How Do We Learn Mathematics? A Framework for a Theoretical and Practical Model
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quantityAbstract
The purpose of this paper is to propose an effective learning environment for the initial stages of mathematical learning. Basic numerical skills and the objects and actions that trigger those skills are conceptualized as a mathematics-learning environment. We discuss numerical learning mechanism and the basic skills and environments we use to learn numbers briefly within the human cognition system. The three subsystems of number, i.e., exact number system, approximate number system and access to symbol system, are explained with reference to basic number competencies. They are discussed within the framework of "number sense” by drawing evidences from the neuroscience and mathematics education literature. Finally, how to manipulate the components of these subsystems for an effective learning of number is exemplified in a proposed model of mathematical learning environment.
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References
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Butterworth, B., & Laurillard, D. (2010). Low numeracy and dyscalculia: identification and intervention. Zdm, 42(6), 527-539. doi:10.1007/s11858-010-0267-4
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Cohen Kadosh, R., Dowker, A., Heine, A., Kaufmann, L., & Kucian, K. (2013). Interventions for improving numerical abilities: Present and future. Trends in Neuroscience and Education, 2(2), 85-93. doi:10.1016/j.tine.2013.04.001
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Desoete, A., Ceulemans, A., Roeyers, H., & Huylebroeck, A. (2009). Subitizing or counting as possible screening variables for learning disabilities in mathematics education or learning? Educational Research Review, 4(1), 55-66. doi:10.1016/j.edurev.2008.11.003
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Olkun, S., & Özdem, Ş. (2015). The Effect of Conceptual Subitizing Training on Calculation Performance. Baskent University Journal of Education, 2(1), 1-9.
Ozdem, S., & Olkun, S. (2019). Improving mathematics achievement via conceptual subitizing skill training. International Journal of Mathematical Education in Science and Technology, 1-15. doi:10.1080/0020739x.2019.1694710
Piazza, M., Mechelli, A., Butterworth, B., & Price, C. J. (2002). Are subitizing and counting implemented as separate or functionally overlapping processes? Neuroimage, 15(2), 435-446. doi:10.1006/nimg.2001.0980
Sankey, M. D., Birch, D., & Gardiner, M. W. (2011). The impact of multiple representations of content using multimedia on. International Journal of Education and Development using Information and Communication Technology (IJEDICT), 7(3), 18-35.
Schleifer, P., & Landerl, K. (2011). Subitizing and counting in typical and atypical development. Dev Sci, 14(2), 280-291. doi:10.1111/j.1467-7687.2010.00976.x
Schmithorst, V. J., & Brown, R. D. (2004). Empirical validation of the triple-code model of numerical processing for complex math operations using functional MRI and group Independent Component Analysis of the mental addition and subtraction of fractions. Neuroimage, 22(3), 1414-1420. doi:10.1016/j.neuroimage.2004.03.021
Spelke, E. S. (2017). Core Knowledge, Language, and Number. Language Learning and Development, 13(2), 147-170. doi:10.1080/15475441.2016.1263572
Spelke, E. S., & Kinzler, K. D. (2007). Core knowledge. Dev Sci, 10(1), 89-96. doi:10.1111/j.1467-7687.2007.00569.x
Vogel, S. E., Goffin, C., & Ansari, D. (2015). Developmental specialization of the left parietal cortex for the semantic representation of Arabic numerals: an fMR-adaptation study. Dev Cogn Neurosci, 12, 61-73. doi:10.1016/j.dcn.2014.12.001
Wakeley, A., Rivera, S., & Langer, J. (2000). Can young infants add and subtract. Child Development, 71(6), 1525–1534.
Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358(6389), 749-750. doi:10.1038/358749a0
Balakrishnan, J. D., & Ashby, F. G. (1992). Subitizing: magical numbers or mere superstition? Psychol Res, 54(2), 80-90. doi:10.1007/BF00937136
Benoit, L., Lehalle, H., & Jouen, F. (2004). Do young children acquire number words through subitizing or counting? Cognitive Development, 19(3), 291-307. doi:10.1016/j.cogdev.2004.03.005
Booth, J. L., & Siegler, R. S. (2006). Developmental and individual differences in pure numerical estimation. Dev Psychol, 42(1), 189-201. doi:10.1037/0012-1649.41.6.189
Butterworth, B., & Laurillard, D. (2010). Low numeracy and dyscalculia: identification and intervention. Zdm, 42(6), 527-539. doi:10.1007/s11858-010-0267-4
Clements, D. H., Sarama, J., & MacDonald, B. L. (2019). Subitizing: The Neglected Quantifier. In Constructing Number (pp. 13-45).
Cohen Kadosh, R., Dowker, A., Heine, A., Kaufmann, L., & Kucian, K. (2013). Interventions for improving numerical abilities: Present and future. Trends in Neuroscience and Education, 2(2), 85-93. doi:10.1016/j.tine.2013.04.001
Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44(1-2), 1-42. doi:10.1016/0010-0277(92)90049-n
Dehaene, S. (2009). Origins of mathematical intuitions: the case of arithmetic. Ann N Y Acad Sci, 1156, 232-259. doi:10.1111/j.1749-6632.2009.04469.x
Dehaene, S., Molko, N., Cohen, L., & Wilson, A. J. (2004). Arithmetic and the brain. Curr Opin Neurobiol, 14(2), 218-224. doi:10.1016/j.conb.2004.03.008
Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cogn Neuropsychol, 20(3), 487-506. doi:10.1080/02643290244000239
Desoete, A., Ceulemans, A., Roeyers, H., & Huylebroeck, A. (2009). Subitizing or counting as possible screening variables for learning disabilities in mathematics education or learning? Educational Research Review, 4(1), 55-66. doi:10.1016/j.edurev.2008.11.003
Groffman, S. (2009). Subitizing: Vision Therapy for Math Deficits. Optom Vis Dev, 40(4), 229-238.
