Competitive Landscape: Math Learning Apps for Ages 8-11 (Grades 3-5)

Research date: 2026-09-24. Method: web search plus direct fetches of product pages, help-center articles, Evidence for ESSA entries, ERIC/IES records, journal abstracts, and review sites. Where a claim could not be verified from a primary source in this session it is marked "(not verified)" or "(recalled, not verified)". Note that Common Sense Education's teacher-facing edtech reviews were paused in January 2026 and removed from the site in February 2026; where I cite them I am relying on search-snippet summaries of the old reviews, not the live pages. Common Sense Media's consumer (parent) reviews are still live and are cited directly.

Framing: the parent's complaint about Times Tables Rock Stars (TTRS) is that it is a rock-themed skin over calculator-style drilling with no conceptual teaching, visualisation, animation, or explanation. This report checks that claim against the product and then maps the rest of the market on the same axes.

1. Times Tables Rock Stars (TTRS) - Maths Circle Ltd, UK

What it is. A web/iOS/Android times-tables recall trainer created in 2010 by secondary maths teacher Bruno Reddy, published by Maths Circle Ltd (Bedford, England). Multiplication and division facts 1x1 to 12x12 only. Queen's Award for Enterprise 2020, BETT Award 2022 (Wikipedia). Used in "over 15,000 primary and secondary schools worldwide" (ttrockstars.com).

Target age. "Suitable for all learners aged 6 years and up" (ttrockstars.com/parents). In practice UK Years 2-6 (ages 6-11), with a heavy Year 4 focus because of the MTC.

Platform / price. Web app plus native apps. School subscription GBP 116.80/year for unlimited pupils and teachers, with optional "Stats" (+GBP 40.15) and "Sessions" (+GBP 36.50) bolt-ons and a GBP 10.95 discount if the school also has NumBots (ttrockstars.com/schools). Family subscription GBP 6/year for two adults and three children; tutor plans about GBP 7/year plus per-tutee fees; 4-week free school trial (Educational App Store; ttrockstars.com/purchase). No ads or in-app purchases.

Core loop (minute to minute). Pick a mode, answer a stream of typed-answer multiplication/division questions against a clock (1-3 minutes, or 25 questions for Soundcheck), earn coins, spend them in the avatar shop, check leaderboards. There is no lesson, worked example, array, number line, or other representation anywhere in the play surface; the parent's "calculator-style drilling" characterisation is accurate. Maths Circle positions it as a retrieval-practice tool, and the Structural Learning review justifies it purely from the spaced-retrieval literature (Rohrer & Pashler 2007; Dunlosky et al. 2013).

Modes. (Sources: two school parent guides, Harewood Junior and Acomb Primary, plus Maths Circle help-center articles.)

Note the incentive design: Garage (adaptive, teacher-set tables) pays 10x the coins of Studio/Festival, which steers children toward targeted practice rather than random-table speed runs.

Adaptivity. The "Adaptive Questions Algorithm" help article explains that the platform builds a per-fact heatmap of recall time; before every Garage and Arena game it recomputes "the 6 highest priority facts from WITHIN the selected tables", swaps a fact out once the child gets faster on it, and scans for the next batch. "Automatic Training Mode" uses monthly Gig results to choose which table the child is on. Facts slower than 6 seconds or repeatedly wrong keep coming back (Acomb parent guide). Studio, Festival and Rock Slam are explicitly non-adaptive ("random weighting of all the tables"). There is no concept of interleaving with related facts, fact families, or derived-fact strategies (e.g. 6x7 from 5x7+7) - it is per-fact recall time only.

Rock Status (speed ranking). Studio Speed = mean recall time over the last 10 Studio games. Thresholds (Maths Circle help center):

"Rock Hero" requires one-second recall including typing, six times faster than the 6-second MTC standard. Schools layer certificates onto the tiers by year group (Harewood guide: Year 3 targets up to Unsigned Act; Year 5-6 targets Rock Star and above).

Coins and avatar shop. Coins buy avatar clothing, instruments and gear and feed class totals on the school leaderboard; physical badges are sold separately. The "Sustaining Enthusiasm" help article recommends Rock Status certificates, "Gone Green" certificates (all facts under 4 s), weekly Arena, "Battle of the Bands" class competitions, "Rock Wrangle" knockouts and "Red Carpet Weeks". There is no in-app narrative; the rock theme is skin plus vocabulary.

Heatmap for teachers and parents. The Stats > Fluency heatmap shows each fact's current recall time, green for fast, red/orange for slower than 5 seconds (search summary; parent page confirms parents can see it). Teachers also get usage graphs, speed/accuracy downloads, printable leaderboards, and, with the Sessions bolt-on, homework-completion tracking.

