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MAPPING CONCEPTUAL UNDERSTANDING OF ALGEBRAIC CONCEPTS: AN EXPLORATORY INVESTIGATION INVOLVING GRADE 8 CHINESE STUDENTS

  • Haiyue JinEmail author
  • Khoon Yoong Wong
Article

Abstract

Conceptual understanding is a major aim of mathematics education, and concept map has been used in non-mathematics research to uncover the relations among concepts held by students. This article presents the results of using concept map to assess conceptual understanding of basic algebraic concepts held by a group of 48 grade 8 Chinese students. The concept maps constructed by these students were scored by the number of links and propositions based on linking phrases, and these scores were analyzed to yield three types of results as follows: (a) relations associated with individual concepts, (b) relations between pairs of concepts, and (c) relations among all the given concepts at the whole class level to reveal the structure encompassing these concepts. It was found that the students tended to link from superordinate concepts to subordinate concepts. They seemed to hold different ideas about the relations among the concepts since there were more weak links than moderate and strong links in the collective map. A gap in the students’ understanding of equations and functions was captured. The future use of concept map to study conceptual understanding of specific mathematics topics should deal with the issues of training and translation of findings into classroom practices.

Key words

algebra concept map conceptual understanding China social network analysis 

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References

  1. Afamasaga-Fuata’I, K. (2006). Developing a more conceptual understanding of matrices and systems of linear equations through concept mapping and Vee diagrams. Focus on Learning Problems in Mathematics, 28(3&4), 58–89.Google Scholar
  2. Afamasaga-Fuata’I, K. (Ed.). (2009a). Concept mapping in mathematics: Research into practice. New York, NY: Springer.Google Scholar
  3. Afamasaga-Fuata’I, K. (2009b). Analyzing the “measurement” strand using concept maps and Vee diagrams. In K. Afamasaga-Fuata’I (Ed.), Concept mapping in mathematics: Research into practice (pp. 19–46). New York, NY: Springer.CrossRefGoogle Scholar
  4. Afamasaga-Fuata’I, K. (2009c). Using concept maps and Vee diagrams to analyze the “fractions” strand in primary mathematics. In K. Afamasagax-Fuata’I (Ed.), Concept mapping in mathematics: Research into practice (pp. 59–86). New York, NY: Springer.CrossRefGoogle Scholar
  5. An, S. (2004). Capturing the Chinese way of teaching: The learning-questioning and learning-reviewing instructional model. In L. Fan, N. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: Perspective from insiders (pp. 462–482). Singapore: World Scientific.CrossRefGoogle Scholar
  6. Bereiter, C., & Scardamalia, M. (1998). Beyond Bloom’s taxonomy: Rethinking knowledge for the knowledge age. In A. Hargreaves, A. Lieberman, M. Fullan, & D. Hopkins (Eds.), International handbook of educational change (pp. 675–692). London: Kluwer Academic Publishers.Google Scholar
  7. Degenne, A., & Forsé, M. (1999). Introducing social networks (A. Borges, trans). London: SAGE Publications. Original work published 1994).Google Scholar
  8. Durland, M. M. (2006). Exploring and understanding relationships. In M. M. Durland & K. A. Fredericks (Eds.), Social network analysis in program evaluation (pp. 25–40). Fairhaven: American Evaluation Association & Wiley.Google Scholar
  9. Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: An effective way of mathematics teaching in China. In L. Fan, N. Y. Wong, J. Cai, & S. Li (Eds.), How Chinese learn mathematics: Perspectives from insiders (pp. 309–345). Singapore: World Scientific.CrossRefGoogle Scholar
  10. Hasemann, K., & Mansfield, H. (1995). Concept mapping in research on mathematical knowledge development: Background, methods, findings, and conclusions. Educational Studies in Mathematics, 29, 45–72.CrossRefGoogle Scholar
  11. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–17). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  12. Hough, S., O’Rode, N., & Terman, N. (2007). Using concept maps to assess change in teachers’ understanding of algebra: A respectful approach. Journal of Mathematics Teacher Education, 10, 23–41.CrossRefGoogle Scholar
