1. Huizinga MM, Beech BM, Cavanaugh KL, Elasy TA, Rothman RL. Low numeracy skills are associated with higher BMI. Obesity (Silver Spring). 2008;16(8): 1966–68.
2. Parsons S. Basic skills and crime. London: The Basic Skills Agency; 2002.
3. Smith JP, McArdle JJ, Willis R. Financial decision-making and cognition in a family context. Econ J. 2010;120(549): F363–80.
4. OECD Statistics Canada. Literacy for life: Further results from the adult literacy and life skills survey. Paris: OECD Publishing; 2011.
5. Noonan R. STEM Jobs: 2017 Update [Internet]. Washington D.C.: U.S. Department of Commerce, Economics and Statistics Administration; 2017 p.4–5. Available from http://www.esa.gov/reports/stem-jobs-2017-update.
6. OECD. PISA 2012 results: What students know and can do—student performance in mathematics, reading and science. Paris: OECD Publishing; 2014.
7. NCES. The nation's report card: A first look: mathematics and reading [Internet]. Washington D.C.: Institute of Education Sciences, U.S. Department of Education; 2013 p. 7. Available from: https://nces.ed.gov/nationsreportcard/subject/publications/main2013/pdf/2014451.pdf
8. ACT. The Condition of College and Career Readiness [Internet]. Iowa City: 2012 p. 4–5. Available from: https://www.act.org/content/dam/act/unsecured/documents/CCCR-2014-Iowa.pdf
9. Duncan GJ, Dowsett CJ, Claessens A, Magnuson K, Huston AC, Klebanov P, et al. School readiness and later achievement. Dev Psychol. 2007;43(6):1428–46. doi: 10.1037/0012-16188.8.131.528 18020822
10. Duncan GJ, Magnuson K. The nature and impact of early achievement skills, attention skills, and behavior Problems. In: Duncan GJ, Richard J, editors. Whither Opportunity: Rising Inequality, Schools, and Children's Life Chances. New York: Russell Sage; 2011. pp. 47–69.
11. Rose H, Betts JR. Math matters: The links between high school curriculum, college graduation, and earnings [Internet]. San Francisco: Public Policy Institute of California, 2001. Available from: https://www.ppic.org/content/pubs/report/R_701JBR.pdf
12. Schoenfeld AH. The math wars. Educ Policy. 2004;18(1):253–86.
13. Banilower ER, Smith PS, Weiss IR, Malzahn KA, Campbell KM, Weis AM. Report of the 2012 national survey of science and mathematics education [Internet]. Chapel Hill: Horizon Research Inc., 2013. Available from: http://www.horizon-research.com/report-of-the-2012-national-survey-of-science-and-mathematics-education
14. Hill HC, Shih JC. Examining the quality of statistical mathematics education research. J Res Math Educ. 2009;40(3):241–50.
15. Slavin RE, Lake C. Effective programs in elementary mathematics: a best-evidence synthesis. Rev Educ Res. 2008;78(3):427–515.
16. ies.ed.gov [Internet]. Washington D.C.: Institute of Education Sciences; c2019 [cited 2019 March 31]. Available from: https://ies.ed.gov/ncee/wwc/FWW/Results?filters=,Math
17. evidenceforessa.org [Internet]. Baltimore: Center for Research and Reform in Education, Johns Hopkins University; c2019 [cited 2019 March 31]. Available from: https://www.evidenceforessa.org/programs/math/
18. Mighton J, Sabourin S, Klebanov A. JUMP Math. Toronto: JUMP Math; 2008.
19. Mighton J. For the love of math. Sci Am Mind. 2013;24(4):60–7.
20. This description is intended to give a general sense of the problem-based approach to math instruction and does not refer to any particular commerical math program.
21. National Council of Teachers of Mathematics. Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics; 2000.
22. Ontario Ministry of Education. The Ontario Curriculum Grades 1–8 Mathematics Rev. Toronto: Ontario Ministry of Education, 2005.
23. Kirschner P, Sweller J and Clark R. Why minmal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teachiang. Educ Psychol. 2006; 41 (2):75–86.
