Skip to main content
SpringerLink
Log in

Tracking practice effects in computation estimation

  • Original Article
  • Published:
Psychological Research Aims and scope Submit manuscript

Abstract

The present study investigated college students’ ability to estimate the results of multi-digit multiplication problems and the extent to which this ability improves with practice. Participants judged whether the results of multiplication problems composed of two-digit numbers were larger or smaller than a given reference number. The reference numbers were either close or far from the exact answer. The effects of practice, size, and distance of the reference number from the exact answer were examined using four measures of performance: speed, accuracy, eye movements, and strategy use. The results show that together with enhanced speed and accuracy with practice, participants also changed the pattern of eye movements and the strategies they used. The eye movement analysis showed longer dwell time and more frequent first fixations toward the reference number with practice, suggesting that participants relied more on the reference number to solve the task with practice. The strategy analysis revealed that with practice participants reduced their use of the approximate calculation strategy, which involves multiplying the rounded operands and comparing the product to the reference number, and increased their reliance on the sense of magnitude strategy which does not involve any calculation, but is grounded in the ANS. This was done especially for trials in which the reference number was far from the exact answer, thus exhibiting enhanced adaptivity in strategy choice with practice.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Anderson, J. R., & Lebiere, C. (1998). The atomic components of thought. Mahwah: LEA.

    Google Scholar 

  • Ansari, D. (2008). Effects of development and enculturation on number representation in the brain. Nature Reviews Neuroscience, 9(4), 278–291.

    Article  PubMed  Google Scholar 

  • Ashcraft, M. H., & Stazyk, E. H. (1981). Mental addition: a test of three verification models. Memory and Cognition, 9, 185–196.

    Article  PubMed  Google Scholar 

  • Cantlon, J., Platt, M. L., & Brannon, E. M. (2009). Beyond the number domain. Trends in Cognitive Sciences, 13(2), 83–91.

    Article  PubMed  PubMed Central  Google Scholar 

  • Corbetta, M., Akbudak, E., Conturo, T. E., Snyder, A. Z., Ollinger, J. M., Drury, H. A., et al. (1998). A common network of functional areas for attention and eye movements. Neuron, 21(4), 761–773.

    Article  PubMed  Google Scholar 

  • De Corte, E., Verschaffel, L., & Pauwels, A. (1990). Influence of the semantic structure of word problems on second graders’ eye movements. Journal of Educational Psychology, 82(2), 359.

    Article  Google Scholar 

  • Dehaene, S. (1997). The number sense. New York: Oxford University Press.

    Google Scholar 

  • Dehaene, S., Dehaene-Lambertz, G., & Cohen, L. (1998). Abstract representations of numbers in the animal and human brain. Trends in Neurosciences, 21(9), 355–361.

    Article  PubMed  Google Scholar 

  • Delazer, M., Domahs, F., Bartha, L., Brenneis, C., Lochy, A., Trieb, T., et al. (2003). Learning complex arithmetic—an fMRI study. Cognitive Brain Research, 18, 76–88.

    Article  PubMed  Google Scholar 

  • Dowker, A. (1997). Young children’s addition estimates. Mathematical Cognition, 3, 141–154.

    Article  Google Scholar 

  • Dowker, A., Flood, A., Griffiths, H., Harriss, L., & Hook, L. (1996). Estimation strategies of four groups. Mathematical Cognition, 2, 113–135.

    Article  Google Scholar 

  • Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314.

    Article  PubMed  Google Scholar 

  • Fias, W., Lammertyn, J., Reynvoet, B., Dupont, P., & Orban, G. A. (2003). Parietal representation of symbolic and nonsymbolic magnitude. Journal of Cognitive Neuroscience, 15(1), 47–56.

    Article  PubMed  Google Scholar 

  • Ganor-Stern, D. (2015a). Can dyscalculics estimate the results of arithmetic problems? Journal of Learning Disabilities.

  • Ganor-Stern, D. (2015b). When you don’t have to be exact—investigating computation estimation skills with a comparison task. Acta Psychologica, 154, 1–9.

    Article  PubMed  Google Scholar 

  • Geary, D. C. (1994). Children’s mathematical development. Washington, DC: American Psychological Association.

