learnbits.app· Spelling

The evidence

How automating works, and why

This is not a times-tables quiz with an animal stuck on it. It is one didactic idea, from start to finish: how you move a number fact from working it out to knowing it. Below is that idea, with the research underneath it.

The idea

Automating is more than learning by heart

Automating arithmetic is the ability to solve a sum quickly, accurately and almost by itself, without thinking about it for long Kennisrotonde, 2023. The literature distinguishes two things inside that, and the difference is the heart of this whole programme Gerrits & Noteboom, 2018:

  • Working it outYou can work it out. You have a method — a strategy — that gets you to the right answer, even if it costs a few seconds.
  • Knowing itYou know it instantly. The answer comes out of memory, with no working out. This is what memorisation means.

A fact is only truly automated when you can do both: work it out and know it directly. Which is why practising speed alone (drilling without understanding) or strategy alone (understanding without ever becoming fluent) is not enough either way. The order is: first be able to work it out, then know it by heart.

This four-state model comes straight out of Dutch primary maths teaching — the idea that a fact is first «worked out» and only later «known by heart» Gerrits & Noteboom, 2018; Danhof et al., 2013. Spelling uses exactly these four states: in your account you can see, per table, which sums are grey, red, orange or green.

Why it matters

A full head cannot think

Working memory — the place where you «hold» something while you think — is small. If a child still has to work out 7 × 8, that small memory is already full before the actual problem (a division, a fraction, a word problem) even begins. A child who has automated the basic facts keeps that memory free for the real thinking. Working on automation therefore has a positive effect on arithmetic in general Kennisrotonde, 2023.

It is not a side effect that arrives on its own. In one study, children who trained deliberately for speed and automation had measurably better arithmetic two months later than children who practised only with the right strategy without training for speed Singer-Dudek & Greer, 2005. And it keeps going all through primary school: addition tends to settle around age 7–8, subtraction and multiplication around 9–10, and division reaches fluency only after primary school for many children Gliksman et al., 2022.

Understand first

You do not learn a table on its own, you derive it

Programmes that teach the strategy first and automate after turn out to be more effective than programmes that do only one or the other Ok & Bryant, 2016, Carr et al., 2011, Woodward, 2006. Knowledge you understand can also be applied flexibly to new sums Boaler et al., 2015. So the app teaches the tables in the order in which they hold each other up — the «anchor tables» first (×1, ×2, ×10, ×5), because every other one can be derived from those SLO — kerndoelen.

Times 10

7 × 10

A zero goes on the end. 7 → 70. The easiest anchor table, and the basis for nearly every other one.

Times 5 is half of times 10

6 × 5

6 × 10 = 60, and half of that is 30.

Times 9 is times 10, minus one lot

8 × 9

8 × 10 = 80, minus 8, is 72.

Split the hard tables

7 × 6

5 × 6 = 30 (you know that one), plus 2 × 6 = 12, is 42.

Where Spelling does this. Every table carries its strategy on the child’s own path — «this is how you work it out», with a worked example. And every time a child gets something wrong, what appears is not just the right answer but that same way of reaching it, applied to that sum.

How you practise

Four things the research shows over and over

1

Short, daily and spread out

Ten to fifteen minutes a day is more effective than one long stretch. It works better still when that time is spread across the day with generous gaps, for instance three times five minutes.

In the app: 3 short sessions a day, with rest in between — never one long one.

Inspectie van het Onderwijs, 2011; Powell et al., 2022; Schutte et al., 2015

2

A good mix of easy and hard

Effective practice keeps a good ratio between sums a child already knows and sums that are still hard. Enough footing to keep going, enough resistance to learn something.

In the app: Every session is 10 sums that already work and 5 that do not yet.

Burns, 2005; Codding et al., 2011; Fuchs et al., 2008

3

See straight away whether it is right — and why

Children need to see immediately whether their answer is right, and a worked example helps them repair the mistake instead of repeating it. Modelling and immediate feedback are constants in what works.

In the app: A mistake stops the session: the screen turns, the right answer appears, and so does the explanation.

Codding et al., 2011; Fuchs et al., 2008

4

Produce the answer yourself

Recognising (pointing at the right answer among options) is not the same as remembering (calling the answer up yourself). Only the second trains the memory that automating needs.

In the app: No multiple choice anywhere: your child types or taps the answer.

didactic principle

Fitted to the child

One continuous line, and the child finds its place on it

Mental arithmetic is not a loose collection of facts but a continuous line: the bonds to ten make bridging the ten possible, that makes the tables possible, and division is a table read backwards. Every step rests on the one before, and every step has a threshold that has to be crossed before the next one makes any sense Danhof et al., 2015. Spelling puts that whole line on one ladder — from the first splits to the big tables, division and telling the time — in the order the curriculum prescribes SLO — kerndoelen.

