Potential of measuring non-symbolic numerical abilities for early diagnosis of dyscalculia
Abstract
The aim of the study is to contribute to the understanding of the links between non-symbolic numerical abilities and developmental dyscalculia. Therefore, we deal with the numerical functions already at the level of innate dispositions preceding the acquiring of numerical apparatus, which are ensured by the instance of the so-called approximate number system (ANS) and are manifested in the quantitative estimation skill. If the relationship exists, it might be possible to use it for the practical diagnosis of dyscalculia at an early age before showing any difficulties in numerical skills. Our pilot study compares the level of non-symbolic numerical abilities (counting) in children with dyscalculia and control groups. The complete research group consisted of 75 pupils aged between 6.6 and 17.8 years (M = 12.03; SD = 2.68), with 25 (33 %) showing mathematical difficulties in 17 cases (23 %) explained by the diagnosis of dyscalculia. Data was collected using an electronic “numerical estimate test” of the own construction, constituted by the so-called approximation tasks based on the principle of differentiation of quantities. The results suggest that the control group discriminates the quantity better than children with mathematical difficulties. However, after checking the age of the respondents, the relationship between dyscalculia and numerical test output is weak and statistically insignificant, beta = –0.232, p = 0.056. The results are discussed in relation to the possibilities of their use in practice
https://doi.org/10.29364/epsy.369
(Fulltext in Czech)
Keywords
non-symbolic numerical systems, approximate number system, ANS, quantity discrimination, diagnostics of dyscalculiaLiterature
Agrillo, C., Petrazzini, M. E. M., & Bisazza, A. (2015). At the root of math: Numerical abilities in fish. In D. C. Geary, D. B. Berch, & K. Mann Koepke (Eds.), Evolutionary origins and early development of number processing (Vol. 1, Mathematical cognition and learning, pp. 3–33). San Diego, CA: Elsevier Academic.
Aunio, P., & Niemivirta, M. (2010). Predicting children's mathematical performance in grade one by early numeracy. Learning and Individual Differences, 20, 427–435. https://doi.org/….2010.06.003
Anderson, M. L., & Penner-Wilger, M. (2013). The relation between finger gnosis and mathematical ability: why redeployment of neural circuits best explains the finding. Frontiers in Psychology, 4, 877 11. https://doi.org/…g.2013.00877
Anobile, G., Castaldi, E., Turi, M., Tinelli, F., & Burr, D. C. (2016). Numerosity but not texture density discrimination correlates with math ability in children. Developmental Psychology, 52(8), 1206–1216. https://doi.org/…7/dev0000155
Anobile, G., Cicchini, G. M., & Burr, D. C. (2015). Number as a primary perceptual attribute: a review. Perception, 45, 5–31. https://doi.org/…006615602599
Ansari, D. (2007). Does the parietal cortex distinguish between “10,” “ten,” and ten dots? Neuron, 53(2), 165–167. https://doi.org/….2007.01.001
Antell, S. E., & Keating, D. P. (1983). Perception of numerical invariance in neonates. Child Development, 54(3), 695–701. https://doi.org/10.2307/1130057
Bednářová, J. (2015). Diagnostika matematických schopností a dovedností. Brno: Pedagogickopsychologická poradna Brno.
Binterová, H., & Hošpesová, A. (2003). Objevování v matematickém vyučování podporované Excelem. University of South Bohemia České Budějovice Department of Mathematics Report Series 11, 267–273.
Binterová, H., Milota, J., & Vaníček, J. (2005). Global School – virtuální prostředí pro výuku matematiky na ZŠ formou e-learningu. University of South Bohemia České Budějovice Department of Mathematics Report Series, 13.
