Symbolic and Non-Symbolic Representation of Quantity: Specific Ratio and Relationship with Success in Math

Abstract

The results of the study of the relationship between the accuracy of the estimation of quantities, expressed in symbolic and non-symbolic form, with the success of completing tasks in mathematics, including in the format of state exams, at different stages of schooling, are presented. A study with the participation of 493 schoolchildren in grades 4, 9 and 11 shows the specificity of the ratio of the accuracy of symbolic and non-symbolic representation of quantity at the primary, basic and complete levels of general education. A conclusion is made about the dependence of the relationship between the accuracy of symbolic and non-symbolic representation of quantity and the success of state examinations on the level of school education and, accordingly, the type of examination or test work.

References

1. Kuz'mina YuV, Voronin IA, Tikhomirova TN, Malykh SB. Vozrastnyye izmeneniya tochnosti reprezentatsii chisla na chislovoy linii. Teoreticheskaya i eksperimental'naya psikhologiya 2020; 13(4):18–30 (in Russian).
2. Tikhomirova TN, Malykh SB. Chuvstvo chisla i uspeshnost' v obuchenii matematike v mladshem shkol'nom vozraste: perekrestno-longityudnyy analiz. Psikhologicheskiy zhurnal 2018; 39(6):47–58 (in Russian).
3. Tikhomirova TN, Malykh SB, Tosto MG, & Kovas YuV. Kognitivnyye kharakteristiki i uspeshnost' v reshenii matematicheskikh zadaniy v starshem shkol'nom vozraste: krosskul'turnyy analiz. Psikhologicheskiy zhurnal 2014; 35(1):41–53 (in Russian).
4. Tikhomirova TN, Malykh SB. Kognitivnyye osnovy individual'nykh razlichiy v uspeshnosti obucheniya. Moscow; St. Petersburg: Nestor-Istoriya, 2017: 312 (in Russian).
5. Booth JL, Siegler RS. Developmental and individual differences in pure numerical estimation. Developmental Psychology 2006; 42(1):189–201.
6. De Smedt B, Noël 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 in Neuroscience and Education 2013; 2(2):48–55.
7. Dehaene S. The number sense: How the mind creates mathematics. USA, NY: Oxford University Press, 2011: 267.
8. Ebersbach M, Luwel K, Frick A, Onghena P, & Verschaffel L. The relationship between the shape of the mental number line and familiarity with numbers in 5-to 9-year old children: Evidence for a segmented linear model. Journal of Experimental Child Psychology 2008; 99(1):1–17.
9. Geary DC. Cognitive predictors of achievement growth in mathematics: a 5-year longitudinal study. Developmental Psychology 2011; 47(6):1539–1552.
10. Gebuis T, Van Der Smagt MJ. False approximations of the approximate number system?. PloS one 2011; 6(10): e25405. doi: 10.1371/ journal.pone.0025405.
11. Gilmore C.K., McCarthy S.E., Spelke E.S. Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling. Cognition 2010; 115(3):394–406.
12. Halberda J, Mazzocco MMM, Feigenson L. Individual differences in non-verbal number acuity correlate with maths achievement. Nature 2008; 455(7213):665–668.
13. Halberda J, Ly R, Wilmer J, Naiman D & Germine L. Number sense across the lifespan as revealed by a massive internet-based sample. Proceedings of the National Academy of Sciences of USA 2012; 109(28):11116–11120.
14. Kuzmina Y, Tikhomirova T, Lysenkova I, & Malykh S. Domain-general cognitive functions fully explained growth in nonsymbolic magnitude representation but not in symbolic representation in elementary school children. PloS one 2020; 15(2): e0228960. doi: 10.1371/ journal.pone.0228960.
15. Libertus ME, Brannon EM. Stable individual differences in number discrimination in infancy. Developmental Science 2010; 13(6):900–906.
16. Merritt DJ, DeWind NK, Brannon EM. Comparative cognition of number representation. TR Zentall, EA Wasserman (Eds). The Oxford Handbook of Comparative Cognition. New York, NY: Oxford University Press, 2012:451–476.
17. Nys J, Content A. Judgement of discrete and continuous quantity in adults: Number counts!. Quarterly Journal of Experimental Psychology 2012; 65(4):675–690.
18. Opfer JE, Siegler RS. Development of quantitative thinking. In KJ Holyoak & RG Morrison (Eds). The Oxford handbook of thinking and reasoning. USA, NY: Oxford University Press: 585–605.
19. Opfer JE, Siegler RS. Representational change and children’s numerical estimation. Cognitive Psychology 2007; 55(3):169–195.
20. Rodic M, Zhou X, Tikhomirova T, Wei W, Malykh S, Ismatulina V, Sabirova E, Davidova Y, Tosto M, Lemelin J-P, Kovas Y. Crosscultural investigation into cognitive underpinnings of individual differences in early arithmetic. Developmental Science 2015; 18(1):165–174.
21. Sasanguie D, Van den Bussche E, Reynvoet B. Predictors for mathematics achievement? Evidence from a longitudinal study. Mind, Brain, and Education 2012; 6(3):119–128.
22. Siegler RS. Magnitude knowledge: The common core of numerical development. Developmental Science 2016; 19(3):341–361.
23. Siegler RS, Ramani GB. Playing linear number board games – but not circular ones – improves low-income preschoolers’ numerical understanding. Journal of Educational Psychology 2009; 101(3):545–560.
24. Slusser EB, Santiago RT, Barth HC. Developmental change in numerical estimation. Journal of Experimental Psychology: General 2013; 142(1):193–208.
25. Slusser E, Barth H. Intuitive proportion judgment in number-line estimation: Converging evidence from multiple tasks. Journal of Experimental Child Psychology 2017; 162:181– 198.
26. Tikhomirova T, Kuzmina Y, Lysenkova I, & Malykh S. Development of approximate number sense across the elementary school years: A cross‐cultural longitudinal study. Developmental Science 2019; 22(4): e12823. doi: 10.1111/desc.12823.
27. Tikhomirova T, Kuzmina Y, Lysenkova I, & Malykh S. The relationship between non-symbolic and symbolic numerosity representations in elementary school: The role of intelligence. Frontiers in Psychology 2019; 10: 2724. doi: 10.3389/fpsyg.2019.02724.
28. Tikhomirova T, Malykh A, Malykh S. Predicting academic achievement with cognitive abilities: Cross-sectional study across school education. Behavioral Sciences 2020; 10(10): 158. doi: 10.3390/bs10100158.
29. Tikhomirova TN, Misozhnikova EB, Malykh AS, Gaydamashko IV, Malykh SB. Mathematical fluency in high school students. Psychology in Russia 2017; 10(1): 193. doi: 10.11621/ pir.2017.0107.
30. Tosto MG, Garon-Carrier G, Gross S, Petrill SA, Malykh S, Malki K, Hart SA, Thompson L, Karadaghi RL, Yakovlev N, Tikhomirova T, Opfer JE, Mazzocco MMM, Dionne G, Brendgen M, Vitaro F, Tremblay RE, Boivin M, Kovas Y. The nature of the association between number line and mathematical performance: An international twin study. British Journal of Educational Psychology 2019; 89(4):787–803.
31. Wang JJ, Odic D, Halberda J, & Feigenson L. Changing the precision of preschoolers’ approximate number system representations changes their symbolic math performance. Journal of Experimental Child Psychology 2016; 147:82–99.