Research 10: Motivation, Emotion, and Habits in Learning Mathematics

Compiled 2026-09-27 for Numberkit. It covers psychology and education research on sustaining effortful practice and positive feelings about mathematics; game and gamification studies are excluded. The streak trial is an economics field experiment on reminder messages. Citations were checked against Crossref or publisher records. Figures I could not re-read this session are marked "(not re-verified)".

Summary

  1. Competence comes first. Real success on just-reachable material is the best-supported driver of continued effort and the best-supported cure for math anxiety. In children, performance drives later self-efficacy more than the reverse (Talsma et al. 2018), and skill-building tutoring reduced children's math anxiety (Supekar et al. 2015).
  2. Rewards are the wrong tool for an activity that can be interesting. Expected tangible rewards undermine later free-choice engagement, more so for children (Deci et al. 1999). Grades lower motivation compared with comments (Koenka et al. 2021). This is the strongest evidence for Numberkit's no-points rule.
  3. Value can be raised lightly and indirectly. Self-generated relevance helps low-expectancy students, and the parent version raised STEM course-taking (Hulleman & Harackiewicz 2009; Harackiewicz et al. 2012; Gaspard et al. 2015). Asserting usefulness can backfire, and effects shrink at scale online (Canning & Harackiewicz 2015; Kizilcec et al. 2020).
  4. A question before the answer evokes curiosity, which aids memory. The effect is strongest when learners feel close to the answer (Loewenstein 1994; Kang et al. 2009; Gruber et al. 2014; Wade & Kidd 2019). The evidence is mostly from adults.
  5. Math anxiety starts by age 6 and can be reduced. Interventions average g = -0.47 on anxiety and g = 0.50 on performance, through cognitive support or emotion regulation (Sammallahti et al. 2023). Single-session expressive writing failed to replicate (O'Meara & Lovett 2025). Anxious parents harm children mainly when they help often with homework (Maloney et al. 2015).
  6. Habits take months and survive a missed day. The median was about 66 days, with a range of 18 to 254 (Lally et al. 2010). If-then plans help (d = 0.65; Gollwitzer & Sheeran 2006). Weekly streak messages and personalised reminders raised use of a children's maths platform without discouraging those who missed a week (Aulagnon et al. 2025). Encouraging a return after a miss was the top intervention in a 61,000-person megastudy (Milkman et al. 2021).
  7. Goals should be near, specific, and about mastery (Bandura & Schunk 1981; Senko et al. 2011).
  8. Autonomy support is teachable and consistently linked to better motivation. That means choice, reasons, non-controlling language, and acknowledging feelings, combined with clear structure (Reeve 2009; Jang et al. 2010; Howard et al. 2021).
  9. Language includes or excludes. Stereotypes appear by age 6 to 7 (Cvencek et al. 2011; Bian et al. 2017). "Doing" framing beats identity framing for girls' persistence (Rhodes et al. 2019).
  10. Self-paced learning loses most learners (Reich & Ruipérez-Valiente 2019). A supporting person, a goal, and visible progress sustain adults (Comings 2007). For children, the parent is the supporter. Value talk helps; homework help does not, on average (Hill & Tyson 2009).

Detailed findings

1. Interest development and relevance

Four-phase model. Hidi and Renninger (2006, Educational Psychologist, 41:111-127, doi:10.1207/s15326985ep4102_4) describe four phases of interest: triggered situational interest, maintained situational interest, emerging individual interest, and well-developed individual interest. Early phases need outside support: novelty, challenge, and a sense of meaning. Later phases are self-sustaining and include stored knowledge and value. This is a theoretical synthesis, not an experiment. Its key design claim is that triggering alone does not produce lasting interest. Maintenance needs meaningful involvement and growing knowledge (Renninger & Hidi 2016, The Power of Interest for Motivation and Engagement, Routledge, doi:10.4324/9781315771045).

How it could translate to an app. Faithful: treat the "hook" (a surprising pattern, a question) as a trigger only. Put the design effort into maintenance, meaning a steady rise in real competence and in knowing why things work. Stretch: claiming an app alone builds individual interest. Fits no-points: yes.

Utility-value intervention (students). Hulleman and Harackiewicz (2009, Science, 326:1410-1412, doi:10.1126/science.1177067) randomly assigned 262 US ninth-grade science students to write, several times across a semester, either about how the material related to their lives or a summary of it. Students with low expectations of success who wrote about relevance earned higher grades and reported more interest. There was no effect for high-expectancy students. It replicated in college courses, partly through raised confidence, most strongly for low-performing men (final exam d = 0.76; Hulleman et al. 2017, J. Educ. Psych., 109:387-404, doi:10.1037/edu0000146). It also narrowed gaps for first-generation minority students in college biology (Harackiewicz et al. 2016, JPSP, 111:745-765, doi:10.1037/pspp0000075). Across motivation interventions in education generally, the mean effect is d = 0.49, smaller in randomised designs (Lazowski & Hulleman 2016). Gaspard et al. (2015, Developmental Psychology, 51:1226-1240) ran a cluster-randomised trial in 82 German ninth-grade mathematics classrooms (1,916 students). A 90-minute relevance session raised utility value up to five months later. The version where students evaluated interview quotations also raised attainment and intrinsic value, and effects were stronger for girls. Brisson et al. (2017, AERJ, 54:1048-1078, doi:10.3102/0002831217716084) reported that the same trial produced sustained gains in math competence beliefs, effort, and achievement. Canning and Harackiewicz (2015, Motivation Science, 1:47-71, doi:10.1037/mot0000015) found that telling low-confidence students that material is useful can backfire, while self-generated connections do not ("teach it, don't preach it").

