How Keynum Math Lab teaches

1. Who this page is for

This document is written for two audiences who share a common need: the *motivated parent* who wants to understand what their child is encountering before they hand them a screen, and the *teacher or pedagogue* evaluating whether the app belongs alongside their own practice. We assume no maths background on the parent side; for the pedagogue side, the research foundations are named explicitly so you can verify the claims directly.

Keynum Math Lab is a curated, child-safe, mathematically-correct learning environment for primary-school children with a real appetite for mathematics. It is **not** a replacement for a teacher, a textbook, or a real-world manipulative collection. It is a carefully-designed sandbox in which a child can explore a well-formed slice of mathematics on their own time, at their own pace, with their own choices about what to look at next.

2. The pedagogical foundations

The design rests on a small set of well-tested principles from the mathematics-education and cognitive-science literature. We name them here so teachers can hold the app accountable to what it claims to do.

Concrete → Pictorial → Abstract (Bruner; the Singapore model)

The most empirically validated sequence in primary-school maths. Every concept the child meets in the app is introduced in a *visualisation* (the Tower of Places, the Adding Forge, the Frost Line, the Strip of Pieces, the Function Mill) before the symbolic form is asked of them. The Singapore Primary Mathematics Syllabus (2021, updated 2025) operationalises this as its central organising principle; we follow it deliberately. Where a metaphor breaks at a generalisation (the balance model famously breaks at negative coefficients — Vlassis 2002), we *replace the metaphor*, not the maths.

Conceptual metaphor (Lakoff and Núñez)

Abstract mathematical objects are not learned by definitions. They are learned by *grounding metaphors* that map them onto concrete experience: numbers as positions on a line, addition as motion along it, fractions as parts of a strip, functions as machines. Every lab in the app is built around one such grounding metaphor; our internal `METAPHOR_AUDIT.md` evaluates each against the literature before any visual is shipped. The standard we hold ourselves to: the metaphor must be an *isomorphism* with the maths, not a decoration around it.

Reification (Sfard 1991)

A new mathematical idea is first encountered as a *process* (compute this, walk this, slide this), then later experienced as an *object* (the function, the fraction, the limit) that can be reasoned about in its own right. Each lab respects this: challenges run the child through the process; the grown-up-maths panel names the object. The transition is the deep cognitive move — Sfard's term for it is *reification*, and we design the challenge sequence to support it.

No hidden skill model (Pelánek 2017)

Pelánek's 2017 survey of intelligent tutoring systems is unambiguous: probabilistic skill models (BKT, IRT, PFA) are brittle, overfit easily, and produce hidden numbers that do not correspond cleanly to anything the child can introspect on. We refuse them. Mastery in the app is a *binary* flag, set deterministically when the child solves a curated set of core challenges (or affirms the concept). The flag is visible to the child. There is no derived score, no average, no skill estimate. We treat this as a feature, not a limitation.

Child autonomy (Self-Determination Theory)

SDT identifies autonomy, competence, and relatedness as the conditions for sustained engagement that does not require external rewards. Every recommendation surface in the app is accompanied by a "show me everything" affordance: the child can always see the full territory and pick from it. We never gate exploration on mastery. Children who want to wander a hard lab can wander it.

3. The curriculum: structure and current state

The curriculum is organised as a *concept-map*: 51 concepts grouped into 9 thematic *threads*, with hard prerequisites between them. Concepts are atomic ("place value", "negative numbers", "fractions" — see the diagram below); threads bundle related concepts ("Numbers and the number line", "Geometry", and so on). The whole structure is encoded in code at content/curriculum/ and is the single source of truth for the diagram below and for the rest of the app.

The current state of the curriculum is captured by the diagram on this page. Each column is a thread; each box is a concept; each line is a hard prerequisite. A *filled* box is a concept the child can master *today* by playing a lab in the app. An *outlined* box is a concept that exists in the registry but whose lab is still in the planned build pipeline. The diagram refreshes whenever the registry changes; it is not hand-maintained.