24. otffeo.on.ca [Internet]. Toronto: Ontario Teachers Federation; c2019 [cited 2019 March 31]. Available from: https://www.otffeo.on.ca/en/wp-content/uploads/sites/2/2014/06/6_Grade-3-sample.pdf
25. Alfieri L, Brooks PJ, Aldrich NJ and Tenenbaum HR. Does discovery-based instruction enhance learning? J Educ Psychol. 2008;103 (1):1–18.
26. Stockard J, Wood TS, Coughlin C, Khoury C. The effectiveness of direct instruction curricula: A meta-analysis of half a century of research. Rev Educ Res. 2018; 88 (4):479–507.
27. Dweck CS. The secret to raising smart kids. Sci Am Mind. 2007;18(6):36–43.
28. Leong YH, Ho WK, Cheng LP. Concrete-pictorial-abstract: Surveying its origins and charting its future. The Math Educ. 2015; 16(1): 1–18.
29. Fyfe ER, McNeil NM, Son JY, Goldstone RL. Concreteness fading in mathematics and science instruction: A systematic review. Educ Psychol Rev. 2014;26(1):9–25.
30. Kaminski JA, Sloutsky VM, Heckler A. Transfer of Mathematical Knowledge: The Portability of Generic Instantiations. Child Dev Perspect. 2009;3(3):151–5.
31. Rittle-Johnson B, Kmicikewycz AO. When generating answers benefits arithmetic skill: The importance of prior knowledge. J Exp Child Psychol. 2008;101(1):75–81. doi: 10.1016/j.jecp.2008.03.001 18439617
32. Jordan NC, Kaplan D, Ramineni C, Locuniak MN. Development of number combination skill in the early school years: When do fingers help? Dev Sci. 2008;11(5):662–8. doi: 10.1111/j.1467-7687.2008.00715.x 18801121
33. Sweller J. Cognitive load during problem solving: Effects on learning. Cogn Sci. 1988;12:257–85.
34. Anderson JR, Reder LM, Simon HA. Applications and misapplications of cognitive Psychol. to mathematics instruction. Texas Educ Rev. 2000;1(2):1–42.
35. Fuchs LS, Fuchs D, Prentice K. Responsiveness to mathematical problem-solving instruction: Comparing students at risk of mathematics disability with and without risk of reading disability. J Learn Disabil. 2004;37(4):293–306. doi: 10.1177/00222194040370040201 15493402
36. van Merienboer J, Kirschner PA, Kester L. Taking the load off a learner's mind: Instructional design for complex learning. Educ Psychol. 2003;38(1):5–13.
37. Nelson Education [Internet]. Toronto: Nelson Education; c2019. What is PRIME?; 2019 March 31 [cited 2019 March 31];[about 1 screen]. Available from: http://www.prime.nelson.com/teacher/index.html
38. rti4success.org [Internet]. Washington, DC: American Institutes for Research; c2019 [cited 2019 March 31]. Available from: http://www.rti4success.org/.
39. Cooper H., Nye B., Charlton K., Lindsay J., & Greathouse S. The effects of summer vacation on achievement test scores: A narrative and meta-analytic review. Rev Educ Res. 1996;3:227–68.
40. Sweller J., Ayres P. & Kalyuga S. Cognitive Load Theory. New York: Springer-Verlag; 2011.
41. Following current recommendations by the CONSORT organization for randomized-controlled trials we did not compare baseline performance for the two groups statistically. See 42–44.
42. Moher D, Hopewell S, Schulz KF, Montori V, Gøtzsche PC, Devereaux PJ, et al. CONSORT 2010 explanation and elaboration: Updated guidelines for reporting parallel group randomised trials. J Clin Epidemiol. 2010;63:e1–37. doi: 10.1016/j.jclinepi.2010.03.004 20346624
43. Altman DG. Comparability of randomised groups. J R Stat Soc. 1985;34:125–36.
44. Altman, Douglas G, Doré CJ. Randomisation and baseline comparisons. Lancet. 1990;335:149–53. doi: 10.1016/0140-6736(90)90014-v 1967441
45. Woodcock RW, McGrew KS, Mather N. Woodcock-Johnson III. Rolling Meadows: Riverside Publishing; 2001.
46. Kaufman AS, Kaufman NL. Kaufman BriESIntelligence Test, 2nd ed. Circle Pines: American Guidance Service; 2004.
47. Kaplan E, Fein D, Kramer D, Delis C, Morris R. WISC-III as a Process Instrument. San Antonio: Psychological Corporation; 1999.
