The Mathematical Education (한국수학교육학회지시리즈A:수학교육)

Korean Society of Mathematical Education (한국수학교육학회)
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The Impact of Children's Understanding of Fractions on Problem Solving

분수의 하위개념 이해가 문제해결에 미치는 영향

  • Published : 2009.08.31
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Abstract

The purpose of the study was to investigate the influence of children's understanding of fractions in mathematics problem solving. Kieren has claimed that the concept of fractions is not a single construct, but consists of several interrelated subconstructs(i.e., part-whole, ratio, operator, quotient and measure). Later on, in the early 1980s, Behr et al. built on Kieren's conceptualization and suggested a theoretical model linking the five subconstructs of fractions to the operations of fractions, fraction equivalence and problem solving. In the present study we utilized this theoretical model as a reference to investigate children's understanding of fractions. The case study has been conducted with 6 children consisted of 4th to 5th graders to detect how they understand factions, and how their understanding influence problem solving of subconstructs, operations of fractions and equivalence. Children's understanding of fractions was categorized into "part-whole", "ratio", "operator", "quotient", "measure" and "result of operations". Most children solved the problems based on their conceptual structure of fractions. However, we could not find the particular relationships between children's understanding of fractions and fraction operations or fraction equivalence, while children's understanding of fractions significantly influences their solutions to the problems of five subconstructs of fractions. We suggested that the focus of teaching should be on the concept of fractions and the meaning of each operations of fractions rather than computational algorithm of fractions.