How it could translate to an app. Faithful: an occasional, optional prompt, mainly for older learners, asking them to type where they might use today's idea, or to pick from short quotes by real people who use it. Stretch: using it with 7-to-9-year-olds. The evidence is from adolescents and adults, and a young child's writing burden is high. Rule of thumb from the evidence: never lecture about usefulness, especially to a learner who is struggling. Fits no-points: yes.

Utility value via parents. Harackiewicz, Rozek, Hulleman and Hyde (2012, Psychological Science, 23:899-906, doi:10.1177/0956797611435530) mailed two brochures to Wisconsin parents of high-school students and gave them a website on the usefulness of maths and science. Treated students took nearly one extra semester of maths and science in grades 11 and 12. Rozek et al. (2017, PNAS, 114:909-914, doi:10.1073/pnas.1607386114) followed up the same cohort. Treated students had higher maths and science college-entrance scores and more STEM career interest years later. Replications come mostly from the original team.

How it could translate to an app. Faithful: the parent screen can offer short, concrete talking points about where this week's maths shows up in daily life. The idea is to help parents convey value (academic socialization), not to supervise drills. Fits no-points: yes.

Limits. Kizilcec et al. (2020, PNAS, 117:14900-14905, doi:10.1073/pnas.1921417117) tested light-touch interventions, including a value-relevance prompt, with about 250,000 learners in 247 online courses. Benefits were small and depended on context, not the medium-to-large effects expected. Rosenzweig, Wigfield and Eccles (2022, Educational Psychologist, 57:11-30, doi:10.1080/00461520.2021.1984242) argue that utility-value work needs to go beyond one-shot writing exercises.

2. Curiosity

Information-gap theory. Loewenstein (1994, Psychological Bulletin, 116:75-98, doi:10.1037/0033-2909.116.1.75) proposed that curiosity arises when attention is drawn to a gap between what one knows and what one wants to know. Curiosity increases as the gap narrows, so people are most curious when they feel close to the answer. This is a theoretical review.

Curiosity and memory. Kang et al. (2009, Psychological Science, 20:963-973, doi:10.1111/j.1467-9280.2009.02402.x) showed adults trivia questions. Curiosity followed an inverted U with confidence. People spent scarce resources (tokens, waiting time) to learn answers they were curious about. Curiosity activated reward regions, and it was associated with better memory for answers that corrected a wrong guess. Gruber, Gelman and Ranganath (2014, Neuron, 84:486-496, doi:10.1016/j.neuron.2014.08.060) found better memory, both immediately and a day later, for answers people were curious about. They also found better memory for unrelated faces shown during high-curiosity waits, linked to midbrain and hippocampal activity. Samples were small adult fMRI groups (roughly 20 per study; not re-verified). Wade and Kidd (2019, Psychonomic Bulletin & Review, 26:1377-1387, doi:10.3758/s13423-019-01598-6) found curiosity peaked when people felt they had some but not full knowledge, and curiosity predicted learning. Metcalfe, Schwartz and Eich (2020, Curr. Opin. Behav. Sci., 35:40-47, doi:10.1016/j.cobeha.2020.06.007) review the same "just beyond what is known" pattern.

Questions before answers. Trying to answer before being taught (prequestions, pretests) generally improves later memory for the tested material, even when the first attempt fails (Pan & Carpenter 2023, review, PsyArXiv doi:10.31234/osf.io/9rqpm; peer-reviewed venue not re-verified).

How it could translate to an app. Faithful: open a Learn segment with a question the learner can nearly answer, such as "What do you think 7 x 8 is, if 7 x 7 is 49?", then show the answer on the picture. Keep the gap small and the reveal quick. The memory benefit depends on the learner wanting the answer and getting it soon. Stretch: using curiosity to justify withholding answers for long periods, or assuming the adult trivia effects transfer directly to children's arithmetic facts. Limit: when the gap feels too big, curiosity drops and an anxious learner may feel threatened. Frame the question as a guess and never score it. Fits no-points: yes.

