Abstract
Recent research into math cognition has identified areas of the brain that are involved in number processing (Dehaene, Piazza, Pinel, & Cohen, 2003) and complex problem solving (Anderson, 2007). Much of this research assumes that participants use a single strategy; yet, behavioral research finds that people use a variety of strategies (LeFevre et al., 1996; Siegler, 1987; Siegler & Lemaire, 1997). In the present study, we examined cortical activation as a function of two different calculation strategies for mentally solving multidigit multiplication problems. The school strategy, equivalent to long multiplication, involves working from right to left. The expert strategy, used by “lightning” mental calculators (Staszewski, 1988), proceeds from left to right. The two strategies require essentially the same calculations, but have different working memory demands (the school strategy incurs greater demands). The school strategy produced significantly greater early activity in areas involved in attentional aspects of number processing (posterior superior parietal lobule, PSPL) and mental representation (posterior parietal cortex, PPC), but not in a numerical magnitude area (horizontal intraparietal sulcus, HIPS) or a semantic memory retrieval area (lateral inferior prefrontal cortex, LIPFC). An ACT-R model of the task successfully predicted BOLD responses in PPC and LIPFC, as well as in PSPL and HIPS.
Article PDF
Similar content being viewed by others
References
Alivisatos, B., & Petrides, M. (1997). Functional activation of the human brain during mental rotation. Neuropsychologia, 35, 111–118.
Anderson, J. R. (2007). Using brain imaging to guide the development of a cognitive architecture. In W. D. Gray (Ed.), Integrated models of cognitive systems (pp. 49–62). New York: Oxford University Press.
Anderson, J. R., Carter, C. S., Fincham, J. M., Qin, Y., Ravizza, S. M., & Rosenberg-Lee, M. (2008). Using f MRI to test models of complex cognition. Cognitive Science, 32, 1323–1348. doi:10.1080/03640210802451588
Anderson, J. R., Fincham, J. M., & Douglass, S. (1997). The role of examples and rules in the acquisition of a cognitive skill. Journal of Experimental Psychology: Learning, Memory, & Cognition, 23, 932–945. doi:10.1037/0278-7393.23.4.932
Anderson, J. R., Qin, Y., Sohn, M.-H., & Stenger, V. A. (2003). An information-processing model of the BOLD response in symbol manipulation tasks. Psychonomic Bulletin & Review, 10, 241–261.
Anderson, J. R., Qin, Y., Stenger, V. A., & Carter, C. (2004). The relationship of three cortical regions to an information-processing model. Journal of Cognitive Neuroscience, 16, 637–653.
Buckner, R. L., Kelley, W. M., & Petersen, S. E. (1999). Frontal cortex contributes to human memory formation. Nature Neuroscience, 2, 311–314.
Cabeza, R., Dolcos, F., Graham, R., & Nyberg, L. (2002). Similarities and differences in the neural correlates of episodic memory retrieval and working memory. NeuroImage, 16, 317–330.
Carpenter, P. A., Just, M. A., Keller, T. A., Eddy, W., & Thulborn, K. (1999). Graded functional activation in the visuospatial system with the amount of task demand. Journal of Cognitive Neuroscience, 11, 9–24.
Carroll, W. M., & Isaacs, A. (2003). Achievement of students using the University of Chicago School Mathematics Project’s Everyday Mathematics. In S. L. Senk & D. R. Thompson (Eds), Standardsbased school mathematics curricula: What are they? What do students learn? (pp. 79–108). Mahwah, NJ: Erlbaum.
Chochon, F., Cohen, L., van de Moortele, P., & Dehaene, S. (1999). Differential contributions of the left and right inferior parietal lobules to number processing. Journal of Cognitive Neuroscience, 11, 617–630.
Cipolotti, L., Warrington, E. K., & Butterworth, B. (1995). Selective impairment in manipulating Arabic numerals. Cortex, 31, 73–86.
Clark, D., & Wagner, A. D. (2003). Assembling and encoding word representations: fMRI subsequent memory effects implicate a role for phonological control. Neuropsychologia, 41, 304–317.
Corbetta, M., Kincade, J. M., Ollinger, J. M., McAvoy, M. P., & Shulman, G. L. (2000). Voluntary orienting is dissociated from target detection in human posterior parietal cortex. Nature Neuroscience, 3, 292–297.
Cox, R. W., & Hyde, J. S. (1997). Software tools for analysis and visualization of fMRI data. NMR in Biomedicine, 10, 171–178.
Culham, J. C., & Kanwisher, N. G. (2001). Neuroimaging of cognitive functions in human parietal cortex. Current Opinion in Neurobiology, 11, 157–163.