    Google Scholar 

  • Geary, D. C., Brown, S., & Samaranayake, V. A. (1991). Cognitive addition: a short longitudinal study of strategy choice and speed-of-processing differences in normal and mathematically disabled children. Developmental Psychology, 27(5), 787–797.

    Article  Google Scholar 

  • Green, H. J., Lemaire, P., & Dufau, S. (2007). Eye movement correlates of younger and older adults’ strategies for complex addition. Acta Psychologica, 125(3), 257–278.

    Article  PubMed  Google Scholar 

  • Hoffman, J. E., & Subramaniam, B. (1995). The role of visual attention in saccadic eye movements. Perception and Psychophysics, 57(6), 787–795.

    Article  PubMed  Google Scholar 

  • Huber, S., Cornelsen, S., Moeller, K., & Nuerk, H.-C. (2015). Toward a model framework of generalized parallel componential processing of multi-symbol numbers. Journal of Experimental Psychology. Learning, Memory, and Cognition, 41(3), 732.

    Article  PubMed  Google Scholar 

  • Huber, S., Mann, A., Nuerk, H.-C., & Möller, K. (2014a). Cognitive control in number magnitude processing: evidence from eye-tracking. Psychological Research, 78(4), 539–548.

    Article  PubMed  Google Scholar 

  • Huber, S., Moeller, K., & Nuerk, H. C. (2014b). Adaptive processing of fractions—evidence from eye-tracking. Acta Psychologica, 148, 37–48.

    Article  PubMed  Google Scholar 

  • Imbo, I., & Vandierendonck, A. (2008). Practice effects on strategy selection and strategy efficiency in simple mental arithmetic. Psychological Research, 72(5), 528–541.

    Article  PubMed  Google Scholar 

  • Kallai, A., & Tzelgov, J. (2012). The place-value of a digit in multi-digit numbers is processed automatically. Journal of Experimental Psychology. Learning, Memory, and Cognition, 38(5), 1221–1233.

    Article  PubMed  Google Scholar 

  • Kirk, E. P., & Ashcraft, M. H. (2001). Telling stories: the perils and promise of using verbal reports to study math strategies. Journal of Experimental Psychology. Learning, memory, and cognition, 27, 157–175.

    Article  PubMed  Google Scholar 

  • LeFevre, J.-A., Greenham, S. L., & Waheed, N. (1993). The development of procedural and conceptual knowledge in computational estimation. Cognition and Instruction, 11, 95–132.

    Article  Google Scholar 

  • Lemaire, P., & Lecacheur, M. (2002). Children’s strategies in computational estimation. Journal of Experimental Child Psychology, 82, 281–304.

    Article  PubMed  Google Scholar 

  • Lemaire, P., & Lecacheur, M. (2010). Strategy switch costs in arithmetic problem solving. Memory and Cognition, 38(3), 322–332.

    Article  PubMed  Google Scholar 

  • Lemaire, P., Lecacheur, M., & Farioli, F. (2000). Children’s strategy use in computational estimation. Canadian Journal of Experimental Psychology, 54(2), 141–148.

    Article  PubMed  Google Scholar 

  • Lemaire, P., & Siegler, R. (1995). Four aspects of strategic change: contributions to children’s learning of multiplication. Journal of Experimental Psychology: General, 124(1), 83–97.

    Article  Google Scholar 

  • Logan, G. (1988). Toward an instance theory of automatization. Psychological Review, 91, 295–327.

    Article  Google Scholar 

  • Lyons, I. M., Ansari, D., & Beilock, S. L. (2015a). Qualitatively different coding of symbolic and nonsymbolic numbers in the human brain. Human Brain Mapping, 36(2), 475–488.

    Article  PubMed  Google Scholar 

  • Lyons, I. M., Nuerk, H. C., & Ansari, D. (2015b). Rethinking the implications of numerical ratio effects for understanding the development of representational precision and numerical processing across formats.

  • McEldoon, K. L., Durkin, K. L., & Rittle-Johnson, B. (2013). Is self-explanation worth the time? A comparison to additional practice. British Journal of Educational Psychology, 83(4), 615–632.

