Workings
Coming soon

The science

Harder in the right places. Easier everywhere else.

Workings isn’t built on ed-tech fashion. Each rule in the engine comes from the cognitive science of learning, and each one is written into the code rather than left as a setting. Here is the science, in plain language, and exactly what it makes the product do.

Start with the four pillars
Gus
Great work, Jarrah. You kept at it today. See you next time.
Kit
You said 33. Have a look at the ones column with me. What is 2 take away 5?
Marlow
You’ve got Faces, edges and corners. That’s yours now.

Four pillars of learning

Four mechanisms, and what each one forces the product to do.

The neuroscientist Stanislas Dehaene compresses the learning sciences into four mechanisms.1 In Workings they are the structure of the engine, not a reading list at the back of a brochure.

Productive struggle · the generation effect

Never the answer. Always the scaffold.

The science

Memory is built by retrieving and generating, not by reading someone else’s solution. Answers you produce yourself are remembered better than answers you are shown: the generation effect.2 And curiosity is not just a mood: it helps memory. In an experiment with adults, curiosity about a question improved memory for its answer.3

Handing a child the answer removes the very thing that was making them curious.

What Workings does

  • The answer is never revealed, in any mode. That was decided, and it is enforced by software checks, not by good intentions.
  • When a child is stuck, help comes as a ladder of smaller steps, and the child always takes the last one.
  • Every question and hint is checked so it can never contain the answer, and the answer is never sent to the child’s browser.
  • Help notices first: when a question is taking longer than usual for this child, Pip offers to think it through together, before frustration sets in.
  • Sometimes a child guesses first (more or less than 30?), which opens a gap the working then closes. The guess is never counted.
  • If a child disengages, the question closes unanswered. It isn’t counted, and its maths returns another day in a new form.

The hint ladder

  1. An offer before frustration

    Pip notices when a question is taking longer than usual for this child, or their working keeps getting cleared, and asks: “Want to think this one through together?” A no is respected.

  2. Where did you get to?

    Four taps: not sure what it’s asking, don’t know how to start, got stuck partway, or I’ve got an answer. The reply decides who helps next.

  3. A question aimed at the mistake

    If a wrong answer matches a known misconception, Kit asks the one question that exposes it.

  4. If the reading is the snag

    When the words are the problem, Bramble unpacks the question instead of giving a maths hint.

  5. Help me: one small step at a time

    Each press shows one more line of a way in: smaller steps, a times-table fact rescued from ones they know, a friendly number (“49 is nearly 50”) or something to see. No line gives the answer.

  6. Step back to the prerequisite

    Deeper still, a few questions from the prerequisite underneath, and a similar question comes back a little later to see if it’s landed.

  7. The last step is always theirs

    A child who has an answer is told to pop it in and check. If they walk away, the question closes unanswered and its maths comes back another day in a new form.

Desirable difficulties · error feedback

The child must sometimes be wrong.

The science

Learning runs on prediction error: no discrepancy, no signal, no update. Practice that feels easy often produces less lasting learning than practice with the right amount of difficulty: what Elizabeth and Robert Bjork call desirable difficulties.4

Research on how people learn suggests the most efficient training sits at roughly 85% success, or about 15% errors.5 Workings allows extra headroom for young and anxious children.

What Workings does

  • Before serving a question, the engine predicts this child’s chance of success and picks one in the 70–85% band, aiming for 78%.
  • Ability and question difficulty are both estimated from every answer, in the style of Elo ratings used by adaptive systems such as Math Garden.6
  • A session where a child gets everything right is treated as mis-pitched. Sustained success above 90% pushes the difficulty up.
  • The session opens with confident warm-ups and always closes on a question the engine expects the child to get right.

Predicted chance of getting it right

Every question is chosen to land between 70% and 85%, aiming for 78%. Below it, practice hurts. Above it, nothing new is being learned.

Cognitive load theory · working memory

One stretch at a time.

The science

Working memory holds only a handful of items at once, about four chunks.7 Whatever competes for those slots doesn’t get learned. If most of them are spent decoding a sentence, the arithmetic has nowhere to run. That’s cognitive load theory in a line.8

Vocabulary adds load too. Researchers sort words into everyday words, general academic words and subject-specific words,9 and the last two cost a young reader far more.

What Workings does

  • Every question sits on a five-level reading scale, from no reading at all to inference, with measured limits on words, vocabulary and distractors.
  • A question may push the maths or the reading, never both at once.
  • Each child has two abilities, maths and reading in maths, so a wrong answer is attributed to whichever was more likely the cause.
  • Hints are never harder to read than the child is comfortable with, and read-aloud never counts against them.

The same maths, five reading loads

  1. Reading level 1: No reading

    72 − 45 = ?

  2. Reading level 2: One plain idea

    Jarrah had 72 cards. She gave away 45. How many are left?

  3. Reading level 3: A short situation

    Jarrah had 72 footy cards. She gave 45 of them to her cousin. How many cards does Jarrah have left?

  4. Reading level 4: A situation with noise

    Jarrah’s team has 18 players. She had 72 footy cards and gave 45 to her cousin. How many does she have now?