Danker, J. F., & Anderson, J. R. (2007). The roles of prefrontal and posterior parietal cortex in algebra problem solving: A case of using cognitive modeling to inform neuroimaging data. NeuroImage, 35, 1365–1377. doi:10.1016/j.neuroimage.2007.01.032
Davachi, L., Maril, A., & Wagner, A. D. (2001). When keeping in mind supports later bringing to mind: Neural markers of phonological rehearsal predict subsequent remembering. Journal of Cognitive Neuroscience, 13, 1059–1070.
Dehaene, S., & Cohen, L. (1997). Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33, 219–250.
Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506. doi:10.1080/02643290244000239
Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brainimaging evidence. Science, 284, 970–974.
Delazer, M., Domahs, F., Bartha, L., Brenneis, C., Locky, A., Trieb, T., & Benke, T. (2003). Learning complex arithmetic—An fMRI study. Cognitive Brain Research, 18, 76–88.
Delazer, M., Ischebeck, A., Domahs, F., Zamarian, L., Koppelstaetter, F., Siedentopf, C. M., et al. (2005). Learning by strategies and learning by drill—Evidence from an fMRI study. Neuro-Image, 25, 838–849.
Dobbins, I. G., & Wagner, A. D. (2005). Domain-general and domainsensitive prefrontal mechanisms for recollecting events and detecting novelty. Cerebral Cortex, 15, 1768–1778. doi:10.1093/cercor/bhi054
Glover, G. H. (1999). Deconvolution of impulse response in eventrelated BOLD fMRI. NeuroImage, 9, 416–429.
Grabner, R. H., Ansari, D., Reishofer, G., Stern, E., Ebner, F., & Neuper, C. (2007). Individual differences in mathematical competence predict parietal brain activation during mental calculation. Neuro Image, 38, 346–356. doi:10.1016/j.neuroimage.2007.07.041
Greicius, M. D., Krasnow, B., Reiss, A. L., & Menon, V. (2003). Functional connectivity in the resting brain: A network analysis of the default mode hypothesis. Proceedings of the National Academy of Sciences, 100, 253–258. doi:10.1073/pnas.0135058100
Jackson, M., & Warrington, E. K. (1986). Arithmetic skill in patients with unilateral cerebral-lesions. Cortex, 22, 611–620.
Kahn, H. J., & Whitaker, H. A. (1991). Acalculia—An historical review of localization. Brain & Cognition, 17, 102–115.
Kwong, T. E., & Varnhagen, C. K. (2005). Strategy development and learning to spell new words: Generalization of a process. Developmental Psychology, 41, 148–159. doi:10.1037/0012-1649.41.1.148
Lee, K. M. (2000). Cortical areas differentially involved in multiplication and subtraction: A functional magnetic imaging study with a case of selective acalculia. Annals of Neurology, 48, 657–661.
LeFevre, J.-A., Bisanz, J., Daley, K. E., Buffone, L., Greenham, S. L., & Sadesky, G. S. (1996). Multiple routes to solution of singledigit multiplication problems. Journal of Experimental Psychology: General, 125, 284–306.
Lovett, M. C., Daily, L. Z., & Reder, L. M. (2001). A source activation theory of working memory: Cross-task prediction of performance in ACT-R. Cognitive Systems Research, 1, 99–118.
Menon, V., Rivera, S. M., White, C. D., Glover, G. H., & Reiss, A. L. (2000). Dissociating prefrontal and parietal cortex activation during arithmetic processing. NeuroImage, 12, 357–365.
Naus, M. J., & Ornstein, P. A. (1983). Development of memory strategies: Analysis, questions, and issues. In M. T. H. Chi (Ed.), Trends in memory development research. New York: Karger.
Pesenti, M. (2005). Calculation abilities in expert calculators. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 413–430). New York: Psychology Press.
Qin, Y., Carter, C. S., Silk, E. M., Stenger, V. A., Fissell, K., Goode, A., & Anderson, J. R. (2004). The change of the brain activation patterns as children learn algebra equation solving. Proceedings of the National Academy of Sciences, 101, 5686–5691. doi:10.1073/pnas.0401227101
Qin, Y., Sohn, M. H., Anderson, J. R., Stenger, V. A., Fissell, K., Goode, A., et al. (2003). Predicting the practice effects on the blood oxygenation level-dependent (BOLD) function of fMRI in a symbolic manipulation task. Proceedings of the National Academy of Sciences, 100, 4951–4956. doi:10.1073/pnas.0431053100
Raftery, A. E. (1995). Bayesian model selection in social research. In P. V. Marsden (Ed.), Sociological methodology (Vol. 25, pp. 111–63). Boston: Blackwell.