    Article  PubMed  Google Scholar 

  • Merkley, R., & Ansari, D. (2010). Using eye tracking to study numerical cognition: the case of the ratio effect. Experimental Brain Research, 206(4), 455–460.

    Article  PubMed  Google Scholar 

  • Moeller, K., Fischer, M., Nuerk, H., & Willmes, K. (2009). Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking. Quarterly Journal of Experimental Psychology, 62(2), 323–334.

    Article  Google Scholar 

  • Moyer, R. S., & Landauer, T. K. (1967). Time required for judgments of numerical inequality. Nature, 215, 1519–1520.

    Article  PubMed  Google Scholar 

  • Pinhas, M., Pothos, E. M., & Tzelgov, J. (2013). Zooming in and out from the mental number line: evidence for a number range effect. Journal of Experimental Psychology. Learning, Memory, and Cognition, 39(3), 972–976.

    Article  PubMed  Google Scholar 

  • Rayner, K. (1998). Eye movements in reading and information processing: 20 years of research. Psychological Bulletin, 124(3), 372–422.

    Article  PubMed  Google Scholar 

  • Rayner, K. (2009). Eye movements and attention in reading, scene perception, and visual search. The Quarterly Journal of Experimental Psychology, 62(8), 1457–1506.

    Article  PubMed  Google Scholar 

  • Rehder, B., & Hoffman, A. B. (2005). Eyetracking and selective attention in category learning. Cognitive Psychology, 51(1), 1–41.

    Article  PubMed  Google Scholar 

  • Schneider, W., Eschman, A., & Zuccolotto, A. (2002). E-Prime user’s guide. Pittsburgh: Psychology Software Tools Inc.

    Google Scholar 

  • Siegler, R. S. (1996). Emerging minds. New York: Oxford University Press.

    Google Scholar 

  • Siegler, R. S., & Booth, J. (2005). Development of numerical estimation: a review. In J. Campbell (Ed.), Handbook of mathematical cognition (pp. 197–212). New York: Psychology Press.

    Google Scholar 

  • Siegler, R. S., & Lemaire, P. (1997). Older and younger adults’ strategy choices in multiplication: testing predictions of ASCM via the choice/no choice method. Journal of Experimental Psychology: General, 126, 71–92.

    Article  Google Scholar 

  • Siegler, R. S., & Shrager, J. (1984). A model of strategy choice. In C. Sophian (Ed.), Origins of cognitive skills (pp. 229–293). Hillsdale: Erlbaum.

    Google Scholar 

  • Smith-Chant, B. L., & LeFevre, J. (2003). Doing as they are told and telling it like it is: Self-reports in mental arithmetic. Memory & Cognition, 31(4), 516–528.

    Article  Google Scholar 

  • Suppes, P. (1990). Eye-movement models for arithmetic and reading performance. Eye Movements and Their Role in Visual and Cognitive Processes, 4, 455–477.

    Google Scholar 

  • 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 and Review, 15(2), 419–425.

    Article  PubMed  Google Scholar 

Download references

Acknowledgments

This research was supported by the Israel Science Foundation to D.G.S. (Grant No. 906/12).

Author information

Author notes

    Authors and Affiliations

    1. Achva Academic College, Achva, Israel

      Dana Ganor-Stern

    2. Ben-Gurion University of the Negev, Beersheba, Israel

      Nilly Weiss

    Authors
    1. Dana Ganor-Stern
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. Nilly Weiss
      View author publications

      You can also search for this author in PubMed Google Scholar

    Corresponding author

    Correspondence to Dana Ganor-Stern.

    Ethics declarations

    Ethical standards

    All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendment or comparable ethical standards.

    Additional information

    D. Ganor-Stern and N. Weiss contributed equally to this manuscript.

    Rights and permissions

    Reprints and permissions

    About this article

    Check for updates. Verify currency and authenticity via CrossMark

    Cite this article

    Ganor-Stern, D., Weiss, N. Tracking practice effects in computation estimation. Psychological Research 80, 434–448 (2016). https://doi.org/10.1007/s00426-015-0720-7

    Download citation

    • Received:

    • Accepted:

    • Published:

    • Issue Date:

    • DOI: https://doi.org/10.1007/s00426-015-0720-7

    Keywords

    Search

    Navigation