Pre-service Teachers' Conceptualization of Arithmetic Mean

산술 평균에 대한 예비교사들의 개념화 분석

  • Joo, Hong-Yun (Dept. of Curriculum and Instruction, Graduate School of Korea University) ;
  • Kim, Kyung-Mi (Center for Curriculum and Instruction studies, Korea University) ;
  • Whang, Woo-Hyung (Dept. of Math. Education, Korea University)
  • Received : 2010.02.01
  • Accepted : 2010.05.07
  • Published : 2010.05.31


The purpose of the study were to investigate how secondary pre-service teachers conceptualize arithmetic mean and how their conceptualization was formed for solving the problems involving arithmetic mean. As a result, pre-service teachers' conceptualization of arithmetic mean was categorized into conceptualization by "mathematical knowledge(mathematical procedural knowledge, mathematical conceptual knowledge)", "analog knowledge(fair-share, center-of-balance)", and "statistical knowledge". Most pre-service teachers conceptualized the arithmetic mean using mathematical procedural knowledge which involves the rules, algorithm, and procedures of calculating the mean. There were a few pre-service teachers who used analog or statistical knowledge to conceptualize the arithmetic mean, respectively. Finally, we identified the relationship between problem types and conceptualization of arithmetic mean.



  1. 김우철 외. (1991). 현대통계학, 영지문화사.
  2. 이영하.남주현 (2005). 통계적 개념 발달에 관한 인식론적 고찰, 한국수학교육학회지 시리즈 A <수학교육>, 44(3), pp.457-475.
  3. Aufmann, R. N., Lockwood, J. S., Nation, R. D., & Clegg, D. K. (2007). Mathematical excursions. Boston, MA: Houghton Mifflin Company.
  4. Bakker, A. (2004). Design research in statistics education : on symbolizing and computer tools. Unpublished Doctoral dissertation.
  5. Bakker, A., & Gravemeijer, K. P. E. (2006). An historical phenomenology of mean and median. Educational Studies in Mathematics, 62, pp.149-168.
  6. Boyer, C. B. (1991). A history of mathematics. NY: John Wiley and Sons, Inc.
  7. Cai, J. (1998). Exploring students' conceptual understanding of the averaging algorithm. School Science and Mathematics, 98(2), pp.93-98.
  8. Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., & Reys, R. E. (1981). Results from the second mathematics assessment of the national assessment of educational progress. National Council of Teachers of Mathematics, Reston. VA.
  9. Cortina, J., Saldanha, L., & Thompson, P. W. (1999). Multiplicative conceptions of arithmetic mean. In F. Hitt (Ed.). Proceedings of the Twenty-first Annual Meeting of the International Group for the Psychology of Mathematics Education. Cuernavaca, Mexico:Centro de Investigaciὁn y de Estudios Avanzados.
  10. Cortina, J. L. (2002). Developing instructional conjectures about how to support student understanding of the arithmetic mean as a ratio. International Conference on Teaching Statistics 6. Retrieved September 10, 2009 from 2_cort.pdf
  11. Fischbein, E., Deri, M., Nello, M., & Marino, M. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal of Research in Mathematics Education, 16, pp. -17.
  12. Goodchild, S. (1988). School pupils' understanding of average. Teaching Statistics. 10, pp. 7-81.
  13. Groth, R. E. (2005). An investigation of statistical thinking in two different contexts: Detecting a signal in a noisy process and determining a typical value. Journal of Mathematical Behavior, 24, pp.109-124.
  14. Groth, R. E., & Bergner, J. A. (2006). Preservice elementary teachers' conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 8, pp. 7-63.
  15. Hardiman, P., Well, A., & Pollatsek, A. (1984). Usefulness of a balance model in understanding the mean. Journal of Educational Psychology, 76, pp.792-801.
  16. Hardiman, P., Pollatsek, A., & Well, A. (1986). Learning to understand the balance beam. Cognition and Instruction, 3, pp.63-86.
  17. Heath, T. H. (1981). A history of greek mathematics. Dover, NY.
  18. 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.1-27). Hillsdale, NJ: Lawrence Erlbaum Associates.
  19. Hunting, R. P., & Sharpley, C. F. (1988). Preschoolers' cognitions of fraction units. British Journal of Educational Psychology, 58, pp.172-183.