Bugden, S. & Ansari, D. (2016). Probing the nature of deficits in the ‚Approximate Number System‘ in children with persistent developmental dyscalculia. Developmental Science, 19, 817–33. https://doi.org/…1/desc.12324
Bugden, S., Price, G. R., McLean, D. A., & Ansari, D. (2012). The role of the left intraparietal sulcus in the relationship between symbolic number processing and children's arithmetic competence. Developmental Cognitive Neuroscience, 2, 448–457. https://doi.org/….2012.04.001
Burr, D., & Ross, J. (2008). A visual sense of number. Current Biology, 18, 425–428. https://doi.org/….2008.02.052
Butterworth, B. (2003). Dyscalculia Screener. London: nferNelson. Dostupné z: http://www.dyscalculie.com/…r_manual.pdf
Butterworth, B. (2010). Foundational numerical capacities and the origins of dyscalculia. Trends in Cognitive Sciences, 14(12), 534–541. https://doi.org/….2010.09.007
Butterworth, B., & Laurillard, D. (2010). Low numeracy and dyscalculia: identification and intervention. ZDM Mathematics Education, 42, 527–539. https://doi.org/…8-010-0267-4
Butterworth, B., Varma, S., & Laurillard, D. (2011). Dyscalculia: From brain to education. Science, 27(6033), 1049–1053. https://doi.org/…ence.1201536
Butterworth, B. & Walsh, V. (2011). Neural basis of mathematical cognition. Current Biology, 21, 1337–1420. https://doi.org/….2011.07.005
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(5), e125. https://doi.org/…pbio.0040125
Cantrell, L., Boyer, T. W., Cordes, S., & Smith, L. B. (2015). Signal clarity: An account of variability in infant quantity discrimination tasks. Developmental Science, 18, 877–893. https://doi.org/…1/desc.12283
Chesney, D. (2018). Numerical distance effect size is a poor metric of approximate number system acuity. Attention, Perception, & Psychophysic, 80, 1057–1063. https://doi.org/…4-018-1515-x
Cígler, H. (2018). Matematické schopnosti : teoretický přehled a jejich měření. Brno: Masarykova univerzita.
Chu, F. W., van Marle, K., & Geary, D. C. (2015). Early numerical foundations of young children’s mathematical development. Journal of Experimental Child Psychology, 132, 205–212. https://doi.org/….2015.01.006
Castelli, F., Glaser, D. E., & Butterworth, B. (2006). Discrete and analogue quantity processing in the parietal lobe: a functional MRI study. Proceeding of the National Academy of Sciences of the USA, 103, 4693–4698. https://doi.org/…s.0600444103
Castronovo, J., & Göbel, S. M. (2012). Impact of high mathematics education on the number sense. Plos One, 7(4), 1–16. https://doi.org/…pone.0033832
Coubart, A., Streri, A., de Hevia, M. D., & Izard, V. (2015). Crossmodal discrimination of 2 vs. 4 objects across touch and vision in 5-month-old infants. PLoS One, 10(3), e0120868. https://doi.org/…pone.0120868
Cutini, S., & Bonato, M. (2012). Subitizing and visual short-term memory in human and non-human species: a common shared system? Frontiers in Psychology, 3, 469. https://doi.org/…g.2012.00469
Dehaene, S. (2011). The number sense: How the mind creates mathematics (2nd ed.). New York: Oxford University Press.
Deloche, G., Souza, L., Braga, L. W., & Dellatolas, G. (1999). A calculation and number processing battery for clinical application in illiterates and semi-literates. Cortex, 35, 503–521. doi.org/10.1016/S0010–9452(08)70815–3
Desoete, A. (2015). Predictive indicators for mathematical learning disabilities/dyscalculia in kindergarten children (pp 90–100). In S. Chinn (Ed.), The International handbook for mathematical difficulties and dyscalculia. London & New York: Routledge.
De Smedt, B., Gilmore, C. K. (2011). Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties. Journal of Experimental Child Psychology, 108, 278–292. https://doi.org/….2010.09.003
De Smedt, B., Noël, M. P., Gilmore, C., & Ansari, D. (2013). How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children’s mathematical skills? A review of evidence from brain and behaviour. Trends in Neuroscience and Education, 2(2), 48–55. https://doi.org/….2013.06.001
DeWind, N. K., & Brannon, E. M. (2012). Malleability of the approximate number system: effects of feedback and training. Frontiers in Human Neuroscience, 6, 68. https://doi.org/…m.2012.00068
Durgin, F. H. (1995). Texture density adaptation and the perceived numerosity and distribution of texture. Journal of Experimental Psychology-Human Perception and Performance, 21, 149–169. https://doi.org/…523.21.1.149
Durgin, F. H. (2008). Texture density adaptation and visual number revisited. Current Biology, 18, R855–R856. https://doi.org/….2008.07.053
Elmore, L. C., Ma, W. J., Magnotti, J. F., Leising, K. J., Passaro, A. D., Katz, J. S., Wright, A. A. (2011). Visual short-term memory compared in rhesus monkeys and humans. Current Biology, 21, 975–979. https://doi.org/….2011.04.031
Fazio, L. K., Bailey, D. H., Thompson, C. A., & Siegler, R. S. (2014). Relations of different types of numerical magnitude representations to each other and to mathematics achievement. Journal of Experimental Child Psychology, 123, 53–72. https://doi.org/….2014.01.013
Feigenson, L., Dehaene S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Science, 8, 307–314. https://doi.org/….2004.05.002
Furman, T., & Rubinsten, O. (2012). Symbolic and non-symbolic numerical representation in adults with and without developmental dyscalculia. Behavioral & Brain Functions, 8(1), 55–69. https://doi.org/…44-9081-8-55
Geary, D. C., Bailey, D. H., & Hoard, M. K. (2009). Predicting mathematical achievement and mathematical learning disability with a simple screening tool the number sets test. Journal of Psychoeducational Assessment, 27(3), 265–279. https://doi.org/…282908330592
Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.