Relationship to the Year 4 Multiplication Tables Check (MTC). The MTC is a statutory 25-question, 6-seconds-per-question online check for English Year 4 pupils. Soundcheck is an explicit emulator, and the "MTC Hub" adds three mock MTCs, a prep schedule and webinars. Maths Circle claims pupils scoring 25/25 typically use TTRS "21 minutes per week" in Year 4 - a usage correlation, not a controlled comparison. DfE: national mean 20.6/25 in 2024, 21.0 in 2025, 37% full marks in 2025. The MTC is the main structural driver of TTRS adoption in the UK.

Evidence of effectiveness. I found no randomised or quasi-experimental study of TTRS, no EEF trial, and no WWC or Evidence for ESSA entry. The evidence base is: (a) company case studies and testimonials; (b) a 2017 single-class, 4-week, no-control study by a Year 3 teacher (educationponderingsblog.wordpress.com): 67% improved completion time, 38% improved speed and accuracy, 91% enjoyed it, 9% used it at home regularly; (c) a British Psychological Society DECP Debate article, "Times Tables Rock Stars: An academic critique" (explore.bps.org.uk/content/bpsdeb/1/189/14), which search summaries characterise as stating that as of December 2024 "there have not been any formal research studies exploring the effectiveness" of TTRS. I could not access the BPS full text. The Guardian (via Wikipedia) noted adoption "despite schools not necessarily knowing whether they're actually helping students".

Praise. Educational App Store 4/5: "fun and engaging", competitive multiplayer, no ads, parent tracking. Common Sense Education (snippet): "The rock music theme is a fun hook for practicing multiplication facts." Mumsnet defenders cite children who became "addicted" and improved.

Criticism. Common Sense Education (snippet): "Doesn't develop conceptual understanding; music doesn't play unless teachers use a playlist." Mumsnet "AIBU to hate Times Tables Rockstars": "Way to put kids under pressure and destroy a child's confidence"; "The timer just adds too much pressure"; "Kids are named and shamed in class if they don't use it"; misconfigured table settings expose children to untaught facts. Teacher forums: struggling pupils "sit for 90 seconds getting wrong answer after wrong answer". The 2017 study saw speed gains accompanied by more errors.

Assessment. A well-executed retrieval-practice engine with a clever priority-fact algorithm and strong school-side social mechanics, but zero instruction, zero representation, and a speed ladder far beyond any curricular standard. The parent's critique holds.

2. Prodigy Math - Prodigy Education, Canada

What it is. A free-to-play fantasy RPG (grades 1-8) where casting spells in turn-based monster battles requires answering curriculum-aligned math questions. Wikipedia and Common Sense Media describe it as covering grades 1-8; Common Sense Media rates it age 7+, learning value "Some".

Price. Free, with memberships (NBC News 2021: USD 59.88-107.40/year per child; Modulo review: USD 6.26-99.95/year depending on tier). Schools are told it is "completely free"; the paid upsell targets home play.

Core loop. Walk the overworld, encounter a monster, choose a spell, answer a math question in a modal, spell fires if correct, collect loot, level up, spend on pets/gear/house. Questions are text problems with an answer box, hints, text-to-speech, and virtual manipulatives available (Prodigy pedagogy blog). Common Sense Media: "the math part of the game is an afterthought" with "standard dry math drill"; questions are "somewhat disconnected from the narrative".

Concept teaching vs drill. A placement test ("stealth assessment") sets level; teacher Plans override. A wrong answer shows the correct answer with a short explanation (Modulo). JHU evaluators observed students who "consistently answered questions incorrectly were still able to move about the world, engage in battles" and were re-served questions they had just been given the answer to; "Researchers and teachers noted the lack of remediation and actual teaching provided by Prodigy" (JHU CRRE Key Findings, Jan 2020).

Motivation. Narrative world, pets, avatar customisation, battles against classmates, leveling. JHU: student engagement "consistently described as a key strength"; teachers' main criticism was children "wander about" in shops and customisation.

Evidence. JHU CRRE (Morrison et al., 2020; ERIC ED611319), commissioned by Prodigy: correlational mixed-methods, 7 schools, 577 K-5 students in Texas. Each additional question answered was associated with +0.00113 STAAR points (p<.05) for grade 4; "roughly 888 questions" per one-point gain; home questions not significant; no dose-response above mid usage. Prodigy claims ESSA Tier 3 for this. Its research page adds state-level correlational studies (Texas 16,000 third graders; Florida 3,700 grade 4-6) with no control design. No RCT, no WWC/ESSA independent rating found.