A few notes on reading the diagram. First, the dependency direction is *downward*: a concept at the top of a column has no prerequisites in its thread; a concept lower down depends on what sits above it. Cross-thread edges (Pythagoras depends on shapes, angles, exponents, *and* square roots — bridging the geometry and algebra threads) are drawn at lower opacity so they don't drown the in-thread chains. Second, the per-thread ordering is by *topological depth* in the whole graph, not by syllabus year — a child who works through a column from top to bottom traverses an unambiguous chain of prerequisites. Third, the map is intentionally rich (51 concepts, ~80 edges): primary-school mathematics is a real subject, not a coloured-in worksheet, and the child should see it as such once they're ready to look.

The mathematicians' Library

Every concept's lab is opened with a one-paragraph vignette about a real human being who worked on the idea. The Library page collects the longer biographies. We chose this design after kid- tester feedback consistently asked "who came up with this?" — mathematics is a *human* activity, and the absence of human faces from school maths is one of the things that makes the subject feel inert. The Library catalogue is curated with two explicit diversity targets: representation of women in mathematics (an under-told history), and representation of mathematicians from outside the European canon (whose contributions are often the actual origin of what schoolchildren learn — place value from India, the carry algorithms from China, algebra from the Islamic world, fractions in poems from Bhaskara II's *Lilavati*).

4. How learning works in the app

A child opens a lab, reads the vignette, and meets the visualisation. They work through 3–5 challenges, each scaling in difficulty; each challenge has a 3-tier hint system the child can tap through at their own discretion. A challenge that has not yet been solved offers a misconception-aware nudge based on the specific wrong answer — we don't just say "try again"; we say "you've left the tens column empty — `105` needs a `0` there as a placeholder" or "the borrow took `1` from the tens column; you forgot to reduce it."

A concept becomes *mastered* in one of two ways: either the child solves a curated subset of the lab's challenges (typically 3 of 5, without using the highest hint level), or the child explicitly taps "I've got this" on the concept's intro screen after opening the lab. Both paths are deliberate: the first prevents the affirmation from being a frictionless skip; the second honours the child's own judgement of when they're done.

Mastered concepts unlock their dependent concepts on the Plaza's concept-map view (the adaptive UI lands in Phase 5.5 of our internal build plan). The map is the *territory*; the home page's three recommended doors are *suggestions* into that territory. When a concept becomes mastered, the app offers three follow-on options — *deepening* (go further in the same thread), *branching* (start a sibling thread), or *returning* (revisit something the child hasn't visited in a while) — each with a one-sentence description. The child can always tap "show me everything" and pick from any available concept.

5. What we deliberately refuse

The mechanisms below are common in mainstream educational software. We refuse them, on principle, because they degrade the learning we are trying to support. This list is enforced in our internal guardrails and reviewed before any feature ships.

• No streaks. Streak-based engagement converts learning into a guilt-management problem. A child who misses a day should not be punished for it.
• No XP, badges, levels, or leaderboards. All of these substitute extrinsic motivation for the intrinsic kind we are trying to cultivate. The literature on self-determination theory is clear on the cost.
• No timed challenges. Time pressure narrows cognition. Mathematics rewards slow looking.
• No failure states. A wrong answer surfaces a hint, not a loss screen.
• No notifications, no daily nudges, no "don't lose your progress" messages. The app is something the child opens when they want to.
• No LLM-generated maths shown to the child. Every formula, every example, every hint, every Journal entry is authored or generated by deterministic templates. AI is optional and confined to *rephrasing* existing curated hints in the child's preferred register.
• No open-ended chat. The app cannot send free-form messages to your child.
• No social features. No public profiles, no comparison to other children, no shared scores. The app is local to your device unless you explicitly opt in to sync.
• No real-name collection. The child profile asks for a nickname and an avatar; codenames are fine. We don't ask for school, address, age beyond a coarse age band, or anything else we don't strictly need.
• No hidden skill estimates. No probabilistic mastery model decides for the child what they should do next. The recommendations they see are explicit; the territory is always visible alongside.

