Concept Mapping in Mathematics pp 279-297 | Cite as

# Concept Mapping and Vee Diagramming “Differential Equations”

The chapter presents a case study of a student’s (Nat’s) developing understanding of differential equations as reflected through his progressive concept maps and Vee diagrams (maps/diagrams). Concept mapping and Vee diagramming made Nat realized that there was a need to deeply reflect on *what* he really knows, determine *how* to use what he knows, identify *when* to use *which* knowledge, and be able to justify *why* using valid mathematical arguments. Simply knowing formal definitions and mathematical principles verbatim did not necessarily guarantee an in-depth understanding of the complexity of inter-connections between mathematical concepts and procedures. The presentations of his ideas and understanding of differential equations, Nat found, was greatly facilitated by using his individually constructed concept maps and Vee diagrams. The external projection of his ideas visually on maps/diagrams also facilitated social critiques and mathematical communication during seminar presentations and one-on-one consultations with the researcher. The chapter discusses some implications for teaching mathematics.

## Keywords

Conceptual Understanding Meaningful Learning Auxiliary Equation Order Linear Differential Equation Progressive Differentiation## References

- Afamasaga-Fuata’i, K. (1998).
*Learning to solve mathematics problems through concept mapping and Vee mapping.*Samoa: National University of Samoa.Google Scholar - Afamasaga-Fuata’i, K. (1999). Teaching mathematics and science using the strategies of concept mapping and Vee mapping.
*Problems, Research, and Issues in Science, Mathematics, Computing and Statistics*,*2*(1), 1–53. Journal of the Science Faculty at the National University of Samoa.Google Scholar - Afamasaga-Fuata’i, K. (2001).
*Enhancing students’ understanding of mathematics using concept maps & Vee diagrams.*Paper presented at the International Conference on Mathematics Education (ICME), Northeast Normal University of China, Changchun, China, August 16–22, 2001.Google Scholar - Afamasaga-Fuata’i, K. (2002).
*A Samoan perspective on Pacific mathematics education.*Keynote Address. Proceedings of the 25th annual conference of the Mathematics Education Research Group of Australasia (MERGA-25), July 7–10, 2002 (pp. 1–13). New Zealand: University of Auckland.Google Scholar - Afamasaga-Fuata’i, K. (2004). Concept maps and Vee diagrams as tools for learning new mathematics topics. In A. J. Canãs, J. D. Novak, & F. M. Gonázales (Eds.),
*Concept maps: Theory, methodology, technology.*Proceedings of the First International Conference on Concept Mapping (Vol. 1, pp. 13–20). Navarra, Spain: Dirección de Publicaciones de la Universidad Pública de Navarra.Google Scholar - Afamasaga-Fuata’i, K. (2005). Students’ conceptual understanding and critical thinking? A case for concept maps and Vee diagrams in mathematics problem solving. In M. Coupland, J. Anderson, & T. Spencer (Eds.),
*Making mathematics vital*. Proceedings of the Twentieth Biennial Conference of the Australian Association of Mathematics Teachers (AAMT) (Vol. 1, pp. 43–52). Sydney, Australia: University of Technology.Google Scholar - Afamasaga-Fuata’i, K. (2006a). Developing a more conceptual understanding of matrices and systems of linear equations through concept mapping.
*Focus on Learning Problems in Mathematics*,*28*(3 & 4), 58–89.Google Scholar - Afamasaga-Fuata’i, K. (2006b). Innovatively developing a teaching sequence using concept maps. In A. Canas & J. Novak (Eds.),
*Concept maps: Theory, methodology, technology*. Proceedings of the Second International Conference on Concept Mapping (Vol. 1, pp. 272–279). San Jose, Costa Rica: Universidad de Costa Rica.Google Scholar - Afamasaga-Fuata’i, K. (2007a). Communicating students’ understanding of undergraduate mathematics using concept maps. In J. Watson, & K. Beswick (Eds.),
*Mathematics: Essential research, essential practice.*Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 73–82). University of Tasmania, Australia: MERGA.Google Scholar - Afamasaga-Fuata’i, K. (2007b). Using concept maps and Vee diagrams to interpret “area” syllabus outcomes and problems. In K. Milton, H. Reeves, & T. Spencer (Eds.),