Raichle, M. E., MacLeod, A. M., Snyder, A. Z., Powers, W. J., Gusnard, D. A., & Shulman, G. L. (2001). A default mode of brain function. Proceedings of the National Academy of Sciences, 98, 676–682.
Reichle, E. D., Carpenter, P. A., & Just, M. A. (2000). The neural bases of strategy and skill in sentence-picture verification. Cognitive Psychology, 40, 261–295.
Rogers, W. A., Hertzog, C., & Fisk, A. D. (2000). An individual differences analysis of ability and strategy influences: Age- related differences in associative learning. Journal of Experimental Psychology: Learning, Memory, & Cognition, 26, 359–394. doi:10.1037/0278-7393.26.2.359
Siegler, R. S. (1987). The perils of averaging data over strategies: An example from children’s addition. Journal of Experimental Psychology: General, 116, 250–264.
Siegler, R. S., & Alibali, M. W. (2005). Children’s thinking (4th ed.). Upper Saddle River, NJ: Prentice Hall.
Siegler, R. S., & Lemaire, P. (1997). Older and younger adults’ strategy choices in multiplication: Testing predictions of ASCM using the choice/no-choice method. Journal of Experimental Psychology: General, 126, 22.
Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), The origins of cognitive skills (pp. 229–293). Hillsdale, NJ: Erlbaum.
Simon, O., Mangin, J. F., Cohen, L., Le Bihan, D., & Dehaene, S. (2002). Topographical layout of hand, eye, calculation, and languagerelated areas in the human parietal lobe. Neuron, 33, 475–487.
Sohn, M.-H., Goode, A., Koedinger, K. R., Stenger, V. A., Fissell, K., Carter, C. S., & Anderson, J. R. (2004). Behavioral equivalence, but not neural equivalence—Neural evidence of alternative strategies in mathematical thinking. Nature Neuroscience, 7, 1193–1194. doi:10.1038/nn1337
Sohn, M.-H., Goode, A., Stenger, V. A., Jung, K.-J., Carter, C., & Anderson, J. R. (2005). An information-processing model of three cortical regions: Evidence in episodic memory retrieval. NeuroImage, 25, 21–33. doi:10.1016/j.neuroimage.2004.11.001
Staszewski, J. J. (1988). Skilled memory and expert mental calculation. In M. T. H. Chi, R. Glaser, & M. J. Farr (Eds.), The nature of expertise (pp. 71–128). Hillsdale, NJ: Erlbaum.
van Harskamp, N. J., Rudge, P., & Cipolotti, L. (2002). Are multiplication facts implemented by the left supramarginal and angular gyri? Neuropsychologia, 40, 1786–1793.
Wagenmakers, E.-J. (2007). A practical solution to the pervasive problems of p values. Psychonomic Bulletin & Review, 14, 779–804.
Warrington, E. K. (1982). The fractionation of arithmetical skills—A single case-study. Quarterly Journal of Experimental Psy chology: Human Experimental Psychology, 34A, 31–51. doi:10.1080/14640748208400856
Whalen, J., McCloskey, M., Lesser, R. P., & Gordon, B. (1997). Localizing arithmetic processes in the brain: Evidence from a transient deficit during cortical stimulation. Journal of Cognitive Neuroscience, 9, 409–417.
Wojciulik, E., & Kanwisher, N. (1999). The generality of parietal involvement in visual attention. Neuron, 23, 747–764.
Woods, R. P., Cherry, S. R., & Mazziotta, J. C. (1992). Rapid automated algorithm for aligning and reslicing PET images. Journal of Computer Assisted Tomography, 16, 620–633.
Woods, R. P., Grafton, S. T., Holmes, C. J., Cherry, S. R., & Mazziotta, J. C. (1998). Automated image registration: I. General methods of intrasubject, intramodality variation. Journal of Computer Assisted Tomography, 22, 139–152.
Author information
Authors and Affiliations
Corresponding author
Additional information
The present work was supported by Grant REC-0087396 from NSF to J.R.A., and a National Science and Engineering Research Council postgraduate scholarship to M.R.-L.
Rights and permissions
About this article
Cite this article
Rosenberg-Lee, M., Lovett, M.C. & Anderson, J.R. Neural correlates of arithmetic calculation strategies. Cognitive, Affective, & Behavioral Neuroscience 9, 270–285 (2009). https://doi.org/10.3758/CABN.9.3.270
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.3758/CABN.9.3.270