Numina: Research and Method

Numina was built, from the first tap, on the science of how children learn math. This page explains the research principles behind the app, how Numina puts each one into practice, and the sources we drew on. It is written for parents and teachers who want to know that screen time is actually time well spent.

Numina is for children aged 8 to 11 (roughly grades 3 to 6). The short version: most apps drill, Numina teaches. Every design decision below traces back to a body of published research, listed in full at the end.

1. Number sense and the mental number line

Long before they memorise facts, children represent numbers as magnitudes, positions along a mental "number line." How accurately a child places numbers on a line is one of the strongest early predictors of later math achievement, and games built on a physical number line measurably improve young children's number knowledge (Siegler & Booth, 2004; Ramani & Siegler, 2008).

In Numina: placing and comparing numbers on a number line is a core mechanic, used to build genuine number sense rather than rote recall.

2. Teach first, then practise

Working memory is limited. New ideas are learned best when instruction reduces unnecessary mental load and presents a concept clearly before a child is asked to perform under pressure (Sweller, 1988).

In Numina: every new idea opens with a short, friendly concept moment before any question is asked. Understanding comes first, practice second.

3. Concrete before abstract

Children understand abstract symbols more deeply when they first meet an idea in a concrete, hands-on form, then a pictorial one, then the symbols. This enactive to iconic to symbolic progression is a foundation of how mathematical understanding develops (Bruner, 1966).

In Numina: place value is built with "bundles" of tens and ones, and multiplication with visual "arrays," so the meaning is felt before the notation is used.

4. Learning by recalling (the testing effect)

Actively retrieving an answer from memory strengthens learning far more than re-reading or watching, an effect repeatedly demonstrated in controlled studies (Roediger & Karpicke, 2006; Karpicke & Roediger, 2008).

In Numina: the heart of the loop is active recall. Children produce answers, which is exactly the act that builds durable memory.

5. Spaced practice that fights forgetting

Memory fades on a predictable curve, and practice spread out over time produces much stronger long-term retention than the same practice crammed together (Ebbinghaus, 1885; Cepeda, Pashler, Vul, Wixted & Rohrer, 2006).

In Numina: a spaced review scheduler, based on the modern open-source Free Spaced Repetition Scheduler (FSRS), brings each skill back for review right as it is about to be forgotten.

6. Interleaving, not blocking

When math problems of different types are mixed together ("interleaved") rather than practised in long single-type blocks, children learn to choose the right strategy and retain more, even though it feels harder in the moment (Rohrer & Taylor, 2007; Rohrer, Dedrick & Stershic, 2015).

In Numina: the session planner deliberately mixes skills and avoids long runs of the same problem type, building flexible, transferable understanding.

7. The right level, always (adaptive difficulty)

Children learn most in the zone just beyond what they can already do (Vygotsky, 1978), and a moderate, "desirable" level of difficulty improves long-term learning (Bjork, 1994). The Elo rating system, originally from chess, is a proven, efficient way to match item difficulty to a learner's current skill in adaptive learning systems (Elo, 1978; Pelanek, 2016).

In Numina: an Elo-based engine continuously estimates each child's skill and selects questions that are challenging but achievable, so they are rarely bored and rarely overwhelmed.

8. Mistakes are information (misconception modelling)

Children's errors are usually not random. They often follow systematic "bugs," consistent but mistaken rules a child has inferred (Brown & Burton, 1978). Spotting the pattern behind an error is far more useful than simply marking it wrong.

In Numina: a misconception model watches for these recurring error patterns and responds with a targeted hint aimed at the underlying misunderstanding.

9. Mastery, growth, and kind feedback

Given the right conditions and feedback, most children can reach mastery, not just the few (Bloom, 1984). Feedback works best when it focuses on the task and the child's progress rather than on the child (Hattie & Timperley, 2007), and praising effort and strategy rather than fixed "smartness" supports a growth mindset and resilience (Dweck, 2006).

In Numina: goals, missions, and medals celebrate genuine mastery and effort rather than speed, and mistakes are framed gently and constructively, never punished.

10. Fractions, decimals, and percentages on one number line

Whole numbers and fractions are best understood as one connected system of magnitudes on a single number line, rather than as separate, disconnected topics. A child's grasp of fractions in primary school is one of the strongest predictors of overall mathematics achievement years later (Siegler, Thompson & Schneider, 2011; Siegler et al., 2012; National Mathematics Advisory Panel, 2008).

In Numina: fractions, decimals, and percentages are taught on the same number line that underpins whole numbers, including the harder step of comparing fractions with unlike denominators.

11. Designed to be loved, not addictive

Educational games work best when the learning is the play itself, not a quiz bolted onto an unrelated game, an idea called intrinsic integration (Habgood & Ainsworth, 2011). We also believe a children's app carries an ethical duty: to engage without exploiting.

In Numina: the math is the game. There are no ads, no loot boxes, no manipulative streak-pressure or engagement traps, and no data collection. Numina is built to be a calm, trustworthy place to learn.

A note on our claims

Numina is designed around well-established principles from learning science, and we describe it that way. The studies below establish those principles in general; they are not studies of Numina itself. Numina is a supplement to good teaching, not a replacement for it, and no app can guarantee a particular grade or outcome.

References

1. Bjork, R. A. (1994). Memory and metamemory considerations in the training of human beings. In J. Metcalfe & A. Shimamura (Eds.), Metacognition: Knowing about Knowing. MIT Press.
2. Bloom, B. S. (1984). The 2 Sigma Problem: The search for methods of group instruction as effective as one-to-one tutoring. Educational Researcher, 13(6), 4-16.
3. Brown, J. S., & Burton, R. R. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2(2), 155-192.
4. Bruner, J. S. (1966). Toward a Theory of Instruction. Harvard University Press.
5. 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.
6. Dweck, C. S. (2006). Mindset: The New Psychology of Success. Random House.
7. Ebbinghaus, H. (1885). Memory: A Contribution to Experimental Psychology (translated 1913).
8. Elo, A. E. (1978). The Rating of Chessplayers, Past and Present. Arco.
9. Free Spaced Repetition Scheduler (FSRS). Open-source spaced-repetition algorithm. github.com/open-spaced-repetition
10. Habgood, M. P. J., & Ainsworth, S. E. (2011). Motivating children to learn effectively: Exploring the value of intrinsic integration in educational games. Journal of the Learning Sciences, 20(2), 169-206.
11. Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77(1), 81-112.
12. Karpicke, J. D., & Roediger, H. L. (2008). The critical importance of retrieval for learning. Science, 319(5865), 966-968.
13. National Mathematics Advisory Panel (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education.
14. Pelanek, R. (2016). Applications of the Elo rating system in adaptive educational systems. Computers & Education, 98, 169-179.
15. Ramani, G. B., & Siegler, R. S. (2008). Promoting broad and stable improvements in low-income children's numerical knowledge through playing number board games. Child Development, 79(2), 375-394.
16. Roediger, H. L., & Karpicke, J. D. (2006). Test-enhanced learning: Taking memory tests improves long-term retention. Psychological Science, 17(3), 249-255.
17. Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35, 481-498.
18. Rohrer, D., Dedrick, R. F., & Stershic, S. (2015). Interleaved practice improves mathematics learning. Journal of Educational Psychology, 107(3), 900-908.
19. Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75(2), 428-444.
20. Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273-296.
21. Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691-697.
22. Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257-285.
23. Vygotsky, L. S. (1978). Mind in Society: The Development of Higher Psychological Processes. Harvard University Press.