Gilmore, C., Attridge, N., & Inglis, M. (2011). Measuring the approximate number system. The Quarterly Journal of Experimental Psychology, 64, 2009–2109. https://doi.org/….2011.574710
Gilmore, C. K., McCarthy, S. E., & Spelke, E. S. (2010). Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling. Cognition, 115, 394–406. https://doi.org/….2010.02.002
Gliga, F., & Gliga, T. (2012). Romanian screening instrument for dyscalculia. Procedia-Social and. Behavioral Sciences, 33, 15–19. https://doi.org/….2012.01.074
Goebel, S. M., Watson, S. E., Lervag, A., & Hulme, C., (2014). Children's arithmetic development it is number knowledge, not the approximate number sense, that counts. Psychological Science, 25 (3), pp. 789–798. https://doi.org/…797613516471
Gross, J., Hudson, C., & Price, D. (2009). The long-term costs of numeracy difficulties. London, UK: Every Child a Chance Trust.
Haist, F., Wazny, J. H., Toomarian, E., & Adamo, M. (2015). Development of brain systems for nonsymbolic numerosity and the relationship to formal math academic achievement. Human Brain Mapping, 36, 804–826. https://doi.org/…02/hbm.22666
Halberda, J., Mazzocco, M. M., & Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455(7213), 665–668. https://doi.org/…/nature07246
Harvey, B. M., Klein, B. P., Petridou, N., & Dumoulin, S. O. (2013). Topographic representation of numerosity in the human parietal cortex. Science (New York, N.Y.), 341(6150), 1123–1126. https://doi.org/…ence.1239052
He, L. X., Zhang, J., Zhou, T. G., & Chen, L. (2009). Connectedness affects dot numerosity judgment: Implications for configural processing. Psychonomic Bulletin & Review, 16, 509–517. https://doi.org/…PBR.16.3.509
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), 17–29. https://doi.org/….2008.04.001
Hyde, C. D. (2011). Two Systems of Non-Symbolic Numerical Cognition. Frontiers in Human Neuroscience, 5, 150. https://doi.org/…m.2011.00150
Iuculano, T., Tang, J., Hall, C. W., & Butterworth, B. (2008). Core information processing deficits in developmental dyscalculia and low numeracy. Developmental Science, 11, 669–680. https://doi.org/…2008.00716.x
Izard, V., Dehaene-Lambertz, G., & Dehaene, S. (2008). Distinct cerebral pathways for object identity and number in human infants. PLoS Biology, 6, e11. https://doi.org/…pbio.0060011
Izard, V., Sann, C., Spelke, E. S., & Streri, A. (2009). Newborn infants perceive abstract numbers. Proceedings of the National Academy of Sciences of the United States of America, 106(25), 10382–10385. https://doi.org/…s.0812142106
Kadosh, R. C., Lammertyn, J., & Izard, V. (2008). Are numbers special? An overview of chronometric, neuroimaging, developmental and comparative studies of magnitude representation. Progress in Neurobiology, 84(2), 132–147. https://doi.org/….2007.11.001
Kaufman, E. L., Lord, M. W., Reese, T. W, & Volkmann, J. (1949). The discrimination of visual number. American Journal of Psychology, 62(4), 498−525. https://doi.org/10.2307/1418556
Kaufmann, L., Vogel, S., Starke, M., Kremser, C., Schocke, M., & Wood, G. (2009). Developmental dyscalculia: compensatory mechanisms in left intraparietal regions in response to nonsymbolic magnitudes. Behavioral and Brain Functions, 5(1), 35. https://doi.org/…44-9081-5-35
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, 2, 31. https://doi.org/…44-9081-2-31
Kucian, K., Loenneker, T., Martin, E., & von Aster, M. (2011). Nonsymbolic numerical distance effect in children with and without developmental dyscalculia: a parametric FMRI study. Developmental Neuropsychology, 36(6), 741–762. https://doi.org/….2010.549867
