, Volume 48, Issue 3, pp 291–303 | Cite as

Exploring mental representations for literal symbols using priming and comparison distance effects

  • Courtney PollackEmail author
  • Sibylla Leon Guerrero
  • Jon R. Star
Original Article


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.


Literal 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.


  1. Ansari, D. (2008). Effects of development and enculturation on number representation in the brain. Nature Reviews Neuroscience, 9(4), 278–291. doi: 10.1038/nrn2334.CrossRefGoogle Scholar
  2. 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
  3. Bartelet, D., Vaessen, A., Blomert, L., & Ansari, D. (2014). What basic number processing measures in kindergarten explain unique variability in first-grade arithmetic proficiency? Journal of Experimental Child Psychology, 117, 12–28. doi: 10.1016/j.jecp.2013.08.010.CrossRefGoogle Scholar
  4. 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
  5. 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
  6. Cohen Kadosh, R., & Walsh, V. (2009). Numerical representation in the parietal lobes: abstract or not abstract? Behavioral and Brain Sciences, 32(3–4), 313. doi: 10.1017/S0140525X09990938.CrossRefGoogle Scholar
  7. De Smedt, B., Verschaffel, L., & Ghesquière, P. (2009). The predictive value of numerical magnitude comparison for individual differences in mathematics achievement. Journal of Experimental Child Psychology, 103(4), 469–479. doi: 10.1016/j.jecp.2009.01.010.CrossRefGoogle Scholar
  8. Defever, E., Sasanguie, D., Gebuis, T., & Reynvoet, B. (2011). Children’s representation of symbolic and nonsymbolic magnitude examined with the priming paradigm. Journal of Experimental Child Psychology, 109(2), 174–186. doi: 10.1016/j.jecp.2011.01.002.CrossRefGoogle Scholar
  9. Dehaene, S. (1997). The number sense: how the mind creates mathematics. New York: Oxford University Press.Google Scholar
  10. Dehaene, S., & Akhavein, R. (1995). Attention, automaticity, and levels of representation in number processing. Journal of Experimental Psychology. Learning, Memory, and Cognition, 21(2), 314–326.CrossRefGoogle Scholar
  11. 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.
  12. den Heyer, K., & Briand, K. (1986). Priming single digit numbers: automatic spreading activation dissipates as a function of semantic distance. The American Journal of Psychology, 99(3), 315–340. doi: 10.2307/1422488.CrossRefGoogle Scholar
  13. 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
  14. Ganor-Stern, D., & Tzelgov, J. (2008). Across-notation automatic numerical processing. Journal of Experimental Psychology. Learning, Memory, and Cognition, 34(2), 430–437. doi: 10.1037/0278-7393.34.2.430.CrossRefGoogle Scholar
  15. 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.
  16. 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
  17. 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: 10.1037/0096-1523.25.6.1882.Google Scholar
  18. Küchemann, D. E. (1981). Algebra. Children’s understanding of mathematics: 11-16 (pp. 82–87). London: Athenaeum Press Ltd.Google Scholar
  19. Lyons, I. M., & Ansari, D. (2009). The cerebral basis of mapping nonsymbolic numerical quantities onto abstract symbols: an fmri training study. Journal of Cognitive Neuroscience, 21(9), 1720–1735.CrossRefGoogle Scholar
  20. McNeil, N. M., Weinberg, A., Hattikudur, S., Stephens, A. C., Asquith, P., Knuth, E. J., & Alibali, M. W. (2010). A is for apple: mnemonic symbols hinder the interpretation of algebraic expressions. Journal of Educational Psychology, 102(3), 625–634. doi: 10.1037/a0019105.CrossRefGoogle Scholar
  21. Moyer, R. S., & Landauer, T. K. (1967). Time required for judgments of numerical inequality. Nature, 215(5109), 1519–1520.CrossRefGoogle Scholar
  22. Naccache, L., & Dehaene, S. (2001). Unconscious semantic priming extends to novel unseen stimuli. Cognition, 80(3), 215–229.CrossRefGoogle Scholar
  23. 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
  24. Peirce, J. W. (2007). PsychoPy—Psychophysics software in Python. Journal of Neuroscience Methods, 162(1–2), 8–13. doi: 10.1016/j.jneumeth.2006.11.017.CrossRefGoogle Scholar
  25. Philipp, R. A. (1992). A study of algebraic variables: beyond the student-professor problem. Journal of Mathematical Behavior, 11(2), 161–176.Google Scholar
  26. 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
  27. Restle, F. (1970). Speed of adding and comparing numbers. Journal of Experimental Psychology, 83(2, Pt.1), 274–278.
  28. Reynvoet, B., Brysbaert, M., & Fias, W. (2002a). Semantic priming in number naming. The. Quarterly Journal of Experimental Psychology. A, Human Experimental Psychology, 55(4), 1127–1139. doi: 10.1080/02724980244000116.CrossRefGoogle Scholar
  29. Reynvoet, B., Caessens, B., & Brysbaert, M. (2002b). Automatic stimulus-response associations may be semantically mediated. Psychonomic Bulletin & Review, 9(1), 107–112.CrossRefGoogle Scholar
  30. Reynvoet, B., De Smedt, B., & Van den Bussche, E. (2009). Children’s representation of symbolic magnitude: the development of the priming distance effect. Journal of Experimental Child Psychology, 103(4), 480–489. doi: 10.1016/j.jecp.2009.01.007.CrossRefGoogle Scholar
  31. Rosnick, P. (1982). Students’ symbolization processes in algebra. Retrieved July 29, 2014.
  32. 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
  33. 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
  34. 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,%20sonia,%20algebra&f=false.
  35. 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
  36. 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
  37. Van Dijck, J.-P., & Fias, W. (2011). A working memory account for spatial–numerical associations. Cognition, 119(1), 114–119. doi: 10.1016/j.cognition.2010.12.013.CrossRefGoogle Scholar
  38. Van Dijck, J.-P., Gevers, W., & Fias, W. (2009). Numbers are associated with different types of spatial information depending on the task. Cognition, 113(2), 248–253. doi: 10.1016/j.cognition.2009.08.005.CrossRefGoogle Scholar
  39. Van Opstal, F., Gevers, W., De Moor, W., & Verguts, T. (2008). Dissecting the symbolic distance effect: comparison and priming effects in numerical and nonnumerical orders. Psychonomic Bulletin & Review, 15(2), 419–425. doi: 10.3758/PBR.15.2.419.CrossRefGoogle Scholar
  40. Verguts, T., & Fias, W. (2004). Representation of number in animals and humans: a neural model. Journal of Cognitive Neuroscience, 16(9), 1493–1504. doi: 10.1162/089892904256849.CrossRefGoogle Scholar
  41. Verguts, T., Fias, W., & Stevens, M. (2005). A model of exact small-number representation. Psychonomic Bulletin & Review, 12(1), 66–80.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2015

Authors and Affiliations

  • Courtney Pollack
    • 1
    Email author
  • Sibylla Leon Guerrero
    • 1
  • Jon R. Star
    • 1
  1. 1.Harvard Graduate School of EducationCambridgeUSA

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