Concept Mapping in Mathematics pp 299-326 | Cite as

# Using Concept Maps to Mediate Meaning in Undergraduate Mathematics

The chapter presents the concept map data from a study, which investigated the use of concept maps and Vee diagrams (maps/diagrams) to illustrate the conceptual structure of a topic, its relevant problems and common procedures. Students were required to construct comprehensive topic maps/diagrams as ongoing exercises throughout the semester and to present these for critique before individuals finalized them. With improved mapping proficiency and on-going social critiques, students’ mathematical understanding deepened, becoming more conceptual as a result of continually revising their work as the validity of each map is dependent on how effective it illustrated the intended meanings and correct mathematics structure. Students also developed an appreciation of the crucial inter-linkages between mathematical principles, common procedures and formulas, and how all of these mutually reinforce each other conceptually and methodologically. Incorporating concept mapping as a normal mathematical practice in classrooms can potentially alter the learning of mathematics, making it more meaningful and conceptual to supplement the predominantly procedural proficiency practised in many mathematics classrooms.

## Keywords

Concept Hierarchy Nodal Entry Concept Label Progressive Differentiation Valid Node## References

- Afamasaga-Fuata’i, K. (1998).
*Learning to solve mathematics problems through concept mapping & 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). New challenges to mathematics education in Samoa.
*Measina A Samoa 2000,**1*, 90–97. Institute of Samoan Studies, National University of Samoa.Google Scholar - Afamasaga-Fuata’i, K. (2002). A Samoan perspective on Pacific mathematics education. In B. Barton, K. C. Irwin, M. Pfannkuch, & M. O. J. Thomas (Eds.),
*Mathematics education in the South Pacific.*Proceedings of the 25th annual conference of the Mathematics Education Research Group of Australasia (MERGA-25) (Vol. 1, pp. 1–13). Auckland, 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. Cañas, 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. (2005a). Mathematics education in Samoa: From past and current situations to future directions.
*Journal of Samoan Studies,**1*, 125–140.Google Scholar - Afamasaga-Fuata’i, K. (2005b). 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). Innovatively developing a teaching sequence using concept maps. In A. Cañas & 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. (2006b). 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. (2007). 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. (2007a). 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. (2007b). 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., Meyer, P., & Falo, N. (2007). Primary students’ diagnosed mathematical competence in semester one of their studies. 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. 83–92). University of Tasmania, Australia: MERGA.Google Scholar - Afamasaga-Fuata’i, K., Meyer, P., Falo, N., & Sufia, P. (2007). Future teachers’ developing numeracy and mathematical competence as assessed by two diagnostic tests. Published on AARE’s website. http://www.aare.edu.au/06pap/afa06011.pdf
- Australian Association of Mathematics Teachers (AAMT) (2007). The AAMT standards for excellence in teaching mathematics in Australian schools. Retrieved from http://www.aamt.edu.au/standards, on July 18, 2007.
- Ausubel, D. P. (2000).
*The acquisition and retention of knowledge: A cognitive view*. Dordrecht, Boston: Kluwer Academic Publishers.Google Scholar - Baroody, A. J., & Bartel, B. H. (2000). Using concept maps to link mathematical ideas.
*Mathematics Teaching in Middle Schools*,*5*(9), 604–609.Google Scholar - Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge.
*Journal of Research in Mathematics Education, 38*(2), 115–131.Google Scholar - Brahier, D. J. (2005).
*Teaching secondary and middle school mathematics*(2nd ed.). New York: Pearson Education, Inc.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 pscychology of mathematics education*(Vol. 3, pp. 97–104). Melbourne: PME.Google Scholar - Hatano, G. (2003). Foreword. In A. J. Baroody & A. Dowker (Eds.),
*The development of arithmetic concepts and skills: Constructing adaptive expertise*(pp. 11–13). Mahwah, NJ: Erlbaum.Google Scholar - Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. G Grouws (Ed.),
*Handbook of research in mathematics teaching and learning*(pp. 65–97). New York: Macmillan.Google Scholar - Liyanage, S., & Thomas, M. (2002). Characterising secondary school mathematics lessons using teachers’ pedagogical concept maps. In B. Barton, K. C. Irwin, M. Pfannkuch, & M. O. J. Thomas (Eds.),
*Mathematics education in the South Pacific.*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 - Marsh, G. E., & Ketterer, J. K. (2005). Situating the zone of proximal development.
*Online Journal of Distance Learning Administration, Volume VIII, Number II Summer, 2005, University of West Georgia, Distance Education Center*. Retrieved on August 20, 2007 from http://www.westga.edu/~distance/ojdla/summer82/marsh82.htm - Mintzes, J. J., Wandersee, J. H., & Novak, J. D. (Eds.). (2000).
*Assessing science understanding: A human constructivist view.*San Diego, CA, London: Academic.Google Scholar - National Council of Teachers of Mathematics (NCTM) (2007).
*2000 Principles and Standards.*Retrieved on July 28, 2007 from http://my.nctm.org/standards/document/index.htm - Novak, J. D. (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., & Cañas, A. J. (2006). The theory underlying concept maps and how to construct them. Technical Report IHMC Map Tools 2006-01, Florida Institute for Human and Machine Cognition, 2006, available at: http://cmap.ihmc.us/publications/ResearchPapers/TheoryUnderlyingConceptMaps.pdf
- Novak, J. D., & Gowin, D. B. (1984).
*Learning how to learn*. Cambridge: Cambridge University Press.Google Scholar - Piaget, J. (1969).
*Science of education and the psychology of the child*. New York: Grossman Publishers.Google Scholar - Pratt, N., & Kelly, P. (2005). Mapping mathematical communities: Classrooms, research communities and master classes. Retrieved on August 20, 2007 from http://orgs.man.ac.uk/projects/include/experiment/communities.htm
- 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. (2004).
*Examining concept maps as an assessment tool.*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 September 14–17, 2004 (pp. 555–562). Navarra: Dirección de Publicaciones de la Universidad Pública de Navarra, Spain.Google Scholar - Schoenfeld, A. H. (1996). In fostering communities of inquiry, must it matter that the teacher knows “the answer."
*For the Learning of Mathematics*,*16*(3), 569–600. Google 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 - Sternberg, R. J., & Williams, W. (Eds.). (1998).
*Intelligence, instruction, and assessment theory into practice.*Mahwah, NJ, London: LEA Lawrence Erlbaum Associates.Google Scholar - Swarthout, M. B. (2001). The impact of the instructional use of concept maps on the mathematical achievement, confidence levels, beliefs, and attitudes of preservice elementary teachers. Dissertation Abstracts International, AADAA-I3039531, The Ohio State University, Ohio.Google Scholar
- Vygotsky, L. S. (1978).
*Mind in society*. Cambridge, MA: Harvard University Press.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