Hyde, D. C., Khanum, S., & Spelke, E. S. (2014). Brief non-symbolic, approximate number practice enhances subsequent exact symbolic arithmetic in children. Cognition, 131(1), 92-107. doi:10.1016/j.cognition.2013.12.007
Kinzler, K. D., & Spelke, E. S. (2007). Core systems in human cognition. In From Action to Cognition (pp. 257-264).
Krajcsi, A., Szabo, E., & Morocz, I. A. (2013). Subitizing is sensitive to the arrangement of objects. Exp Psychol, 60(4), 227-234. doi:10.1027/1618-3169/a000191
Kucian, K., Grond, U., Rotzer, S., Henzi, B., Schonmann, C., Plangger, F., . . . von Aster, M. (2011). Mental number line training in children with developmental dyscalculia. Neuroimage, 57(3), 782-795. doi:10.1016/j.neuroimage.2011.01.070
Lipton, J. S., & Spelke, E. S. (2003). Origins of number sense. Large-number discrimination in human infants. Psychol Sci, 14(5), 396-401. doi:10.1111/1467-9280.01453
Mandler, G., & Shebo, B. J. (1982). Subitizing: an analysis of its component processes. J Exp Psychol Gen, 111(1), 1-22. doi:10.1037//0096-3445.111.1.1
Mazzocco, M. M., Feigenson, L., & Halberda, J. (2011). Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia). Child Dev, 82(4), 1224-1237. doi:10.1111/j.1467-8624.2011.01608.x
Olkun, S., Altun, A., & Göçer-Şahin, S. (2015). Beyond Subitizing: Symbolic Manipulations of Numbers. International Journal of Learning, Teaching and Educational Research, 10(1), 93-103.
Olkun, S., Altun, A., Göçer Şahin, S., & Akkurt Denizli, Z. (2015). Deficits in Basic Number Competencies May Cause Low Numeracy in Primary School Children. Egitim ve Bilim, 40(177). doi:10.15390/eb.2015.3287
Olkun, S., Karslı-Çalamak, E., Sözen-Özdoğan, S., Solmaz, G., & Haşlaman, T. (2018). Subitizing and beyond: Perception of set cardinality from different spatial representations. Paper presented at the 4. Cyprus International Congress of Educational Research, Kyrenia, TRNC.
Olkun, S., & Özdem, Ş. (2015). The Effect of Conceptual Subitizing Training on Calculation Performance. Baskent University Journal of Education, 2(1), 1-9.
Ozdem, S., & Olkun, S. (2019). Improving mathematics achievement via conceptual subitizing skill training. International Journal of Mathematical Education in Science and Technology, 1-15. doi:10.1080/0020739x.2019.1694710
Piazza, M., Mechelli, A., Butterworth, B., & Price, C. J. (2002). Are subitizing and counting implemented as separate or functionally overlapping processes? Neuroimage, 15(2), 435-446. doi:10.1006/nimg.2001.0980
Sankey, M. D., Birch, D., & Gardiner, M. W. (2011). The impact of multiple representations of content using multimedia on. International Journal of Education and Development using Information and Communication Technology (IJEDICT), 7(3), 18-35.
Schleifer, P., & Landerl, K. (2011). Subitizing and counting in typical and atypical development. Dev Sci, 14(2), 280-291. doi:10.1111/j.1467-7687.2010.00976.x
Schmithorst, V. J., & Brown, R. D. (2004). Empirical validation of the triple-code model of numerical processing for complex math operations using functional MRI and group Independent Component Analysis of the mental addition and subtraction of fractions. Neuroimage, 22(3), 1414-1420. doi:10.1016/j.neuroimage.2004.03.021
Spelke, E. S. (2017). Core Knowledge, Language, and Number. Language Learning and Development, 13(2), 147-170. doi:10.1080/15475441.2016.1263572
Spelke, E. S., & Kinzler, K. D. (2007). Core knowledge. Dev Sci, 10(1), 89-96. doi:10.1111/j.1467-7687.2007.00569.x
Vogel, S. E., Goffin, C., & Ansari, D. (2015). Developmental specialization of the left parietal cortex for the semantic representation of Arabic numerals: an fMR-adaptation study. Dev Cogn Neurosci, 12, 61-73. doi:10.1016/j.dcn.2014.12.001
Wakeley, A., Rivera, S., & Langer, J. (2000). Can young infants add and subtract. Child Development, 71(6), 1525–1534.
Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358(6389), 749-750. doi:10.1038/358749a0
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Published
2022-03-16
How to Cite
Olkun, S. (2022). How Do We Learn Mathematics? A Framework for a Theoretical and Practical Model. International Electronic Journal of Elementary Education, 14(3), 295–302. Retrieved from https://www.iejee.com/index.php/IEJEE/article/view/1706