3. Expectancy-value, self-efficacy, attribution, and feedback

Expectancy-value theory. Wigfield and Eccles (2000, Contemporary Educational Psychology, 25:68-81, doi:10.1006/ceps.1999.1015) and Eccles and Wigfield (2020, CEP, 61:101859, doi:10.1016/j.cedpsych.2020.101859) model choice and persistence as the product of expectancy of success and subjective task value. Value has four parts: intrinsic, attainment, utility, and cost. Jacobs et al. (2002, Child Development, 73:509-527, doi:10.1111/1467-8624.00421) tracked children from grades 1 to 12. Competence beliefs and values fell across school, and math value fell sharply. The gender gap in math competence beliefs narrowed with age.

How it could translate to an app. Faithful: design to protect both terms. Raise expectancy through achievable next steps. Lower perceived cost, for example with short sessions, no penalty for stopping, and no public failure. Fits no-points: yes.

Self-efficacy and its sources. Bandura (1977, Psychological Review, 84:191-215) names four sources: mastery experiences, vicarious experience (seeing similar others succeed), verbal persuasion, and physiological and emotional states. Usher and Pajares (2008, Review of Educational Research, 78:751-796, doi:10.3102/0034654308321456) reviewed school studies and found mastery experience the most consistent predictor. Talsma et al. (2018, Learning and Individual Differences, 61:136-150, doi:10.1016/j.lindif.2017.11.015) meta-analysed longitudinal data (k = 11, N = 2,688). Performance predicted later self-efficacy (beta = 0.205) more strongly than self-efficacy predicted later performance (beta = 0.071). The effect was reciprocal in adults. In children, performance predicted later self-efficacy but not the reverse.

How it could translate to an app. Faithful: do not try to talk children into confidence; arrange genuine success and make it visible. Persuasion should be specific and credible ("You got all of the 6s right today"). A peer-aged character modelling a solution is a stretch on the vicarious source. Fits no-points: yes.

Attribution and praise. Weiner (1985, Psychological Review, 92:548-573) argues that attributing outcomes to controllable, unstable causes (effort, strategy) sustains persistence better than attributing them to fixed ability. Mueller and Dweck (1998, JPSP, 75:33-52, doi:10.1037/0022-3514.75.1.33) ran studies with US fifth-graders. Praise for intelligence after success led to less persistence, less enjoyment, and worse performance after a later failure than praise for effort. Gunderson et al. (2013, Child Development, 84:1526-1541, doi:10.1111/cdev.12064) found that parents' process praise at 14 to 38 months predicted children's incremental beliefs five years later (N = 53; correlational). Brummelman et al. (2014, JEP: General, 143:9-14, doi:10.1037/a0031917) found that inflated person praise backfired in children with low self-esteem.

How it could translate to an app. Faithful: feedback names the strategy or the fact ("You used 5 x 8 to get 6 x 8"), never the person ("You're a maths genius"). Avoid inflated praise entirely. Fits no-points: yes.

Feedback. Kluger and DeNisi (1996, Psychological Bulletin, 119:254-284, doi:10.1037/0033-2909.119.2.254) meta-analysed 607 effect sizes. Feedback improved performance on average (d about 0.41), but over a third of feedback interventions lowered performance (figures not re-verified). Feedback that draws attention to the self rather than the task tends to hurt. Hattie and Timperley (2007, Review of Educational Research, 77:81-112, doi:10.3102/003465430298487) reach the same conclusion: task, process, and self-regulation feedback help, and "self"-level praise does little. Koenka et al. (2021, Educational Psychology, 41:922-947, doi:10.1080/01443410.2019.1659939) meta-analysed K-12 studies. Compared with comments, grades produced poorer achievement and less optimal motivation. Butler (1988, British Journal of Educational Psychology, 58:1-14) found that comments sustained fifth and sixth graders' interest and performance, while grades, or grades with comments, did not.

How it could translate to an app. Faithful: feedback is about the task and immediate. No score out of 100 and no grade-like label on a session. Fits no-points: yes, directly.

4. Math anxiety interventions

Anxiety starts early and travels. Ramirez et al. (2013, Journal of Cognition and Development, 14:187-202, doi:10.1080/15248372.2012.664593) found math anxiety in US first and second graders, and it was negatively related to achievement, especially among children with high working memory. Beilock et al. (2010, PNAS, 107:1860-1863) found that female first- and second-grade teachers' math anxiety predicted girls' end-of-year endorsement of the belief that boys are good at maths and girls at reading, and lower maths achievement.

Tutoring reduces anxiety in children. Supekar, Iuculano, Chen and Menon (2015, Journal of Neuroscience, 35:12574-12583, doi:10.1523/jneurosci.0786-15.2015) gave 46 US third graders an 8-week one-to-one cognitive tutoring programme on number knowledge and arithmetic strategies (session count not re-verified). High-anxious children's math anxiety fell, and their elevated amygdala response to maths disappeared after tutoring. The size of the amygdala reduction predicted the size of the anxiety reduction. There was no untreated anxious control group, so a practice or time effect cannot be fully excluded.