Criticism. Fairplay/CCFC complaint to the FTC (Feb 2021): in 19 minutes of play observers counted 16 membership ads versus 4 math problems; non-members see dull chests, members "float on clouds". Kid and parent reviews on Common Sense Media describe it as "pay-to-win". Equity concern in classrooms when some children are members.

3. DragonBox (Numbers, Big Numbers, Algebra 5+) - now "Kahoot! DragonBox"

What it is. A Norwegian studio (WeWantToKnow, founded by former maths teacher Jean-Baptiste Huynh) acquired by Kahoot! in May 2019 for USD 18M in cash and shares (TechCrunch, EdSurge). Apps now: Numbers (ages 4-8), Big Numbers (6-9), Multiplication (8+), Algebra 5+ (5+), Algebra 12+ (9+ per Kahoot page), Geometry (9+), Chess. Distributed through a Kahoot!+ family subscription (kahoot.com); previously one-off purchases.

Core loops. Numbers: "Nooms" are stackable bar-like creatures (Cuisenaire-rod logic); the child feeds, slices, and stacks Nooms to make numbers, in sandbox, puzzle and ladder modes. Big Numbers (Common Sense Media, age 6+, learning "Some"): collect apples and coins, group into ones and tens, progress to multi-digit addition/subtraction with carrying and borrowing through a farming/resource loop across six worlds; "no instructions" by design. Algebra 5+: isolate a box (the unknown) by dragging cards, cancelling opposite cards, and dividing; the picture cards gradually become numbers and letters. (The Numbers/Algebra descriptions are from product knowledge and the Kahoot page; only Big Numbers was independently reviewed in this session.)

Concept teaching. Concept-first with no text: the mechanic is the concept (place-value bundling, equation balancing). Common Sense Media praised Big Numbers for math that is "well-integrated" and "increases in difficulty at a good pace", and criticised slow early gameplay (30+ second waits), inconsistent handwriting recognition, repetitive tapping, and no instructions.

Adaptivity. Linear level progression, not adaptive.

Evidence. The Washington "Algebra Challenge" (Center for Game Science, U. Washington, June 2013): 4,192 K-12 students; 92.9% of those who played 1.5 hours reached "mastery" (solving x*a/d + b = c/e in-game); Norway 2014: 36,106 students. Mastery was in-game; the page offers no transfer-to-paper data. Peer-reviewed: Chan et al. 2023, BJET, "Keep DRAGging ON" (DragonBox 12+ problems solved vs performance during COVID) - abstract not accessible in this session. NYT 2013 called Algebra 5+ "the most impressive math education app I've seen". Evidence strength: moderate for engagement and in-game mastery, weak for transfer.

4. ST Math - MIND Research Institute (JiJi the penguin)

What it is. A PreK-8 supplemental program of visual puzzles. JiJi must cross the screen; the child manipulates objects so that the path works. "Concepts before symbols", "visual feedback in the moment", "productive struggle", "mastery before progression", "access beyond language barriers" (mindeducation.org). The 2021 RCT paper describes "dynamic imagery that minimizes, at least initially, mathematical symbols and terminology", with students "moving to higher levels only after mastering each current level".

Price. School licences; a homeschool edition exists (price not found).

Core loop. Select an objective, solve a sequence of wordless puzzles; wrong answers play out visually (the bridge is too short, JiJi falls) so the child sees why. "Many Tries" alert after 10+ failed attempts. Recommended dosage: 90 min/week for grades 2-8; sessions of 20-30 minutes. No points, no avatar shop; motivation is the puzzle itself and level completion.

Evidence. Evidence for ESSA "Strong" (Tier 1): 4 studies, 346,248 students, mean ES +0.07. The key RCT (IES grant R305A090527; 52 Southern California schools; 16,307 grade 3-5 students) has been analysed twice: Rutherford et al. 2014 (JREE 7:358-383) found "a negligible effect of ST Math on mathematics scores" and a non-significant two-year effect (d = 0.15), while MIND's own WWC-aligned reanalysis (Bodner & Coulson 2021) reports significant effects of g = 0.062 (grade 4) and 0.096 (grade 5). Evidence for ESSA lists +0.09/+0.03 (years 1/2) and +0.04. A separate RCT found higher math self-beliefs, strongest for students who were behind. Robust design, small effects.

Praise / criticism. Trustpilot student reviews: "boring, repetitive", "unsolvable puzzles with no explanation", "no rewards at all for completing levels", JiJi "staring blankly". MIND frames this as productive struggle. Teachers generally value the language-free access for ELL students (RCT sample was 56% ELL).

