Browse > Article

Teaching Proportional Reasoning in Elementary School Mathematics  

Chong, Yeong Ok (Gyeongin National University of Education)
Publication Information
Journal of Educational Research in Mathematics / v.25, no.1, 2015 , pp. 21-58 More about this Journal
Abstract
The aim of this study is to look into the didactical background for teaching proportional reasoning in elementary school mathematics and offer suggestions to improve teaching proportional reasoning in the future. In order to attain these purposes, this study extracted and examined key ideas with respect to the didactical background on teaching proportional reasoning through a theoretical consideration regarding various studies on proportional reasoning. Based on such examination, this study compared and analyzed textbooks used in the United States, the United Kingdom, and South Korea. In the light of such theoretical consideration and analytical results, this study provided suggestions for improving teaching proportional reasoning in elementary schools in Korea as follows: giving much weight on proportional reasoning, emphasizing multiplicative comparison and discerning between additive comparison and multiplicative comparison, underlining the ratio concept as an equivalent relation, balancing between comparisons tasks and missing value tasks inclusive of quantitative and qualitative, algebraic and geometrical aspects, emphasizing informal strategies of students before teaching cross-product method, and utilizing informal and pre-formal models actively.
Keywords
ratio; proportion; proportional reasoning; multiplicative comparison; comparisons task; missing value task;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 고은성.이경화(2007). 초등학교 6학년 학생의 비례 추론 능력 분석. 수학교육학연구, 17(4), 359-380.
2 교육과학기술부(2011a). 수학 6-1. 서울: 두산동아(주).
3 교육과학기술부(2011b). 수학과교육과정. 교육과학기술부 고시 제2011-361호 [별책 8]. 교육과학기술부.
4 교육과학기술부(2012). 수학 5-2. 서울: 두산동아(주).
5 교육인적자원부(2007). 수학과교육과정. 교육인적자원부 고시 제 2007-79호 [별책 8]. 교육인적자원부.
6 김경선.박영희(2007). 초등학생의 비례 추론 지도에 관한 연구. 학교수학, 9(4), 447-466.
7 김경희.백희수(2010). 비와 비율 영역에 대한 우리나라와 싱가포르 교육과정 및 교과서 비교-TIMSS 평가 목표와 공개문항을 중심으로. 학교수학, 12(4), 473-491.
8 김수현.나귀수(2008). 비와 비율 지도에 대한 연구. 수학교육학연구, 18(3), 309-333.
9 나귀수.황혜정.한경혜(2001). 수학과 교육목표 및 내용체계 연구(II). 연구보고 RRC 2001-9. 한국교육과정평가원.
10 박정숙(2008). 비와 비례 과제에서 가법적 전략을 사용하는 학생의 문제해결 특징. 학교수학, 10(4), 603-623.
11 안숙현.방정숙(2008). 5, 6, 7학년 학생들의 비례 추론 능력 실태 조사. 수학교육학연구, 18(1), 103-121.
12 유현주(1995). 유리수 개념의 교수현상학적 분석과 학습-지도 방향에 관한 연구. 서울대학교 대학원 박사학위논문.
13 이경화.정영옥.나귀수.김성여(2009). 수학 국정교과서 개발 사전 기획연구. 한국과학창의 재단.
14 정은실(2003a). 비 개념에 대한 교육적 분석. 수학교육학연구, 13(3), 247-265.
15 정은실(2003b). 비 개념에 대한 역사적, 수학적, 심리적 분석. 학교수학, 5(4), 421-440.
16 정은실(2013). 초등학교 수학교과서에서의 비례 추론에 대한 연구. 수학교육학연구, 23(4), 505-516.
17 홍지연.김민경(2013). 초등수학 비구조화된 문제해결 과정에서의 비례적 추론. 학교수학, 15(4), 723-742.
18 Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws(Ed), Handbook of Research on Mathematics Teaching and Learning. A Project of the National Council of Teachers of Mathematics (pp. 296-331). New York: Macmillan Publishing Company.
19 Bell, M., Bell, J., Bretzlauf, J., Dillard, A., Flanders, J., Hartfield, R., Issacs, A., Leslie, D. A., McBride, J., Pitvorec, K., & Saecker, P. (2007). Everyday Mathematics. The University of Chicago School Mathematics Project. Student Reference Book. Chicago: Wright Group/McGrawHill.
20 Ben-Chaim, D., Keret, Y., & Ilany, B. S. (2012). Ratio and Proportion. Research and Teaching in Mathematics Teachers' Education (Pre-and In-Service Mathematics Teachers of Elementary and Middle School Classes). Rotterdam, AW: Sense Publishers.
21 Broekman, H., van der Valk, T., & Wijers, M. (2000). Teacher knowledge needed to teach ratio and proportion in secondary school mathematics. Paper presented to the 25th ATEE Conference, Barcelona, Spain.
22 Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. T. Owens(Ed.), Research Ideas for the Classroom: Middle Grades Mathematics (pp. 159-178). New York: Macmillan Publishing Company.
