Exploring mental representations for literal symbols using priming and comparison distance effects
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Higher-level mathematics requires a connection between literal symbols (e.g., ‘x’) and their mental representations. The current study probes the nature of mental representations for literal symbols using both the priming distance effect, in which ease of comparing a target number to a fixed standard is a function of prime-target distance, and the comparison distance effect, in which ease of comparing two numbers depends on the distance between them. Can literal symbols that have been assigned magnitude access mental representations of quantity to produce distance effects? Forty participants completed number comparison tasks involving Arabic numerals and literal symbols, a training task, and a working memory task. While both distance effects were present with Arabic numerals, there was no evidence of either with literal symbols. Results suggest that literal symbols may not share the same mental representations of magnitude as other number formats or may access them differently. Additional research is needed to understand mental representations utilized in higher-level mathematics (e.g., algebra), which includes both Arabic numerals and literal symbols.
KeywordsLiteral symbols Priming Distance effects Mental representations
We thank Dr. Gigi Luk for her assistance with research design, analysis, and feedback on previous versions of this manuscript. We also thank Dr. Kurt Fischer, George Spencer, Janine de Novais, and three anonymous reviewers for feedback on prior versions of this manuscript. This research was funded by the Graduate Student Award from the Mind, Brain, Behavior Interfaculty Initiative at Harvard University to CP and the Harvard Graduate School of Education Dean’s Summer Fellowship to CP.
- Bardini, C., Radford, L., & Sabena, C. (2005). Struggling with variables, parameters, and indeterminate objects or, how to go insane in mathematics. In Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 129–136). Melbourne, Australia.Google Scholar
- Booth, L. (1999). Children’s difficulties in beginning algebra. Algebraic thinking, grades K-12: Readings from the NCTM’s school-based journals and other publications (pp. 299–307). National Council of Teachers of Mathematics: Reston, VA.Google Scholar
- Christou, K. P., & Vosniadou, S. (2005). How students interpret literal symbols in algebra: a conceptual change approach. In Proceedings of the XXVII Annual Conference of the Cognitive Science Society (Vol. Italy, pp. 453–458).Google Scholar
- Dehaene, S. (1997). The number sense: how the mind creates mathematics. New York: Oxford University Press.Google Scholar
- Dehaene, S., Naccache, L., Le Clec’H, G., Koechlin, E., Mueller, M., Dehaene-Lambertz, G., Le Bihan, D. (1998). Imaging unconscious semantic priming. Nature, 395(6702), 597–600. doi: 10.1038/26967.
- Fias, W., van Dijck, J.-P., & Gevers, W. (2011). Chapter 10—How is Number Associated with Space? The Role of Working Memory. In S. Dehaene & E. M. Brannon (Eds.), Space, Time and Number in the Brain (pp. 133–148). San Diego: Academic Press. Retrieved from http://www.sciencedirect.com/science/article/pii/B9780123859488000104.
- Holloway, I. D., & Ansari, D. (2008). Domain-specific and domain-general changes in children’s development of number comparison. Developmental Science, 11(5), 644–649. 10.1111/j.1467-7687.2008.00712.x.
- Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: building meaning for symbols and their manipulation. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (2nd ed., pp. 707–762). Charlotte: Information Age Pub.Google Scholar
- Koechlin, E., Naccache, L., Block, E., & Dehaene, S. (1999). Primed numbers: exploring the modularity of numerical representations with masked and unmasked semantic priming. Journal of Experimental Psychology: Human Perception and Performance, 25(6), 1882–1905. doi: 10.1037/0096-15220.127.116.112.Google Scholar
- Küchemann, D. E. (1981). Algebra. Children’s understanding of mathematics: 11-16 (pp. 82–87). London: Athenaeum Press Ltd.Google Scholar
- Nie, B., Cai, J., & Moyer, J. (2009). How a standards-based mathematics curriculum differs from a traditional curriculum: with a focus on intended treatments of the ideas of variable. ZDM—The International Journal on Mathematics Education, 41(6), 777–792. doi: 10.1007/s11858-009-0197-1.CrossRefGoogle Scholar
- Philipp, R. A. (1992). A study of algebraic variables: beyond the student-professor problem. Journal of Mathematical Behavior, 11(2), 161–176.Google Scholar
- Philipp, R. (1999). The many uses of algebraic variables. Algebraic thinking, grades K-12: Readings from the NCTM’s school-based journals and other publications (pp. 157–162). National Council of Teachers of Mathematics: Reston, VA.Google Scholar
- Restle, F. (1970). Speed of adding and comparing numbers. Journal of Experimental Psychology, 83(2, Pt.1), 274–278. http://doi.org/10.1037/h0028573.
- Rosnick, P. (1982). Students’ symbolization processes in algebra. Retrieved July 29, 2014. http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=ED300230&site=ehost-live&scope=site.
- Sasanguie, D., Göbel, S. M., Moll, K., Smets, K., & Reynvoet, B. (2013). Approximate number sense, symbolic number processing, or number–space mappings: what underlies mathematics achievement? Journal of Experimental Child Psychology, 114(3), 418–431. doi: 10.1016/j.jecp.2012.10.012.CrossRefGoogle Scholar
- Schoenfeld, A. H., & Arcavi, A. (1999). On the meaning of variable. Algebraic thinking, grades K-12: Readings from the NCTM’s school-based journals and other publications (pp. 150–156). National Council of Teachers of Mathematics: Reston, VA.Google Scholar
- Trigueros, M., & Ursini, S. (2003). First-year undergraduates’ difficulties in working with different uses of variable. In Research in collegiate mathematics education, Volume 5 (pp. 1–29). Providence, RI: American Mathematical Society. Retrieved July 29, 2014, from http://books.google.com/books?id=foJJvXneF5sC&lpg=PA1&ots=xKHM66djYv&dq=ursini%2C%20sonia%2C%20algebra&lr&pg=PA1#v=onepage&q=ursini,%20sonia,%20algebra&f=false.
- Unsworth, N., Redick, T. S., Heitz, R. P., Broadway, J. M., & Engle, R. W. (2009). Complex working memory span tasks and higher-order cognition: a latent-variable analysis of the relationship between processing and storage. Memory, 17(6), 635–654. doi: 10.1080/09658210902998047.CrossRefGoogle Scholar
- Usiskin, Z. (1999). Conceptions of school algebra and uses of variables. Algebraic thinking, grades K-12: Readings from the NCTM’s school-based journals and other publications (pp. 7–13). National Council of Teachers of Mathematics: Reston, VA.Google Scholar