5. Zearn Math

What it is. A K-8 digital curriculum aligned to Eureka Math/EngageNY, used alongside teacher-led small groups. Free for individual teachers/families (recalled, not verified); school-wide licences are paid.

Core loop (Independent Digital Lesson). Fluency warm-up, guided practice with an on-screen teacher video plus interactive number lines / place-value discs / tape diagrams, then the "Tower of Power" mastery check, then an exit ticket. On a Tower mistake a "Boost" "breaks down the question into smaller steps with more supportive manipulatives", then a fresh problem; all Tower questions must be correct to unlock the next lesson; repeated struggle fires a "Tower Alert" to the teacher (help.zearn.org).

Concept teaching. Strongest concrete-pictorial-abstract implementation in this list: manipulatives are the input method, not a decoration.

Evidence. Evidence for ESSA "Strong": RAND school-level RCT 2022-24, 64 Texas schools, 10,000+ grade 3-5 students, +0.11 SD on NWEA MAP (+0.13 for below-proficient), +0.07 on STAAR (ns); JHU quasi-experiment in Louisiana grades 4-8, +0.20 SD on LEAP; an earlier pilot with no overall effect but usage-linked gains.

Criticism. Teacher and student reviews: the Tower of Power "requires students to answer perfectly", is "very frustrating", "weirdly-worded word problems"; students report being reset (the help page says a new problem, not a full restart - the perception of reset is still a UX problem). Trustpilot/SmartCustomer ratings around 1.7/5 are dominated by students forced to use it at school.

6. Khan Academy (Kids, main platform grade 3-5, Khanmigo)

Khan Academy Kids (ages 2-8): free, no ads; five characters led by Kodi Bear; books, videos, and adaptive activities in literacy and math; a UMass Amherst peer-reviewed study showed pre-literacy gains (khanacademy.org/kids). It tops out around grade 2, so it is below a 9-year-old.

Main platform, grade 3 math: 14 units, 142 skills, each with videos/articles plus hinted exercises; mastery levels Attempted / Familiar / Proficient / Mastered; unit quizzes, unit tests, course challenge. Free. Exercises are open-response, not games; visual models appear in videos and some widgets.

Khanmigo: AI tutor that "guides learners to find the answer themselves"; paid monthly/annual for families (price not shown on page; recalled as about USD 4/month, not verified).

Evidence. Khan's Nov 2024 efficacy post: ~350K grade 3-8 students in 2022-23; 30+ min/week associated with "~20% greater-than-expected" MAP growth (ES 0.36 reported); only ~9% of students reached that dosage; a 221K-student longitudinal design supports a causal reading per Khan. A University of Toronto RCT in 18 Tennessee middle schools (2024-26) found Khan Academy helped remediation students modestly but "the addition of Khanmigo provided no measurable learning boost"; students abandoned the tutor when it would not give answers (Hechinger Report).

7. Beast Academy Online - Art of Problem Solving

What it is. Comic-book curriculum for ages 6-13 (Levels 1-5; Level 3 = ages 8-10, Level 4 = ages 9-12). Online: USD 15/month or USD 96/year; USD 160/year with printed books (Cathy Duffy; Brighterly). "20,000+ problems, puzzles, and activities across more than 1,000 lessons", 1,000+ videos by Richard Rusczyk, puzzles by Palmer Mebane, a Puzzle Lab (beastacademy.com/online).

Core loop. Read a comic-guide chapter (monsters discover the idea), watch a short video, then work interactive problem sets that turn from red to green as mastered; starred problems have hints; parent dashboard with difficulty settings.

Concept teaching. High: every chapter builds the idea through characters reasoning aloud; problems "require creativity, not just computation". Not adaptive in the algorithmic sense; difficulty is chosen by parent/child.

Evidence. No efficacy studies found. Reviewers (Cathy Duffy, Smarter Learning Guide, Modulo) praise rigour and problem-solving; caution that "kids experiencing learning difficulties don't progress well" and that it needs moderate parent involvement.

8. Reflex - ExploreLearning (fact fluency)

What it is. Grades 2+ fact-fluency system for +, -, x, /. Common Sense Media age 7+. Historically about USD 35/student/year (older grant listing); pricing now quoted per school.

Core loop. "Crabby's Fact Fair" warm-up assesses fluency, Coach Penny teaches a fact family with its commutative/inverse relationships, then fluency games with increasingly fast decision-making; a daily "Green Light" marks enough practice; tokens buy avatar items; a Progress Tree shows mastered facts.

Teaching vs drill. Uses fact families and inverse relationships explicitly ("builds on and reinforces ... the commutative property and the relationship between operations"), which is a step above TTRS, but it is still recall-oriented. Common Sense Education (snippet) criticised "a purely memorization-based approach without pattern-based deep understanding".

