Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-03T08:42:52.219Z Has data issue: false hasContentIssue false

Non-abstract numerical representations in the IPS: Further support, challenges, and clarifications

Published online by Cambridge University Press:  27 August 2009

Roi Cohen Kadosh
Affiliation:
Institute of Cognitive Neuroscience and Department of Psychology, University College London, 17 Queen Square, London WC1N 3AR, United Kingdom. [email protected]://www.ucl.ac.uk/neuroscience/Page.php?ID=12&[email protected]://www.icn.ucl.ac.uk/Research-Groups/Visual-Cognition-Group/index.php
Vincent Walsh
Affiliation:
Institute of Cognitive Neuroscience and Department of Psychology, University College London, 17 Queen Square, London WC1N 3AR, United Kingdom. [email protected]://www.ucl.ac.uk/neuroscience/Page.php?ID=12&[email protected]://www.icn.ucl.ac.uk/Research-Groups/Visual-Cognition-Group/index.php

Abstract

The commentators have raised many pertinent points that allow us to refine and clarify our view. We classify our response comments into seven sections: automaticity; developmental and educational questions; priming; multiple representations or multiple access(?); terminology; methodological advances; and simulated cognition and numerical cognition. We conclude that the default numerical representations are not abstract.

Type
Authors' Response
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ainsworth, S., Bibby, P. & Wood, D. (2002) Examining the effects of different multiple representational systems in learning primary mathematics. Journal of Learning Sciences 11:2561.CrossRefGoogle Scholar
Algom, D., Dekel, A. & Pansky, A. (1996) The perception of number from the separability of the stimulus: The Stroop effect revisited. Memory and Cognition 24:557–72.CrossRefGoogle ScholarPubMed
Ansari, D., Dhital, B. & Siong, S. C. (2006a) Parametric effects of numerical distance on the intraparietal sulcus during passive viewing of rapid numerosity changes. Brain Research 1067:181–88.CrossRefGoogle ScholarPubMed
Barth, H., Kanwisher, N. & Spelke, E. (2003) The construction of large number representations in adults. Cognition 86:201–21.CrossRefGoogle ScholarPubMed
Blankenberger, S. & Vorberg, D. (1997) The single-format assumption in arithmetic fact retrieval. Journal of Experimental Psychology: Learning, Memory and Cognition 23:721–38.Google Scholar
Booth, J. L. & Siegler, R. S. (2008) Numerical magnitude representations influence arithmetic learning. Child Development 79:1016–31.CrossRefGoogle ScholarPubMed
Campbell, J. I. D. (1994) Architectures for numerical cognition. Cognition 53(1):144.CrossRefGoogle ScholarPubMed
Campbell, J. I. D. & Clark, J. M. (1988) An encoding complex view of cognitive number processing: Comment on McCloskey, Sokol, & Goodman (1986). Journal of Experimental Psychology: General 117:204–14.CrossRefGoogle Scholar
Campbell, J. I. D. & Epp, L. J. (2004) An encoding-complex approach to numerical cognition in Chinese-English bilinguals. Canadian Journal of Experimental Psychology (Revue Canadienne De Psychologie Experimentale) 58(4):229–44.CrossRefGoogle ScholarPubMed
Campbell, J. I. D. & Epp, L. J. (2005) Architectures for arithmetic. In: The handbook of mathematical cognition, ed. Campbell, J. I. D., pp. 347–60. Psychology Press.CrossRefGoogle Scholar
Cantlon, J. F., Brannon, E. M., Carter, E. J. & Pelphrey, K. A. (2006) Functional imaging of numerical processing in adults and 4-y-old children. PLoS Biology 4:e125.CrossRefGoogle ScholarPubMed
