Abstract
When we add or subtract, do the corresponding quantities “move” along a mental number line? Does this internal movement lead to spatial biases? A new method was designed to investigate the psychophysics of approximate arithmetic. Addition and subtraction problems were presented either with sets of dots or with Arabic numerals, and subjects selected, from among seven choices, the most plausible result. In two experiments, the subjects selected larger numbers for addition than for subtraction problems, as if moving too far along the number line. This operational momentum effect was present in both notations and increased with the size of the outcome. Furthermore, we observed a new effect of spatial-numerical congruence, related to but distinct from the spatial numerical association of response codes effect: During nonsymbolic addition, the subjects preferentially selected numbers at the upper right location, whereas during subtraction, they were biased toward the upper left location. These findings suggest that approximate mental arithmetic involves dynamic shifts on a spatially organized mental representation of numbers. Supplemental materials for this study may be downloaded from app.psychonomic-journals.org/content/supplemental.
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Barth, H., Kanwisher, N., & Spelke, E. (2003). The construction of large number representations in adults. Cognition, 86, 201–221.
Barth, H., La Mont, K., Lipton, J., Dehaene, S., Kanwisher, N., & Spelke, E. (2006). Nonsymbolic arithmetic in adults and young children. Cognition, 98, 199–222.
Beran, M. J. (2007). Rhesus monkeys (Macaca mulatta) enumerate large and small sequentially presented sets of items using analog numerical representations. Journal of Experimental Psychology: Animal Behavior Processes, 33, 42–54.
Brannon, E. M., & Terrace, H. S. (1998). Ordering of the numerosities 1 to 9 by monkeys. Science, 282, 746–749.
Brannon, E. M., & Terrace, H. S. (2000). Representation of the numerosities 1-9 by rhesus monkeys (Macaca mulatta). Journal of Experimental Psychology: Animal Behavior Processes, 26, 31–49.
Cantlon, J. F., & Brannon, E. M. (2006). Shared system for ordering small and large numbers in monkeys and humans. Psychological Science, 17, 401–406.
Cordes, S., Gallistel, C. R., Gelman, R., & Latham, P. (2007). Nonverbal arithmetic in humans: Light from noise. Perception & Psychophysics, 69, 1185–1203.
Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44, 1–42.
Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press.
Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and numerical magnitude. Journal of Experimental Psychology: General, 122, 371–396.
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.
Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284, 970–974.
Fischer, M. H., Warlop, N., Hill, R. L., & Fias, W. (2004). Oculomotor bias induced by number perception. Experimental Psychology, 51, 91–97.
Freyd, J. (1993). Five hunches about perceptual processes and dynamic representations. In D. Meyer & S. Kornblum (Eds.), Attention and performance XIV: Synergies in experimental psychology, artificial intelligence, and cognitive neuroscience (pp. 99–119). Cambridge, MA: MIT Press.
Freyd, J., & Finke, R. A. (1984). Representational momentum. Journal of Experimental Psychology: Learning, Memory, & Cognition, 10, 126–132.
Freyd, J., & Finke, R. A. (1985). A velocity effect for representational momentum. Bulletin of the Psychonomic Society, 26, 443–446.
Freyd, J., & Pantzer, T. M. (1995). Static patterns moving in the mind. In S. M. Smith, T. B. Ward, & R. A. Finke (Eds.), The creative cognition approach (pp. 181–204). Cambridge, MA: MIT Press.
Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44, 43–74.
Gilmore, C. K., McCarthy, S. E., & Spelke, E. S. (2007). Symbolic arithmetic knowledge without instruction. Nature, 447, 589–591.
Halpern, A. R., & Kelly, M. H. (1993). Memory biases in left versus right implied motion. Journal of Experimental Psychology: Learning, Memory, & Cognition, 19, 471–484.
Hooge, I. T., & Erkelens, C. J. (1998). Adjustment of fixation duration in visual search. Vision Research, 38, 1295–1302.