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References

  1. 강홍규. 고정화 (2003). 양의 측정을 통한 자연수와 분수 지도의 교수학적 의의. 대한수학교육학회지 <학교수학>, 5(3), pp.385-399
  2. 권성룡 (2003). 초등학생의 분수이해에 관한 연구. 대한수학교육학회지 <학교수학>, 5(2), pp.259-273
  3. 박교식. 송상헌. 임재훈 (2004). 우리나라 예비 초등 교사들의 분수 나눗셈의 의미 이해에 관한 연구. 대한수학교육학회지 <학교수학>, 6(3), pp.235-249
  4. 방정숙 . Yeping Li (2008). 예비 초등 교사들의 분수 나눗셈에 대한 지식 분석. 한국수학교육학회지 시리즈 A <수학교육>, 47(3), pp.291-310
  5. 백선수. 김원경 (2005). 분수의 곱셈에서 비형식적 지식의 형식화 사례 연구. 대한수학교육학회지 <학교수학>, 7(2), pp.139-168
  6. 오영열 (2004). 초등수학에 대한 예비교사들의 이해: 분수으 곱셈을 중심으로. 대한수학교육학회지 <학교수학>, 6(3), pp.267-281
  7. 오유경. 김진호 (2009). 분수 개념에 대한 초등학생들의 비형식적 지식 분석: 1-3학년 중심으로. 한국수학교육학회지 시리즈 E <수학교육 논문집>, 23(1), pp.145-174
  8. 임재훈. 김수미. 박교식 (2005). 분수 나눗셈 알고리즘 도입 방법 연구: 남북한, 중국, 일본의 초등학교 수학 교과서의 내용 비교를 중심으로. 대한수학교육학회지 <학교수학>, 7(2), pp.103-121
  9. 정은실 (2006). 분수 개념의 의미 분석과 교육적 시사점 탐구. 대한수학교육학회지 <학교수학>, 8(2), pp.123-138
  10. 서동엽 (2005). 분수의 역사발생적 지도 방안. 대한수학교육학회지 <수학교육학연구>, 15(3), pp.233-249
  11. 한혜숙 (2009). Lesh 표상 변환(Translation) 모델을 적용한 3학년 학생들의 분수개념 학습, 한국수학교육학회지 시리즈 E <수학교육 논문집>, 23(1), pp.129-144
  12. Amato, S. A. (2005). Developing students' understanding of the concept of fractions as numbers. In H. L. Chick & J. L. Vincent (Eds.). Proceedings 29th Conference of the International Group for the Psychology of Mathematics Education, 2, pp.49-56. Melbourne: PME.
  13. Amato, S. A. (2006). Improving student teachers' understanding of fractions. In J, Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 2, pp. 41-48. Prague: PME.
  14. Behr, M., Lesh, R., Post, T., & Silver, E. (1983). Rational number concepts. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes(pp.91-125), NY: Academic Press.
  15. Behr, M, Wachsmuth, I., Post, T., & Lesh, R. (1984). Order and equivalence of rational numbers. Journal for Research in Mathematics Education, 15, pp.323-341. https://doi.org/10.2307/748423
  16. Behr, M. J., Hard, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning(pp.296-333). Reston, Virginia : NCTM.
  17. Behr, M. J., Hard, G., Post, T., & Lesh, R. (1993). Rational numbers: Toward a semantic analysis-emphasis on the operator construct. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp.13-47). Hillsdale, NJ: Lawrence Erlbaum Associates.
  18. Behr, M. J, Khoury, H. A, Harel, G., Post, T., & Lesh, R. (1997). Conceptual units analysis of preservice elementary school teachers' strategies on a rational-number-as-operator task. Journal for Research in Mathematics Education, 28(1), pp.48-69. https://doi.org/10.2307/749663
  19. Brousseau, G, Brousseau, K, & Warfield, V. (2004). Rationals and decimals as required in the school curriculum. Part 1: Rationals as measurements. Journal of Mathematical Behavior, 23, pp.32-42.
  20. Carraher, D. W. (1996). Learning about fraction. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Thories of mathematical learning(pp. 241-266). NJ: Lawrence Eribaum Associates.
  21. Charalambous, C. Y. & Pitta-Pantazi, D. (2006). Drawing on a theoretical model to study students' understandings of fractions. Educational Studies in Mathematics, 64, pp.293-316.
  22. Charles, K., & Nason, R. (2001). Young children's partitioning strategies. Educational Studies in Mathematics, 43, pp. 91-221.
  23. Clarke, D. M., & Sukenik, M. (2006). Assessing fraction understanding using task-based interviews. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova, (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 2, pp,337-344. Prague: PME.
  24. Cramer, K. A., Post, T. R, & del Mas, R. C. (2002). Initial fraction learning by fourth- and fifth-grade students : A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum, Journal for Research in Mathematics Education, 33(2), pp. 111-144. https://doi.org/10.2307/749646
  25. Empson, S., Junk, D., Dominguez, H., & Turner, E. (2005), Fractions as the coordination of multiplicatively related quantities: A cross-sectional study of children's thinking. Educational Studies in Mathematics, 63, pp.1-28. https://doi.org/10.1007/s10649-005-9000-6
  26. English, L. D., & Halford, G. S. (1995). Mathematics education models and processes. NY: Lawrence Erlbaum Associates.
  27. Freudentahl, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht : D. Reidel Publishing Company.
  28. Gearhart, M., Saxe, G. B., Seltzer, M., Schlackman, J., Cing, C. C, Nasir, N, Fall, R, Bennett, T, Rhine, S., & Sloan, T. F. (1999). Opportunities to learn fractions in elementary mathematics classrooms. Journal for Research in Mathematics Education, 30(3), pp.286-315. https://doi.org/10.2307/749837
  29. Home, M., & Mitchell, A. (2008). Fraction number line tasks and the additively concept of length measurement. In M. Goos, R. Brown, & K. Makar (Eds.) Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia(pp.353-360). MERGA.
  30. Izsak, A., Tillema, E., & Tung-Pekkan, Z. (2008). Teaching and learning fraction addition on number lines. Journal for Research in Mathematics Education, 39(1), pp.33-62.
  31. Jaslow, L, & Vik, T. (2006). Using children's understandings of linear measurement to inform instruction. In S. Z. Smith, & M. E. Smith (Eds.), Teachers engaged in research: Inquiry into mathematics classrooms, grades pre-K-2. Greenwich, CT: Information Age Publishing.
  32. Johanning, D. I. (2008). Learning to use fractions: Examining middle school students' emerging fraction literacy. Journal for Research in Mathematics Education, 39(3), pp.281-310.
  33. Keijzer, R, & Terwel, J. (2002). Audrey's acquisition of fractions: A case study into the learning of formal mathematics. Educational Studies in Mathematics, 47, pp.53-73. https://doi.org/10.1023/A:1017971912662
  34. Kerslake, D. (1986). Fractions-' Children's strategies and errors: A report of the strategies and errors in secondary mathematics project. Windsor: Berkshire.