48. Tallmadge GK. The Joint Dissemination Review Panel Idea Book. Washington, DC: National Institute of Education; 1977.
49. Lipsey MW, Puzio K, Yun C, Herbert MA, Steinka-Fry K, Cole MW. Translating the statistical representation of effects of education interventions into more readily iterpretable forms [Internet]. Washington D.C: National Center for Special Education Research, Institute of Education Sciences, U.S. Department of Education; 2012 p. 33–37. Available from: https://ies.ed.gov/ncser/pubs/20133000/pdf/20133000.pdf
50. Based on demographic data derived from a database of the results from a compulsory, regional assessment in academic progress written annually at the end of grades 3 and 6, for the year preceding the study.
51. Derived from eqao.com [Internet]. Toronto: Education Quality and Accountability Office; c2019 [cited 2019 March 31]. Available from: https://eqaoweb.eqao.com/eqaoweborgprofile/profile.aspx?_Mident=84&Lang=E
52. The David Johnson/C. D. Howe Institute Ontario School Performance Database; 2009. Available from: https://www.cdhowe.org/public-policy-research/ontario%E2%80%99s-best-public-schools-2009-2011
53. Viera AJ, Garrett JM. Understanding inter-observer agreement: the kappa statistic. Fam Med 2005;37(5):360–3.
54. Altman DG. Practical Statistics for Medical Research. London: Chapman and Hall; 1991.
55. Fuchs LS, Hamlett CL, Fuchs D. Monitoring Basic Skills Progress: Basic Math Computation. Austin: PRO-ED; 1998.
56. Wolf M. Denckla MB. Rapid Automatized Naming and Rapid Alternating Stimulus Tests. Austin: PRO-ED; 2005.
57. Mean estimates of change by curricula, grade and time period for each measure are summarized in Tables F, G and H in S1 Text.
58. Agodini R, Harris B. An experimental evaluation of four elementary school math curricula. J Res Educ Eff. 2010; 3(3): 199–253.
59. P-values were not adjusted for multiple comparisons. Given the correlated nature of the data, any adjustment for multiple comparisons (which are based on independent measures) would be overly conservative and increase the risk of Type II errors. See 60.
60. Perneger T. V. What’s wrong with Bonferroni adjustments? Br Med J. 1998;316:1236–8.
61. Ramirez G, Gunderson E, Levine S, Beilock S. Math anxiety, working memory and math achievement in early elementary school. J Cogn Dev. 2013;14(2):187–202.
62. Mueller C. Dweck CS. Praise for intelligence can undermine children’s motivation and performance. J Pers Soc Psychol. 1998;75(1):33–52. doi: 10.1037//0022-35184.108.40.206 9686450
63. Geary DC. Mathematics and learning disabilities. J Learn Dis. 2004; 37(1): 4–15.
64. Geary D, Brown SC, Samaranayake VA. Cognitive addition: A short longitudinal study of strategy choice and speed-of-processing differences in normal and mathematically disabled children. Dev Psychol. 1991;27:787–97.
65. Rivera S, Reiss AL, Eckert MA, Menon V. Developmental changes in mental arithmetic: Evidence for increased functional specialization in the left inferior parietal cortex. Cereb Cortex. 2005;15:1779–90. doi: 10.1093/cercor/bhi055 15716474
66. Price GR, Mazzocco MM, Ansari D. Why mental arithmetic counts: Brain activation during single digit arithmetic predicts high school math scores. J Neurosci. 2013;33(1):156–63. doi: 10.1523/JNEUROSCI.2936-12.2013 23283330
67. McGrew KS, Schrank FA, Woodcock RW. Technical Manual. Woodcock-Johnson III Normative Update. Rolling Meadows: Riverside Publishing; 2007.
68. De Smedt B, Noel MP, Gilmore C, Ansari D. How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children's mathematical skills? A review of evidence from brain and behavior. Trends Neurosci Educ. 2013;2(2):48–55.
69. Booth JL, Siegler RS. Numerical magnitude representations influence arithmetic learning. Child Dev. 2008;79(4):1016–31. doi: 10.1111/j.1467-8624.2008.01173.x 18717904