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Summit Maths11+ maths practice

The Science

The science behind every question

Summit Maths is built on three well-studied ideas from learning science. Here’s what the research found, how we use each one, and where the evidence stops.

Three ideas, one engine

References

Flow: the right challenge

Each question is chosen so your child has about a 3-in-4 chance of getting it right: stretched, but rarely stuck.

Explore flow

Spaced repetition: practice that sticks

Skills come back for review at growing intervals, mixed with other topics, so they’re recalled rather than re-read.

Explore spaced repetition

Gamification, carefully

A companion that grows with skill gives your child feedback on real progress, and never shrinks after a bad day.

Explore gamification

1 · Flow

Challenge matched to skill

Psychologist Mihaly Csikszentmihalyi described flow as the state of being fully absorbed in a task. One of its best-studied conditions is balance: the challenge is high enough to demand attention, but within reach of the person’s skill. Too hard and people get anxious; too easy and they get bored.

For a child, that balance shifts every week as they learn. A fixed worksheet can’t keep up. An adaptive system can: it re-estimates your child’s level after every answer and raises the challenge as their skill grows, keeping practice inside the “flow channel”.

Skill →Challenge →AnxietyToo hard: mostly wrongBoredomToo easy: little to learnFlow channel
The flow channel, after Csikszentmihalyi. Each white dot is a question: as your child’s skill grows, the next question gets harder, so practice stays in the channel.
Research

Balance predicts flow

A meta-analysis of 28 studies found a moderate link between challenge–skill balance and flow. The link was weaker in education settings, and most studies rely on questionnaires.

Csikszentmihalyi, 1990 · Fong, Zaleski & Leach, 2015

Research

The 85% rule

A mathematical analysis found that, for a broad class of learning algorithms on two-choice tasks, learning is fastest at about 85% correct. It was tested on computer models, not children: a guide, not a law.

Wilson, Shenhav, Straccia & Cohen, 2019

Research

Success keeps children practising

207 primary school children practised on an adaptive maths program for six weeks at different target success rates. Higher success rates led to more problems attempted and bigger gains.

Jansen et al., 2013

Why we aim for 3 in 4

We aim for about a 3-in-4 chance of a right answer. That’s the same target used by Math Garden, a maths practice site that researchers have studied with thousands of children in the Netherlands. No study has found one perfect success rate for children, so 3 in 4 is a sensible middle. We’ll watch how often children practise and come back, and adjust if our own data points to something better.

Klinkenberg et al., 2011 · Jansen et al., 2013 · Papoušek et al., 2016

How we measure challenge and skill

We use an adaptation of the Elo rating system, first built for chess and now used in adaptive learning. Each answer is a match between your child and a question, and both ratings update after every answer.

The bigger the surprise, the bigger the change. Updates start large, so a new child settles quickly, then shrink, so one bad day can’t undo weeks of progress.

From the two ratings we work out the chance of a right answer, then pick questions where it’s about 3 in 4.

Elo, 1978 · Klinkenberg et al., 2011 · Pelánek, 2016

Questions learn too

If children a question should suit keep getting it wrong, its difficulty rises, so it isn’t given to a child too early.

  1. Starts with an estimate we set by hand, from easiest to hardest in each topic.
  2. Adjusts with every answer, quickly at first, then settling.
  3. Ignores guesses: answers under 2 seconds don’t count.
  4. Gets sharper over time: every child’s answers help, so estimates improve as more families practise.

Klinkenberg et al., 2011

Try it: how likely is a right answer?

500 · Secure at Year 5
A typical Year 5 question

75% chance your child gets this question right

Just right: the zone we aim for

20%50%75%100%Our aim: about 3 in 4A guess is right 1 time in 5← Question much harderQuestion much easier →
“Secure” at a year level means a 3-in-4 chance on that year’s typical question: exactly the kind of question we pick. Uses our starting difficulties.