*Mathematics essential for learning, essential for life*. Proceedings of the 21st biennial conference of the Australian Association of Mathematics Teachers, Inc. (pp. 102–111). University of Tasmania, Australia: AAMT.Google Scholar - Afamasaga-Fuata’i, K., & Cambridge, L. (2007). Concept maps and Vee diagrams as tools to understand better the “division” concept in primary mathematics. In K. Milton, H. Reeves, & T. Spencer (Eds.),
*Mathematics essential for learning, essential for life*. Proceedings of the 21st biennial conference of the Australian Association of Mathematics Teachers, Inc. (pp. 112–120). University of Tasmania, Australia: AAMT.Google Scholar - Ausubel, D. P. (2000).
*The acquisition and retention of knowledge: A cognitive view*. Dordrecht: Kluwer Academic.Google Scholar - Ausubel, D. P., Novak, J. D., & Hanesian, H. (1978).
*Educational psychology: A cognitive view*. New York: Holt, Rhinehart and Winston. Reprinted 1986, New York: Werbel and Peck.Google Scholar - Brahier, D. J. (2005).
*Teaching secondary and middle school mathematics*(2nd ed.). New York: Pearson Education, Inc.Google Scholar - Bransford, J., Brown, A., & Cocking, R. (2000).
*How people learn: Brain, mind, experience, and school.*Washington, DC: National Academy Press.Google Scholar - Ernest, P. (1999). Forms of knowledge in mathematics and mathematics education: Philosophical and rhetorical perspectives.
*Educational Studies in Mathematics*,*38*, 67–83.CrossRefGoogle Scholar - Gowin, D. B. (1981).
*Educating*. Ithaca, NY: Cornell University Press.Google Scholar - Hansson, O. (2005). Preservice teachers’ view on y=x+5 and y=πr
^{2}expressed through the utilization of concept maps: A study of the concept of function. In H. Chick & J. L. Vincent (Eds.),*Proceedings of the 29th conference of the international group for the psychology of mathematics education*(Vol. 3, pp. 97–104). Melbourne: PME.Google Scholar - Knuth, E., & Peressini, D. (2001). A theoretical framework for examining discourse in mathematics classrooms.
*Focus on Learning Problems in Mathematics*, 23(2 & 3), 5–22.Google Scholar - Liyanage, S., & Thomas, M. (2002).
*Characterising secondary school mathematics lessons using teachers’ pedagogical concept maps.*Proceedings of the 25th annual conference of the Mathematics Education Research Group of Australasia (MERGA-25), July 7–10, 2002 (pp. 425–432). New Zealand: University of Auckland.Google Scholar - Mintzes, J. J., Wandersee, J. H., & Novak, J. D. (Eds.). (1998).
*Teaching science for understanding. A human constructivistic view.*San Diego, CA, London: Academic Press.Google Scholar - Moschkovitch, J. (2004). Using two languages when learning mathematics. Retrieved on February 20, 2008 from http://math.arizona.edu/Ecemela/spanish/content/workingpapers/UsingTwoLanguages.pdf
- Novak, J. D. (1985). Metalearning and metaknowledge strategies to help students learn how to learn. In L. H. West & A. L. Pines (Eds.),
*Cognitive structure and conceptual change*(pp. 189–209). Orlando, FL: Academic Press.Google Scholar - Novak, J. D. (1998).
*Learning, creating, and using knowledge: Concept maps as facilitative tools in schools and corporations.*Mahwah, NJ: Academic Press.Google Scholar - Novak, J. (2002). Meaningful learning: the essential factor for conceptual change in limited or appropriate propositional hierarchies (LIPHs) leading to empowerment of learners.
*Science Education*,*86*(4), 548–571.CrossRefGoogle Scholar - Novak, J. D., & Gowin, D. B. (1984).
*Learning how to learn*. Cambridge: Cambridge University Press.Google Scholar - Richards, J. (1991). Mathematical discussions. In E. von Glaserfeld (Ed.),
*Radical constructivism in mathematics education*(pp. 13–51). London: Kluwer Academic Publishers.Google Scholar - Ruiz-Primo, M. A., & Shavelson, R. J. (1996). Problems and issues in concept maps in science assessment.
*Journal of Research in Science Teaching*,*33*(6), 569–600.CrossRefGoogle Scholar - Steffe, L. P., & D’Ambrosio, B. S. (1996). Using teaching experiments to enhance understanding of students’ mathematics. In D. F. Treagust, R. Duit, & B. F. Fraser (Eds.),
*Improving teaching and learning in science and mathematics*(pp. 65–76). New York: Teachers College Press, Columbia University.Google Scholar - Williams, C. G. (1998). Using concept maps to access conceptual knowledge of function.
*Journal for Research in Mathematics Education*,*29*(4), 414–421.CrossRefGoogle Scholar