Evidence. ExploreLearning cites "a randomized control trial ... in rural and urban Title I schools, grades 2-4" with 2.6x post-test advantage for Reflex users, a 35,000-student usage study (grade-level fluency 20.4% to 84.8%), and Digital Promise certification. All hosted by the vendor; I found no independent WWC/ESSA rating. Common Sense Media parents: "largely viewed as ineffective and anxiety-inducing", stereotyped characters, repetitive games.

9. Duolingo Math

What it is. Launched Aug 2022 as a separate iOS app, now folded into the main Duolingo app (recalled). Elementary track covers grades 2-12 per duolingo.com/math ("Arithmetic, Geometry, Pre-Algebra, Algebra I..."); The 74 reported the elementary course starts at grade 3 fractions because they "predict your math performance". Free; Super Duolingo USD 12.99/month removes ads and adds unlimited hearts.

Core loop. Bite-sized lessons of tap/drag/draw exercises (pizza-slice fraction models, number-line tapping, hand-drawn digit input); hearts, XP, streaks, leagues. "Guided lessons break down math problems into simple steps."

Concept teaching. Visual models are used but as answer formats; explanations are short. Keith Devlin (Stanford): for most games "do games produce better ability to solve a mathematical problem that is more than a basic skill? ... the answer is no." A parent review noted no long division for a 10-year-old. No efficacy studies found.

10. Math Academy (Justin Skycak, Jason Roberts)

What it is. Fully automated adaptive course system, USD 49/month (USD 500/year), courses from "4th Grade Math" to university (mathacademy.com). Book "The Math Academy Way" (Skycak & Roberts, 2024; free PDF at justinmath.com/files/the-math-academy-way.pdf) documents the pedagogy: mastery learning, spaced repetition via "Fractional Implicit Repetition" (FIRe) over a knowledge graph, interleaving, retrieval practice, "layering", minimising cognitive load, targeted remediation on prerequisites rather than easier problems.

Core loop. Diagnostic places the student on the knowledge-graph frontier; each day the system serves a mix of new lessons (worked example then exercises, split into "knowledge points" e.g. two-digit addition with vs without carrying), spaced reviews, quizzes and multistep tasks; XP = effort-minutes, with leagues and promotion/relegation; perfect scores earn bonuses, poor work loses XP (Frank Hecker's 11-part review).

Concept teaching. Explicit worked examples, text-heavy, deliberately "without distractions" - no animation, no game world. For a 9-year-old this is a serious step; several reviewers suggest starting with Beast Academy or Khan and moving over around age 9-11.

Evidence. Company claims "4X the speed of a traditional math class" and transfer to MAP/AP scores; no independent study found. The original Pasadena in-school program's results are anecdotal. Skycak's own framing on justinmath.com is that they unify well-replicated cognitive-science findings rather than having trialled the product itself.

11. Matific

What it is. K-9 (Grade 9 on the parent site) library of "episodes" - short game-like activities, each targeting one concept with about five questions of rising challenge, plus worksheets, word problems and workshops. Home tiers monthly/annual (price not shown), 7-day trial; home modes are "Adaptive Adventure", "Training Zone", and a multiplayer arena.

Concept teaching. Episodes are built as interactive models (balance scales, area models, fraction bars). The Western Sydney University evaluation (Attard, 2016; sponsored by Slate Science/Matific) reported that "rather than simply state when an answer was incorrect, Matific provided assistance that promoted the development of conceptual understanding" and students "found it was fun because they felt they were learning".

Evidence. Attard 2016: 8 schools, 16 teachers, case-study design, pre/post "Improvement Index" 34%; the author cautions "it is not possible to claim whether or how much the use of Matific influenced this outcome", yet the 34% figure headlines Matific's marketing. The research page lists 18 further items (Plan Ceibal Uruguay, 2,700 fifth graders, ES 0.33-0.76; SEG Measurement ~1,500 students; many small action-research studies), mostly vendor-hosted; no WWC/ESSA rating found. EdTech Tulna (IIT Bombay) rated it "Exemplary" for grades 3-5. Students in the WSU study asked for more levels per episode and clearer explanations.

12. Mathigon Polypad and Brilliant

Polypad (Mathigon, acquired by Amplify 2021): free browser canvas of virtual manipulatives - number tiles, fraction bars, algebra tiles, Cuisenaire rods, tens frames, number lines, geometry, probability, plus a JS embed API (Polypad.create(), optional apiKey; licensing terms not stated on the API page). Mathigon's interactive textbooks target ages 12-18. It is a tool, not a course: no adaptivity, no rewards, no progression. Reviewers: "excellent for exploring ... but works best when paired with a structured curriculum".