Kucian, K., Ashkenazi, S. S., Hanggi, J., Rotzer, S., Jancke, L., Martin, E., & von Aster, M. (2013). Developmental dyscalculia: a dysconnection syndrome? Brain Structure Function, 219(5), 1721–1733. https://doi.org/…9-013-0597-4
Kucian, K. & von Aster, M. (2015). Developmental dyscalculia. European Journal of Pediatrics, 174, 1–13. https://doi.org/…1-014-2455-7
Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: a study of 8–9-year-old students. Cognition, 93, 99–125. https://doi.org/….2003.11.004
Landerl, K., Fussenegger, B., Moll, K., & Willburger, E. (2009). Dyslexia and dyscalculia: Two learning disorders with different cognitive profiles. Journal of Experimental Child Psychology, 103(3), 309–324. https://doi.org/….2009.03.006
Libertus, M. E., & Brannon, E. M. (2010). Stable individual differences in number discrimination in infancy. Developmental Science, 13(6), 900–906. https://doi.org/…2009.00948.x
Libertus, M. E., Feigenson, L., & Halberda, J. (2013). Is approximate number precision a stable predictor of math ability? Learning and Individual Differences, 25, 126–133. https://doi.org/….2013.02.001
Libertus, M. E., Odic, D., & Halberda, J. (2012). Intuitive sense of number correlates with math scores on college-entrance examination. Acta Psychologica, 141(3), 373–379. https://doi.org/….2012.09.009
Lipton, J. S., Spelke, E. S. (2004). Discrimination of large and small numerosities by human infants. Infancy, 5(3), 271–290. https://doi.org/…7078in0503_2
Lyons, I. M., Price, G. R., Vaessen, A., Blomert, L., & Ansari, D. (2014). Numerical predictors of arithmetic success in grades 1–6. Developmental Science, 17, 714–726. https://doi.org/…1/desc.12152
Mazzocco, M. M. M. (2005). Challenges in identifying target skills for math disability screening and intervention. Journal of Learning Disabilities, 38(4), 318–323. https://doi.org/…050380040701
Mazzocco, M. M. M., Feigenson, L., & Halberda, J. (2011). Preschoolers’ precision of the approximate number system predicts later school mathematics performance. PLoS ONE, 6, Article e23749. https://doi.org/…pone.0023749
McLean, J. F., and Hitch, G. J. (1999). Working memory impairments in children with specific arithmetic learning difficulties. Journal of Experimental Child Psychology, 74(3), 240–260. https://doi.org/…cp.1999.2516
Meck, W. H., & Church, R. M. (1983). A mode control model of counting and timing process. Journal of Experimental Psychology. Animal Behavior Processes, 9(3), 320–334.
Mussolin, C., De Volder, A., Grandin, C., Schlogel, X., Nassogne, MC., & Noel, M. P. (2010). Neural correlates of symbolic number comparison in developmental dyscalculia. Journal of Cognitive Neuroscience, 22(5), 860–874. https://doi.org/…n.2009.21237
Nieder, A. (2013). Coding of abstract quantity by ‘number neurons’ of the primate brain. Journal of Comparative Physiology A, 199, 1–16. https://doi.org/…9-012-0763-9
Nieder, A. & Dehaene, S. (2009). Representation of number in the brain. Annual Review of Neuroscience, 32, 185–208. https://doi.org/…51508.135550
Nieder, A., & Miller, E. K. (2004). A parieto-frontal network for visual numerical information in the monkey. Proceedings of the National Academy of Sciences of the United States of America, 101(19), 7457–7462. https://doi.org/…s.0402239101
Novák, J. (2001). Barevná kalkulie. Brno: Psychodiagnostika.
Novák, J. (2002). Kalkulie IV. Brno: Psychodiagnostika.
