of classical times, wrote a paper in the form of a letter to the king of his city explaining how to write such very large numbers. Archimedes, however, did not go so far as to invent the decimal system, with its potential for extending indefinitely.


Knuth, 1974, p. 323.


Steen, 1990. See Morrow and Kenney, 1998, for more perspectives on algorithms.


The ellipsis points “…” in the expression are a significant piece of abstract mathematical notation, compactly designating the omission of the terms needed (to reach m, in this case).


Behr, M.J., Harel, 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: Macmillan.

Bruner, J.S. (1966). Toward a theory of instruction. Cambridge, MA: Belknap Press.

Cuoco, A. (Ed.). (2001). The roles of representation in school mathematics (2001 Yearbook of the National Council of Teachers of Mathematics). Reston, VA: NCTM.

Duvall, R. (1999). Representation, vision, and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F.Hitt & M.Santos (Eds.), Proceedings of the twenty-first annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 3–26). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. (ERIC Document Reproduction Service No. ED 433 998).

Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Reidel.

Greeno, J.G., & Hall, R. (1997). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 78, 1–24. Available: http://www.pdkintl.org/kappan/kgreeno.htm. [July 10, 2001].

Kaput,]. (1987). Representation systems and mathematics. In C.Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 19–26). Hillsdale, NJ: Erlbaum.

Knuth, D.E. (1974). Computer science and its relation to mathematics. American Mathematical Monthly, 81, 323–343.

Lakoff, G., & Núñez, R.E. (1997). The metaphorical structure of mathematics: Sketching out cognitive foundations for a mind-based mathematics. In L.D.English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 21–89). Mahwah, NJ: Erlbaum.

Morrow, L.J., & Kenney, M.J. (Eds.). (1998). The teaching and learning of algorithms in school mathematics (1998 Yearbook of the National Council of Teachers of Mathematics). Reston, VA: NCTM.

Pimm, D. (1995). Symbols and meanings in school mathematics. London: Routledge.

Russell, B. (1919). Introduction to mathematical philosophy. New York: Macmillan.

Sfard, A. (1997). Commentary: On metaphorical roots of conceptual growth. In L.D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 339–371). Mahwah, NJ: Erlbaum.

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