  20. Jackson, S. (1965). The growth of logical thinking in normal and subnormal children. British Journal of Educational Psychology, 35, pp.255-258.
  21. Kieren, T. E. (1988). Personal knowledge of rational numbers. In M. Behr & J. Hiebert (Eds.), Number concepts and operations in the middle grades (pp.162-181). Reston, VA: Erlbaum.
  22. Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33, pp.259-289.
  23. Leavy, A. M., & O'Loughlin, N. (2006). Moving beyond the arithmetic average: Preservice teachers understanding of the mean. Journal of Mathematics Teacher Education, 9(1), pp.53-90.
  24. Leon, M., & Zawojewski, J. (1990). Use of the arithmetic mean: An investigation of four properties Issues and preliminary results. Paper presented at The Third International Conference on Teaching Statistics (ICOTS III) . Dunedin, New Zealand.
  25. Lovell, K. (1961). A follow-up study of Inhelder and Piaget's 'The growth of logical thinking.'. British Journal of Psychology, 52, pp.143-153.
  26. MacCullough, D. (2007). A study of expert's understanding of the arithmetic mean. Unpublished Doctoral dissertation, Pennsylvania State University.
  27. Marnich, M. (2008). A knowledge structure for the arithmetic mean: relationships between statistical conceptualizations and mathematical concepts. Unpublished Doctoral dissertation, University of Pittsburgh.
  28. McGatha, M., Cobb, P., & McClain, K. (2002). An analysis of students' initial statistical understandings: Developing a conjectured learning trajectory. Journal of Mathematical Behavior, 16, pp.339-355.
  29. Mevarech, A. R. (1983). A deep structure model of students' statistical misconceptions. Educational Studies in Mathematics, 14, pp.415-429.
  30. Miller, K. (1984). Child as measurer of all things: Measurement procedures and the development of quantitative concepts. In C. Sophian (Ed.), Origins of cognitive skills Hillsdale, NJ : Lawrence Earlbaum Associates.
  31. Mokros, J., & Russell, S. (1995). Children's concepts of average and representativeness. Journal for Research in Mathematics Education, 26(1), pp.20-39.
  32. Murray, F. B., & Holm, J. (1982). The absence of lag in conservation of discontinuous and continuous materials, Journal of Genetic Psychology, 141, pp.213-217.
  33. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
  34. Plackett, R. L. (1970). The principle of the arithmetic mean, In E. Pearson & M. G. Kendall (Eds.), Studies in the history of statistics and probability, 1, Griffin, London.
  35. Pollatsek, A., Lima, S., & Well, A. D. (1981), Concept or Computation: Students' Misconceptions of the Mean, Educational Studies in Mathematics, 12, pp.191-204.
  36. Porter, T. M. (1986). The Rise of Statistical Thinking, 1820-1900. Princeton University Press, Princeton.
  37. Russell, S., & Mokros, J. (1996). Research into practice: What do children understand about average? Teaching Children Mathematics, 2(6), pp.360-364.
  38. Schwartzman, S. (1994). The words of mathematics: An etymological dictionary of math terms used in English. Washington, DC: Mathematical Association of America.
  39. Siegler, R. S. (1976). Three aspects of cognitive development. Cognitive Psychology, 8, pp.481-520.
  40. Stigler, S. M. (1986). The History of Statistics. The Measurement of Uncertainty Before 1900, Harvard University Press, Cambridge, MA.
  41. Strauss, S., & Bichler, E. (1988). The development of children's concept of arithmetic average. Journal for Research in Mathematics Education, 19, pp.64-80.
  42. Tukey, J. W. (1977). Exploratory data analysis. Reading, MA: Addison-Wesley.
  43. Van deWalle, J. A., & Lovin, L. H., (2006). Teaching student centered mathematics. Boston, MA: Pearson Education, Inc.
  44. Watson, J. M., & Moritz, J. B. (2000). The longitudinal development of understanding of average. Mathematical Thinking and Learning, 2, pp.11-50.
  45. Zawojewski, J. S. (1986) The Teaching and Learning Processes of Junior High Studeas Under Different Modes of Instruction in Measures of Central Tendency. Unpublished doctoral dissertation, Northwestern University. Evanston, Illinois.