The evidence
Number Path makes narrow claims on purpose. This page is the full receipt: every study the product leans on, what each one actually shows, and the one popular idea we examined and left out. Every entry below was verified by a human against the published record before it shipped.
How we label evidence in the app
Each game card carries two labels: how strong the evidence is — randomized trial, replicated trial, strong correlation, or theory-based — and how much the field agrees: high consensus, moderate consensus, or emerging. Most early-math products stop at “research-based.” We’d rather you see which kind of research, per game.
The question engine, certified
The check-in’s question engine is a deterministic state machine, and every release must pass an automated certification suite before the app will even load its content. In the certified acceptance tiers it runs against 31,500 simulated children across every rung and three levels of toddler-style noise — and thousands more sessions of stress tests on top. It must finish within 18 questions (typically about a dozen), and when noise does push it off the true rung, its errors are required to land low more often than high. An over-placed toddler gets frustrating games for a week; an under-placed one gets slightly easy ones. We tuned for the second, and the test suite enforces it.
The ladder and the check-in
The counting ladder is the knower-level framework, and Feed-the-Bear is a home adaptation of the titrated Give-N task these labs built and refined.
Krajcsi et al. (2023, Developmental Science)
Five- and six-knowers still behave like subset-knowers; CP should be identified with strict criteria
Krajcsi, A., et al. (2023). Miscategorized subset-knowers: Five- and six-knowers can compare only the numbers they know. Developmental Science, 27(2), e13430.
Le Corre & Carey (2007, Cognition)
Titrated Give-N; subset-knower to cardinal-principle progression
Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105(2), 395-438.
Sarnecka & Carey (2008, Cognition)
Cardinal principle as inductive leap; what CP-knowers know
Sarnecka, B. W., & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108(3), 662-674.
Wynn (1990, Cognition); Wynn (1992)
Give-N task; knower-level stages in number word learning
Wynn, K. (1990). Children's understanding of counting. Cognition, 36(2), 155-193. And: Wynn, K. (1992). Children's acquisition of the number words and the counting system. Cognitive Psychology, 24(2), 220-251.
Number talk
Why a daily ten-second prompt is on the plan: how much parents talk about number — and which numbers — tracks and causally shifts what children learn.
Carrazza, Gibson & Levine (2024, Developmental Psychology)
Counting books that embed number in goal-based narratives beat sparse counting drills for cardinal-knowledge gains
Carrazza, C., Gibson, D. J., & Levine, S. C. (2024). Less is not always more: Rich and meaningful counting books lead to greater gains in number understanding than sparse counting books. Developmental Psychology, 61(4).
Gibson, Gunderson & Levine (2020, Child Development)
RCT: parent-child number-book reading causally increases 2-4-year-olds' number knowledge; small-number (1-3) input most effective overall, larger numbers (4-6) for children further along
Gibson, D. J., Gunderson, E. A., & Levine, S. C. (2020). Causal effects of parent number talk on preschoolers' number knowledge. Child Development, 91(5), e1162-e1177.
Gunderson & Levine (2011, Developmental Science)
Talk about sets (especially larger, present sets) drives the number-talk effect
Gunderson, E. A., & Levine, S. C. (2011). Some types of parent number talk count more than others: Relations between parents' input and children's cardinal-number knowledge. Developmental Science, 14(5), 1021-1032.
Levine et al. (2010, Developmental Psychology)
Parent number talk at 14-30 months predicts later cardinal-number knowledge
Levine, S. C., Suriyakham, L. W., Rowe, M. L., Huttenlocher, J., & Gunderson, E. A. (2010). What counts in the development of young children's number knowledge? Developmental Psychology, 46(5), 1309-1319.
The games
The linear board game in your printable pack is the single most-replicated activity in this literature; spatial-language play earns its slot too.
James-Brabham et al. (2024, J. Educational Psychology)
Preregistered school RCT: no evidence linear number board games improve numerical skills beyond teaching-as-usual
James-Brabham, E., et al. (2024). No evidence that playing a linear number board game improves numerical skills beyond teaching as usual: A randomized controlled trial in 4- to 5-year-old primary school children. Journal of Educational Psychology.
Nelson et al. (2025, Review of Educational Research)
Meta-analysis of linear number board games: pooled g = 0.21 [0.08, 0.34] on mathematics outcomes
Nelson, G., et al. (2025). Investigating main effects and moderators of linear number board games: A meta-analytic review. Review of Educational Research. 18 studies, 123 effect sizes.