Meta-analysis. Sammallahti, Finell, Jonsson and Korhonen (2023, Journal of Numerical Cognition, 9:346-362, doi:10.5964/jnc.8401) pooled 50 studies (75 effect sizes). Interventions reduced math anxiety (g = -0.47) and improved performance (g = 0.50). Cognitive-support interventions (g = -0.53 on anxiety, 0.82 on performance) and emotion-regulation interventions (g = -0.52 on anxiety, 0.37 on performance) both worked. Motivation-focused interventions did not significantly reduce anxiety (g = -0.25) or raise performance (g = 0.20). Effects on anxiety were larger for students over 12 (g = -0.64) than under 12 (g = -0.22), and longer interventions beat short ones. Hembree (1990, Journal for Research in Mathematics Education, 21:33-46, doi:10.2307/749455) earlier found that behavioural and cognitive-behavioural treatments reduced anxiety and that curricular changes alone did not (not re-verified in detail).

Expressive writing: a striking finding that did not hold. Ramirez and Beilock (2011, Science, 331:211-213, doi:10.1126/science.1199427) reported that writing about worries for 10 minutes before a high-stakes test improved scores of anxious students in the lab and in ninth-grade biology. Park, Ramirez and Beilock (2014, JEP: Applied, 20:103-111, doi:10.1037/xap0000013) extended it to math anxiety in adults. But the Social Sciences Replication Project (Camerer et al. 2018, Nature Human Behaviour, 2:637-644, doi:10.1038/s41562-018-0399-z) lists Ramirez and Beilock (2011) among the studies that did not replicate. That replication was of the lab study, not the classroom one. A conceptual replication of Park et al. (Scheibe, Was & Thompson; eScholarship; venue not verified) with 168 college students found no benefit. O'Meara and Lovett (2025, Anxiety, Stress, & Coping, 39:21-34, doi:10.1080/10615806.2025.2552857) meta-analysed 21 studies (N = 1,457). The effect on anxiety was negligible and non-significant (r = -0.05), and the effect on performance was non-significant once an outlier was removed (r = 0.06).

Reappraisal of arousal. Jamieson, Mendes, Blackstock and Schmader (2010, Journal of Experimental Social Psychology, 46:208-212, doi:10.1016/j.jesp.2009.08.015) told US adults preparing for the GRE that arousal can improve performance. They scored higher on practice GRE maths and, one to three months later, on the real GRE maths section (small sample; not re-verified). Jamieson et al. (2016, Social Psychological and Personality Science, 7:579-587, doi:10.1177/1948550616644656) found reduced evaluation anxiety and better exam scores in a community-college maths class. Rozek, Ramirez, Fine and Beilock (2019, PNAS, 116:1553-1558, doi:10.1073/pnas.1808589116) ran a trial in US ninth-grade biology (about 1,175 students; not re-verified). Before exams, students did expressive writing, reappraisal, both, or a control. For lower-income students, course failure fell from 39% in the control group to 18% with an intervention, and exam scores rose. Note that this includes expressive writing, which has since failed to replicate on its own. The field evidence rests on few studies, mostly with adolescents and adults.

Parents' anxiety. Maloney, Ramirez, Gunderson, Levine and Beilock (2015, Psychological Science, 26:1480-1488, doi:10.1177/0956797615592630) followed US first and second graders (about 438; not re-verified). Children of math-anxious parents learned less maths and became more anxious over the year, but only when those parents helped with maths homework often. Berkowitz et al. (2015, Science, 350:196-198, doi:10.1126/science.aac7427) gave families of first graders an app with short maths conversations for parent and child at bedtime. Frequent use was linked to greater gains, especially for children of math-anxious parents. Schaeffer et al. (2018, JEP: General, 147:1782-1790, doi:10.1037/xge0000490) found the benefit persisted through third grade even after app use dropped. The mechanism appeared to be changes in parents' expectations and in the value they placed on their child's maths success.

What does not work. Single-session expressive writing, interventions aimed only at motivation, very short programmes, and frequent homework help from an anxious parent.

How it could translate to an app. Faithful, and a strong fit: the app's main anxiety intervention is competence. That means careful sequencing, explaining on a representation, success rates kept high, no visible timer, and no penalty display. This is the "cognitive support" route with the best evidence, and it matches the Supekar tutoring result, though an app is not a human tutor, so that part is a stretch. For older learners and adults, an optional short "your heartbeat is your body getting ready" reappraisal before an opt-in timed sprint is faithful to Jamieson and Rozek, but the evidence in children is thin. Do not ship expressive writing as an anxiety feature. For parents: offer brief, warm shared activities rather than homework help, and tell anxious parents they only need to show interest, not teach. Fits no-points: yes.