Cantlon, J. F., Libertus, M. E., Pinel, P., Dehaene, S., Brannon, E. M. & Pelphrey, K. A., (in press) The neural development of an abstract concept of number. Journal of Cognitive Neuroscience. [Epub ahead of print]Google Scholar
Cantlon, J. F., Platt, M. L. & Brannon, E. M. (2009) Beyond the number domain. Trends in Cognitive Sciences 13:8391.CrossRefGoogle ScholarPubMed
Carey, S. (2004) Bootstrapping and the origin of concepts. Daedalus 133(1):5968.CrossRefGoogle Scholar
Cohen, D. J. (2009) Integers do not automatically activate their quantity representations. Psychonomic Bulletin and Review 16:332–36.CrossRefGoogle ScholarPubMed
Cohen, Kadosh K., & Johnson, M. H. (2007) Developing a cortex specialized for face perception. Trends in Cognitive Sciences 11:367–69.CrossRefGoogle Scholar
Cohen, Kadosh R., (2008a) Numerical representation: Abstract or non-abstract? Quarterly Journal of Experimental Psychology 61(8):1160–68.Google Scholar
Cohen, Kadosh R., Bahrami, B., Walsh, V., Butterworth, B., & Price, C. J. (2008a) Numerical specialisation: Within and between dimensions. Poster presented at the 14th annual meeting of the Organization for Human Brain Mapping, Melbourne, Australia, June 15–19, 2008.Google Scholar
Cohen, Kadosh R., Brodsky, W., Levin, M., & Henik, A. (2008b) Mental representation: What can pitch tell us about the distance effect? Cortex 44:470–77.Google Scholar
Cohen, Kadosh R., Cohen, Kadosh K., Henik, A., & Linden, D. E. J. (2008d) Processing conflicting information: Facilitation, interference, and functional connectivity. Neuropsychologia 46:2872–79.CrossRefGoogle Scholar
Cohen, Kadosh R., Cohen, Kadosh K., Kaas, A., Henik, A., & Goebel, R. (2007b) Notation-dependent and -independent representations of numbers in the parietal lobes. Neuron 53(2):307–14.CrossRefGoogle Scholar
Cohen, Kadosh R., Cohen, Kadosh K., Linden, D. E. J., Gevers, W., Berger, A. & Henik, A. (2007c) The brain locus of interaction between number and size: A combined functional magnetic resonance imaging and event-related potential study. Journal of Cognitive Neuroscience 19:957–70.Google Scholar
Cohen, Kadosh R., Gevers, W., & Notebaert, W. (submitted a) Behavioural evidence for separate representations of different magnitudesGoogle Scholar
Cohen, Kadosh R., Henik, A., & Rubinsten, O. (2007d) The effect of orientation on number word processing. Acta Psychologica 124:370–81.Google Scholar
Cohen, Kadosh R., Henik, A., & Rubinsten, O. (2008e) Are Arabic and verbal numbers processed in different ways? Journal of Experimental Psychology: Learning, Memory and Cognition 34(6):1377–91.Google Scholar
Cohen, Kadosh R., Henik, A., Rubinsten, O., Mohr, H., Dori, H., Van de Ven, V., Zorzi, M., Hendler, T., Goebel, R. & Linden, D. E. J. (2005) Are numbers special? The comparison systems of the human brain investigated by fMRI. Neuropsychologia 43:1238–48.Google Scholar
Cohen, Kadosh R., Henik, A., & Walsh, V. (2009) Synaesthesia: Learned or lost? Developmental Science 12:484–91.Google Scholar
Cohen, Kadosh R., Lammertyn, J., & Izard, V. (2008f) Are numbers special? An overview of chronometric, neuroimaging, developmental and comparative studies of magnitude representation. Progress in Neurobiology 84:132–47.Google Scholar
Dehaene, S. (1996) The organization of brain activations in number comparison: Event-related potentials and the additive-factors method. Journal of Cognitive Neuroscience 8:4768.CrossRefGoogle ScholarPubMed