6. How we frame examples (and why gold coins)

A surprising number of primary-school maths examples are money problems. They are a natural fit for percent, ratio, fractions, and four-operation arithmetic — money is a thing children encounter, and the operations they want to do with it match the maths we are trying to teach. We use the framing because the underlying mathematics genuinely is what shopkeeping arithmetic looks like. We just refuse to name a real currency.

Concretely: in the Hundred Lab (percent and ratio), the shopkeeping examples are framed in gold coins — a deliberate fantasy currency. The price tag on a wizard's cloak is 40 gold coins; a 25% discount makes it 30 gold coins; the conversion to pounds, euros, dollars, rupees, or yen is left to the child and their grown-up.

We made this choice for three reasons. First, a real currency dates the example the moment we ship it: a "£10 toy" or "$5 lunch" reads as wrong to a child in any country where those numbers don't match local life, and reads as quaint to anyone reading it five years later. Second, a real currency implicitly endorses a consumer framing — *what can you buy* — which is not the cognitive habit we want to anchor when a child is first meeting proportional reasoning. The mathematics is about the relationship between quantities; the consumer subtext is a distractor. Third, our audience is international from day one; gold coins are equally legible in every country and every era, and we don't have to ship localised currency strings to honour that.

The fantasy framing has a small cost: a child who is already comfortable with their local currency loses a moment of familiarity. We judged the trade worth it; the gold-coin examples live alongside non-money framings (recipe scaling, marble bags, stained-glass tiles) so the percent and ratio ideas are never tied to *one* application. Parents who want to anchor the arithmetic in their own currency are warmly invited to do so at the kitchen table; the app deliberately does not.

7. What we are not claiming

We do not yet have controlled efficacy data. The design is grounded in well-validated principles, and the app is being shaped by kid-tester feedback at every iteration, but we make no claim that a child who plays it will outperform a child who does not. We are building it to be *good*, and the early signal is encouraging, but please treat it as one tool in a varied mathematical diet — not as a replacement for handling beans, counting fingers, drawing on paper, or arguing with a teacher.

Some labs in the app are first-cut and will iterate as kid-tester feedback lands. We track this openly in our internal CHANGELOG.md and ROADMAP.md; the same dependency diagram above will continue to expand as new labs ship. If you spot a maths bug, a confusing framing, or a metaphor that breaks down for your child, we want to hear about it.

8. Sources (selected)

References are listed as plain text rather than clickable links to keep this page faithful to the same no-outbound-tracking rule the child-facing surfaces follow. You can find each through any academic search.

• Bruner, J. S. (1966). *Toward a Theory of Instruction.* Harvard University Press. [Concrete–Pictorial–Abstract foundation.]
• Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. *Educational Studies in Mathematics,* 22(1), 1–36.
• Lakoff, G. and Núñez, R. (2000). *Where Mathematics Comes From.* Basic Books. [Conceptual metaphor in mathematics.]
• Vlassis, J. (2002). The balance model: hindrance or support for the solving of linear equations with one unknown. *Educational Studies in Mathematics,* 49, 341–359.
• Pelánek, R. (2017). Bayesian knowledge tracing, logistic models, and beyond: an overview of learner modeling techniques. *User Modeling and User-Adapted Interaction,* 27, 313–350.
• Singapore Ministry of Education (2021, updated 2025). *Primary Mathematics Syllabus P1–P6.*
• Tall, D. and Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. *Educational Studies in Mathematics,* 12, 151–169.
• Cornu, B. (1991). Limits. In D. Tall (Ed.), *Advanced Mathematical Thinking* (pp. 153–166). Kluwer.