# Introduction

## Abstract

In writing this book, I have definitely stood on the shoulders of giants whose works on various aspects of mathematical visualization have enriched our understanding of how students actually learn mathematics. The impressive critical syntheses of research studies on visualization in mathematics by Presmeg (2006) and Owens and Outhred (2006), which have been drawn from the annual peer-reviewed proceedings of the International Group for the Psychology of Mathematics Education (IGPME) over a period that spans three decades (1976–2006), provide a comprehensive list of important contributors whose thoughts are reflected in various places in this book. Twenty years ago, the Mathematical Association of America published a visual-driven monograph edited by Zimmermann and Cunningham (1991) that consists of reflective essays by, including references to other, researchers who then began the exciting task of exploring ways to visualize abstract mathematical objects via the power of computer software tools that could support and mediate in the development of advanced mathematical concepts and processes.

## Keywords

Mathematics Education Visual Image Mathematical Knowledge Mathematical Activity Polynomial Inequality## Notes

### Acknowledgment and Dedication

I wish to express my gratitude to the National Science Foundation (NSF) that provided funding for me to engage in longitudinal classroom work from 2005 to 2010 (under NSF Career Grant #0448649). Results that are reported in this book are all mine and do not reflect the views of the foundation. This book is dedicated to the students and teachers in my 2005–2010 study.