Brilliant: interactive lessons, mostly aimed at teens and adults; the privacy policy allows under-13 users only with verifiable parental consent, and the content library has no grade 3-5 track. Not a realistic competitor for a 9-year-old.

13. NumBots (Maths Circle) and Hit the Button (Topmarks)

NumBots (ages 6+; free download; school/family subscriptions - family price not verified): TTRS's stablemate for addition and subtraction. Story Mode upgrades "RustyBot from Rust to Diamond" through stages of subitising, number bonds to 5/10/20/100, then bridging-10 and partitioning strategies, moving "from iconic image representations to abstract numbers"; Challenge Mode is timed practice. It cites NCETM and White Rose Maths pedagogy. The Good Play Guide praised the structured progression and tens-frame/number-line visuals, and criticised that "speed requirements for top scores frustrated some children, even when they knew answers". NumBots shows that Maths Circle can build representation-first content; TTRS simply does not.

Hit the Button (ages 6-11; free on web, paid app with untimed mode and profiles): 60-second rounds tapping the answer among candidates across number bonds, times tables, doubling/halving, division facts, square numbers - 166 mode/difficulty combinations. Pure recall; the Educational App Store notes "not a great deal of variety". Structural Learning: "a supplement that sharpens recall, rather than a complete teaching scheme".

14. Concept-first exemplars: Motion Math, Slice Fractions, Numberblocks

Motion Math (Fractions, Zoom, Wings, Hungry Fish...): Stanford School of Education origin, embodied-cognition premise - tilt the device to land a falling fraction on a number line, pinch-zoom number lines for estimation. Superb Bundle USD 35.99 (2015). The controlled study (Riconscente 2013, Games and Culture; abstract paywalled) - recalled as ~122 fifth graders in a crossover design, five 20-minute sessions, roughly +15% fractions test scores and +10% attitude, not verified here. EdSurge's 2015 review scored the bundle B/A/B+ and criticised accelerometer controls that "unnecessarily conflict with the game's educational goals" and difficulty curves that jump "from boredom to anxiety". Many of the apps are no longer maintained.

Slice Fractions (Ululab, Montreal): USD 3.99, age 4+ on the App Store; the child slices ice and lava so pieces fall onto a mammoth's path, learning parts of a whole, equivalence and comparison with no words; App Store Best of 2014, Serious Play Gold. Ululab cites an "independent study by UQAM" showing Slice Fractions "significantly boosts students' performance"; the press kit names the UQAM advisers (Cyr, Houle, Riopel) but gives no design or sample details, so this is unverified. Successor: Math Makers (subscription, UQAM/McGill collaboration).

Numberblocks apps (Blue Zoo / Alphablocks Ltd, from the BBC CBeebies series): Meet the Numberblocks (free; morph blocks, count 1-20), Hide and Seek (number bonds), Card Fun (60 arithmetic levels), Numberblocks World (subscription, "100+ episodes", made with NCETM experts). Each number is a character built from unit cubes, so composition, part-whole and the shapes of numbers are visible. Aimed at ages ~3-7; the design lesson (numbers as manipulable, animated objects with personalities) matters more than the products for a 9-year-old.

15. Bedtime Math

What it is. Nonprofit founded by Laura Overdeck (2012). A daily story followed by three tiers of questions ("Wee ones", "Little kids", "Big kids"); free app, books, and the Crazy 8s after-school club. The app is a parent-child conversation scaffold, not a game.

Evidence. Berkowitz et al., Science 350:196-198 (2015): RCT, 587 first-graders, 22 Chicago-area schools, iPad with the math app vs a reading app. Frequent users gained about three months of math achievement; even weekly use helped; effects were largest for children of high-math-anxious parents. Frank (2016, Science) reanalysed the open data and argued the intent-to-treat effect was not significant; the authors replied that his reanalysis "reported largely similar results" and stood by their conclusions. WWC reviewed it (Study 81031). Schaeffer et al. 2018 (JEP: General) found benefits for children of math-anxious parents "persisted even after app usage declined significantly", mediated by parents' expectations. The strongest home-product evidence in this list is for a low-tech, parent-involving, talk-based intervention.

Comparison table

Scale: 0 = absent, 1 = minimal, 2 = present, 3 = core strength. "Fun source" = whether enjoyment comes from the mathematics itself or from a wrapper (points, avatars, story) around it.

