A way to work it out first, then practice that brings it back

Where your child starts

• Before the first round, Pip asks a few questions to find a starting point. It is twelve questions at most, and it starts near your child's age: stage 2 for ages 4 to 5, stage 9 for ages 6 to 7, stage 20 for ages 8 to 9 and stage 27 for ages 10 to 12.
• Your child sees no score and no clock. The screen says: "This isn't a test. Pip is just finding your starting point."
• An answer counts only when it is right and in good time. A wrong or slow answer, or "I'm not sure yet", moves the search down.
• Your child can skip it, or do it again later. A grown-up can also move a child to any open stage.

A strategy before a drill

• Every fact is taught with a way to work it out: fill the ten, double it, count in fives, turn the array. The method grows with your child. Each stage teaches a way to work its facts out, and later stages teach quicker ways for the same facts. 3 × 5 is first 5 + 5 + 5, and a later stage teaches it as counting in fives.
• The first time your child meets a new kind of fact, a short card comes first. It works a similar fact in three steps, with a picture. Then Pip says "Now you" and asks the new fact, with no clock.
• A round teaches at most two new methods. After that, a new fact worked the same way gets a one-line reminder instead of the whole card.
• Hints start with the method, never the answer. Hint shows the line first, then the picture.

Facts come back before they fade

• A fact counts as known after two quick right answers, or four slower ones, with no hint.
• Then it comes back for a check a day later, then four days after that, then two weeks after that. A fact that passes all three is mastered, and after that it comes back about once a month.
• A slow right answer at a check is still right. The fact waits the same gap again before it moves on.
• One miss at a check brings the fact back the next day, and its checks start again from the first. A second miss puts it back into everyday practice. Nothing your child has learned is lost over one slip.
• Practice and Journey rounds make room for the checks that are due.

No clock unless they want one

• A question stays on the screen until your child answers it. Nothing times out, and a slow answer is never marked wrong.
• The timer is a ring that slowly empties, never numbers counting down. It starts hidden for ages 4 to 7 and shown for ages 8 to 12. Your child can hide or show it with one tap during a round.
• When the ring runs out, it fills with a soft amber and says "Take your time."
• Time still matters a little. A right answer in good time counts twice as much toward knowing the fact as a slower one. Good time is 15 seconds a question on the early stages, 12 in the middle and 10 at the top.
• Extra time makes the ring up to three times as long. Like every accessibility setting, it is free on every plan.

Mistakes get the strategy back, not a red X

• Nothing in your child's app turns red, and there is no buzzer. A wrong answer gets an amber card that says "Not yet — here it is", with the answer and the method, step by step, beside its picture.
• Your child taps "Got it" and tries the same question again, with the answer hidden. That try is not scored. They can keep trying, or tap "Next one".
• The missed fact comes back later in the same round.
• Placement and the Tables Check show no card. They move straight on. A race shows no card either: it shows the answer for a moment, then moves on.
• Asking for a hint is not a mistake. On the round's route, an answer after a hint shows as a light bulb, not a "Not yet".

What we don't claim

• Numbula covers math facts: adding, subtracting, multiplying and dividing, up to 12 × 12. It does not cover reasoning, word problems, geometry or measurement.
• It is not a test or a diagnosis. Placement finds a starting point, nothing more.
• We have not run a study of Numbula. The research below is about how people learn and remember in general. It shaped how Numbula is built. It does not show that Numbula will work for your child.
• If your child is already quick at number facts, Numbula will feel slow to them. It is built for the child who freezes.

Research

Each source is published in a research journal. After each one: what it found, and where it shows up in Numbula.

• Baroody, A. J., Bajwa, N. P., & Eiland, M. (2009). Why can't Johnny remember the basic facts? Developmental Disabilities Research Reviews, 15(1), 69–79. DOI 10.1002/ddrr.45. Argues that fluency with the basic facts grows out of number sense: patterns and relations, not answers memorized one by one. In Numbula: a strategy before a drill.
• Roediger, H. L., & Karpicke, J. D. (2006). Test-enhanced learning: Taking memory tests improves long-term retention. Psychological Science, 17(3), 249–255. DOI 10.1111/j.1467-9280.2006.01693.x. Students who practiced recalling a passage remembered more of it days later than students who studied it again. In Numbula: every fact is asked, not shown.
• 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), 354–380. DOI 10.1037/0033-2909.132.3.354. A review of 317 memory experiments on spacing practice out over time. The longer something has to last, the longer the best gap between sessions. In Numbula: checks a day, four days and two weeks apart.
• Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2013). Math anxiety, working memory, and math achievement in early elementary school. Journal of Cognition and Development, 14(2), 187–202. DOI 10.1080/15248372.2012.664593. Found math anxiety in first and second graders, linked to lower math scores in children with more working memory. In Numbula: nothing times out, and nothing turns red.
• Pashler, H., Cepeda, N. J., Wixted, J. T., & Rohrer, D. (2005). When does feedback facilitate learning of words? Journal of Experimental Psychology: Learning, Memory, and Cognition, 31(1), 3–8. DOI 10.1037/0278-7393.31.1.3. In a study of learning word pairs, giving the correct answer after a wrong one made it far more likely to be remembered a week later. In Numbula: a wrong answer gets the answer and the method back.