It is a common belief by mathematics educators that students, who understand mathematical concepts, are expected to be able to talk about, explain and represent the concepts (Mousely, 1999).

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It is a common belief by mathematics educators
that students, who understand mathematical
concepts, are expected to be able to talk about,
explain and represent the concepts (Mousely,
1999).
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Contrast is comparing the experience of
something else in order to make sure what the
learner has experienced (Marton et al, 2004).
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Hiebert and Carpenter (1992) explain that a
particular mathematical idea can often be
represented and communicated in different forms,
such as physical objects, diagrams, written
symbols and spoken language, students can
construct meanings through connecting different
representations of the same mathematical idea or
between related ideas within the same form.
Lesh (1979) developed a translation model for
instructions. It suggests that learning is
enhanced if students can use multiple ways to
investigate the same mathematical concept and
make translations between and within different
modes of representation. Students can use
manipulatives, pictures, written symbols, verbal
symbols and real-life contexts to explore and
represent the concept.
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References
Hiebert, J. Carpenter, T.P. (1992). Learning and teaching with understanding. In Grouws D.A. (ed.). Handbook of research on mathematics teaching and learning, NCTM, Macmillan, Inc., New York.
Lesh, R. (1979). Mathematical learning disabilities Considerations for identification, diagnosis, and remediation. In R. Lesh, D. Mierkiewicz, M. G. Kantowski (Eds.), Applied mathematical problem solving (pp. 111-180). Columbus, OH ERIC/SMEAC.
Marton, F. and Tsui, A.B.M. with P. Chik, P. Y. Ko, M. L. Lo, I. Mok, D. Ng, M. F. Pang, W. Y. Pong and U. Runesson (2004). Classroom Discourse and the Space of Learning, N. J. Lawrence Erlbaum.
Mousley, J. (1999). Perceptions of mathematical understanding. In J. M. Truran K. M.. Truran (Eds.), Making the difference (Proceedings of the 22nd annual conference of the Mathematics Education Research Group of Australasia, Adelaide, pp. 388-395). Sydney MERGA.
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