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Review Article

Extending the Mental Number Line

A Review of Multi-Digit Number Processing

Published Online:https://doi.org/10.1027/2151-2604/a000041

Multi-digit number processing is ubiquitous in our everyday life – even in school, multi-digit numbers are computed from the first year onward. Yet, many problems children and adults have are about the relation of different digits (for instance with fractions, decimals, or carry effects in multi-digit addition). Cognitive research has mainly focused on single-digit processing, and there is no comprehensive review of the different multi-digit number processing types and effects. The current review aims to fill this gap. First, we argue that effects observed in single-digit tasks cannot simply be transferred to multi-digit processing. Next, we list 16 effect types and processes which are specific for multi-digit number processing. We then discuss the development of multi-digit number processing, its neurocognitive correlates, its cultural or language-related modulation, and finally some models for multi-digit number processing. We finish with conclusions and perspectives about where multi-digit number processing research may or should be heading in following years.

References

  • Ashcraft, M. H. , Stazyk, E. H. (1981). Mental addition: A test of three verification models. Memory & Cognition, 9, 185–196. First citation in articleCrossrefGoogle Scholar

  • Ashkenazi, S. , Mark-Zigdon, N. , Henik, A. (2009). Numerical distance effect in developmental dyscalculia. Cognitive Development, 24, 387–400. First citation in articleCrossrefGoogle Scholar

  • Barbey, A. K. , Sloman, S. A. (2007). Base-rate respect: From ecological rationality to dual processes. Behavioral and Brain Sciences, 30, 241–254. First citation in articleGoogle Scholar

  • Barrouillet, P. , Camos, V. , Perruchet, P. , Seron, X. (2004). ADAPT: A developmental, asemantic, and procedural model for transcoding from verbal to Arabic numerals. Psychological Review, 111, 368–394. First citation in articleCrossrefGoogle Scholar

  • Barrouillet, P. , Thevenot, C. , Fayol, M. (2010). Evidence for knowledge of the syntax of large numbers in preschoolers. Journal of Experimental Child Psychology, 105, 264–271. First citation in articleCrossrefGoogle Scholar

  • Behr, M. J. , Herel, G. , Post, T. , Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws, (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). New York, NY: Macmillan. First citation in articleGoogle Scholar

  • Blanken, G. , Dorn, M. , Sinn, H. (1997). Inversion errors in Arabic number reading: Is there a nonsemantic route?. Brain & Cognition, 34, 404–423. First citation in articleCrossrefGoogle Scholar

  • Bonato, M. , Fabbri, S. , Umiltà, C. , Zorzi, M. (2007). The mental representation of numerical fractions: Real or integer? Journal of Experimental Psychology: Human Perception and Performance, 33, 1410–1419. First citation in articleCrossrefGoogle Scholar

  • Brysbaert, M. (1995). Arabic number reading: On the nature of the numerical scale and the origin of phonological recoding. Journal of Experimental Psychology: General, 124, 434–452. First citation in articleCrossrefGoogle Scholar

  • Brysbaert, M. , Fias, W. , Noël, M. P. (1998). The Whorfian hypothesis and numerical cognition: Is “twenty-four” processed in the same way as “four-and-twenty”? Cognition, 66, 51–77. First citation in articleGoogle Scholar

  • Campbell, J. I. D. (1994). Architectures for numerical cognition. Cognition, 53, 1–44. First citation in articleCrossrefGoogle Scholar

  • Campbell, J. I. D. (1997). On the relation between skilled performance of simple division and multiplication. Journal of Experimental Psychology: Learning, Memory, and Cognition, 23, 1140–1159. First citation in articleCrossrefGoogle Scholar

  • Campbell, J. I. D. (2005). Handbook of mathematical cognition. New York, NY: Psychology Press. First citation in articleGoogle Scholar

  • Chen, Q. , Verguts, T. (2010, January). Spatial intuition in elementary arithmetic: A neurocomputational account of the operational momentum effect. Poster presented at the 28th European Workshop of Cognitive Neuropsychology Bressanone, Italy. First citation in articleGoogle Scholar

  • Cipolotti, L. , Butterworth, B. (1995). Towards a multiroute model of number processing: Impaired number transcoding with the preservation of calculation skills. Journal of Experimental Psychology: General, 124, 375–390. First citation in articleCrossrefGoogle Scholar

  • Colome, A. , Laka, I. , Sebastiaen-Galles, N. (2010). Language effects in addition: How you say it counts. The Quarterly Journal of Experimental Psychology, 63, 965–983. First citation in articleCrossrefGoogle Scholar

  • Dehaene, S. (1989). The psychophysics of numerical comparison: A re-examination of apparently incompatible data. Perception and Psychophysics, 45, 557–566. First citation in articleCrossrefGoogle Scholar

  • Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44, 1–42. First citation in articleCrossrefGoogle Scholar

  • Dehaene, S. , Bossini, S. , & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122, 371–396. First citation in articleCrossrefGoogle Scholar

  • Dehaene, S. , Changeux, J.-P. (1993). Development of elementary numerical abilities: A neuronal model. Journal of Cognitive Neuroscience, 5, 390–407. First citation in articleCrossrefGoogle Scholar

  • Dehaene, S. , Cohen, L. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1, 83–120. First citation in articleGoogle Scholar

  • Dehaene, S. , Cohen, L. (1997). Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33, 219–250. First citation in articleCrossrefGoogle Scholar

  • Dehaene, S. , Dupoux, E. , Mehler, J. (1990). Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology: Human Perception and Performance, 16, 626–641. First citation in articleCrossrefGoogle Scholar

  • 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, 1217–1220. First citation in articleCrossrefGoogle Scholar

  • Dehaene, S. , Mehler, J. (1992). Cross-linguistic regularities in the frequency of number words. Cognition, 43, 1–29. First citation in articleCrossrefGoogle Scholar

  • Dehaene, S. , Piazza, M. , Pinel, P. , Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506. First citation in articleCrossrefGoogle Scholar

  • Dehaene, S. , Spelke, E. , Pinel, P. , Stanescu, R. , Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284, 970–974. First citation in articleCrossrefGoogle Scholar

  • Deloche, G. , Seron, X. (1982). From one to 1: An analysis of a transcoding process by means of neuropsychological data. Cognition, 12, 119–149. First citation in articleCrossrefGoogle Scholar

  • Deloche, G. , Seron, X. (1987). Numerical transcoding: A general production model. In G. Deloche, X. Seron, (Eds.), Mathematical disabilities: A cognitive neuropsychological perspective (pp. 137–170). Hillsdale, NJ: Erlbaum. First citation in articleGoogle Scholar

  • Deschuyteneer, M. , De Rammelaere, S. , Fias, W. (2005). The addition of two-digit numbers: Exploring carry versus no-carry problems. Psychology Science, 47, 74–83. First citation in articleGoogle Scholar

  • Desmet, L. , Grégoire, J. , Mussolin, C. (2010). Developmental changes in the comparison of decimal fractions. Learning and Instruction, 20, 521–532. First citation in articleCrossrefGoogle Scholar

  • Domahs, F. , Delazer, M. , & Nuerk, H.-C (2006). What makes multiplication facts difficult. Problem size or neighbourhood consistencey? Experimental Psychology, 53, 275–286. First citation in articleLinkGoogle Scholar

  • Domahs, F. , Domahs, U. , Schlesewsky, M. , Ratinckx, E. , Verguts, T. , Willmes, K. , Nuerk, H.-C. (2007). Neighborhood consistency in mental arithmetic: Behavioral and ERP evidence. Behavioral and Brain Functions, 3, 66. First citation in articleCrossrefGoogle Scholar

  • Domahs, F. , Moeller, K. , Huber, S. , Willmes, K. , Nuerk, H.-C. (2010). Embodied numerosity: Implicit hand-based representations influence symbolic number processing across cultures. Cognition, 116, 251–266. First citation in articleCrossrefGoogle Scholar

  • Doricchi, F. , Merola, S. , Aiello, M. , Guariglia, P. , Bruschini, M. , Gevers, W. , …, Tomaiuolo, F. (2009). Spatial orienting biases in the decimal numeral system. Current Biology, 19, 682–687. First citation in articleCrossrefGoogle Scholar

  • Ebersbach, M. , Luwel, K. , Frick, A. , Onghena, P. , & Verschaffel, L. (2008). The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: Evidence for a segmented linear model. Journal of Experimental Child Psychology, 99, 1–17. First citation in articleCrossrefGoogle Scholar

  • English, L. D. , Halford, G. S. (1995). Mathematics education: Models and processes. New York, NY: Erlbaum. First citation in articleGoogle Scholar

  • Fischer, M. H. (2008). Finger counting habits modulate spatial-numerical associations. Cortex, 44, 386–392. First citation in articleCrossrefGoogle Scholar

  • Fischer, M. H. , Campens, H. (2009). Pointing to numbers and grasping magnitudes. Experimental Brain Research, 192, 149–153. First citation in articleCrossrefGoogle Scholar

  • Fischer, M. H. , Mills, R. , & Shaki, S. (2010). How to cook a SNARC: Number placement in text rapidly changes spatial-numerical associations. Brain and Cognition, 72, 333–336. First citation in articleCrossrefGoogle Scholar