Dehaene, S. & Akhavein, R. (1995) Attention, automaticity, and levels of representation in number processing. Journal of Experimental Psychology: Learning, Memory, and Cognition 21(2):314–26.Google ScholarPubMed
Dehaene, S. & Cohen, L. (2007) Cultural recycling of cortical maps. Neuron 56(2):384–98.CrossRefGoogle ScholarPubMed
Dehaene, S., Dehaene-Lambertz, G. & Cohen, L. (1998a) Abstract representations of numbers in the animal and human brain. Trends in Neurosciences 21:355–61.CrossRefGoogle ScholarPubMed
Dehaene, S., Izard, V., Spelke, E. & Pica, P. (2008) Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science 320(5880):1217–20.CrossRefGoogle ScholarPubMed
Dehaene, S., Molko, N., Cohen, L. & Wilson, A. J. (2004) Arithmetic and the brain. Current Opinion in Neurobiology 14:218–24.CrossRefGoogle ScholarPubMed
Dehaene, S., Naccache, L., Le Clec'H, G., Koechlin, E., Mueller, M., Dehaene-Lambertz, G., , Van, deMoortele, P. F. & Le Bihan, D., (1998b) Imaging unconscious semantic priming. Nature 395:597600.CrossRefGoogle ScholarPubMed
Dehaene, S., Piazza, M., Pinel, P. & Cohen, L. (2003) Three parietal circuits for number processing. Cognitive Neuropsychology 20 (3–6):487506.CrossRefGoogle ScholarPubMed
Dehaene, S., Spelke, E. S., Pinel, P., Stanescu, R. & Tsivkin, S. (1999) Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science 284(5416):970–74.CrossRefGoogle ScholarPubMed
Diester, I. & Nieder, A. (2007) Semantic associations between signs and numerical categories in the prefrontal cortex. PLoS Biology 5 (11):e294; 2684–95.CrossRefGoogle ScholarPubMed
Droit-Volet, S., Clement, A. & Fayol, M. (2008) Time, number and length: Similarities and differences in discrimination in adults and children. Quarterly Journal of Experimental Psychology 61(12):1827–46.CrossRefGoogle ScholarPubMed
Dulany, D. E. (1996) Consciousness in the explicit (deliberate) and the implicit (evocative). In: Scientific approaches to consciousness, ed. Cohen, J. & Schooler, J., pp. 179212. Erlbaum.Google Scholar
Eger, E., Michel, V., Thirion, B., Amadon, A., Dehaene, S. & Kleinschmidt, A. (submitted) Decoding of individual number information from spatial activation patterns in human intraparietal cortex.Google Scholar
Eger, E., Sterzer, P., Russ, M. O., Giraud, A.-L. & Kleinschmidt, A. (2003) A supramodal number representation in human intraparietal cortex. Neuron 37:719–25.CrossRefGoogle ScholarPubMed
Ganor-Stern, D. & Tzelgov, J. (2008) Across-notation automatic numerical processing. Journal of Experimental Psychology: Learning, Memory and Cognition 34(2):430–37.Google ScholarPubMed
Gebuis, T., Cohen, Kadosh R., de Haan, E., & Henik, A. (2009) Automatic quantity processing in 5-year-olds and adults. Cognitive Processing 10:133–42.CrossRefGoogle ScholarPubMed
Girelli, L., Lucangeli, D. & Butterworth, B. (2000) The development of automaticity in accessing number magnitude. Journal of Experimental Child Psychology 76(2): 104–22.CrossRefGoogle ScholarPubMed
Henik, A. & Tzelgov, J. (1982) Is three greater than five: The relation between physical and semantic size in comparison tasks. Memory and Cognition 10:389–95.CrossRefGoogle ScholarPubMed
Holloway, I. D. & Ansari, D. (2009) Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children's mathematics achievement. Journal of Experimental Child Psychology 103(1):1729.CrossRefGoogle ScholarPubMed
Hung, Y.-H., Hung, D. L., Tzeng, O. J.-L. & Wu, D. H. (2008) Flexible spatial mapping of different notations of numbers in Chinese readers. Cognition 106(3):1441–50.CrossRefGoogle ScholarPubMed
