Atom Learning Model (ALM): how a real classroom got tokenised

Philipp Bogdan

Source-linked AI summary

Atom Learning Model (ALM): how a real classroom got tokenised

Philipp Bogdan

arXiv:2608.21106v1Aug 21, 2026cs.CYcs.AI

TL;DR

The paper asks how classroom instruction can represent individual mathematical abilities finely enough to select work for each child. It presents ALM, which tokenises textbooks into atoms linked by prerequisites and reads questions and mastery on the same graph. In deployment, the structure supported individualized composition, but its central deep-prerequisite premise remained unfalsified because deep questions were rarely served.

Problem
Classroom instruction lacks a representation that reads children at the single-skill resolution of available questions and targets work to individuals rather than attainment bands.
Method
ALM machine-tokenises two mathematics textbooks into atoms and prerequisite links, expressing both question closures and child mastery on the same graph.
Results
£615–£1,230 was the whole-catalogue cost, with prerequisite-link wiring taking 48.8%–68.3% of stage spend across the two runs.
Takeaways & Limitations
The system demonstrates a shared-structure approach to individualized question selection without fitting separate difficulty and ability parameters.
Takeaways & Limitations
98.1% of scored questions had prerequisite-chain depth one or two, leaving the central deep-question premise unfalsified rather than confirmed.

Abstract

from arXiv · show

The Atom Learning Model (ALM) tokenises a school curriculum. Two secondary mathematics textbooks were read by machine into 1,934 atoms, each one thing a learner can do in a single step, ordered by 4,616 machine-written prerequisite links. Both sides of a lesson are then expressed in that one structure: a question is a set of atoms plus everything beneath them, a child's ability is a score between 0 and 1 on every atom of the same graph, and whether a question suits a child is arithmetic over one index, with no difficulty parameter fitted for either side. Nobody wrote an atom, a link or a question. Reading the 757 pages cost £55, building the whole structure cost between £615 and £1,230, and against it the system composed 6,648 questions for 373 children in two English secondary schools over seven weeks, at 26p per composed question. Four measurements went against expectation. The cost is in the links, not the pages. The composer's own difficulty label has a rank correlation of -0.0123 with measured facility, so a language model shown a question cannot say how hard it is. Children stop working when a mark takes seven seconds instead of three. And the deployment never served a question deeper than two prerequisite steps, which is exactly where the central premise becomes testable, leaving it unfalsified rather than confirmed.

1 Thirty children, one hour

ALM addresses the classroom’s resolution problem by representing curriculum knowledge and each child’s position at single-step granularity, enabling individualized question selection.

The problem: Teachers need each child’s small-skill profile and work pitched just beyond that child’s current knowledge.In a class of thirty, this would otherwise require thirty different worksheets.
The problem: English schools commonly narrow this gap by grouping pupils into roughly five attainment bands.Banding improves on cohort-level instruction but stops short of individual resolution.
ALM’s response: ALM tokenises textbook prose into 1,934 named atoms and orders them, putting questions and child abilities on the same structure.The paper develops the representation, construction process, deployment, measurements, and limitations across later sections.
Longer-term horizon: The proposed endpoint is child-paced progression and device-based assessment, but the reported deployment does not claim to achieve it.These are described as the direction of the work rather than results established by the deployment.

2 How school mathematics is already divided

Existing school mathematics structures organize content for planning and reporting, but they do not represent the individual abilities that questions test.

The existing tree: A syllabus tree nests qualifications into modules, topics, subtopics, and microtopics as a filing system.This structure supports planning, publishing, material search, and reporting.
The representational gap: A tree level names a location rather than a learner action, so one shelf can contain several distinguishable abilities.A child may hold some abilities on a subtopic while lacking others.
The representational gap: The gap is between 191 subtopic-level shelves and work targeted to one item on one shelf.Nothing in the existing tree directly addresses that item-level target.

3 Atoms

ALM replaces syllabus locations with machine-generated atoms, explicit actions, and prerequisite relations that define what questions ask and how knowledge is connected.

