# Visual Roots of Mathematical Symbols

Chapter
Part of the Mathematics Education Library book series (MELI, volume 49)

## Abstract

In my Algebra 1 class, solving for the unknown in a linear equation occurred when they had to deal with reversal tasks in patterning situations such as item 4 in Fig. 4.1. Figure 4.2 shows the written work of Dung (eighth grader, Cohort 1), who understood the process of solving for the unknown in the context of finding a particular stage number p whose total number of objects is known. As shown in the figure, he initially took 1 away from 73 and then divided the result by 3. For Dung and his classmates, this particular process of undoing has been drawn from their everyday experiences in which doing and undoing form a natural and intuitive action pair. When my Cohort 1 participated in a teaching experiment on patterning and generalization in sixth grade, the first time I saw them use the undoing strategy occurred in the context of the patterning activity shown in Fig. 4.3. When they were confronted with the situation in item 21, the first thought that came to them was to take away the height of the original cup hold and then divide the result by 3.

## Keywords

Teaching Experiment Number Line Seventh Grade Stage Number Mathematical Symbol
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Alexander, A. (2006). Tragic mathematics: Romantic narratives and the refounding of mathematics in the early nineteenth century. Isis, 97, 714–726.
2. Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24–35.Google Scholar
3. Atkin, A. (2005). Peirce on the index and indexical reference. Transactions of the Charles S. Peirce Society, XLI(I), 161–188.Google Scholar
4. Barwise, J., & Etchemendy, J. (1991). Visual information and valid reasoning. In W. Zimmerman & S. Cunningham (Eds.), Visualizing in teaching and learning mathematics (pp. 9–24). Washington, DC: Mathematical Association of America.Google Scholar
5. Beckmann, S. (2008). Activities manual: Mathematics for elementary teachers (3rd ed.). Boston, MA: Pearson and Addison Wesley.Google Scholar
6. Berkeley, I. (2008). What the <0.70, 1.17, 0.99, 1.07> is a symbol? Minds and Machines, 18, 93–105.
7. Bialystok, E., & Codd, J. (1996). Developing representations of quantity. Canadian Journal of Behavioural Science, 28(4), 281–291.Google Scholar
8. Brown, J. R. (1997). Proofs and pictures. British Journal for the Philosophy of Science, 48, 161–180.
9. Brown, J. R. (2005). Naturalism, pictures, and platonic intuitions. In P. Mancosu, K. Jorgensen, & S. Pedersen (Eds.), Visualization, explanation, and reasoning styles in mathematics (pp. 57–73). Dordrecht, Netherlands: Springer.Google Scholar
10. Brown, J. R. (1999). Philosophy of mathematics: An introduction to the world of proofs and pictures. New York: Routledge.Google Scholar
11. Cellucci, C. (2008). The nature of mathematical explanation. Studies in the History and Philosophy of Science, 39, 202–210.
12. Davydov, V. (1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
13. Fey, J. (1990). Quantity. In L. Steen (Ed.), On the shoulders of giants: New approaches to numeracy (pp. 61–94). Washington, DC: National Academy Press.Google Scholar
14. Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.Google Scholar
15. Freudenthal, H. (1981). Major problems of mathematics education. Educational Studies in Mathematics, 12, 133–150.
16. Giaquinto, M. (2005). Mathematical activity. In P. Mancosu, K. Jorgensen, & S. Pedersen (Eds.), Visualization, explanation, and reasoning styles in mathematics (pp. 75–90). Dordrecht, Netherlands: Springer.Google Scholar
17. Harel, G. (2007). The DNR system as a conceptual framework for curriculum development and instruction. In R. Lesh, J. Kaput, & E. Hamilton (Eds.), Foundations for the future in mathematics education (pp. 263–280). Mahwah, NJ: Erlbaum.Google Scholar
18. Hoffmann, M. (2007). Learning from people, things, and signs. Studies in Philosophy Education, 26, 185–204.
19. Jaffe, A., & Quinn, F. (1993). Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics. Bulletin of the American Mathematical Society, 29(1), 1–13.
20. Legg, C. (2008). The problem of the essential icon. American Philosophical Quarterly, 45(3), 208–232.Google Scholar
21. Mason, J. (1980). When is a symbol symbolic? For the Learning of Mathematics, 1(2), 8–12.Google Scholar
22. Mason, J. (1987). What do symbols represent? In C. Janvier (Ed.), Problems of representation on the teaching and learning of mathematics (pp. 73–81). Hillsdale, NJ: Erlbaum.Google Scholar
23. Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21(2), 33–49.Google Scholar
24. Mitchell, W. (1986). Iconology: Image, text, ideology. Chicago, IL: University of Chicago Press.Google Scholar
25. Nickson, M. (1992). The culture of the mathematics classroom: An unknown quantity? In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 101–114). New York: Simon & Schuster Macmillan and National Council of Teachers of Mathematics.Google Scholar
26. Parker, T., & Baldridge, S. (2004). Elementary mathematics for teachers. Okemos, MI: Sefton-Ash Publishing.Google Scholar
27. Peirce, C. (1957). In V. Tomas (Ed.), Essays in the philosophy of science. New York: The Bobbs-Merrill CompanyGoogle Scholar
28. Peirce, C. (1958b). Lessons of the history of science. In P. Wiener (Ed.), Charles S. Peirce: Selected writings (Values in a universe of chance) (pp. 227–232). New York: Dover.Google Scholar
29. Perkins, D. (1997). Epistemic games. International Journal of Educational Research, 27, 49–61.
30. Ploetzner, R., Lippitsch, S., Galmbacher, M., Heuer, D., & Scherrer, S. (2009). Students’ difficulties in learning from dynamic visualizations and how they may be overcome. Computers in Human Behavior, 25, 56–65.
31. Presmeg, N. (1992). Prototypes, metaphors, metonymies, and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23, 595–610.
32. Presmeg, N. (2008). An overarching theory for research in visualization in mathematics education. Paper presented at the 11th international congress in mathematical education, Merida, Mexico.Google Scholar
33. Radford, L. (2002). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42, 237–268.
34. Rakoczy, H., Tomasello, M., & Striano, T. (2005). How children turn objects into symbols: A cultural learning account. In L. Namy (Ed.), Symbol use and symbolic representation: Developmental and comparative perspectives (pp. 69–97). Mahwah, NJ: Erlbaum.Google Scholar
35. Thurston, W. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177.
36. Uttal, D., & DeLoache, J. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18, 37–54.
37. Vlassis, J. (2008). The role of mathematical symbols in the development of number conceptualization: The case of the minus sign. Philosophical Psychology, 21(4), 555–570.
38. Wittgenstein, L. (1961). Tractacus logico-philosophicus (D. Pears & B. McGuiness, Trans.). London: Routledge.Google Scholar
39. Wittgenstein, L. (1973). In G. H. von Wright (Ed.), Letters to C. K. Ogden. Oxford, UK: Blackwell.Google Scholar
40. Katz, V. (2007). Stages in the history of algebra with implications for teaching. Educational Studies in Mathematics , 66(2), 185–201.
41. DeLoache, J. (2005). The Pygmalion problem in early problem use. In L. Namy (Ed.), Symbol use and symbolic representation (pp. 47–67). Mahwah, NJ: Erlbaum.Google Scholar