5. Habit formation and consistency

How long habits take. Lally, van Jaarsveld, Potts and Wardle (2010, European Journal of Social Psychology, 40:998-1009, doi:10.1002/ejsp.674) had 96 UK adults repeat a chosen daily eating, drinking, or exercise behaviour in the same context for 12 weeks. Of those, 82 gave usable data. Median time to 95% of peak automaticity was about 66 days, with a range of 18 to 254 days. Missing one opportunity did not materially affect habit formation, though many misses reduced the plateau. Exercise took about 1.5 times as long as eating or drinking behaviours. Buyalskaya et al. (2023, PNAS, doi:10.1073/pnas.2216115120) used machine learning on large gym and hospital handwashing datasets. They found habits form on very different timescales, typically months for gym attendance and weeks for handwashing (timescales not re-verified), with context cues such as time since last visit and day of week predicting behaviour. Habit is a context-response link, so stable cues matter more than motivation once it forms (Wood & Neal 2007, Psychological Review, 114:843-863, doi:10.1037/0033-295x.114.4.843; Gardner, Lally & Wardle 2012, BJGP, 62:664-666, doi:10.3399/bjgp12x659466).

How it could translate to an app. Faithful: help the family choose a fixed cue ("after breakfast", "when I get home from school") and treat the first two to three months as the habit-building period. Tell parents a missed day does not undo it. The cue belongs to the family routine (a mild stretch from adult samples). Fits no-points: yes.

Implementation intentions. Gollwitzer and Sheeran (2006, Advances in Experimental Social Psychology, 38:69-119, doi:10.1016/s0065-2601(06)38002-1) meta-analysed 94 independent tests. If-then plans ("When X happens, I will do Y") improved goal attainment over goal intentions alone (d = 0.65). Samples were mostly adults. Duckworth, Kirby, Gollwitzer and Oettingen (2013, Social Psychological and Personality Science, 4:745-753, doi:10.1177/1948550613476307) taught mental contrasting with implementation intentions to low-income US schoolchildren (a small sample; not re-verified) and improved attendance, grades, and conduct. In MOOCs, Kizilcec and Cohen (2017, PNAS, 114:4348-4353, doi:10.1073/pnas.1611898114) found a brief plan-making exercise raised completion in individualist but not collectivist countries. At larger scale the effects shrank (Kizilcec et al. 2020).

How it could translate to an app. Faithful: a one-time setup step where the child, with a parent, completes "When ___, I will do my maths," and an obstacle plan such as "If I feel stuck, I will open a picture." The app can show the plan back at session start. Fits no-points: yes.

Streaks and reminders. Aulagnon, Cristia, Cueto and Malamud (2025, Economics of Education Review, 109:102721, doi:10.1016/j.econedurev.2025.102721; NBER w34173) randomised 60,000 Peruvian households of grade 4 to 6 pupils over a six-week summer break. The platform offered 30 maths exercises a week. The arms were: a Thursday app notification highlighting the child's weekly streak, which always framed a new streak as worth starting; personalised reminders with congratulation; a generic Monday reminder; or no messages. Take-up was very low: 5.3% of the control group used the platform at all. Personalised reminders raised the share who connected at least once more (3.8 vs 2.8 percentage points). Streak messages raised the share of weeks connected among users more (9.4 vs 6.9 points). The authors looked for longer runs of non-use after a broken streak and found no sign of a discouragement effect. Effects were concentrated on message days. Streak-arm students scored about 0.13 to 0.17 SD higher on the endline test, but only 2.3% (about 1,500) took it, and the difference from the other message arms was not significant. The authors note Khan Academy retired its streak feature because broken streaks can demotivate. Milkman et al. (2021, Nature, 600:478-483, doi:10.1038/s41586-021-04128-4) tested 54 four-week programmes with 61,293 gym members. 45% increased visits (by 9% to 27%). The top one gave micro-rewards for returning after a missed workout. Only 8% had effects measurable after the programme ended.

How it could translate to an app. Faithful: one or two gentle, well-timed reminders a week, personalised to what the child did ("New 7s practice is ready; last time you finished the 6s"), sent to the parent's device. A weekly consistency view ("You practised on 3 days this week") is closer to the tested design than a daily count. Stretch: extrapolating to daily streaks with loss framing. That is not what was tested, and the Khan Academy concern and loss-aversion logic both argue against it. Fits no-points: a daily streak counter with loss framing is a volume badge by another name and conflicts with the rule. A self-referenced weekly calendar with no count to lose, and with a warm "welcome back" after a gap, fits. The megastudy's "return after a miss" result suggests the moment after a lapse is where encouragement pays off most.

Avoiding shame when a habit breaks. Dai, Milkman and Riis (2014, Management Science, 60:2563-2582) documented the "fresh start effect": people start aspirational behaviour more after temporal landmarks such as new weeks, months, and birthdays. Beshears et al. (2021, Management Science) found that rewarding flexible-time exercise produced more gym visits than rewarding a fixed routine (not re-verified). Lally et al. (2010) show one miss is harmless.

How it could translate to an app. Faithful: after a gap, open with "Welcome back. Here's where you left off," frame the new week or month as a fresh start, and never show a broken count or "you missed X days." Allow flexible practice times rather than insisting on one slot. Fits no-points: yes.