Hurewitz, F., Gelman, R. & Schnitzer, B. (2006) Sometimes area counts more than number. Proceedings of the National Academy of Science USA 103:19599–604.CrossRefGoogle ScholarPubMed
Ito, Y. & Hatta, T. (2003) Semantic processing of Arabic, Kanji, and Kana numbers: Evidence from interference in physical and numerical size judgments. Memory and Cognition 31(3):360–68.CrossRefGoogle ScholarPubMed
Izard, V. & Dehaene, S. (2008) Calibrating the mental number line. Cognition 106:1221–47.CrossRefGoogle ScholarPubMed
Johnson, M. H. (2001) Functional brain development in humans. Nature Reviews Neuroscience 2(7):475–83.CrossRefGoogle ScholarPubMed
Johnson, M. H., Grossman, T. & Cohen, Kadosh K., (2009) Mapping functional brain development: Building a social brain through interactive specialization. Developmental Psychology 45:151–59.CrossRefGoogle ScholarPubMed
Kiesel, A., Kunde, W & Hoffman, J. (2007) Unconscious priming according to multiple S-R rules. Cognition 104:89105.CrossRefGoogle ScholarPubMed
Knops, A., Thirion, B., Hubbard, E. M., Michel, V., & Dehaene, S. (2009) Recruitment of an area involved in eye movements during mental arithmetic. Science 324:1583–85.CrossRefGoogle ScholarPubMed
Knops, A., Viarouge, A. & Dehaene, S. (2009) Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect. Attention, Perception, and Psychophysics 71:803–21.CrossRefGoogle ScholarPubMed
Kozma, R., Chin, C., Russel, J. & Marx, N. (2000) The roles of representations and tools in the chemistry laboratory and their implications for chemistry learning. Journal of the Learning Sciences 9:105–43.CrossRefGoogle Scholar
Kunde, W., Kiesel, A. & Hoffmann, J. (2003) Conscious control over the content of unconscious cognition. Cognition 88:223–42.CrossRefGoogle ScholarPubMed
Kunde, W., Kiesel, A. & Hoffmann, J. (2005) On the masking and disclosure of unconscious elaborate processing. A reply to Van Opstal, Reynvoet, and Verguts (2005). Cognition 97:99105.CrossRefGoogle Scholar
Lakoff, G. (2008) The neural theory of metaphor. In: The Cambridge handbook of metaphor and thought, ed. Gibbs, R. W. Jr., Cambridge University Press.Google Scholar
Landauer, T. K. & Dumais, S. T. (1997) A solution to Plato's problem: The latent semantic analysis theory of acquisition, induction, and representation of knowledge. Psychological Review 104:211–40.CrossRefGoogle Scholar
Libertus, M. E., Woldorff, M. G. & Brannon, E. M. (2007) Electrophysiological evidence for notation independence in numerical processing. Behavioral Brain Function 3(1). (Online journal).CrossRefGoogle ScholarPubMed
Link, S. (1990) Modeling imageless thought: The relative judgment theory of numerical comparisons. Journal of Mathematical Psychology 34:241.CrossRefGoogle Scholar
Lyons, I. M. & Ansari, D. (2009) The cerebral basis of mapping non-symbolic numerical quantities onto abstract symbols: An fMRI training study. Journal of Cognitive Neuroscience 21:1720–35.CrossRefGoogle Scholar
Naccache, L. & Dehaene, S. (2001a) The priming method: Imaging unconscious repetition priming reveals an abstract representation of number in the parietal lobes. Cerebral Cortex 11(10):966–74.CrossRefGoogle ScholarPubMed
Noël, M.-P. & Seron, X. (1993) Arabic number reading deficit: A single case study. Cognitive Neuropsychology 10:317–39.Google Scholar
Nuerk, H.-C., Weger, U. & Willmes, K. (2002) A unit-decade compatibility effect in German number words. Current Psychology Letters: Behaviour, Brain, and Cognition 2:1938.Google Scholar