  • Fischer, M. H. , Shaki, S. , & Cruise, A. (2009). It takes just one word to quash a SNARC. Experimental Psychology, 56, 361–366. First citation in articleLinkGoogle Scholar

  • Frank, M. C. , Everett, D. L. , Fedorenko, E. , Gibson, E. (2008). Number as a cognitive technology: Evidence from Pirahã language and cognition. Cognition, 108, 819–824. First citation in articleCrossrefGoogle Scholar

  • Fürst, A. J. , Hitch, G. J. (2000). Separate roles for executive and phonological components of working memory in mental arithmetic. Memory & Cognition, 28, 774–782. First citation in articleCrossrefGoogle Scholar

  • Galfano, G. , Rusconi, E. , & Umiltà, C. (2003). Automatic activation of multiplication facts: Evidence from the nodes adjacent to the product. The Quarterly Journal of Experimental Psychology, 56, 31–61. First citation in articleCrossrefGoogle Scholar

  • Ganor-Stern, D. , Pinhas, M. , Tzelgov, J. (2009). Comparing two-digit numbers: The importance of being presented together. The Quarterly Journal of Experimental Psychology, 62, 444–452. First citation in articleCrossrefGoogle Scholar

  • Ganor-Stern, D. , Tzelgov, J. (2011). Across-notation automatic processing of two-digit numbers. Experimental Psychology, 58, 147–153. First citation in articleLinkGoogle Scholar

  • Ganor-Stern, D. , Tzelgov, J. , & Ellenbogen, R. (2007). Automaticity of two-digit numbers. Journal of Experimental Psychology: Human Perception and Performance, 33, 483–496. First citation in articleCrossrefGoogle Scholar

  • García-Orza, J. , Damas, J. (2011). Sequential processing of two-digit numbers: Evidence of decomposition from a perceptual number matching task. Journal of Psychology, 219. First citation in articleGoogle Scholar

  • Gazellini, S. , Laudanna, A. (2011). Digit repetition effect in two-digit number comparison. Journal of Psychology, 219. First citation in articleGoogle Scholar

  • Geary, D. C. , Bow-Thomas, C. C. , Liu, F. , Siegler, R. S. (1996). Development of arithmetical competencies in Chinese and American children: Influence of age, language, and schooling. Child Development, 67, 2022–2044. First citation in articleCrossrefGoogle Scholar

  • Gigerenzer, G. , Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102, 684–704. First citation in articleCrossrefGoogle Scholar

  • Goebel, S. M. , Johansen-Berg, H. , Behrens, T. , Rushworth, M. F. S. (2004). Response-selection-related parietal activation during number comparison. Journal of Cognitive Neuroscience, 16, 1536–1551. First citation in articleCrossrefGoogle Scholar

  • Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306, 496–499. First citation in articleCrossrefGoogle Scholar

  • Grégoire, J. , Meert, G. (2005). L’apprentissage des nombres rationnels et ses obstacles. In M. P. Noël, (Ed.), La dyscalculie. Trouble du développement numérique chez l’enfant (pp. 223–251). Marseille, France: Solal. First citation in articleGoogle Scholar

  • Grossberg, S. , Repin, D. (2003). A neural model of how the brain represents and compares multi-digit numbers: Spatial and categorical processes. Neural Networks, 16, 1107–1140. First citation in articleCrossrefGoogle Scholar

  • Hartnett, P. M. , Gelman, R. (1998). Early understandings of number: Paths or barriers to the construction of new understandings? Learning and Instruction, 8, 341–374. First citation in articleCrossrefGoogle Scholar

  • Healy, A. F. , Nairne, J. S. (1985). Short-term memory processes in counting. Cognitive Psychology, 17, 417–444. First citation in articleCrossrefGoogle Scholar

  • Helmreich, I. , Zuber, J. , Pixner, S. , Kaufmann, L. , Nuerk, H.-C. , Moeller, K. (in press). Language effects on children’s mental number line: How cross-cultural differences in number word systems affect spatial mappings of numbers in a non-verbal task. Journal of Cross-Cultural Psychology. First citation in articleGoogle Scholar

  • Henik, A. , Tzelgov, J. (1982). Is three greater than five: The relation between physical and semantic size in comparison tasks. Memory & Cognition, 10, 389–395. First citation in articleCrossrefGoogle Scholar

  • Hines, T. M. (1990). An odd effect: Lengthened reaction times for judgments about odd digits. Memory & Cognition, 18, 40–46. First citation in articleCrossrefGoogle Scholar

  • Hoeckner, S. , Moeller, K. , Zauner, H. , Wood, G. , Haider, C. , Gaßner, A. , Nuerk, H.-C. (2008). Impairments of the mental number line for two-digit numbers in neglect. Cortex, 44, 429–438. First citation in articleCrossrefGoogle Scholar

