How automating works, and why

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.

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.

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.

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. Geography 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. Anyone who does not know where to start can therefore take a short placement test per area. It is a choice, not a gate: questions from every topic in that area, around the level of the child’s year, 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.

Benjamin Bloom saw that pupils get much further when they only start the next step once the previous one is truly mastered, with extra explanation and practice for those who are not there yet Bloom, 1968. A review of dozens of studies on this mastery learning found clearly better results, above all for the pupils who struggled Kulik et al., 1990. The heart of it is not being stricter, but letting nobody move on with a gap underneath Guskey, 2007.

In Geography this happens by itself: a new step only opens once the previous one comes from memory, and whatever has not stuck comes back until it does. A teacher can set this even stricter per subject, so the pupil cannot jump ahead on their own, and sees in the class heatmap, down to the single sum, who has mastered it and who has not. Whatever a child needs, the teacher or parent sets with one tap.

Formative assessment is teaching on what you have just seen: finding out where each child stands, and choosing the next step on that straight away. A much-cited review of the research found that exactly this feedback during learning — and not a mark afterwards — moves pupils forward, and the weaker pupils most of all Black & Wiliam, 1998.

In Geography the looking happens while children practise. A teacher sees a heatmap per child and per class: every step grey, red, orange or green, down to the single sum. Who has mastered something and who has not is visible at a glance, and whatever a child or a small group needs, you set with one tap. A short test — the placement round, or a quiz with the class — shows where a new lesson should start, without turning it into a mark.

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.