Pansky, A. & Algom, D. (1999) Stroop and Garner effects in comparative judgments of numerals: The role of attention. Journal of Experimental Psychology: Human Perception and Performance 25:3958.Google Scholar
Pansky, A. & Algom, D. (2002) Comparative judgment of numerosity and numerical magnitude: Attention preempts automaticity. Journal of Experimental Psychology: Learning, Memory, and Cognition 28:259–74.Google ScholarPubMed
Piazza, M. & Dehaene, S. (2004) From number neurons to mental arithmetic: The cognitive neuroscience of number sense. In: The cognitive neurosciences, 3rd edition, ed. Gazzaniga, M. S., pp. 865–77. The MIT Press.Google Scholar
Piazza, M., Izard, V., Pinel, P., Le Bihan, D., & Dehaene, S. (2004) Tuning curves for approximate numerosity in the human intraparietal sulcus. Neuron 44(3):547–55.CrossRefGoogle ScholarPubMed
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(2):293305.CrossRefGoogle ScholarPubMed
Pica, P., Lemer, C., Izard, V. & Dehaene, D. (2004) Exact and approximate arithmetic in an Amazonian indigene group. Science 306:496–99.CrossRefGoogle Scholar
Pinel, P., Dehaene, S., Rivière, D. & Le Bihan, D., (2001) Modulation of parietal activation by semantic distance in a number comparison task. NeuroImage 14:1013–26.CrossRefGoogle Scholar
Pinel, P., Piazza, M., Le Bihan, D., & Dehaene, S. (2004) Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments. Neuron 41(6):983–93.CrossRefGoogle ScholarPubMed
Roggeman, C., Verguts, T. & Fias, W. (2007) Priming reveals differential coding of symbolic and non-symbolic quantities. Cognition 105(2):380–94.CrossRefGoogle ScholarPubMed
Rourke, B. P. (1993) Arithmetic disabilities, specific and otherwise: A neuropsychological perspective. Journal of Learning Disabilities 26:214–26.CrossRefGoogle ScholarPubMed
Rousselle, L., Palmers, E. & Noël, M. P. (2004) Magnitude comparison in preschoolers: What counts? Influence of perceptual variables. Journal of Experimental Child Psychology 87:5784.CrossRefGoogle ScholarPubMed
Rubinsten, O. & Henik, A. (2002) Is an ant larger than a lion? Acta Psychologica 111:141154.CrossRefGoogle ScholarPubMed
Rubinsten, O. & Henik, A. (2009) Developmental dyscalculia: Heterogeneity may not mean different mechanisms. Trends in Cognitive Sciences 13:9299.CrossRefGoogle Scholar
Rubinsten, O., Henik, A., Berger, A. & Shahar-Shalev, S. (2002) The development of internal representations of magnitude and their association with Arabic numerals. Journal of Experimental Child Psychology 81:7492.CrossRefGoogle ScholarPubMed
Sanders, A. F. (1998) Elements of human performance: Reaction processes and attention in human skill. Erlbaum.Google Scholar
Santens, S., Roggeman, C., Fias, W., & Verguts, T. (in press) Number processing pathways in human parietal cortex. Cerebral Cortex. doi:10.1093/cercor/bhp080.Google Scholar
Sawamura, H., Orban, G. A. & Vogels, R. (2006) Selectivity of neuronal adaptation does not match response selectivity: A single-cell study of the fMRI adaptation paradigm. Neuron 49(2):307–18.CrossRefGoogle Scholar
Schwarz, W. & Heinze, H. J. (1998) On the interaction of numerical and size information in digit comparison: A behavioral and event-related potential study. Neuropsychologia 36:1167–79.CrossRefGoogle ScholarPubMed
Schwarz, W. & Ischebeck, A. (2000) Sequential effects in number comparison. Journal of Experimental Psychology: Human Perception and Performance 26:1606–21.Google ScholarPubMed