  • Hoffrage, U. , Gigerenzer, G. , Krauss, S. , Martignon, L. (2002). Representation facilitates reasoning: What natural frequencies are and what they are not. Cognition, 84, 343–352. First citation in articleCrossrefGoogle Scholar

  • Imbo, I. , Vandierendonck, A. , De Rammelaere, S. (2007). The role of working memory in the carry operation of mental arithmetic: Number and value of the carry. The Quarterly Journal of Experimental Psychology, 60, 708–731. First citation in articleCrossrefGoogle Scholar

  • Ischebeck, A. , Schocke, M. , & Delazer, M. (2009). The processing and representation of fractions within the brain: An fMRI investigation. NeuroImage, 47, 403–413. First citation in articleCrossrefGoogle Scholar

  • Iuculano, T. , Karolis, V. , Cheng, Butterworth, B. (2010, January). Constructing the mental number line using rational numbers. Poster presented at the 28th European Workshop of Cognitive Neuropsychology, Bressanone, Italy. First citation in articleGoogle Scholar

  • Jacob, S. N. , Nieder, A. (2009). Notation-independent representation of fractions in the human parietal cortex. Journal of Neuroscience, 29, 4652–4657. First citation in articleCrossrefGoogle Scholar

  • Kallai, A. , Tzelgov, J. (2009). A generalized fraction: An entity smaller than one on the mental number line. Journal of Experimental Psychology: Human Perception and Performance, 35, 1845–1864. First citation in articleCrossrefGoogle Scholar

  • Klein, E. , Moeller, K. , Dressel, K. , Domahs, F. , Wood, G. , Willmes, K. , & Nuerk, H.-C. (2010). To carry or not to carry – Is this the question? Disentangling the carry effect in multi-digit addition. Acta Psychologica, 135, 67–76. First citation in articleCrossrefGoogle Scholar

  • Klein, E. , Nuerk, H.-C. , Wood, G. , Knops, A. , & Willmes, K. (2009). The exact versus approximate distinction in numerical cognition may not be exact, but only approximate: How different processes work together in multi-digit addition. Brain and Cognition, 2, 369–381. First citation in articleCrossrefGoogle Scholar

  • Knops, A. (2006). On the structure and neural correlates of the numerical magnitude representation and its influence in the assessment of verbal working memory. Germany: University of Aachen Unpublished doctoral dissertation. First citation in articleGoogle Scholar

  • Knops, A. , Nuerk, H.-C. , Sparing, R. , Foltys, H. , Willmes, K. (2006). On the functional role of Human parietal cortex in number processing: How gender mediates the impact of a virtual lesion induced by rTMS. Neuropsychologia, 44, 2270–2283. First citation in articleCrossrefGoogle Scholar

  • 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, 1583–1585. First citation in articleCrossrefGoogle Scholar

  • 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, 803–821. First citation in articleCrossrefGoogle Scholar

  • Kong, J. , Wang, C. , Kwong, K. , Vangel, M. , Chua, E. , Gollub, R. (2005). The neural substrate of arithmetic operations and procedure complexity. Cognitive Brain Research, 22, 397–405. First citation in articleCrossrefGoogle Scholar

  • Korvorst, M. , Damian, M. F. (2008). The differential influence of decades and units on multi-digit number comparison. The Quarterly Journal of Experimental Psychology, 61, 1250–1264. First citation in articleCrossrefGoogle Scholar

  • Kucian, K. , Loenneker, T. , Dietrich, T. , Dosch, M. , Martin, E. , von Aster, M. (2006). Impaired neural networks for approximate calculation in dyscalculic children: A functional MRI study. Behavioral and Brain Functions, 5, 1–17. First citation in articleGoogle Scholar

  • Landerl, K. , Kölle, C. (2009). Typical and atypical development of basic numerical skills in elementary school. Journal of Experimental Child Psychology, 103, 546–565. First citation in articleCrossrefGoogle Scholar

  • LeFevre, J. A. , Bisanz, J. , & Mrkonjic, L. (1988). Cognitive arithmetic: Evidence for obligatory activation of arithmetic facts. Memory & Cognition, 16, 45–53. First citation in articleCrossrefGoogle Scholar

  • Lemer, C. , Dehaene, S. , Spelke, E. , & Cohen, L. (2003). Approximate quantities and exact number words: Dissociable systems. Neuropsychologia, 41, 1942–1958. First citation in articleCrossrefGoogle Scholar

  • Lindemann, O. , Alipour, A. , & Fischer, M. H. (in press). Finger counting habits in Middle-Eastern and Western individuals: An online survey. Journal of Cross-Cultural Psychology. . First citation in articleGoogle Scholar