## References

- Artigue, M. (2003). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work.
*Proceedings of the CAME 2001 symposium: communicating mathematics through computer algebra systems.*Retrieved from http://ltsn.mathstore.ac.uk/came/events/freudenthal/theme1.htm - Bakker, A. (2007). Diagrammatic reasoning and hypostasized abstraction in statistics education.
*Semiotica, 164*(1/4), 9–29.CrossRefGoogle Scholar - Bartolini-Bussi, M., & Boni, M. (2003). Instruments for semiotic mediation in primary school classrooms.
*For the Learning of Mathematics, 23*(2), 15–22.Google Scholar - 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 - Brown, R. (2002). Forum discussion: How art can help the teaching of mathematics. In C. Bruter (Ed.),
*Mathematics and art: Mathematical visualization in art and education*(pp. 155–159). Berlin: Springer.Google Scholar - Bruner, J. (1978). Prologue to the English edition. In R. Rieber & A. Carton (Eds.),
*The collected works of L. S. Vygotsky, volume 1: Problems of general psychology*(pp. 1–16). New York: Plenum Press.Google Scholar - Cobb, P. (2007). Putting philosophy to work: Coping with multiple theoretical perspectives. In F. Lester, Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 3–38). Charlotte, NC: National Council of Teachers of Mathematics and Information Age Publishing.Google Scholar - Davis, P. (1974). Visual geometry, computer graphics, and theorems of perceived type.
*Proceedings of the symposia in applied mathematics: Volume 20*.Google Scholar - Donaghue, E. (2003). Algebra and geometry textbooks in twentieth-century America. In G. Stanic & J. Kilpatrick (Eds.),
*A history of school mathematics*(Vol. 1, pp. 329–398). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.
*Educational Studies in Mathematics, 61*(1–2), 103–131.CrossRefGoogle Scholar - Eisenberg, T., & Dreyfus, T. (1991). On the reluctance to visualize in mathematics. In W. Zimmermann & S. Cunningham (Eds.),
*Visualization in teaching and learning mathematics*(pp. 25–38). Washington, DC: Mathematical Association of America.Google Scholar - Goldin, G. (2002). Representation in mathematical learning and problem solving. In L. English (Ed.),
*Handbook of International Research in Mathematics Education*(pp. 197–218). Mahwah, NJ: Erlbaum.Google Scholar - Haciomeroglu, E., Aspinwall, L., & Presmeg, N. (2010). Contrasting cases of calculus students’ understanding of derivative graphs.
*Mathematical Thinking and Learning, 12*, 152–176.CrossRefGoogle Scholar - Hutchins, E. (1995).
*Cognition in the wild.*Cambridge, MA: MIT Press.Google Scholar - Leushina, A. (1991).
*Soviet studies in mathematics education (volume 4): The development of elementary mathematical concepts in preschool children.*(J. Teller, Trans.). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Levy, S. (1997). Peirce’s theoremic/corollarial distinction and the interconnections between mathematics and logic. In N. Houser, D. Roberts, & J. Van Evra (Eds.),
*Studies in the logic of Charles Sanders Peirce*(pp. 85–110). Bloomington, IN: Indiana University Press.Google Scholar - Maienschein, J. (1991). From presentation to representation in E. B. Wilson’s “The Cell.”
*Biology and Philosophy, 6*, 227–254.CrossRefGoogle Scholar - Mariotti, M. A. (2000). Introduction to proof: The mediation of a dynamic software environment.
*Educational Studies in Mathematics, 44*, 25–53.CrossRefGoogle Scholar - Mariotti, M. A. (2002). The influence of technological advances on students’ mathematics learning. In L. English (Ed.),
*Handbook of international research in mathematics education*(pp. 695–724). Mahwah, NJ: Erlbaum.Google Scholar - Mason, J. (1980). When is a symbol symbolic?
*For the Learning of Mathematics, 1*(2), 8–12.Google Scholar - McCormick, B., DeFanti, T., & Brown, M. (1987). Definition of visualization.
*ACM SIGGRAP Computer Graphics, 21*(6), 1–3.Google Scholar - Michalowicz, K., & Howard, A. (2003). Pedagogy in text: An analysis of mathematics texts from the nineteenth century. In G. Stanic & J. Kilpatrick (Eds.),
*A history of school mathematics*(Vol. 1, pp. 77–112). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Millar, S. (1994).
*Understanding and representing space: Theory and evidence from studies with blind and sighted children.*Oxford, UK: Oxford University Press.Google Scholar - Mitchell, W. (2005).
*What do pictures want? The lives and loves of images.*Chicago, IL: University of Chicago Press.Google Scholar - Moyer, P. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics.
*Educational Studies in Mathematics, 47*, 175–197.CrossRefGoogle Scholar - Ng, S. F., & Lee, K. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems.
*Journal for Research in Mathematics Education, 40*(3), 282–313.Google Scholar - O’Halloran, K. (2005).
*Mathematical discourse: Language, symbolism, and visual images.*New York: Continuum.Google Scholar - Otte, M. (2003). Complementarity, sets, and numbers.
*Educational Studies in Mathematics, 53*, 203–228.CrossRefGoogle Scholar - Owens, K., & Outhred, L. (2006). The complexity of learning geometry and measurement. In A. Gutierrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education: Past, present, and future*(pp. 83–115). Rotterdam, Netherlands: Sense Publishers.Google Scholar - Pedemonte, B. (2007). How can the relationship between argumentation and proof be analyzed?
*Educational Studies in Mathematics, 66*(1), 25–41.CrossRefGoogle Scholar - Peirce, C. (1934). In C. Hartshorne & P. Weiss (Eds.),
*Collected papers of Charles Sanders Peirce: Volume V*. Cambridge, MA: Harvard University Press.Google Scholar - Peirce, C. (1976). In C. Eisele (Ed.),
*The new elements of mathematics: Volume IV mathematical philosophy*. Atlantic Highlands, NJ: Humanities Press.Google Scholar - Poplu, G., Ripoll, H., Mavromatis, S., & Baratgin, J. (2008). How do expert soccer players encode visual information to make decisions in simulated game situations?
*Research Quarterly for Exercise and Sport, 79*(3), 392–398.Google Scholar - Presmeg, N. (2006). Research on visualization in learning and teaching mathematics. In A. Gutierrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education: Past, present, and future*(pp. 205–236). Rotterdam: Sense Publishers.Google Scholar - Rivera, F. (2007a). Accounting for students’ schemes in the development of a graphical process for solving polynomial inequalities in instrumented activity.
*Educational Studies in Mathematics, 65*(3), 281–307.CrossRefGoogle Scholar - Rivera, F. (2010b). There is more to mathematics than symbols.
*Mathematics Teaching, 218*, 42–47.Google Scholar - Skemp, R. (1987/1971).
*The psychology of learning mathematics: Expanded American edition*. Hillsdale, NJ: Erlbaum.Google Scholar - Taussig, M. (2009). What do drawings want?
*Culture, Theory, & Critique, 50*(2–3), 263–274.CrossRefGoogle Scholar - Tucker, J. M. (2010). A lesson on the slopes of perpendicular lines.
*Mathematics Teacher, 103*(8), 603–608.Google Scholar - Wise, M. N. (2006). Making visible.
*Isis, 97*, 75–82.CrossRefGoogle Scholar - Zimmermann, W., & Cunningham, S. (1991).
*Visualization in teaching and learning mathematics. MAA Notes Series 19.*Washington, DC: Mathematical Association of America.Google Scholar - Mitchell, W. (1994).
*Picture theory: Essays on verbal and visual representation.*Chicago, IL: University of Chicago Press.Google Scholar - Galison, P. (2002). Images scatter into data, data gathers into images. In B. Latour, & P. Weibel (Eds.),
*Iconoclash: Beyond the image wars in science, religion, and art*(pp. 300–323). Cambridge, MA: MIT Press.Google Scholar - Shin, S. (1994).
*The logical status of diagrams*. New York: Cambridge University Press.Google Scholar - Presmeg, N. (1986). Visualization in high school mathematics.
*For the Learning of Mathematics, 6*(3), 42–46.Google Scholar - Presmeg, N. (1991). Classroom aspects which influence use of visual imagery in high school mathematics. In F. Furinghetti (Ed.),
*Proceedings of the 15th International group for the psychology of mathematics education*(Vol. 3, pp. 191–198). Assisi, Italy: PME Committee.Google Scholar