Mechanics worth borrowing

1. Priority-fact selection from a recall-time heatmap (TTRS Garage). Six weakest facts within the assigned set, swapped out as they speed up. The one genuinely good idea in TTRS; extend it with fact-family links (Reflex) so 7x8 improves when 7x4 does.
2. Untimed mode as a first-class citizen (TTRS Jamming; Hit the Button app). Every timed product drew anxiety complaints (TTRS, NumBots, Reflex, Zearn). Default untimed; timing opt-in.
3. Pay more for targeted practice (TTRS: Garage 10 coins vs Studio 1). Use the economy to steer toward useful practice.
4. Mechanic-as-model (DragonBox, Slice Fractions, ST Math). The manipulation is the operation: bundle ten ones, slice a whole, isolate the unknown. Wrong answers play out visually (JiJi's bridge) rather than flashing red.
5. Boost on error (Zearn). Re-present the problem decomposed with a more concrete representation, then a fresh problem. Never reset progress.
6. Stuck detection with an adult alert (Zearn Tower Alerts, ST Math "Many Tries").
7. Fact families and inverses before fluency drills (Reflex; NumBots bonds then bridging).
8. Characters that embody the number (Numberblocks, Beast Academy). Shapes and personalities carry structure (square, odd/even) for free.
9. Concrete-pictorial-abstract fading inside a single skill (NumBots "iconic to abstract"; Zearn).
10. Spaced review over a knowledge graph with prerequisite-targeted remediation (Math Academy FIRe). Serve the prerequisite, not an easier copy.
11. Parent-facing fact heatmap and weekly summary (TTRS Stats; Beast Academy dashboard).
12. Asynchronous ghost races (TTRS Rock Slam). Competition without co-presence, including against a teacher's ghost.
13. Parent-child conversation prompts (Bedtime Math). The best-evidenced home intervention here is talk, especially for math-anxious parents.

Mistakes to avoid

1. Speed ladder detached from any standard (TTRS Rock Hero = 1 s vs the MTC's 6 s). Manufactures failure for most children and rewards typing speed.
2. Serving untaught facts by default (TTRS misconfiguration threads on Mumsnet). Default to a taught set, visible to the parent.
3. Public leaderboards and "named and shamed" dynamics (TTRS class boards; Prodigy membership perks visible to classmates). Keep competition opt-in and self-referenced.
4. Math as a toll-gate for unrelated fun (Prodigy). JHU and reviewers found the math "an afterthought" and children cycling wrong answers to keep playing.
5. Upsell aimed at children (Prodigy; Fairplay counted 16 ads per 4 problems).
6. Perfect-run mastery checks that feel like a reset (Zearn Tower of Power).
7. No instruction, then blaming "productive struggle" (ST Math reviews; Big Numbers "no instructions"). Wordless is good; hintless is not.
8. Controls that compete with the concept (Motion Math accelerometer; DragonBox handwriting recognition). Input errors become "math errors".
9. Status from the last 10 games only (TTRS Studio Speed). A bad week drops rank; children avoid the mode.
10. Correlational usage data marketed as efficacy ("21 minutes = 25/25", "34% improvement"). Publish the design with the number.
11. Text-heavy explanation for a 9-year-old (Math Academy; Khan exercises). Right pedagogy, wrong modality.
12. Unbounded reward shops (Prodigy; TTRS coins). JHU's main teacher complaint was children "wandering" in shops; time-box it.

Public APIs, open curricula, open-source components

• Mathigon / Polypad. Public JS embed API (Polypad.create(), optional apiKey) for manipulative canvases; commercial licensing terms are not stated in the API docs and should be confirmed with Amplify. Open-source repos under github.com/mathigon: studio (interactive-course server, TypeScript), textbooks (course content), fermat.js (maths/statistics), boost.js (DOM/animation), hilbert.js (expression parsing/MathML/CAS). CLA required for contributions.
• Khan Academy. Khan/perseus (MIT) is the exercise renderer/editor with widgets (number line, grapher, expression input); Khan states it is "not accepting external contributions" but the packages are published to npm. Khan's public content API was discontinued (recalled, not verified); exercise/video content is CC BY-NC-SA (recalled, not verified), which rules out commercial reuse.
• Zearn. No public API. It is aligned to Eureka Math / EngageNY, whose K-5 materials are published under CC BY-NC-SA (recalled, not verified) - the scope-and-sequence and models (tape diagrams, place-value charts) can be studied freely.
• Illustrative Mathematics K-5 (not reviewed above) is CC BY 4.0 and is the most permissive open elementary curriculum to borrow sequencing and representations from (recalled, not verified this session).
• UK open scaffolds. NCETM Primary Mastery Professional Development materials and White Rose Maths schemes of learning are free to download and define the representations UK schools use (NumBots cites both); the DfE's MTC framework specifies the 25-question, 6-second format exactly.
• TTRS / NumBots, Prodigy, ST Math, Reflex, Beast Academy, DragonBox, Matific, Duolingo Math, Math Academy. No public APIs or open components found. (TTRS integrates with school MIS via Wonde for rostering - recalled, not verified.)
• Bedtime Math. Daily problems are free on the site and app but copyrighted; the intervention design (story + tiered questions + parent) is what is reusable.

