Abstract
Mathematical activity involves work with concepts and problems. Understanding mathematical activity in mathematics education is different for the policy maker, the mathematician, the teacher, and the student. This paper deals with the understanding of a concept in mathematics from the standpoint of the student learner. We make the case for the existence at least five dimensions to this understanding: the skill-algorithm dimension, the property-proof dimension, the use-application (modeling) dimension, the representation-metaphor dimension, and the history-culture dimension. We delineate these dimensions for two concepts: multiplication of fractions, and congruence in geometry.
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Notes
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It is understood that composites of these transformations are allowed. In particular, congruent figures may be related by glide reflections.
References
Cajori, F. (1928). A history of mathematical notations. Vol I: Notations in elementary mathematics. LaSalle, IL: Open Court.
Common Core State Standards. (2010). Mathematics standards. http://www.corestandards.org/the-standards/mathematics. Accessed 9 March 2012.
Flegg, G. (2002). Numbers: Their history and meaning. New York: Dover Publications.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: D. Reidel Publishing Company.
Hadamard, J. (1945). The psychology of invention in the mathematical field. Princeton, NJ: Princeton University Press.
Hardy, G. H. (1940). A mathematician’s apology. London: Cambridge University Press.
Hersh, R., & John-Steiner, V. (2010). Loving and hating mathematics: Challenging the myths of mathematical life. Princeton, NJ: Princeton University Press.
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics learning and teaching (pp. 65–97). New York: Macmillan Publishing Company.
Krathwohl, D., Bloom, B., & Masia, B. (1964). Taxonomy of educational objectives: The classification of educational goals. Handbook I. Cognitive domain. New York: David McKay Company.
Polya, G. (1957). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton, NJ: Princeton University Press.
Polya, G. (1962). Mathematical discovery: On understanding, learning, and teaching problem solving (Vol. I, II). New York: Wiley.
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 16(2), 4–14.
Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20–26. (Reprinted (1978) in Arithmetic Teacher, 26(3), 9–15).
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Usiskin, Z. (2015). What Does It Mean to Understand Some Mathematics?. In: Cho, S. (eds) Selected Regular Lectures from the 12th International Congress on Mathematical Education. Springer, Cham. https://doi.org/10.1007/978-3-319-17187-6_46
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DOI: https://doi.org/10.1007/978-3-319-17187-6_46
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