Hubbard, E. M., Piazza, M., Pinel, P., & Dehaene, S. (2005). Interactions between number and space in parietal cortex. Nature Reviews Neuroscience, 6, 435–448.
Hubbard, T. L. (1995). Cognitive representation of motion: Evidence for friction and gravity analogues. Journal of Experimental Psychology: Learning, Memory, & Cognition, 21, 241–254.
Hubbard, T. L. (2005). Representational momentum and related displacements in spatial memory: A review of the findings. Psychonomic Bulletin & Review, 12, 822–851.
Hubbard, T. L., & Ruppel, S. E. (2000). Spatial memory averaging, the landmark attraction effect, and representational gravity. Psychological Research, 64, 41–55.
Huynh, H., & Feldt, L. S. (1976). Estimation of the box correction for degrees of freedom from sample data in randomized block and splitplot designs. Journal of Educational Statistics, 1, 69–82.
Izard, V., & Dehaene, S. (2008). Calibrating the mental number line. Cognition, 106, 1221–1247.
Jacobs, A. M. (1986). Eye-movement control in visual search: How direct is visual span control? Perception & Psychophysics, 39, 47–58.
Kerzel, D. (2003). Attention maintains mental extrapolation of target position: Irrelevant distractors eliminate forward displacement after implied motion. Cognition, 88, 109–131.
Krueger, L. E. (1984). Perceived numerosity: A comparison of magnitude production, magnitude estimation, and discrimination judgments. Perception & Psychophysics, 35, 536–542.
McCrink, K., Dehaene, S., & Dehaene-Lambertz, G. (2007). Moving along the number line: Operational momentum in nonsymbolic arithmetic. Perception & Psychophysics, 69, 1324–1333.
McCrink, K., & Wynn, K. (2004). Large-number addition and subtraction by 9-month-old infants. Psychological Science, 15, 776–781.
Moyer, R. S., & Landauer, T. K. (1967). Time required for judgments of numerical inequality. Nature, 215, 1519–1520.
Nieder, A., Freedman, D. J., & Miller, E. K. (2002). Representation of the quantity of visual items in the primate prefrontal cortex. Science, 297, 1708–1711.
Nieder, A., & Miller, E. K. (2004). A parieto-frontal network for visual numerical information in the monkey. Proceedings of the National Academy of Sciences, 101, 7457–7462.
Oldfield, R. C. (1971). The assessment and analysis of handedness: The Edinburgh Inventory. Neuropsychologia, 9, 97–113.
Piazza, M., Izard, V., Pinel, P., Le Bihan, D., & Dehaene, S. (2004). Tuning curves for approximate numerosity in the human intraparietal sulcus. Neuron, 44, 547–555.
Piazza, M., Pinel, P., Le Bihan, D., & Dehaene, S. (2007). A magnitude code common to numerosities and number symbols in human intraparietal cortex. Neuron, 53, 293–305.
Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2004). Exact and approximate arithmetic in an Amazonian indigene group. Science, 306, 499–503.
Pinhas, M., & Fischer, M. H. (2008). Mental movements without magnitude? A study of spatial biases in symbolic arithmetic. Cognition, 109, 408–415.
Restle, F. (1970). Speed of adding and comparing numbers. Journal of Experimental Psychology, 91, 191–205.
Sheliga, B. M., Riggio, L., & Rizzolatti, G. (1994). Orienting of attention and eye movements. Experimental Brain Research, 98, 507–522.
Shepard, R. N., Kilpatrick, D. W., & Cunningham, J. P. (1975). The internal representation of numbers. Cognitive Psychology, 7, 82–183.
Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185, 1124–1131.
Verguts, T., & Fias, W. (2004). Representation of number in animals and humans: A neural model. Journal of Cognitive Neuroscience, 16, 1493–1504.
Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-monthold infants. Cognition, 74, B1-B11.
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Knops, A., Viarouge, A. & Dehaene, S. Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect. Attention, Perception, & Psychophysics 71, 803–821 (2009). https://doi.org/10.3758/APP.71.4.803
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DOI: https://doi.org/10.3758/APP.71.4.803