  5. Reading level 5: Inference

    Jarrah now has 45 fewer footy cards than at the start of the season, when she had 72. How many does she have?

Every line asks for 72 − 45. The numbers and the operation never move; only the reading does.

Retrieval practice · the spacing effect · consolidation

Remembered next month, not just today.

The science

Pulling something out of memory strengthens it more than looking at it again (retrieval practice),10 and spreading that practice over widening gaps beats cramming (the spacing effect).11 Both are among the most reliable findings in learning research.12

Sleep matters as well: it’s part of what moves a skill from effortful to automatic.13 Performance during practice is not the same as learning.

What Workings does

  • Mastered skills come back after 1, 3, 7, 16 and 35 days, at the harder end of the stage and in a different question type where possible.
  • The same maths returns in new forms; the identical question never repeats for a child, ever.
  • Crossing the mastery line isn’t enough: it must be confirmed by a correct, unhinted answer after at least one night’s sleep.
  • A kept skill that holds up at its first review, a week or more later, is marked held. A later slip never takes that away.
  • Sessions are short, about 20 minutes, and then they stop.
  1. Mastered
  2. Day 1
  3. Day 3
  4. Day 7
  5. Day 16
  6. Day 35
A mastered skill comes back at widening intervals, in a different form each time. Two missed reviews in a row put it back in practice, never shown as a loss.

Interleaving

Mixed practice, not blocks of the same thing.

The science

Doing twenty of the same problem type in a row feels productive, but mixing problem types forces a child to decide which method a problem needs, and that choice is much of what maths asks. Interleaved maths practice has outperformed blocked practice in experiments with college students.14

What Workings does

  • Practice is interleaved across strands, with no more than three questions in a row from one skill.
  • Roughly 70% of questions are at the child’s frontier, about 20% are spaced review, and remediation steps in when a child stalls.
  • Question types rotate too, so the same idea is met as a picture, a number sentence, a word problem and a riddle.

Concrete, pictorial, abstract

Many ways to see the same number.

Children build understanding by moving between hands-on objects, pictures and symbols: the Concrete–Pictorial–Abstract progression that grows out of Jerome Bruner’s work.15 Workings asks the same maths through ten question types, from hands-on to explain-it, and draws every model precisely from structured components, never with an image model.

Base-ten blocks72 as tens and ones. Take 45 away. What’s left?
7245?
Bar modelA whole of 72, a part of 45, and a part to find.
45507072+5+20+2
Number lineJump from 45 to 72. How far did you go?

Error analysis · metacognition

Every mistake is a clue.

The science

Children’s wrong answers are rarely random. 72 − 45 = 33 is what you get by taking the smaller digit from the larger in each column, a specific, fixable misconception. Asking children to find and explain errors, and to justify their own thinking, builds the reasoning that syllabus writers call working mathematically and that Webb’s Depth of Knowledge rates as higher-order.16

What Workings does

  • The wrong options in multiple-choice and spot-the-mistake questions are real errors, each mapped to a misconception.
  • The bank covers every strand from Kindergarten to Year 10: 121 misconceptions, each with a hinge question for Kit and a teaching move for the teacher.
  • When a child’s answer matches one, Kit asks the one question that exposes it.
  • “Spot the mistake”, “riddle” and “explain it” questions sit alongside the procedural ones, and reasoning types count for more in mastery.
  • Each week, common wrong answers no one anticipated are proposed as new misconceptions, checked against children’s real answers, and added only when a teacher approves.

Spot the mistake

Tom says 72 − 45 = 33. Is he right? What did he do?

Kit
You said 33. Have a look at the ones column with me. What is 2 take away 5?

Kit’s line is from an illustration of a session. The answer is never given.

Prerequisite knowledge · remediation

A curriculum that knows what depends on what.

The science

New knowledge is built on old knowledge. When a child is stuck on a topic, the obstacle is often a prerequisite underneath it, and more practice on the topic itself won’t fix that.

Fractions are a good example: in the Australian syllabus the groundwork is partitioning a length in measurement, which is why fractions are taught on a number line rather than as slices of pizza.

What Workings does

  • The curriculum is a graph of skills by year and strand, with each link marked as required or supporting.
  • A skill unlocks only when its required prerequisites are mastered, so a child progresses per strand and is never capped at their school year.
  • When a child stalls, practice drops to the least secure prerequisite for a few questions, then returns.
  • The graph, the dependencies and the mastery bar are owned by a mathematics teacher, not by the software.
A slice of the curriculum graphYear 2 Geometric measure is required for Year 3 Fractions. Year 3 Multiplicative relations supports Year 3 Fractions. Year 3 Fractions and Year 4 Multiplicative relations are both required for Year 4 Fractions.Year 2Geometric measureYear 3MultiplicativeYear 3FractionsYear 4MultiplicativeYear 4FractionsStalled here?Step back topartitioning a length.
Solid arrow: requiredDashed arrow: supportingGreen: the prerequisite to step back to

Measuring mastery

Mastery is estimated, not counted.

The science

Because Workings deliberately serves questions a child gets right about 78% of the time, a raw hit rate says almost nothing about mastery. The better question is: if this child faced every kind of question this stage asks, how often would they succeed?