Pruden, Levine & Huttenlocher (2011, Developmental Science)
Spatial language input predicts later spatial skills
Pruden, S. M., Levine, S. C., & Huttenlocher, J. (2011). Children's spatial thinking: Does talk about the spatial world matter? Developmental Science, 14(6), 1417-1430.
Ramani & Siegler (2008, Child Development)
Linear number board games improve numerical knowledge in young children (RCT)
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.
Running it at home
Evidence that parent-run and remote assessment holds up, and the protocol adaptations for the youngest children.
Marchand, Lovelett, Kendro & Barner (2022, Cognition)
Give-N test-retest reliability is high overall but varies sharply by knower level; titrated administration yields slightly higher levels than non-titrated
Marchand, E., Lovelett, J. T., Kendro, K., & Barner, D. (2022). Assessing the knower-level framework: How reliable is the Give-a-Number task? Cognition, 222, 104998.
Silver et al. (2021, Frontiers in Psychology)
Point-to-X (2AFC) complements Give-N for 2-3-year-olds, captures emerging knowledge Give-N misses; no in-person vs remote administration difference
Silver, A. M., Elliott, L., Braham, E. J., Bachman, H. J., Votruba-Drzal, E., Tamis-LeMonda, C. S., Cabrera, N., & Libertus, M. E. (2021). Measuring emerging number knowledge in toddlers. Frontiers in Psychology, 12, 703598.
Wege, Trezise & Inglis et al. (2025, J. Numerical Cognition)
Systematic review of 30 years of Give-N: administration and scoring variants matter and should be documented explicitly
Wege, T., et al. (2025). Thirty years of the Give-N task: A systematic review, reflections, and recommendations. Journal of Numerical Cognition.
For math-anxious parents
Why every line is scripted: unstructured help from a math-anxious parent can pass the anxiety on, while structured, low-pressure formats buffer it.
Berkowitz et al. (2015, Science)
Parent-child math-story app RCT (first-graders): weekly-plus use improved math achievement; largest gains for children of high-math-anxious parents
Berkowitz, T., Schaeffer, M. W., Maloney, E. A., Peterson, L., Gregor, C., Levine, S. C., & Beilock, S. L. (2015). Math at home adds up to achievement in school. Science, 350(6257), 196-198.
Maloney, Ramirez, Gunderson, Levine & Beilock (2015, Psychological Science)
When high-math-anxious parents helped frequently with math homework, first/second-graders learned less math and developed more math anxiety
Maloney, E. A., Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2015). Intergenerational effects of parents' math anxiety on children's math achievement and anxiety. Psychological Science, 26(9), 1480-1488.
Schaeffer, Rozek, Berkowitz, Levine & Beilock (2018, JEP: General)
Follow-up of the 2015 cohort: the app intervention disassociated parents' math anxiety from children's math achievement, detectable ~3 years later
Schaeffer, M. W., Rozek, C. S., Berkowitz, T., Levine, S. C., & Beilock, S. L. (2018). Disassociating the relation between parents' math anxiety and children's math achievement: Long-term effects of a math app intervention. Journal of Experimental Psychology: General, 147(12), 1782-1790.
What we deliberately left out
The claims we examined and chose not to build on — listed with the same care as the ones we did.
Position note (Barner-aligned)
Why Number Path does not sell approximate-number-system training: transfer is contested
Position note grounded in: Szucs, D., & Myers, T. (2017). A critical analysis of design, facts, bias and inference in the approximate number system training literature: A systematic review. Trends in Neuroscience and Education, 6, 187-203. Szkudlarek, E., Park, J., & Brannon, E. M. (2020). Failure to replicate the benefit of approximate arithmetic training for symbolic arithmetic fluency in adults. Cognition, 207, 104521 (four experiments, N = 318). Qiu, K., Chen, E. H., Wan, S., & Bailey, D. H. (2021). A multilevel meta-analysis on the causal effect of approximate number system training on symbolic math performance. JEP: Learning, Memory, and Cognition, 47(11), 1820-1835 (g = 0.11, 95% CI [-0.01, 0.22]; PEESE-adjusted g = -0.04). Contrast: Park, J., & Brannon, E. M. (2013). Training the approximate number system improves math proficiency. Psychological Science, 24(10), 2013-2019.
What the effects look like
Honest answer: real and modest. Nothing here turns a two-year-old into a prodigy, and anyone promising that is selling something else. What the research supports is steady, level-matched number talk and play compounding over months — which is exactly the shape of the product: a few relaxed minutes a day, a ladder that moves every couple of months.
This is an enrichment tool, not a medical or developmental screening. For questions about your child's development, talk with your pediatrician.