Novák, J. (2004). Dyskalkulie: metodika rozvíjení základních početních dovedností. Vyd. 3., zcela přeprac. Havlíčkův Brod: Tobiáš.
Olmstead, M., & Kuhlmeier, V. (2015). Comparative cognition. London, UK: Cambridge University Press.
Olsson, L., Östergren, R., & Träff, U. (2016). Developmental dyscalculia: A deficit in the approximate number system or an access deficit? Cognitive Development, 39, 154–167. https://doi.org/….2016.04.006
Park, J., & Brannon, E. M. (2014). Improving arithmetic performance with number sense training: an investigation of underlying mechanism. Cognition, 133(1), 188–200. https://doi.org/….2014.06.011
Park, J., Bermudez, V., Roberts, R. C., & Brannon, E. M. (2016). Non-symbolic approximate arithmetic training improves math performance in preschoolers. Journal of Experimental Child Psychology, 152(12), 278–293. https://doi.org/….2016.07.011
Parsons, S., & Bynner, J. (2005). Does numeracy matter more? London: National Research and Development Centre for Adult Literacy and Numeracy.
Piazza, M. (2010). Neurocognitive start-up tools for symbolic number representations. Trends in Cognitive Sciences, 14, 542–551. https://doi.org/….2010.09.008
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), 547–555. https://doi.org/….2004.10.014
Piazza, M., Facoetti, A., Trussardi, A. N., Berteletti, I., Conte, S., Lucangeli, D., Dehaene, S., & Zorzi, M. (2010). Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia. Cognition, 116(1), 33–41. https://doi.org/….2010.03.012
Piazza, M., Fumarola, A., Chinello, A., & Melcher, D. (2011). Subitizing reflects visuo-spatial object individuation capacity. Cognition, 121, 147–153. https://doi.org/….2011.05.007
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. https://doi.org/….2006.11.022
Piazza, M., Facoetti, A., Trussardi, A. N., Berteletti, I., Conte, S., Lucangeli, D., Dehaene, S., & Zorzi, M. (2010). Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia. Cognition, 116(1), 33–41. https://doi.org/….2010.03.012
Pinel, P., Dehaene, S., Riviere, D., & LeBihan, D. (2001). Modulation of parietal activation by semantic distance in a number comparison task. Neuroimage, 14(5), 1013–1026. https://doi.org/…mg.2001.0913
Pinheiro-Chagas, P., Wood, G., Knops, A., Krinzinger, H., Lonnemann, J., Starling-Alves, I., & Haase, V. G. (2014). In how many ways is the approximate number system associated with exact calculation? PLoS ONE, 9(11):e111155. https://doi.org/…pone.0111155
Plassová, M., Tesař, M., Vavrečka, M., & Valuchová, K. (2016). Approximate number system in children. In M. McGreevy & R. Rita (Eds.), Proceedings of the 6th Biannual CER Comparative European Research Conference (pp. 182–187). London: Science.
Pražáková, K. (2017). Přesnost a rychlost ve vnímání množství u jedinců s dyskalkulií. Diplomová práce. Praha: Univerzita Karlova v Praze.
Price, G. R., Palmer, D., Battista, C., & Ansari, D. (2012). Nonsymbolic numerical magnitude comparison: Reliability and validity of different task variants and outcome measures, and their relationship to arithmetic achievement in adults. Acta Psychologica, 17, 50–57. https://doi.org/….2012.02.008
Purpura, D. J., & Logan, J. A. R. (2015). The nonlinear relations of the approximate number system and mathematical language to early mathematics development. Developmental Psychology, 57(12), 1717–1724. https://doi.org/…7/dev0000055
Rotzer, S., Kucian, K., Martin, E., von Aster, M., Klaver, P., & Loenneker, T. (2008). Optimized voxel-based morphometry in children with developmental dyscalculia. Neuroimage, 39, 417–422. https://doi.org/….2007.08.045
Rotzer, S., Loenneker, T., Kucian, K., Martin, E., Klaver, P., & von Aster, M. (2009). Dysfunctional neural network of spatial working memory contributes to developmental dyscalculia. Neuropsychologia, 47(13), 2859–2865. https://doi.org/….2009.06.009
Rousselle, L., & Noël, M. P. (2007). Basic numerical skills in children with mathematics learning disabilities: a comparison of symbolic vs non-symbolic number magnitude processing. Cognition, 102(3), 361–395. https://doi.org/….2006.01.005
Rykhlevskaia, E., Uddin, L. Q., Kondos, L., & Menon, V. (2009). Neuroanatomical correlates of developmental dyscalculia: Combined evidence from morphometry and tractography. Frontiers in Human Neuroscience, 3, 51. https://doi.org/….09.051.2009
Samková, L. (2013). Využití programu GeoGebra při nácviku odhadů. Sborník 6. konference Užití počítačů ve výuce matematiky (323−336). České Budějovice: Jihočeská univerzita v Č. Budějovicích.