Atoms: An atom is one learner action small enough to count as a single step; the catalogue contains 1,934 atoms across 191 subtopics.This yields roughly ten atoms per syllabus shelf.
Actions: The catalogue uses six actions across 3,058 atom-and-action pairs, making each pair the basis for question writing.Actions include apply, state, identify, justify, interpret, and derive.
Actions: The action axis replaces a fact-versus-routine taxonomy because the distinction reflects how knowledge is asked, not different kinds of knowledge.The same atom can support different demonstrations under different actions.
Prerequisites: 4,616 links connect the atoms through 3,581 prerequisites and 1,035 derivable-from relations.The serving policy treats both relation types alike when expanding an atom into its closure.
Graph integrity: The graph contains three prerequisite cycles, including a power-and-root pair that only a whole-graph pass can detect.Tarjan’s algorithm identifies the cycles among the prerequisite links.
Automation: No person wrote an atom, link, or question; people supplied schemas, answer forms, constraint language, and page-reading prompts.The remaining catalogue and served content are machine output.

4 One structure, read from both sides

ALM expresses questions and child mastery on one atom graph, so serving can compare the question’s closure with the child’s scores without fitting separate difficulty and ability scales.

One representation: A question contains a target atom plus every atom in its prerequisite closure, here eight named atoms in order.The closure makes the question a structured request rather than a flat item.
One representation: A child’s mastery is a 0-to-1 score on every atom in the same 1,934-node graph.The representation is neither a subtopic score nor one overall ability score.
Serving: Serving uses a lookup and threshold to identify missing, reachable atoms in a question’s closure for each child.The worked example selects two missing atoms before the child attempts the larger question.
Design premise: ALM fits neither a question-difficulty parameter nor a child-ability parameter, unlike approaches that estimate and compare separate scales.Both sides are instead read directly through the shared structure.
Design premise: The model assumes questions are hard because of missing prerequisite atoms and chain depth, not because the top atom has an intrinsic difficulty.This makes difficulty a property of the question-child pair that the graph can compute.
Dependency: 62.4% of the system’s inferred prerequisite knowledge came from propagation rather than direct observation.Because the same links select questions and propagate credit, incorrect links affect both operations.

5 What this is built on

ALM combines established ideas about structured knowledge, prerequisite relations, and generated assessment into an end-to-end classroom system without expert authoring for each unit.

What this is built on: ALM’s novelty is composing existing layers—knowledge structure, prerequisite links, question generation, and adaptive tutoring—without expert human authoring per unit.The paper distinguishes its end-to-end composition from the age of the individual ideas.
What this is built on: Knowledge space theory provides the formalism for curricula represented as ordered components and assessed against that structure.The cited lineage includes work from 1985 onward and later large-scale ALEKS data.
What this is built on: Earlier research extracted concepts and prerequisite maps jointly from textbook text, paralleling ALM’s catalogue and prerequisite-building stages.The paper places these textbook-based approaches about a decade before ALM.
What this is built on: Question generation has longstanding foundations in adaptive testing, item models, worked examples, and mathematics template engines.The cited systems include STACK, WeBWorK, Numbas, and Khan Academy.
What this is built on: Recent language-model work scores, tags, trains on, or generates questions, but typically begins from an item, bank, or human-written component.ALM is positioned as starting from curriculum text rather than an existing question resource.
What this is built on: ALM keeps arithmetic out of the language model by having the composer emit templates while a deterministic kernel draws numerical values.This design responds to evidence that changing only numbers can alter frontier-model accuracy.

6 Two textbooks into a catalogue

ALM turns two mathematics textbooks into a shared atom catalogue through six narrow machine agents, then measures repeatability, consolidation, and build cost. The catalogue is useful but incompletely logged, stochastic in extraction, and expensive mainly because of prerequisite and syllabus wiring.

6.2 What reading a book looks like: £55 bought machine reading of 757 textbook pages; the full catalogue build cost between £615 and £1,230.The books were processed sequentially, with the reading cost representing only a fraction of total construction.
6.1 Six agents, six narrow jobs: Six agents divide extraction, deduplication, syllabus mapping, prerequisite linking, question-constraint proposal, and teaching backfill into independently testable jobs.Each agent has a narrow input, structured output, and contract; none sees the whole catalogue at once.
6.1 Six agents, six narrow jobs: 4,616 prerequisite links are produced in two disjoint forms: conceptual prerequisites for teaching order and derivable-from relations for decomposition.The linker keeps these relations separate because they answer different questions about knowledge.
6.2 What reading a book looks like: 5,193 extraction verdicts comprised 1,838 new atoms, 3,095 reuses, and 260 refinements across both books.Reuse was the majority, while refinements could shrink the pool by absorbing multiple entries into one better-stated atom.
6.2 What reading a book looks like: Two reads of the same 35 pages ended at 85 and 96 atoms, showing page-by-page stochasticity despite stable curve shape and verdict shares.The runs diverged as early as page seven.
6.3 Where the money goes: Reading cost rose from 7.1p to 8.6p per page as the starting pool grew from 1,096 to 1,913 atoms.The reported increase is about one fifth as the pool grows from roughly 1,100 to 1,900 atoms.
6.3 Where the money goes: 86% to 90% of instrumented spend went to prerequisite linking and syllabus joining, versus 5.2% to 8.9% for page reading.The two wiring stages were 48.8% and 68.3% for linking, and 37.5% and 21.9% for syllabus joining across the runs.
6.3 Where the money goes: The larger-scale cost slope remains unmeasured because both gap-filling runs operated against pools near 1,900 atoms.The paper reports the cost level at that pool size, not how rapidly linking cost rises as the catalogue expands.