  • Lindemann, O. , Tira, M. D. (2011). Operational momentum in numerosity production judgements of multi-digit number problems. Journal of Psychology, 219, . First citation in articleGoogle Scholar

  • Macizo, P. , Herrera, A. (2008). El efecto del código numérico en la tarea de comparación de números de dos cifras. [The effect of number codes in the comparison task of two-digit numbers.]. Psicologica, 29, 1–34. First citation in articleGoogle Scholar

  • Macizo, P. , Herrera, A. (2010). Two-digit number comparison: Decade-unit and unit-decade produce the same compatibility effect with number words. Canadian Journal of Experimental Psychology, 64, 17–24. First citation in articleCrossrefGoogle Scholar

  • Macizo, P. , Herrera, A. , Paolieri, D. , Román, P. (2010). Is there cross-language modulation when bilinguals process number words?. Applied Psycholinguistics, 31, 651–669. First citation in articleCrossrefGoogle Scholar

  • Mann, A. , Moeller, K. , Pixner, S. , Kaufmann, L. , Nuerk, H.-C. (2011). Attentional strategies in place-value integration: A longitudinal study on two-digit number comparison. Journal of Psychology, 219. First citation in articleGoogle Scholar

  • McCloskey, M. (1992). Cognitive mechanisms in numerical processing: Evidence from acquired dyscalculia. Cognition, 44, 107–157. First citation in articleCrossrefGoogle Scholar

  • McCloskey, M. , Lindemann, A. M. (1992). MATHNET: Preliminary results from a distributed model of arithmetic fact retrieval. In J. I. D. Campbell, (Ed.), The nature and origins of mathematical skills (pp. 365–409). North Holland, Amsterdam: Elsevier. First citation in articleCrossrefGoogle Scholar

  • McCloskey, M. , Macaruso, P. (1995). Representing and using numerical information. American Psychologist, 50, 351–363. First citation in articleCrossrefGoogle Scholar

  • Meert, G. , Grégoire, J , Noël, M. P. (2009). Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task. The Quarterly Journal of Experimental Psychology, 62, 1598–1616. First citation in articleCrossrefGoogle Scholar

  • Meert, G. , Grégoire, J. , Noël, M. P. (2010a). Comparing the magnitude of two fractions with common components: Which representations are used by 10- and 12-year-olds. Journal of Experimental Child Psychology, 107, 244–259. First citation in articleCrossrefGoogle Scholar

  • Meert, G. , Grégoire, J. , Noël, M. P. (2010b). Comparing 5/7 and 2/9: Adults can do it by accessing the magnitude of the whole fraction. Acta Psychologica, 135, 284–292. First citation in articleCrossrefGoogle Scholar

  • Menon, V. , Rivera, S. M. , White, C. D. , Eliez, S. , Glover, G. H. , Reiss, A. L. (2000). Functional optimisation of arithmetic processing in perfect performers. Cognitive Brain Research, 9, 343–345. First citation in articleCrossrefGoogle Scholar

  • Miura, I. T. , Okamoto, Y. , Kim, C. C. , Chang, C.-M. , Steere, M. , Fayol, M. (1994). Comparisons of children’s cognitive representation of number: China, France, Japan, Korea, Sweden and the United States. International Journal of Behavioral Development, 17, 401–411. First citation in articleCrossrefGoogle Scholar

  • Miura, I. T. , Okamoto, Y. , Kim, C. C. , Steere, M. , & Fayol, M. (1993). First graders’ cognitive representation of number and understanding of place value: Cross-national comparisons – France, Japan, Korea, Sweden, and the United States. Journal of Educational Psychology, 85, 24–30. First citation in articleCrossrefGoogle Scholar

  • Moeller, K. (2010). The influence of the place-value structure of the Arabic number system on two-digit number processing Unpublished doctoral dissertation, Tuebingen, Germany, Eberhard Karls University. First citation in articleGoogle Scholar

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

  • Moeller, K. , Huber, S. , Nuerk, H.-C. , Willmes, K. (2010). Two-digit number processing – Holistic, decomposed or hybrid? A computational modelling approach. Psychological Research. Advance online publication. doi: 10.1007/s00426-010-0307-2. First citation in articleGoogle Scholar

  • Moeller, K. , Klein, E. , Nuerk, H.-C. (2011). On the cognitive instantiation of the carry effect in addition – evidence from eye-tracking. Manuscript submitted for publication. First citation in articleGoogle Scholar

  • Moeller, K. , Klein, E. , Nuerk, H.-C. (in press). (No) small adults – Children’s processing of carry addition problems. Developmental Neuropsychology. First citation in articleGoogle Scholar