What Workings does

  • Mastery is calculated across every complexity level, every valid question type and the reading level the stage expects, weighted toward the harder end and the reasoning types.
  • It needs at least 20 responses on a skill, so a lucky run can’t cross the line.
  • Guessing is left out: multiple choice counts for less, because it’s weaker evidence.
  • The mastery threshold is set by the curriculum lead, and a mastery once earned is never taken away retrospectively.

Maths anxiety · motivation

No clock. Nothing to lose.

The science

Maths anxiety isn’t a metaphor. In adults with high maths anxiety, just expecting a maths problem activates brain regions linked to threat and pain,17 and the anxiety consumes the working memory the arithmetic needs.18

The mechanic the research warns about isn’t counting effort. It’s the count you can break.

What Workings does

  • No timers, countdowns, speed scores or “quick!”. Response time is never used as evidence of ability.
  • The 20-minute budget is kept invisibly; a child never sees a clock or a count of questions remaining.
  • Reps and practice weeks only accumulate. A week that falls short takes nothing away, so there is no run to break and no “at risk” warning.
  • Over a term, recognition tapers from counts toward mastery unlocks, so motivation moves into what a child can do.
  • No leaderboards and no comparison with other children. The words are “next” and “frontier”, never “behind”.

Verifiable by design

The AI writes the words. It never decides the answer.

The science

Language models are fluent and occasionally wrong. A model that miscounts is an inconvenience; one that marks a child wrong for a right answer is a betrayal. The only safe design is one where a model can’t decide what’s correct.

What Workings does

  • Every answer is computed by code from the question’s own numbers. Fractions and decimals use exact arithmetic.
  • Question wording is drafted ahead of time, checked by software across fifty sets of numbers and approved by a person. No model writes a question while a child waits.
  • Where help is written live, software checks every reply before a child sees it: never the answer in any form, and only the question’s own numbers.
  • Every question is validated before it is shown (no invented numbers, no leaked answer, no reading load above its level), and anything that fails is never shown.
  • A child’s name, email and school are never sent to a model.

What we don’t claim

Honest about the evidence.

The research above is about how learning works. It isn’t proof that Workings works. That has to be measured.

  • We don’t publish efficacy percentages or “grade levels of growth”. When we have independent results, we’ll share them.
  • We plan an independent, pre-registered evaluation with consented, de-identified data, and the engine checks its own calibration continuously.
  • Workings is not a clinical product. It doesn’t diagnose, treat or remediate any condition.
  • Practice weeks and reps are our weakest-evidence decision, so they ship as an experiment with written criteria for removing them.
Coming soon

References

  1. Dehaene, S. (2020). How we learn: Why brains learn better than any machine… for now. Viking.
  2. Slamecka, N. J., & Graf, P. (1978). The generation effect: Delineation of a phenomenon. Journal of Experimental Psychology: Human Learning and Memory, 4(6).
  3. Gruber, M. J., Gelman, B. D., & Ranganath, C. (2014). States of curiosity modulate hippocampus-dependent learning via the dopaminergic circuit. Neuron, 84(2).
  4. Bjork, E. L., & Bjork, R. A. (2011). Making things hard on yourself, but in a good way: Creating desirable difficulties to enhance learning. In Psychology and the real world (pp. 56–64). Worth Publishers.
  5. Wilson, R. C., Shenhav, A., Straccia, M., & Cohen, J. D. (2019). The Eighty Five Percent Rule for optimal learning. Nature Communications, 10.
  6. Klinkenberg, S., Straatemeier, M., & van der Maas, H. L. J. (2011). Computer adaptive practice of maths ability using a new item response model for on the fly ability and difficulty estimation. Computers & Education, 57(2).
  7. Cowan, N. (2001). The magical number 4 in short-term memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences, 24(1).
  8. Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2).
  9. Beck, I. L., McKeown, M. G., & Kucan, L. (2013). Bringing words to life: Robust vocabulary instruction (2nd ed.). Guilford Press.
  10. Roediger, H. L., & Karpicke, J. D. (2006). Test-enhanced learning: Taking memory tests improves long-term retention. Psychological Science, 17(3).
  11. Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psychological Bulletin, 132(3).
  12. Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students’ learning with effective learning techniques. Psychological Science in the Public Interest, 14(1).
  13. Diekelmann, S., & Born, J. (2010). The memory function of sleep. Nature Reviews Neuroscience, 11(2).
  14. Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35(6).
  15. Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.
  16. Webb, N. L. (1997). Criteria for alignment of expectations and assessments in mathematics and science education (Research Monograph No. 6). National Institute for Science Education.
  17. Lyons, I. M., & Beilock, S. L. (2012). When math hurts: Math anxiety predicts pain network activation in anticipation of doing math. PLoS ONE, 7(10).
  18. Ashcraft, M. H. (2002). Math anxiety: Personal, educational, and cognitive consequences. Current Directions in Psychological Science, 11(5).

Practice that respects how children learn.

Fourteen days free. Twenty minutes a day. Every question pitched for your child, and the last step always theirs.

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