Sasanguie, D., Defever, E., Maertens, B., & Reynvoet, B. (2014). The approximate number system is not predictive for symbolic number processing in kindergarteners. Quarterly Journal of Experimental Psychology, 67, 271–280. https://doi.org/….2013.803581
Sasanguie, D., Göbel, S., & Reynvoet, B. (2013). Left parietal TMS disturbs priming between symbolic and non-symbolic number representations. Neuropsychologia 51(8), 1528–1533. https://doi.org/….2013.05.001
Schneider, M. Schneider, M., Beeres, K., Coban, L., Merz, S., Schmidt, S. S., Stricker, J., & De Smedt, B. (2016). Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis. Developmental Science, 20(3). https://doi.org/…1/desc.12372
Starr, A., Libertus, M. E., & Brannon, E. M. (2013). Number sense in infancy predicts mathematical abilities in childhood. Proceedings of the National Academy of Sciences of the United States of America. https://doi.org/…s.1302751110
Sullivan, J., Frank, M. C., & Barner, D. (2016). Intensive math training does not affect approximate number acuity: Evidence from a three-year longitudinal curriculum intervention. Journal of Numerical Cognition, 2(2), 57–76. https://doi.org/…/jnc.v2i2.19.
Shalev, R.S., Manor, O., Kerem, B., Ayali, M., Badichi, N., Friedlander, Y., & Gross-Tsur, V. (2001). Developmental dyscalculia is a familial learning disability. Journal of Learn Disabilities, 34 (1), 59–65. https://doi.org/…940103400105
Schleger, F., Landerl, K., Muenssinger, J., Draganova, R., Reinl, M., Kiefer-Schmidt, I., Weiss, M., Wacker-Gussmann, A., Huotilaine, M., & Preissl, H. (2014). Magnetoencephalographic signatures of numerosity discrimination in fetuses and neonates. Developmental Neuropsychology, 39(4), 316–329. https://doi.org/….2014.914212.
Schwenk, C., Sasanguie, D., Kuhn, J. T., Kempe, S., Doebler, P., & Holling, H. (2017). (Non-) symbolic magnitude processing in children with mathematical difficulties: A meta-analysis. Research in Developmental Disabilities, 64, 152–167. https://doi.org/….2017.03.003
Soto-Calvo, E., Simmons, F. R., Willis, C., & Adams, A. M. (2015). Identifying the cognitive predictors of early counting and calculation skills: Evidence from a longitudinal study. Journal of Experimental Child Psychology, 140, 16–37. https://doi.org/….2015.06.011
Sousa, D. (2010). Mind, brain, and education: Neuroscience implications for the classroom. Bloomington: Solution Tree.
Taves, E. H. (1941). Two mechanisms for the perception of visual numerousness. Archives of Psychology, 265(47).
Traspe, P., & Skalková, I. (2013). DISMAS : Diagnostika struktury matematických schopností. Praha: Národní ústav pro vzdělávání.
Ven, F. van der, Takashima, A., Segers, P. C. J., Fernandez, G. S. E., Verhoeven, L. T. W. (2016). Non-symbolic and symbolic notations in simple arithmetic differentially involve intraparietal sulcus and angular gyrus activity. Brain Research, 1643, 91 – 102. https://doi.org/….2016.04.050
Von Aster, M., & Shalev, R. (2007). Number development and developmental dyscalculia. Developmental Medicine & Child Neurology, 49, 868–873. https://doi.org/…2007.00868.x
von Aster, M. G. & Weinholdová, M. (2008). ZAREKI: Neuropsychologická batéria testov na spracovávanie čísiel a počítanie u detí. Bratislava, Brno: Psychodiagnostika.
Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74, B1–B11. https://www.harvardlds.org/…u2000b-1.pdf