A child should not start halfway through a school year at whatever point their class happens to be. So every chosen path opens with a short placement test: questions spread across the width of that path, from which the app sees what is already fluent and what is not. On that basis it moves the child a step up if they already have the material, or a step down if the foundation is not there yet — with one line of explanation why. That way nobody practises what they already know, and nobody gets stuck on something whose stepping stone is missing Danhof et al., 2013.

And it does not stop there. Every sum adjusts the level: how fast and how often it goes right decides whether a fact turns green and drops out of sight, or comes back. If a block goes too easily the line shifts up a school year by itself; if a block keeps catching, the app pulls the part underneath it — sometimes a year lower — back in until it holds again. The level is not a setting chosen once, but something that keeps moving with the child. Underneath that colour, the app also watches reaction time and which sub-step — a borrow column, a denominator — is still catching, without that ever changing the colour itself.

The engine

What is doing the sums under the colour

Red, amber and green are what you see; underneath, the app keeps a probability model of three states per fact — new, can work it out, knows it instantly — and moves it after every answer with the same kind of maths that has been used for thirty years to track what someone knows: Bayesian Knowledge Tracing Corbett & Anderson, 1994. Two kinds of answer do not move that model: asking for help (it proves the child can work it out, not that they remember it) and a correct retry after a mistake — that does not count as first evidence.

Alongside those three states per fact, the app also keeps a skill rating per learning line that moves with the child, on the same kind of scale that has ranked chess players against each other for a hundred years: an Elo rating Elo, 1978. Every fact gets its own difficulty from the rung it sits on; a correct answer on something harder than expected pushes the rating up more than a correct answer on something easier. That way the app already has a sense of how likely a child is to know a sum before it has ever been asked — and that is what makes the placement test, and the very first guess at any new fact, land in the right place.

Time counts as a third observation alongside right and wrong: a slow correct answer mostly feeds «can work it out», a fast correct answer mostly feeds «knows it instantly» — exactly the distinction this whole page opens with. And where a question has several steps inside it (a borrow column in subtraction, the minutes on a clock, the denominator of a fraction), the app keeps a separate profile of those too: a block that leans on a sub-step still catching stays lower, even when the rest of the line is doing fine.

This model steers exactly two things, and never a third: which question comes up in a sprint, and where the line sits between «practised enough» and «keeps coming back». It never steers the colour itself — that keeps counting on the same three fields a parent and a teacher see (how often right, how often tried, how fast), so a green tick keeps meaning the same thing, model or no model. The full account, with the maths spelled out, is in docs/motor.md.

Ownership

You practise for yourself, not for a grade

Children who can see for themselves where they stand and what is left to learn practise more purposefully and are more motivated — in one study, self-regulated practice with your own progress in view was more effective than fifteen minutes of drilling rows Sleeman et al., 2021. An educational game adds no miracle of learning on top of that, but it does add something that counts: children feel more confident and keep practising for longer Pan et al., 2022, Cozad & Riccomini, 2016.

Which is why the only number a child sees about themselves in the app is how many days in a row they were there. No score, no leaderboard, no comparison with anyone else. Turning up is the behaviour we reward; the animal and its world are the reason to come back. «You are not doing it for the teacher, not for a mark, but for yourself» Gerrits & Noteboom, 2018.

For facts that are hard to remember, a mnemonic can help Nelson et al., 2013 — and knowledge that is not maintained fades Danhof et al., 2015. So the app keeps asking known facts back among the 10 of 15 «already works» sums, long after they have turned green.

Honest about what this is

This page is a plain-language summary of published research on automating arithmetic. It is not evidence that this app itself has been trialled — that would be a different claim. What we can show is that every choice in the app traces back to a source below, and that a green fact means: 3 seconds or faster, and almost always right. The numbers on this page come from the same code the app runs on.