Schwarz, W. & Ischebeck, A. (2003) On the relative speed account of the number-size interference in comparative judgment of numerals. Journal of Experimental Psychology: Human Perception and Performance 29:507–22.Google ScholarPubMed
Sirotin, Y. B. & Das, A. (2009) Anticipatory haemodynamic signals in sensory cortex not predicted by local neuronal activity. Nature 457(7228):475–79.CrossRefGoogle Scholar
Striedter, G. F. (2005) Principles of brain evolution. Sinauer.Google Scholar
Szűcs, D. & Soltész, F. (2007) Event-related potentials dissociate facilitation and interference effects in the numerical Stroop paradigm. Neuropsychologia 45:3190–202.CrossRefGoogle ScholarPubMed
Szűcs, D., Soltész, F., Jármi, É. & Csépe, V. (2007) The speed of magnitude processing and executive functions in controlled and automatic number comparison in children: An electro-encephalography study. Behavioral and Brain Functions 3(23). (Online journal).CrossRefGoogle ScholarPubMed
Szűcs, D., Soltész, F., & White, S. (in press) Motor conflict in Stroop tasks: Direct evidence from single-trial electro-myography and electro-encephalography. Neuroimage.Google Scholar
Tabachneck, H. J. M., Leonardo, A. M. & Simon, H. A. (1994) How does an expert use a graph? A model of visual & verbal inferencing in economics. In: 16th Annual Conference of the Cognitive Science Society, ed. Ram, A. & Eiselt, K., pp. 842–47 Erlbaum.Google Scholar
Tang, Y., Zhang, W., Chen, K., Feng, S., Ji, Y., Shen, J. & Reiman, E. M. (2006b) Arithmetic processing in the brain shaped by cultures. Proceedings of the National Academy of Sciences USA 103:10775–80.CrossRefGoogle ScholarPubMed
Tsujimoto, S. (2008) The prefrontal cortex: Functional neural development during early childhood. The Neuroscientist 14:345–58.CrossRefGoogle ScholarPubMed
Tudusciuc, O. & Nieder, A. (2007) Neuronal population coding of continuous and discrete quantity in the primate posterior parietal cortex. Proceedings of the National Academy of Sciences USA 104(36):14513–18.CrossRefGoogle ScholarPubMed
Tzelgov, J. & Ganor-Stern, D. (2005) Automaticity in processing ordinal information. In: Handbook of mathematical cognition, ed. Campbell, J. I. D., pp. 5566. Psychology Press.Google Scholar
Tzelgov, J., Meyer, J. & Henik, A. (1992) Automatic and intentional processing of numerical information. Journal of Experimental Psychology: Learning, Memory and Cognition 18:166–79.Google Scholar
Tzelgov, J., Yehene, V., Kotler, L. & Alon, A. (2000) Automatic comparisons of artificial digits never compared: Learning linear ordering relations. Journal of Experimental Psychology: Learning, Memory and Cognition 26:103–20.Google ScholarPubMed
Van Opstal, F., Gevers, W., , De Moor, W., & Verguts, T. (2008a) Dissecting the symbolic distance effect: Comparison and priming effects in numerical and non-numerical orders. Psychonomic Bulletin and Review 15:419–25.CrossRefGoogle Scholar
Van Opstal, F., Verguts, T., Orban, G. A. & Fias, W. (2008b) A hippocampal-parietal network for learning an ordered sequence. NeuroImage 40:333441.CrossRefGoogle Scholar
Verguts, T. & Fias, W. (2004) Representation of number in animals and humans: A neural model. Journal of Cognitive Neuroscience 16(9):1493–504.CrossRefGoogle ScholarPubMed
Walsh, V. (2003) A theory of magnitude: Common cortical metrics of time, space and quantity. Trends in Cognitive Sciences 7(11):483–88.CrossRefGoogle ScholarPubMed
Wood, G., Ischebeck, A., Koppelstaetter, F., Gotwald, T. & Kaufmann, L., (in press) Developmental trajectories of magnitude processing and interference control: An fMRI study. Cerebral Cortex.Google Scholar