar model / Singapore Math as a package. WWC (2015) found no eligible studies; Math in Focus RCT evidence is limited. Schema diagrams in general are strongly supported.
  • Concreteness fading effect sizes in children come mainly from one lab (McNeil, Fyfe) on one topic (equivalence); the 2014 review is a systematic review, not a meta-analysis. A 2019 replication in physics (Jaakkola & Veermans, Learning and Instruction) and a systematic elaboration (ERIC ED630441) suggest results depend on how well the stages are linked.
  • Multimedia principles in children. Most effect sizes are from adult samples; the fourth-grade modality result (Herrlinger et al. 2017) is one study. The seductive-details effect is well replicated but its size in gamified children's apps specifically is not measured.
  • Fuchs fraction intervention effect sizes (up to 2.50) are against a business-as-usual control with at-risk students and researcher-made measures; the NAEP effect (0.58) is the more generalisable number.
  • Working-memory moderation of practice type (Fuchs et al. 2017 Year 2: conceptual practice better at the 10th percentile of working memory, ES 0.61; fluency practice better at the 90th, ES 0.52) is a single interaction and the authors themselves caution against personalising on it yet.
  • Ratio tables, subitizing at age 9, and the specific superiority of the area model over other multiplication models: plausible, no controlled evidence found.
  • Unverified items in this review: exact effect sizes in Berney & Bétrancourt 2016, Schneider et al. 2018, Rey et al. 2019 and Ginns et al. 2013 (citations verified, numbers not re-read); Blöte et al. 2001 (not checked); MTC table weighting (framework PDF not re-read).

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