Accountability

Sources

  1. Kennisrotonde. (2023). Is een kwartier rekenautomatisering per dag effectief voor de rekenautomatisering van basisschoolleerlingen? Welke invulling (qua aanpak en inhoud) is het effectiefst per leerjaar? (KR.1849). NRO.
  2. Gerrits, P., & Noteboom, A. (2018). Rekenen op je basisvaardigheden: automatiseren en memoriseren. JSW, 102(10), 12–15.
  3. Danhof, W., Bandstra, P., & Hofstetter, W. (2015). Rekendrempels nemen. Volgens Bartjens, 34(3), 4–7.
  4. Danhof, W., Bandstra, P., Faber, S., Minnaert, A., & Ruijssenaars, W. (2013). Rapport Rekenproject Leerbaarheid van hoofdrekenen. Groningen: Rijksuniversiteit Groningen.
  5. SLO. Kerndoelen rekenen-wiskunde (o.a. kerndoel 26 en 27: de basisbewerkingen tot 100 uit het hoofd, de tafels van buiten). tule.slo.nl.
  6. Boaler, J., Williams, C., & Confer, A. (2015). Fluency without fear: Research evidence on the best ways to learn math facts. Reflections, 40(2), 7–12.
  7. Ok, M. W., & Bryant, D. P. (2016). Effects of a strategic intervention with iPad practice on the multiplication fact performance of fifth-grade students with learning disabilities. Learning Disability Quarterly, 39(3), 146–158.
  8. Carr, M., Taasoobshirazi, G., Stroud, R., & Royer, J. M. (2011). Combined fluency and cognitive strategies instruction improves mathematics achievement in early elementary school. Contemporary Educational Psychology, 36(4), 323–333.
  9. Woodward, J. (2006). Developing automaticity in multiplication facts: Integrating strategy instruction with timed practice drills. Learning Disability Quarterly, 29(4), 269–289.
  10. Singer-Dudek, J., & Greer, R. D. (2005). A long-term analysis of the relationship between fluency and the training and maintenance of complex math skills. The Psychological Record, 55, 361–376.
  11. Powell, S. L., Duhon, G., Poncy, B. C., Mwavita, M., & Englen, A. J. (2022). Distributed practice in math facts fluency: A comparative analysis of varied intersession intervals. School Psychology Review, 51(5), 517–525.
  12. Schutte, G. M., Duhon, G. J., Solomon, B. G., Poncy, B. C., Moore, K., & Story, B. (2015). A comparative analysis of massed vs. distributed practice on basic math fact fluency growth rates. Journal of School Psychology, 53(2), 149–159.
  13. Fuchs, L. S., Fuchs, D., Powell, S. R., Seethaler, P. M., Cirino, P. T., & Fletcher, J. M. (2008). Intensive intervention for students with mathematics disabilities: Seven principles of effective practice. Learning Disability Quarterly, 31(2), 79–92.
  14. Inspectie van het Onderwijs. (2011). Automatiseren bij rekenen-wiskunde: Een onderzoek naar het automatiseren van basisbewerkingen rekenen-wiskunde in het basisonderwijs. Utrecht.
  15. Burns, M. (2005). Using incremental rehearsal to increase fluency of single-digit multiplication facts with children identified as learning disabled in mathematics computation. Education and Treatment of Children, 28(3), 237–249.
  16. Codding, R. S., Burns, M. K., & Lukito, G. (2011). Meta-analysis of mathematic basic-fact fluency interventions: A component analysis. Learning Disabilities Research & Practice, 26(1), 36–47.
  17. Gliksman, Y., Berebbi, S., & Henik, A. (2022). Math fluency during primary school. Brain Sciences, 12(3), 371.
  18. Sleeman, M., Friesen, M., Tyler-Merrick, G., & Walker, L. (2021). The effects of precision teaching and self-regulated learning on early multiplication fluency. Journal of Behavioral Education, 30, 149–177.
  19. Pan, Y., Ke, F., & Xu, X. (2022). A systematic review of the role of learning games in fostering mathematics education in K-12 settings. Educational Research Review, 36, 100448.
  20. Cozad, L. E., & Riccomini, P. J. (2016). Effects of digital-based math fluency interventions on learners with math difficulties: A review of the literature. Journal of Special Education Apprenticeship, 5(2), 1–19.
  21. Nelson, P. M., Burns, M. K., Kanive, R., & Ysseldyke, J. E. (2013). Comparison of a math fact rehearsal and a mnemonic strategy approach for improving math fact fluency. Journal of School Psychology, 51(6), 659–667.
  22. Corbett, A. T., & Anderson, J. R. (1994). Knowledge tracing: Modeling the acquisition of procedural knowledge. User Modeling and User-Adapted Interaction, 4(4), 253–278.
  23. Elo, A. E. (1978). The rating of chessplayers, past and present. Arco Publishing. (Toegepast op leren in o.a. 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.)

See the theory in practice?

The first week is free. You make an account first — it belongs to the parent and it carries your child’s progress — and then practice can start.

For parentsOpen Spelling