  • Moeller, K. , Nuerk, H.-C. (2011). Psychophysics of numerical representation: Why seemingly logarithmic representations may rather be multilinear. Journal of Psychology, 219. First citation in articleGoogle Scholar

  • Moeller, K. , Nuerk, H.-C. , Willmes, K. (2009). Internal magnitude representation is not holistic, either. European Journal of Cognitive Psychology, 21, 672–685. First citation in articleCrossrefGoogle Scholar

  • Moeller, K. , Pixner, S. , Kaufmann, L. , Nuerk, H.-C. (2009). Children’s early mental number line: Logarithmic or rather decomposed linear? Journal of Experimental Child Psychology, 103, 503–515. First citation in articleCrossrefGoogle Scholar

  • Moyer, R. S. , Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215, 1519–1520. First citation in articleCrossrefGoogle Scholar

  • Myung, I. J. , Pitt, M. A. (1997). Applying Occam’s razor in modeling cognition: A Bayesian approach. Psychonomic Bulletin & Review, 4, 79–95. First citation in articleCrossrefGoogle Scholar

  • Nairne, J. S. , Healy, A. F. (1983). Counting backwards produces systematic errors. Journal of Experimental Psychology: General, 112, 37–40. First citation in articleCrossrefGoogle Scholar

  • Nesher, P. , Peled, I. (1986). Shifts in reasoning: The case of extending number concepts. Educational Studies in Mathematics, 17, 67–79. First citation in articleCrossrefGoogle Scholar

  • Nuerk, H.-C. , Bauer, F. , Krummenacher, J. , Heller, D. , Willmes, K. (2005). The power of the mental number line: How the magnitude of unattended numbers affects performance in an Eriksen task. Psychology Science, 47, 34–50. First citation in articleGoogle Scholar

  • Nuerk, H.-C. , Geppert, B. E. , van Herten, M. , Willmes, K. (2002). On the impact of different number representations in the number bisection task. Cortex, 38, 691–715. First citation in articleCrossrefGoogle Scholar

  • Nuerk, H.-C. , Kaufmann, L. , Zoppoth, S. , Willmes, K. (2004). On the development of the mental number line. More or less or never holistic with increasing age. Developmental Psychology, 40, 1199–1211. First citation in articleCrossrefGoogle Scholar

  • Nuerk, H.-C. , Weger, U. , & Willmes, K. (2001). Decade breaks in the mental number line? Putting the tens and units back in different bins. Cognition, 82, B25–33. First citation in articleCrossrefGoogle Scholar

  • Nuerk, H.-C. , Weger, U. , & Willmes, K. (2002). A unit-decade compatibility effect in German number words. Current Psychology Letters: Behavior, Brain, & Cognition, 2, 19–38. First citation in articleGoogle Scholar

  • Nuerk, H.-C. , Weger, U. , & Willmes, K. (2004). On the perceptual generality of the unit-decade-compatibility effect. Experimental Psychology, 51, 72–79. First citation in articleLinkGoogle Scholar

  • Nuerk, H.-C. , Weger, U. , & Willmes, K. (2005). Language effects in magnitude comparison: Small, but not irrelevant. Brain and Language, 92, 262–277. First citation in articleCrossrefGoogle Scholar

  • Nuerk, H.-C. , Willmes, K. (2005). On the magnitude representations of two-digit numbers. Psychology Science, 47, 52–72. First citation in articleGoogle Scholar

  • Nuerk, H.-C. , Wood, G. , & Willmes, K. (2005). The universal SNARC effect: The association between magnitude and space is amodal. Experimental Psychology, 52, 187–194. First citation in articleLinkGoogle Scholar

  • Opfer, J. E. , Siegler, R. S. (2007). Representational change and children’s numerical estimation. Cognitive Psychology, 55, 169–195. First citation in articleCrossrefGoogle Scholar

  • Pica, P. , Lemer, C. , Izard, V. , Dehaene, S. (2004). Exact and approximate arithmetic in an Amazonian indigene group. Science, 306, 499–503. First citation in articleCrossrefGoogle Scholar

  • Pinhas, M. , Fischer, M. H. (2008). Mental movements without magnitude? A study of spatial biases in symbolic arithmetic. Cognition, 109, 408–415. First citation in articleCrossrefGoogle Scholar

  • Pixner, S. (2009). How do children process two-digit numbers? Influences of verbal, visual-Arabic and spatial representations on two-digit number processing in children. Unpublished doctoral thesis, Salzburg, Austria: Paris Lodron University. First citation in articleGoogle Scholar