  35. Kieren, T. E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh (Ed.), Number and measurement : Papers from a research workshop(pp.101-144). Columbus, OH: ERIC/SMEAC.
  36. Kieren, T. E. (1988). Personal knowledge of rational numbers: Its intuitive and formal development. In J. Hiebert, & M. Behr (Eds.). Number concept and operations in the middle grades(pp. 162-181). Reston, VG: Lawrence Erlbaum Associates.
  37. Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp.49-84), NJ: Lawrence Erlbaum Associates.
  38. Lamon, S. J. (1993). Ratio and proportion: Children's cognitive and metacognitive process. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of researchipp. 131-156). Hillsdale, NJ: Lawrence Erlbaum Associates.
  39. Lamon, S. J. (2001). Presenting and representing: From fractions to rational numbers. In A. A. Cuoco, & F. R. Curcio (Eds.), The roles and representations in school mathematics 2001 yearbook(pp.146-165), National council of teachers of mathematics, Reston : Virginia.
  40. Lamon, S. J. (2002). Part-whole comparisons with unitizing. In B. Litwiller, & G. Bright (Eds.). Making sense of fractions, ratios, and proportions 2002 yearbook(pp.79-86). Reston, VA: NCTM.
  41. Lamon, S. J. (2005). Teaching fractions and ratios for understanding(2th Ed.) Mahwah, NJ: Lawrence Erlbaum Associates.
  42. Mack, N. K. (1990). Learning fractions with understanding: building on informal knowledge. Journal for Research in Mathematics Education, 21(1), pp.16-32. https://doi.org/10.2307/749454
  43. Mack, N. K. (1993). Learning rational numbers with understanding: The case of informal knowledge. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp.85-105). Hillsdale, NJ: Lawrence Erlbaum Associates.
  44. Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), pp.267-295. https://doi.org/10.2307/749828
  45. Merriam, S. B. (1998). Qualitative research and case study applications in education: Revised and expanded from case study research in education. San Francisco: Jossey-Bass Publishers.
  46. Marshall, S. P. (1993). Assessment of rational number understanding: A schema-based approach. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research(pp. 261-288). Hillsdale, NJ: Lawrence Erlbaum Associates.
  47. Moseley, B. (2005). Students' early mathematical representation knowledge: The effects of emphasizing single or multiple perspectives of the rational numbers domain in problem solving. Educational Studies in Mathematics, 60, pp.37-69. https://doi.org/10.1007/s10649-005-5031-2
  48. Moss, J., & Case, R. (1999). Developing children's understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), pp.122-147. https://doi.org/10.2307/749607
  49. Noelting, E. (1978). The development of proportional reasoning in the child and adolescent through combination of logic and arithmetic. Proceeding of the Second Conference of the International Group for the, Psychology of Mathematics Education, pp. 242-277. West Germany.
  50. Pearn, C, & Stephens, M. (2007). Whole number knowledge and number lines help develop fraction concepts. 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(MERGA), Hobart, 2, pp.601-610). Sydney: MERGA.
  51. Saxe, G. B., & Gearhart, M. (1999). Rlations between classroom practices and student learning in the domain of fractions. Cognition and Instruction, 17(1), pp.1-24. https://doi.org/10.1207/s1532690xci1701_1
  52. Saxe, G. B., Taylor, E. V, Mcintosh, C, & Gearhart, M. (2005). Representing fractions with standard notation: A developmental analysis. Journal for Research in Mathematics Education, 36(2), pp. 137-157.
  53. Smith III, J. P. (2002). The development of students' knowledge of fractions and ratios. In B. Litwiller, & G, Bright (Eds.), Making sense of fractions, ratios, and proportions (pp.3-17). Reston, VA: NCTM.
  54. Stafylidou, S, & Vosniadou, S. (2004). The development of students' understanding of the numerical value of fraction. Learning and Instruction, 14(5), pp.503-518. https://doi.org/10.1016/j.learninstruc.2004.06.015
  55. Steffe, L. P. (2003). Fractional commensurate, composition, and adding schemes learning trajectories of Jason and Laura: Grade 5. Journal of Mathematical Behavior, 22, pp.237-295. https://doi.org/10.1016/S0732-3123(03)00022-1
  56. Steffe, L. P. (2004). On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning, 6(2), pp. 129-162. https://doi.org/10.1207/s15327833mtl0602_4
  57. Stephan, ML, & Clements, D. H. (2003). Linear and area measurement in prekindergarten to grade 2. In D. H. Clement, & G. Bright (Eds.), Leaning and teaching measurement 2003 yearbook(pp.3-16). Reston, VA: NCTM.
  58. Tall, D. (1991). Advanced mathematical thinking. Dordrecht Kluwer Academic Publishers.
  59. Toluk, Z., & Middleton, J. A. (2004). The development of children's understanding of the quotient: A teaching experiment. International Journal for Mathematics Teaching and Learning, 5(10), Article available online http:// www.ex.ac.uk/cimt/ijmtl/ijmenu.htm.
  60. Tzur, R. (1999). An integrated study of children's construction of improper fractions and the teacher's role in promoting that learning. Journal for Research in Mathematics Education, 30(4), pp.390-416. https://doi.org/10.2307/749707
  61. Van de Walle, J. (2006). Elementary and middle school mathematics: Teaching developmentally (6th ed.). Boston, MA: Allyn & Bacon.
  62. Vinner, S., & Dreyfus, T. (1989). Image and definition for the concept of function. Journal for Research in Mathematics Education, 20(4), pp.356-366. https://doi.org/10.2307/749441
  63. Yanik, H. B., Helding, B., & Flores, A. (2008). Teaching the concept of unit in measurement interpretation of rational numbers. Elementary Education Online, 7(3), pp.693-705.