6. Goal setting and progress

Goal-setting theory. Locke and Latham (2002, American Psychologist, 57:705-717, doi:10.1037/0003-066x.57.9.705) summarise 35 years of mostly adult work research. Specific, challenging goals with feedback outperform "do your best," provided the person has the ability and commitment. For complex new tasks, learning goals (find strategies) beat outcome goals. Children: Bandura and Schunk (1981, JPSP, 41:586-598, doi:10.1037/0022-3514.41.3.586) worked with children (about 7 to 10 years old; not re-verified) who had large deficits in subtraction. Pursuing proximal subgoals (a few pages per session) produced faster progress, higher subtraction skill, higher self-efficacy, and more intrinsic interest than distal goals or no goals.

How it could translate to an app. Faithful: each session names one near goal about capability ("Today: the 7s you haven't met yet"), and the end of the session shows whether it moved. Fits no-points: yes.

Achievement goals. Senko, Hulleman and Harackiewicz (2011, Educational Psychologist, 46:26-47, doi:10.1080/00461520.2011.538646) review evidence that mastery goals (developing competence) reliably predict interest, deep processing, and persistence, but only weakly predict grades. Performance-approach goals (outperforming others) predict grades in some settings but also anxiety and cheating, and performance-avoidance goals are harmful. Senko and Dawson (2017, J. Educ. Psych., 109:574-598, doi:10.1037/edu0000160) found that effects depend on definition. Normative "outperform others" goals relate positively to achievement, while appearance-focused goals ("look smart") do not.

How it could translate to an app. Faithful: all goals and displays are mastery goals, measured against the learner's own past. That some learners do well under normative goals is not a reason to add comparison in a children's app, given the anxiety and appearance risks. Fits no-points: yes. The no-leaderboard rule is consistent with this literature.

Progress principle. Amabile and Kramer (2011, The Progress Principle, Harvard Business Review Press; HBR article "The power of small wins") analysed about 12,000 daily diary entries from 238 knowledge workers (figures not re-verified). Making progress in meaningful work was the event most often associated with good days. It is a book and practitioner article, not a peer-reviewed test of the principle.

How it could translate to an app. A small, true gain shown at the end of each session. Faithful in spirit, a stretch in population (adult workers). Fits no-points: yes.

Showing improvement honestly. The evidence above (Koenka et al. 2021; Butler 1988; Kluger & DeNisi 1996) favours task-focused, self-referenced information over grades and comparison. Honest progress has three features. It compares the learner with their own earlier self. It uses a real measure (facts now secure, strategies now used, delayed-recall probes) rather than volume. And it shows plateaus and slips without alarm. I could not verify a controlled non-game study of self-referenced progress graphs for children, so this rests on the feedback literature.

7. Autonomy-supportive environments

Reeve's autonomy-supportive teaching. Reeve (2009, Educational Psychologist, 44:159-175, doi:10.1080/00461520903028990) defines autonomy support by concrete behaviours: take the student's perspective, invite rather than demand, give reasons for requests, use non-controlling language, acknowledge and accept negative feelings, and be patient. Su and Reeve (2011, Educational Psychology Review, 23:159-188, doi:10.1007/s10648-010-9142-7) meta-analysed training programmes and found teachers can learn this style, with a large effect (d about 0.63; not re-verified). Jang, Reeve and Deci (2010, J. Educ. Psych., 102:588-600, doi:10.1037/a0019682) found engagement was highest with autonomy support and structure together. Cheon, Reeve and Vansteenkiste (2020, Teaching and Teacher Education, 90:103004, doi:10.1016/j.tate.2019.103004) showed teachers can be trained to give structure in an autonomy-supportive way, with benefits for students.

SDT meta-analyses. Howard et al. (2021, Perspectives on Psychological Science, 16:1300-1323, doi:10.1177/1745691620966789) covered 344 samples (223,209 students). Intrinsic motivation related to success and well-being. Identified regulation (personal value) related most strongly to persistence. External regulation (rewards, punishment) was not associated with performance or persistence and was associated with lower well-being. Bureau et al. (2022, Review of Educational Research, 92:46-72, doi:10.3102/00346543211042426) found autonomy support from teachers and parents predicts autonomous motivation. Vasconcellos et al. (2020, J. Educ. Psych., 112:1444-1469, doi:10.1037/edu0000420; 265 studies) found the same pattern, but in school physical education, not maths.

Choice and rewards. Patall, Cooper and Robinson (2008, Psychological Bulletin, 134:270-300, doi:10.1037/0033-2909.134.2.270) found choice raises intrinsic motivation and related outcomes. Effects were larger with a moderate number of options, and choice paired with rewards lost its benefit (details not re-verified). Deci, Koestner and Ryan (1999, Psychological Bulletin, 125:627-668, doi:10.1037/0033-2909.125.6.627) meta-analysed 128 experiments. Expected tangible rewards undermined free-choice intrinsic motivation (d = -0.36), as did engagement-contingent rewards (d = -0.40). Undermining was significantly greater for children than for college students. Positive verbal feedback enhanced intrinsic motivation for college students (d = 0.43) but not significantly for children (d = 0.11). Undermining was large on interesting tasks (d = -0.68) and absent on dull tasks.