  • Pixner, S. , Moeller, K. , Hermanova, V. , Nuerk, H.-C. (2011). Whorf reloaded: Language effects on non-verbal number processing in 1st grade – A trilingual study. Journal of Experimental Child Psychology, 108, 371–382. First citation in articleCrossrefGoogle Scholar

  • Pixner, S. , Moeller, K. , Zuber, J. , & Nuerk, H.-C. (2009). Decomposed but parallel processing of two-digit numbers in 1st graders. The Open Psychology Journal, 2, 40–48. First citation in articleCrossrefGoogle Scholar

  • Pollman, T. (2003). Some principles involved in the acquisition of number words. Language Acquisition, 11, 1–31. First citation in articleCrossrefGoogle Scholar

  • Posner, M. I. (1980). Orienting of attention. The Quarterly Journal of Experimental Psychology, 32, 3–5. First citation in articleCrossrefGoogle Scholar

  • Power, R. J. D. , Dal Martello, M. F. (1990). The dictation of Italian numerals. Language and Cognitive Processes, 5, 237–254. First citation in articleCrossrefGoogle Scholar

  • Power, R. J. D. , Dal Martello, M. F. (1997). From 834 to eighty thirty four: The reading of Arabic numerals by seven-year-old children. Mathematical Cognition, 3, 63–85. First citation in articleCrossrefGoogle Scholar

  • Proios, H. , Weniger, D. , & Willmes, K. (2002). Number representation deficit: A bilingual case of failure to access written verbal numeral representations. Neuropsychologia, 40, 2341–2349. First citation in articleCrossrefGoogle Scholar

  • Ratinckx, E. , Brysbaert, M. , Fias, W. (2005). Naming two-digit arabic numerals: Evidence from masked priming studies. Journal of Experimental Psychology: Human Perception Performance, 31, 1150–1163. First citation in articleCrossrefGoogle Scholar

  • Ratinckx, E. , Nuerk, H.-C. , van Dijk, J.-P. , Willmes, K. (2006). Effects of interhemispheric communication on two-digit number processing. Cortex, 42, 1128–1137. First citation in articleCrossrefGoogle Scholar

  • Resnick, L. B. , Nesher, P. , Leonard, F. , Magone, M. , Omanson, S. , & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20, 8–27. First citation in articleCrossrefGoogle Scholar

  • Reynvoet, B. , Notebaert, K. , Van den Bussche, E. (2011). The processing of two-digit numbers depends on task instructions. Journal of Psychology, 219. First citation in articleGoogle Scholar

  • Rubinsten, O. , Henik, A. (2009). Developmental dyscalculia: Heterogeneity may not mean different mechanisms. Trends in Cognitive Sciences, 13, 92–99. First citation in articleCrossrefGoogle Scholar

  • Rusconi, E. , Galfano, G. , Rebonato, E. , Umiltà, C. (2006). Bidirectional links in the network of multiplication facts. Psychological Research, 70, 32–42. First citation in articleCrossrefGoogle Scholar

  • Sackur-Grisvard, C. , Leonard, F. (1985). Intermediate cognitive organizations in the process of learning a mathematical concept: The order of positive decimal numbers. Cognition and Instruction, 2, 157–174. First citation in articleCrossrefGoogle Scholar

  • Seron, X. , Fayol, M. (1994). Number transcoding in children: A functional analysis. British Journal of Developmental Psychology, 12, 281–300. First citation in articleCrossrefGoogle Scholar

  • Shaki, S. , Fischer, M. H. , & Petrusic, W. M. (2009). Reading habits for both words and numbers contribute to the SNARC effect. Psychonomic Bulletin & Reviews, 16, 328–331. First citation in articleCrossrefGoogle Scholar

  • Shaki, S. , Goebel, S. , Fischer, M. (2010, January). Multiple reading habits influence counting direction in Israeli children. Poster presented at the 28th European Workshop of Cognitive Neuropsychology, Bressanone, Italy. First citation in articleGoogle Scholar

  • Shepard, R. N. , Kilpatric, D. W. , Cunningham, P. (1975). The internal representation of numbers. Cognitive Psychology, 7, 82–138. First citation in articleCrossrefGoogle Scholar

  • Siegler, R. S. , Opfer, J. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14, 237–243. First citation in articleCrossrefGoogle Scholar

  • Siegler, R. S. , Robinson, M. (1982). The development of numerical understandings. In H. W. Reese, L. P. Lipsett, (Eds.), Advances in child development and behavior:, Vol. 16, (pp. 242–312). New York, NY: Academic Press. First citation in articleGoogle Scholar

  • Stafylidou, S. , Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503–518. First citation in articleCrossrefGoogle Scholar

  • Thibodeau, M. H. , LeFevre, J.-A. , Bisanz, J. (1996). The extension of the interference effect to multiplication. Canadian Journal of Experimental Psychology, 50, 393–396. First citation in articleCrossrefGoogle Scholar