Bottom line for the product decision

Fluency drills (TTRS, Hit the Button, Reflex) have no concept teaching and no independent efficacy evidence; their strength is adaptive fact selection and social mechanics. Concept-first games (DragonBox, ST Math, Slice Fractions, Motion Math, Numberblocks) make the mechanic the model and have small-to-moderate evidence, but have no fluency layer, no spaced review, and weak hinting. Curriculum systems (Zearn, Khan, Beast, Math Academy, Matific) have the strongest evidence and explanation but are schoolwork-shaped for a 9-year-old at home. Nothing combines animated conceptual explanation on demand, mechanic-as-model practice, and heatmap-driven spaced fluency without a public speed ladder. That gap is the opportunity.

Sources (selected)

• ttrockstars.com (home, /parents, /schools, /mtc-hub); Maths Circle help center: Rock Status (article 2028839), Adaptive Questions Algorithm (5628202), Sustaining Enthusiasm (3007664); Harewood Junior and Acomb Primary TTRS parent guides (PDF); Wikipedia "Times Tables Rock Stars"; educationalappstore.com TTRS review; structural-learning.com/post/ttrs; Mumsnet AIBU thread 4041783; educationponderingsblog.wordpress.com (2017 study); explore.bps.org.uk/content/bpsdeb/1/189/14 (not accessed in full); DfE MTC attainment statistics.
• Prodigy: JHU CRRE "Evaluation of Prodigy: Key Findings" (Jan 2020, PDF); ERIC ED611319; commonsensemedia.org Prodigy app review; NBC News on Fairplay complaint (Feb 2021); modulo.app review; prodigygame.com/main-en/research and /blog/prodigy-pedagogical-approaches.
• DragonBox: kahoot.com/home/learning-apps/dragonbox; dragonbox.com/about/algebra-challenge; TechCrunch and EdSurge (May 2019 acquisition); commonsensemedia.org Big Numbers review; Chan et al. 2023 BJET doi:10.1111/bjet.13304.
• ST Math: evidenceforessa.org ST Math entry; Rutherford et al. 2014 JREE doi:10.1080/19345747.2013.856978; Bodner & Coulson 2021 (ERIC ED616922); mindeducation.org/programs/st-math; play.stmath.com best-results module; Trustpilot reviews.
• Zearn: evidenceforessa.org Zearn entry; about.zearn.org/research; help.zearn.org Tower of Power article; teacher and student reviews via Common Sense Education snippets and Trustpilot.
• Khan: blog.khanacademy.org efficacy results Nov 2024; hechingerreport.org Khanmigo Proof Points; khanacademy.org/kids; khanmigo.ai; khanacademy.org/math/cc-third-grade-math; github.com/Khan/perseus.
• Beast Academy: beastacademy.com/online; cathyduffyreviews.com; smarterlearningguide.com; brighterly.com.
• Reflex: reflex.explorelearning.com (home, research pages); commonsensemedia.org Reflex review; Common Sense Education snippets.
• Duolingo Math: the74million.org (2022); duolingo.com/math.
• Math Academy: mathacademy.com; justinmath.com (value proposition; The Math Academy Way PDF); frankhecker.com Math Academy parts 3-5, 9, 11.
• Matific: Attard, C. (2016) Research Evaluation of Matific Mathematics Learning Resources, Western Sydney University (PDF); matific.com research-and-efficacy and parents pages.
• Mathigon: mathigon.io/polypad; github.com/mathigon; brilliant.org/privacy.
• NumBots: numbots.com; goodplayguide.com NumBots review. Hit the Button: topmarks.co.uk; educationalappstore.com.
• Motion Math: edsurge.com "Motion Math: The Definitive Review" (2015); Riconscente 2013 Games and Culture doi:10.1177/1555412013496894; karenmahon.com. Slice Fractions: App Store listing; ululab.com and press kit. Numberblocks: blocksuniverse.tv/numberblocks/apps.
• Bedtime Math: Berkowitz et al. 2015 Science; Frank 2016 Science doi:10.1126/science.aad8008; authors' response doi:10.1126/science.aad8555; Schaeffer et al. 2018 JEP:General doi:10.1037/xge0000490; news.uchicago.edu; WWC Study 81031; Wikipedia "Bedtime Math".