7 Composing a question

ALM separates question writing from arithmetic: a model emits structured templates, while deterministic components instantiate, validate, repair, and either ship or fail them.

Generation: A template specifies typed slots, constraints, answer form, and the atom-level step graph before numbers are drawn.A deterministic kernel then samples values, checks constraints, and computes answers.
Generation: Backward generation draws hidden solutions first and derives right-hand sides, guaranteeing clean integer solutions for every valid draw.The constraint excludes degenerate cases.
Generation: Eight seeds produced eight valid questions with seven distinct answers from one template.The full template, draws, and worked-atom closure records were published for inspection.
Validation: The sampler rejects candidates that violate constraints or answer-form requirements, then selects the first acceptable draw.This lets the composer declare ranges while the mathematics kernel produces valid instances.
Redesign: Cutting redundant verifier arms and tightening the contract reduced median composition time from 83.4 seconds to 26.0 and raised first-pass ship rate from 67.1% to 86.5%.Compose passes per question fell from 1.47 to 1.16, while mean arms per question fell from 7.14 to 6.41.
Validation: 6,574 of 6,648 shipped questions were recomputable, and all 6,574 reproduced the displayed answer.The remaining 74 could not be compared because stored forms lacked support or the kernel errored.

8 From a mark to mastery

ALM converts marked work into atom-level evidence, propagates credit through prerequisite links, and updates time-sensitive mastery estimates; the deployment mostly exercised shallow questions.

Decomposition: Each marked attempt becomes a credit-vector entry for every atom named by the question’s step graph.Entries record the atom, action, credit, signal strength, and sampled range.
Decomposition: Follow-through is assigned by a deterministic pass over the directed step graph rather than by asking a model to interpret the whole response.This preserves method credit when an early mistake propagates into later work.
Propagation: 62.4% of prerequisite beliefs came from propagation rather than direct observation.Credited atoms push weaker credit down prerequisite links, making the mastery map denser but allowing wrong links to corrupt downstream belief.
Decay: Mastery scores use a modified FSRS variant with hand-set spacing and stability changes; no constants were fitted to the corpus.Scores rise with evidence and fall with time rather than functioning as running mark averages.
Deployment: Only 921 of 16,083 credit vectors, or 5.7%, distinguished between atoms within one question.The deployment mostly served questions too shallow to require the decomposition mechanism.
Deployment: Mastery scores were reconstructed after deployment in a four-day burst, so they did not drive what children were served.The state was re-derived from credit vectors in each child’s time order.

9 The knowledge map

The knowledge map presents each child’s atom-level standing across the curriculum, with coarse mastery bands and drill-down from subject territories to individual microtopics.

Map structure: Every atom appears as a point, arranged into six module territories with topics clustered inside their modules.Zooming moves from the whole subject to a single microtopic.
Mastery display: Four colour bands mark not started, just starting from 0%, getting there from 40%, and mastered from 80%.The underlying value is a decayed estimate rather than a mark, and gaps are translated into approximate further questions needed.

10 Seven weeks in two schools

Over seven weeks, ALM ran after a dated changeover across 373 children, with most reported cost arising from question composition and checking rather than catalogue construction. The archive also leaves answered-question cost bounded rather than precisely measured.