How it could translate to an app. Faithful: offer small, real choices, such as which of two open topics to do first, which representation to use, or when to stop after the core block. Give a one-sentence reason for anything required ("We check the 6s again today because they fade after a week"). Accept negative feelings in words ("This one is hard. That's normal for new facts"). Pair all of this with clear structure: a predictable session shape and a visible next step. Fits no-points: yes. The Deci et al. results are the core evidence for the rule, and they are strongest for children and for interesting tasks.

8. Belonging, identity, and stereotypes

Stereotypes form early. Cvencek, Meltzoff and Greenwald (2011, Child Development, 82:766-779, doi:10.1111/j.1467-8624.2010.01529.x) tested US children in grades 1 to 5 (about 247; not re-verified). By second grade, children implicitly associated maths with boys, and boys identified with maths more than girls did. Bian, Leslie and Cimpian (2017, Science, 355:389-391, doi:10.1126/science.aah6524) found that at age 5 children did not link brilliance to gender. By 6, girls were less likely to think their own gender "really, really smart" and avoided games described as for really smart children, but not games described as for children who try hard.

Environment and language. Master, Cheryan and Meltzoff (2016, J. Educ. Psych., 108:424-437, doi:10.1037/edu0000061) showed US high-school students photos of computer science classrooms. Stereotypical rooms lowered girls' interest in the course, through lower anticipated belonging; non-stereotypical rooms did not. Rhodes, Leslie, Yee and Saunders (2019, Psychological Science, 30:455-466, doi:10.1177/0956797618823670) ran four experiments with 501 children aged 4 to 5. Framing science as actions ("Let's do science") instead of an identity ("Let's be scientists") increased girls' persistence.

Belonging interventions. Walton and Cohen (2011, Science, 331:1447-1451, doi:10.1126/science.1198364) gave first-year college students a one-hour intervention framing worries about belonging as common and temporary. Black students' grades rose over three years (small sample; not re-verified). Walton et al. (2023, Science, 380:499-505, doi:10.1126/science.ade4420; erratum 2024) ran a preregistered trial with 26,911 students at 22 institutions. A 30-minute online version raised full-time first-year completion, mainly among groups who historically progressed less. It worked only where the context gave those groups real opportunities to belong.

How it could translate to an app. Faithful: use action language ("Let's work out...", "You figured out...") rather than identity labels ("maths whizz", "genius", "maths person"). Never frame an activity as for smart or gifted children, and describe hard items as for people who "keep trying" (Bian et al.). Word problems and characters should vary in gender, names, and settings, and should not show maths as a boys' or adults' domain. When learners struggle, normalise it ("Lots of people find the 7s tricky at first; it gets easier"); that mirrors belonging messages, but moving it from a college transition to an app is a stretch. Fits no-points: yes.

9. Sustaining learning over years

Dropout from self-paced online learning. Reich and Ruipérez-Valiente (2019, Science, 363:130-131, doi:10.1126/science.aav7958) report that completion among MITx and HarvardX course starters was low and falling, near 3% in 2017-18 (not re-verified), with most learners never returning for a second course. Jordan (2015, IRRODL, 16(3), doi:10.19173/irrodl.v16i3.2112) found median completion around 12.5%, higher for shorter courses (not re-verified).

Adult persistence. Comings (2007, Review of Adult Learning and Literacy, vol. 7, 23-46; reissued doi:10.4324/9781003417996-2) synthesised US adult basic education research. Persistence depends on positive and negative forces, on a "sponsor" (a person who supports the learner), on setting a clear goal, on self-efficacy, and on seeing progress. Adults often "stop out" and return. I found no verifiable trial specific to adult numeracy.

Social commitment and parents. Hill and Tyson (2009, Developmental Psychology, 45:740-763, doi:10.1037/a0015362) meta-analysed 50 studies of parental involvement in middle school. Involvement was positively linked to achievement except direct homework help. "Academic socialization" (communicating the value of education and linking school to future goals) had the strongest association. Patall, Cooper and Robinson (2008, Review of Educational Research, 78:1039-1101, doi:10.3102/0034654308325185) found mixed effects of parent homework involvement, better when parents support autonomy than when they control.

How it could translate to an app. Faithful: the parent is the sponsor. Give parents a light weekly summary of what was learned (not how many minutes or points), one sentence of value talk they can use, and a suggestion to ask the child to show them something. For adult learners later, allow an optional named supporter who receives the same kind of summary, and design for re-entry after a long break ("stop-out, not drop-out"). Stretch: assuming a social feature will fix persistence; the evidence does not show that. Fits no-points: yes, provided nothing is ranked or compared.