From rating to score: 300 to 700

ScoreMeans
300Secure at Year 3
400Secure at Year 4
500Secure at Year 5
600Secure at Year 6
700Secure at 11+ stretch level, beyond the Year 6 curriculum

“Secure” at a year means about a 3-in-4 chance on that year’s typical question: the same zone we pick questions from. A child at 500 mostly gets typical Year 5 questions, rising with their score.

Still settling. The overall score appears after 40 answers, and each topic score after 15.

Never “11+ ready”. Pass marks differ between areas, schools and years. We’ll only describe readiness once real results show what it looks like.

2 · Spaced repetition

Practice that sticks

Getting a question right today doesn’t mean it will still be there in a month. Two of the most reliable findings in learning science are that practice spread out over time beats the same practice crammed together, and that recalling an answer strengthens memory more than reading it again.

Research

317

The spacing effect

A synthesis of 317 experiments found spaced study beats massed study for later recall, and the best gap grows with how long the knowledge must last.

Cepeda et al., 2006 · Dunlosky et al., 2013

Research

d = 0.83

Mixing topics in maths

In a randomised trial with 787 pupils aged about 12, mixed (interleaved) maths practice beat practice grouped by type on a test a month later: 61% versus 37%.

Rohrer, Dedrick, Hartwig & Cheung, 2020

Research

222

Recall beats re-reading

Answering questions helps knowledge last longer than re-reading it. A review of 222 classroom studies found quizzing raised achievement by a medium amount, from primary school up.

Roediger & Karpicke, 2006 · Yang et al., 2021

How Summit Maths does it
  1. Every skill gets a review date. Once your child has practised a question type, it’s scheduled to come back.
  2. The gaps grow. Each correct review pushes the next one further away: after 1 day, then 3, 7, 14 and 30. A miss brings it back the next day.
  3. Reviews stay at the right level. Due reviews get priority, but the question is still matched to your child, so reviews stay in the 3-in-4 zone.
  4. Topics are mixed. Mixed practice draws on every topic and never repeats the same question type twice in a row.
  5. Mistakes become lessons. Wrong answers explain the likely mistake where there’s a clear one, then show worked steps, then offer “Try a similar one” with fresh numbers.

Worked examples: Sweller & Cooper, 1985 · Barbieri et al., 2023

One skill’s review schedule

01371430 daysPractisedEach right answer stretches the gap

Missed a review? It comes back the next day, and the gaps grow again from there.

If your child has clearly outgrown a question type, its reviews simply move further apart rather than taking up practice time.

Why the gaps grow

Spacing research finds the best gap between reviews depends on how long the knowledge needs to last: the longer the goal, the longer the gap. Growing gaps also mean each review takes a little effort to recall, a “desirable difficulty” that strengthens memory, while we aim to bring each skill back before it fades.

Cepeda et al., 2006 · Bjork & Bjork, 2011

3 · Gamification

Motivation without pressure

Game elements can help children learn, but the type of reward matters. The research points towards feedback on growing skill, and away from prizes children come to expect.

Research

g = 0.49

Gamification and learning

A meta-analysis found gamified learning improved learning outcomes, and that effect held in the most rigorous studies. Effects on motivation and behaviour were smaller and less robust.

Sailer & Homner, 2020

Research

128

Which rewards help

Across 128 experiments, expected, tangible rewards tended to reduce people’s own interest in a task, more so for children. Positive feedback about their competence increased it.

Deci, Koestner & Ryan, 1999

How Summit Maths does it

Watch it grow

  1. Stage 1Tiny cub
  2. Stage 2Playful cub
  3. Stage 3Young lion
  4. Stage 4Brave lion
  5. Stage 5Mighty lion
  6. Stage 6Legendary lion

Grows with skill

Your child’s lion, tree or dragon grows through six stages as their level rises: feedback on real progress.

Never shrinks

We keep the best stage reached, so a bad day never takes anything away.

Treats for effort

Flowers, butterflies and balloons appear as practice adds up. Decorations, not prizes.

You choose what they see

Full score, growth only (the default) or effort only, so a dip after a hard session needn’t knock confidence.