Deployment: 373 children made 26,755 attempts after the 27 May–16 July 2026 changeover.The first machine-composed question reached a child on 1 June, and every report count comes after the changeover.
Cost: £1,760 covered composing and checking 6,648 stored questions, or 26p per composed question.This excludes the wider product’s shared diagram and audit stages, which raise the figure to 43p.
Cost: £615 to £1,230 built the catalogue, while the full 46-day window cost £13.90 per child.The one-off catalogue build represented 12–24% of the window total.
Cost: The honest per-question cost bracket was 26p to £1.10 because the archive cannot show how many composed questions children answered.The £1.10 ceiling divides £1,760 by 1,596 distinct questions touched by the 278 end-to-end children.
Cost: 35–70% was the catalogue build’s cost relative to £1,760 of composing and checking, implying payback between the third and fifth week.The result reverses the usual assumption that construction dominates while serving is marginal.
Pipeline: The marking model consumed 191,828 calls and £1,810, while the frontier composing model used 29,871 calls and £1,320.Across fifteen models and three providers, the high-volume marking component ran most often, while composing used the more expensive frontier model.

11 What the measurements showed

The measurements challenged several design expectations: declared difficulty did not track facility, latency reduced continuation, and structural self-reporting required independent checks. Evidence for the depth-based premise remained incomplete because nearly all scored questions were shallow.

Difficulty: -0.0123 was the rank correlation between declared difficulty band and measured facility across 730 sufficiently attempted questions.The three bands covered 3,378 basic, 2,002 moderate and 1,268 advanced questions.
Atom difficulty: The measured facility differences between atoms were reliable, with split-half reliability of 0.877 and a spread of about 0.23.The evidence therefore supported genuine atom differences rather than noise, while leaving the depth comparison difficult to separate.
Depth premise: 98.1% of scored questions were at prerequisite depth 1 or 2, so the deployment could not distinguish the competing explanations of difficulty.The premise predicts differently only when questions reach deeper prerequisite chains.
Latency: 0.941 fell to 0.760 for starting another question when median marking wait rose from 3.2 to 7.3 seconds.The decline stepped down through all five equal groups of the 26,466 marked attempts.
Latency: Doubling the current attempt’s wait was associated with a 13.55-percentage-point fall in the probability of starting another question.The analysis held fixed writing amount, marks, score and session position; the previous attempt’s wait contributed 4.45 points.
Structural validity: 7.9% of a generator’s structural claims were fabricated under an early prompt, increasing with the amount of requested structure.The system therefore measured depth from the artefact and re-derived dependency edges instead of trusting the generator’s claims.

12 Not built, not measured

The deployment did not test several central ALM capabilities or assumptions: it served almost no deep questions, reconstructed mastery offline, and lacked expert catalogue review and human marking checks.

Untested premise: 98.1% of scored questions had prerequisite depth 1 or 2, leaving the central depth-based premise unfalsified rather than confirmed.Only 14 questions reached depth 3 or more, and evidence beyond depth 4 comprised 85 attempts from 7 children.
Scope: The shipped pipeline cannot pose open-ended questions because each question has one fixed answer formula and one fixed step graph.That design supports verifiability but excludes proofs with multiple valid routes and no single answer value.
Deployment scope: Mastery scores were reconstructed after the seven-week window, so serving never used ALM’s per-child state.Making the state live is described as remaining plumbing before the system performs its intended adaptive function.
Validation: No teacher, examiner or subject specialist checked whether atoms, links or questions were correct and aligned to the specification.Shuffled-link replay provided only weaker predictive evidence, not qualified content validation.
Validation: No human re-marked any of the 26,466 attempts, leaving marker accuracy as an unmeasured input to every child estimate.The paper identifies a few hundred hand-marked attempts as the cheapest missing measurement.
Reproducibility: The response corpus requires school authorisation for release, while the catalogue and pipeline remain proprietary.The children’s classroom work is pseudonymous rather than anonymous under the described data-governance arrangement.

13 A school without year groups

ALM’s longer-term vision is to replace cohort-based pacing with learning organized around each child’s atom-level position. The paper explicitly presents this as a destination, not a result established by seven weeks in two schools.

Pacing: Knowing each child’s position at single-step resolution would make year groups levels that children move through at their own pace.The proposed structure replaces cohort containment with individually paced progression.
Pacing: A child could work just past what they hold, with one learner taking nine months and another eighteen months over the same level.Both trajectories are presented as normal when the structure tracks what each child knows.
Assessment: Running exams on lesson devices when children are ready would remove the need for one hall, one morning and one paper for a cohort.Assessment becomes an on-demand activity rather than a single logistical event.
Classroom organization: Classes would become rooms organized around whoever is working on the same material, rather than fixed groups of thirty same-age children.The teacher would regain atom-level knowledge of what each child can do.
Scope: None of this follows from seven weeks in two schools or is claimed as demonstrated by that deployment.The paper frames the vision as the purpose of curriculum tokenisation, not as an established outcome.