  • Tlauka, M. (2002). The processing of numbers in choice reaction tasks. Australian Journal of Psychology, 54, 94–98. First citation in articleCrossrefGoogle Scholar

  • Towse, J. N. , Saxton, M. (1998). Mathematics across national boundaries: Cultural and linguistic perspectives on numerical competence. In C. Donlan, (Ed.), The development of mathematics skills (pp. 129–150). Hove, UK: Psychology Press. First citation in articleGoogle Scholar

  • Verguts, T. , De Moor, W. (2005). Two-digit comparison decomposed, holistic or hybrid? Experimental Psychology, 52, 195–200. First citation in articleLinkGoogle Scholar

  • Verguts, T. , Fias, W. (2004). Representation of number in animals and humans: A neural model. Journal of Cognitive Neuroscience, 16, 1493–1504. First citation in articleCrossrefGoogle Scholar

  • Verguts, T. , Fias, W. (2005). Neighbourhood effects in mental arithmetic. Psychology Science, 47, 132–140. First citation in articleGoogle Scholar

  • Verguts, T. , Fias, W. (2008). Symbolic and nonsymbolic pathways of number processing. Philosophical Psychology, 21, 539–554. First citation in articleCrossrefGoogle Scholar

  • Verguts, T , Fias, W. , Stevens, M. (2005). A model of exact small-number representation. Psychonomic Bulletin & Review, 12, 66–80. First citation in articleCrossrefGoogle Scholar

  • Viscuso, S. R. , Anderson, J. A. , Spoehr, K. T. (1989). Representing simple arithmetic in neural networks. In G. Tiberghien, (Ed.), Advances in Cognitive Science, Vol. 2, Chichester, UK: Ellis Horwood. First citation in articleGoogle Scholar

  • von Aster, M. , Shalev, R. (2007). Number development and developmental dyscalculia. Developmental Medicine & Child Neurology, 49, 868–873. First citation in articleCrossrefGoogle Scholar

  • Wood, G. , Mahr, M. , Nuerk, H.-C. (2005). Deconstructing and reconstructing the base-10 structure of Arabic numbers. Psychology Science, 47, 84–95. First citation in articleGoogle Scholar

  • Wood, G. , Nuerk, H.-C. , Freitas, P. , Freitas, G. , Willmes, K. (2006). What do semi-illiterate adults know about 2-digit Arabic numbers? Cortex, 42, 48–56. First citation in articleCrossrefGoogle Scholar

  • Wood, G. , Nuerk, H.-C. , Moeller, K. , Geppert, B. , Schnitker, R. , Weber, J. , Willmes, K. (2008). All for one but not one for all: How multiple number representations are recruited in one numerical task. Brain Research, 1187, 154–166. First citation in articleCrossrefGoogle Scholar

  • Wood, G. , Nuerk, H.-C. , Willmes, K. (2006). Neural representations of two-digit numbers: A parametric fMRI study. NeuroImage, 46, 358–367. First citation in articleCrossrefGoogle Scholar

  • Wood, G. , Willmes, K. , Nuerk, H.-C. , Fischer, M. (2008). On the cognitive link between space and number: A meta-analysis of the SNARC effect. Psychology Science Quarterly, 59, 489–525. First citation in articleGoogle Scholar

  • Young, C. J. , Opfer, J. E. (2011). Psychophysics of numerical representation: A unified approach to single- and multi-digit magnitude estimation. Journal of Psychology, 219, . First citation in articleGoogle Scholar

  • Zhang, J. , Wang, H. (2005). The effect of external representation on numeric tasks. The Quarterly Journal of Experimental Psychology, 58, 817–838. First citation in articleCrossrefGoogle Scholar

  • Zhou, X. , Chen, C. , Chen, L. , Dong, Q. (2008). Holistic or compositional representation of two-digit numbers? Evidence from the distance, magnitude, and SNARC effects in a number-matching task. Cognition, 106, 1525–1536. First citation in articleCrossrefGoogle Scholar

  • Zorzi, M. , Butterworth, B. (1999). A computational model of number comparison. In M. Shafto, P. Langley, (Eds.), Proceedings of the nineteenth annual conference of the cognitive science society (pp. 772–777). Mahwah, NJ: Erlbaum. First citation in articleGoogle Scholar

  • Zuber, J. , Pixner, S. , Moeller, K. , & Nuerk, H.-C. (2009). On the language-specificity of basic number processing: Transcoding in a language with inversion and its relation to working memory capacity. Journal of Experimental Child Psychology, 102, 60–77. First citation in articleCrossrefGoogle Scholar