What an app can do without points (ranked by evidence and ease)

Ranked by strength of evidence first, then by ease of building. Every item is compatible with a design that has no points, badges, or leaderboards.

  1. Genuine success on just-reachable material. Strong evidence (Talsma et al. 2018; Supekar et al. 2015; Sammallahti et al. 2023; Bandura & Schunk 1981). Easy, because this is what the adaptive engine does.
  2. Feedback on the task that names the strategy, with no grades, scores, or praise of the person. Strong evidence (Kluger & DeNisi 1996; Hattie & Timperley 2007; Koenka et al. 2021; Mueller & Dweck 1998). Easy: copy rules in the locale files.
  3. No expected rewards for engagement. Strong evidence, especially for children (Deci et al. 1999; Howard et al. 2021). Already a product rule.
  4. One near mastery goal per session, and an honest end-of-session "what moved". Good evidence (Bandura & Schunk 1981; Locke & Latham 2002; Senko et al. 2011). Moderate effort: needs a real capability measure, not volume.
  5. Autonomy support in copy and flow. Small real choices, a reason for each requirement, difficulty acknowledged, and a predictable structure. Good evidence (Reeve 2009; Jang et al. 2010; Patall et al. 2008a). Easy to moderate.
  6. Action language, no talk of brilliance, and varied people in problems. Good evidence from young children (Rhodes et al. 2019; Bian et al. 2017; Cvencek et al. 2011). Easy: a copy review plus a validator word list.
  7. A routine tied to a cue, and an if-then plan set up with the parent. Good evidence in adults, some in children (Gollwitzer & Sheeran 2006; Lally et al. 2010; Duckworth et al. 2013). Easy.
  8. One or two personalised reminders a week, sent to the parent. Moderate evidence (Aulagnon et al. 2025). Easy, behind the notification adapter.
  9. A warm welcome back after a gap, with no broken counts and a fresh start each week. Moderate evidence (Milkman et al. 2021; Dai et al. 2014; Lally et al. 2010). Easy.
  10. A question before the answer in Learn, then a quick reveal. Moderate evidence, mostly adult (Kang et al. 2009; Gruber et al. 2014; Wade & Kidd 2019). Moderate authoring cost.
  11. Parent prompts for talking about value and sharing warmth, not policing homework. Tell anxious parents they need not teach. Moderate evidence (Hill & Tyson 2009; Maloney et al. 2015; Schaeffer et al. 2018). Easy.
  12. Rare, optional prompts for older learners to find their own reasons the maths matters. Moderate evidence that depends on context (Hulleman & Harackiewicz 2009; Gaspard et al. 2015; Canning & Harackiewicz 2015). Easy.
  13. An optional line reframing nerves before an opt-in timed sprint, for adolescents and adults. Modest evidence (Jamieson et al. 2010, 2016; Rozek et al. 2019). Easy.

Contested or weak evidence

  • Expressive writing for anxiety. Did not replicate (Camerer et al. 2018). Meta-analytic effects are negligible (O'Meara & Lovett 2025). Do not build it.
  • Growth-mindset interventions. A small effect for lower-achieving ninth graders in one national trial (Yeager et al. 2019, Nature, 573:364-369, doi:10.1038/s41586-019-1466-y). Overall d = 0.05 across 63 studies, near zero in the best ones (Macnamara & Burgoyne 2023). Process feedback is supported; mindset lessons are not.
  • Streaks. There is one field trial, with about 5% take-up, 2.3% endline testing, six weeks, and weekly (not daily) streaks (Aulagnon et al. 2025). No trial tests daily loss-framed streaks for children over the long run.
  • "66 days to a habit." This was a median with a range of 18 to 254 days, from 82 adults doing simple health behaviours (Lally et al. 2010). There are no comparable data for children.
  • Utility value. Robust within the original programme and in the German maths trials. The effect shrinks in light-touch online deployments (Kizilcec et al. 2020) and can backfire if asserted (Canning & Harackiewicz 2015). Independent replications of the parent study are scarce (unverified gap).
  • Curiosity. Mostly small adult trivia studies with fMRI. Applying them to children's fact practice is an inference.
  • Reappraisal. Few trials, and little evidence below age 14. The Rozek et al. (2019) arms included expressive writing.
  • Supekar et al. (2015). Small (46 children), with no untreated anxious control group.
  • Progress principle. Rests on a book and a workplace diary study.
  • Reward undermining. Disputed by Cameron, Pierce, and colleagues (not re-verified). The findings specific to children and to interesting tasks are the most robust parts, and they are the relevant ones here.
  • Study partners and public commitment. Intuitive, but I found no verifiable trial showing they help in self-paced learning. Announcing identity goals publicly may reduce effort (Gollwitzer et al. 2009; not re-verified).
  • Adult numeracy programmes. No rigorous motivation studies specific to adult numeracy were verified. This is an open question.

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