References

Where this comes from

These studies are about how people learn in general. None of them is a study of Summit Maths itself.

  1. Barbieri, C. A., Miller-Cotto, D., Clerjuste, S. N., & Chawla, K. (2023). A meta-analysis of the worked examples effect on mathematics performance. Educational Psychology Review, 35(1). doi.org/10.1007/s10648-023-09745-1
  2. Bjork, E. L., & Bjork, R. A. (2011). Making things hard on yourself, but in a good way: Creating desirable difficulties to enhance learning. In M. A. Gernsbacher, R. W. Pew, L. M. Hough, & J. R. Pomerantz (Eds.), Psychology and the real world (pp. 56–64). Worth Publishers. Free PDF from the Bjork lab
  3. 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.org/10.1037/0033-2909.132.3.354
  4. Csikszentmihalyi, M. (1990). Flow: The psychology of optimal experience. Harper & Row.
  5. Deci, E. L., Koestner, R., & Ryan, R. M. (1999). A meta-analytic review of experiments examining the effects of extrinsic rewards on intrinsic motivation. Psychological Bulletin, 125(6), 627–668. doi.org/10.1037/0033-2909.125.6.627
  6. Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students’ learning with effective learning techniques: Promising directions from cognitive and educational psychology. Psychological Science in the Public Interest, 14(1), 4–58. doi.org/10.1177/1529100612453266
  7. Elo, A. E. (1978). The rating of chessplayers, past and present. Arco Publishing.
  8. Fong, C. J., Zaleski, D. J., & Leach, J. K. (2015). The challenge–skill balance and antecedents of flow: A meta-analytic investigation. The Journal of Positive Psychology, 10(5), 425–446. doi.org/10.1080/17439760.2014.967799
  9. Jansen, B. R. J., Louwerse, J., Straatemeier, M., Van der Ven, S. H. G., Klinkenberg, S., & Van der Maas, H. L. J. (2013). The influence of experiencing success in math on math anxiety, perceived math competence, and math performance. Learning and Individual Differences, 24, 190–197. doi.org/10.1016/j.lindif.2012.12.014
  10. 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), 1813–1824. doi.org/10.1016/j.compedu.2011.02.003
  11. Papoušek, J., Stanislav, V., & Pelánek, R. (2016). Impact of question difficulty on engagement and learning. In Intelligent Tutoring Systems: ITS 2016 (pp. 267–272). Springer. doi.org/10.1007/978-3-319-39583-8_28
  12. Pelánek, R. (2016). Applications of the Elo rating system in adaptive educational systems. Computers & Education, 98. doi.org/10.1016/j.compedu.2016.03.017
  13. Roediger, H. L., & Karpicke, J. D. (2006). Test-enhanced learning: Taking memory tests improves long-term retention. Psychological Science, 17(3), 249–255. doi.org/10.1111/j.1467-9280.2006.01693.x
  14. Rohrer, D., Dedrick, R. F., Hartwig, M. K., & Cheung, C.-N. (2020). A randomized controlled trial of interleaved mathematics practice. Journal of Educational Psychology, 112(1), 40–52. What Works Clearinghouse review
  15. Sailer, M., & Homner, L. (2020). The gamification of learning: A meta-analysis. Educational Psychology Review, 32(1), 77–112. doi.org/10.1007/s10648-019-09498-w
  16. Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89. doi.org/10.1207/s1532690xci0201_3
  17. Wilson, R. C., Shenhav, A., Straccia, M., & Cohen, J. D. (2019). The Eighty Five Percent Rule for optimal learning. Nature Communications, 10(1), 4646. doi.org/10.1038/s41467-019-12552-4
  18. Yang, C., Luo, L., Vadillo, M. A., Yu, R., & Shanks, D. R. (2021). Testing (quizzing) boosts classroom learning: A systematic and meta-analytic review. Psychological Bulletin, 147(4), 399–435. doi.org/10.1037/bul0000309

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