  13. Jin, H. (2013). Conceptual understanding of Grade 8 students about basic algebra and geometric shapes: Using concept map as an assessment technique. Unpublished doctoral thesis. Nanyang Technological University, Singapore.Google Scholar
  14. Jin, H., & Wong, K. Y. (2010). Training on concept mapping skills in geometry. Journal of Mathematics Education, 3(1), 103–118.Google Scholar
  15. Jin, H., & Wong, K. Y. (2011). Assessing conceptual understanding in mathematics with concept mapping. In B. Kaur & K. Y. Wong (Eds.), Assessment in the mathematics classroom (pp. 67–90). Singapore: World Scientific.CrossRefGoogle Scholar
  16. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academic Press.Google Scholar
  17. Knoke, D., & Yang, S. (2008). Social network analysis (2nd ed.). Thousand Oaks: Sage.Google Scholar
  18. Mansfield, H., & Happs, J. (1989a, July). Using concept maps to explore studentsunderstanding in geometry. Paper presented at the thirteenth Annual Conference of the International Group for the Psychology of Mathematics Education, Paris.Google Scholar
  19. Mansfiled, H., & Happs, J. (1989b). Difficulties in achieving long term conceptual change in Geometry. Unpublished paper, Perth, Western Australia.Google Scholar
  20. Mansfield, H., & Happs, J. (1991). Concept maps. The Australian Mathematics Teacher, 47(3), 30–33.Google Scholar
  21. McClure, J. R., Sonak, B., & Suen, H. K. (1999). Concept map assessment of classroom learning: Reliability, validity, and logistical practicality. Journal of Research in Science Teaching, 36(4), 475–492.CrossRefGoogle Scholar
  22. Ministry of Education (MOE), China. (2011). Mathematics curriculum standards (compulsory education) [In Chinese: 义务教育数学课程标准(2011年版)]. Beijing: Beijing Normal University Press.Google Scholar
  23. Novak, J. D., & Gowin, D. B. (1984). Learning how to learn. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  24. Ruiz-Primo, M.A. (2004). Examining concept maps as assessment tool. In A.J. Cañas, J.D. Novak & F.M. Gonzalez (Eds.), Concept maps: Theory, methodology, technology: Proceedings of the First Conference on Concept Mapping. Retrieved from http://cmc.ihmc.us/CMC2004Programa.html
  25. Ruiz-Primo, M. A., Schultz, S. E., Li, M., & Shavelson, R. J. (2001). Comparison of the reliability and validity of scoring from two concept-mapping techniques. Journal of Research in Science Teaching, 3(2), 260–278.CrossRefGoogle Scholar
  26. Schau, C., & Mattern, N. (1997). Use of map techniques in teaching statistics courses. The American Statistician, 51(2), 171–175.Google Scholar
  27. Schmittau, J. (2009). Concept mapping as a means to develop and assess conceptual understanding in secondary mathematics teacher education. In K. Afamasaga-Fuata’I (Ed.), Concept mapping in mathematics: Research into practice (pp. 137–148). New York: Springer.CrossRefGoogle Scholar
  28. Shavelson, R. J., Ruiz-Primo, M. A., & Wiley, E. D. (2005). Windows into the mind. Higher Education, 49, 413–430.CrossRefGoogle Scholar
  29. Skemp, R. (1976). Relational understanding and instrumental understanding. Arithmetic Teacher, 26(3), 9–15.Google Scholar
  30. Skemp, R. (1986). The psychology of learning mathematics. London: Penguin.Google Scholar
  31. Wang, C. (2013). Suggestions for overcoming the teaching and learning difficulty of function in secondary mathematics [In Chinese: 初中函数教学难点的突破策略]. Mathematics Teaching and Learning in the Secondary School, 3, 21–22.Google Scholar
  32. Wasserman, S., & Faust, K. (1994). Social network analysis: Methods and applications. New York: Cambridge University Press.CrossRefGoogle Scholar
  33. Williams, C. G. (1998). Using concept maps to assess conceptual knowledge of function. Journal for Research in Mathematics Education, 29(4), 414–421.CrossRefGoogle Scholar

Copyright information

© National Science Council, Taiwan 2014

Authors and Affiliations

  1. 1.College of Teacher EducationNanjing Normal UniversityNanjingChina
  2. 2.National Institute of EducationNanyang Technological UniversitySingaporeSingapore

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