23 Freudenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. Dordrecht: D. Reidel Publishing Company.
24 De Bock, D., Verschaffel, L., & Janssens, D. (1998). The predominance of the linear model in secondary school students' solutions of word problems involving length and area of similar figures. Educational Studies in Mathematics, 35(1), 65-83.   DOI
25 Dwyer, N. K., Causey-Lee, B. J., & Irby, N. M. (2003). Conceptualizing ratios with look-alike polygons. Mathematics Teaching in the Middle School, 8(8), 426-431.
26 Feijs, E., de Lange, J., van Reeuwijk, M., Spence, M. S., & Brendefur, J. (2004). 어디까지 보이나요? Looking at an Angle. (나온교육연구소 역). 서울: 도서출판 이우. (영어원작은 1998년 출판).
27 Freudenthal, H., Janssen, G. M., & Sweers, W. J. (1976). Five Years IOWO. Educational Studies in Mathematics, 7(3), 186-367.
28 Harel, G., Behr, M., Lesh, R., & Post, T. (1994). Invariance of ratio: The case of children's anticipatory scheme for constancy of taste. Journal for Research in Mathematics Education, 25(4), 324-345.   DOI
29 Heath, T. L. (1956). Euclid The Thirteen Books of the Elements, Vol. 2 (Books III-IX). New York: Dover Publications, Inc.
30 Howe, C., Nunes, T., Bryant, P. (2011). Rational number and proportional reasoning: Using intensive quantities to promote achievement in mathematics and science. International Journal of Science and Mathematics Education, 9(2), 391-417.   DOI
31 Karplus, R., Pulos, S., & Stage, E. (1983). Early adolescents' proportional reasoning on 'rate' problems. Educational Studies in Mathematics, 14(3), 219-233.   DOI
32 Keijer, R., van den Heuvel-Panhuizen, M., Wijers, M., Shew, J., Brinker, L. J., Margaret, A. P., Shafer, M.C., & Brendefur, J. (2003). 늘었다 줄었다. More or Less. (나온교육연구소 역). 서울: 도서출판 이우. (영어원작은 1998년 출판).
33 Kastberg, S. E., D'Ambrosio, B., & Lynch-Davis, K. (2012). Understanding proportional reasoning for teaching. Australian Mathematics Teacher, 68(3), 32-40.
34 Keijer, R., Abels, M. Brinker, L. J., Cole, B. R., & Shew, J. A. (2004). 비와 비율. Ratios and Rate. (나온교육연구소 역). 서울: 도서출판 이우. (영어원작은 1998년 출판).
35 Keijer, R., Abels, M. Brinker, L. J., Cole, B. R., & Shew, J. A. (1998). Ratios and Rate. Teacher Guide. In T. A. Romberg (Ed.), Mathematics in Contexts: A Connected Curriculum for Grades 5-8. Chicago: Encyclopedia Britannica Educational Corporation.
36 Lamon, S. J. (1993). Ratio and proportion: Children's cognitive and metacognitive processes. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational Numbers An Integration of Research(pp. 131-156). New Jersey, Hillsdale: Lawrence Erlbaum Associates, Publishers.
37 Lamon, S. J. (2005). Teaching Fractions and Ratios for Understanding. New Jersey: Lawrence Erbaum Associates, Inc., Publishers.
38 Langrall, C. W., & Swafford, J. (2000). Three balloons for two dollars: Developing proportional reasoning. Mathematics Teaching in the Middle School, 6(4), 254-261.
39 Lanius, C. A., & Williams, S. E. (2003). Proportionality: A unifying theme for the middle grades. Mathematics Teaching in the Middle School, 8(8), 392-396.
40 Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number Concepts and Operations in the Middle Grades(pp. 93-118). Reston, VA: National Council of Teachers of Mathematics, INC.
41 Post, T., Behr, M., & Lesh, R. (1988). Proportionality and the development of pre-algebra understandings. In A. Coxford & A. Shulte (Eds.), The Idea of Algebra K-12 (1988 Yearbook, pp. 78-90). Reston, VA: National Council of Teachers of Mathematics.
42 National Council of Teachers of Mathematics (2007). 학교수학을 위한 원리와 규준. (류희찬.조완영.이경화.나귀수.김남균.방정숙 옮김). 서울: 경문사 (영어 원작은 2000년 출판).
43 Norton, S. J. (2005). The construction of proportional reasoning. In H. L. Chick, & J. L. Vincent (Eds.), Proceedings of the 29th Conference International Group for the Psychology of Mathematics Education (Vol 4, pp. 17-24). Melbourne: PME.
44 Ozgun-Koca, S. A., Altay, M. K. (2009). An investigation of proportional reasoning skills of middle school students. Investigations in Mathematics Learning, 2(1), 26-48.   DOI
45 Reys, R. E., Lindquist, M. M., Lamdin, D. V., & Smith, N. L. (2012). 초등교사를 위한 수학과 교수법. 박성선.김민경.방정숙.권점례 역. 서울; 경문사. (영어 원작은 2009년 출판).
46 Schwarz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert, & M. Behr (Eds.), Number Concepts and Operations in the Middle Grades, Vol. 2 (pp. 41-52). Reston, VA: National Council of Teachers of Mathematics.
47 Shield, M., & Dole, S. (2013). Assessing the potential of mathematics textbooks to promote deep learning. Educational Studies in Mathematics, 82(2), 183-199.   DOI
48 Streefland, L. (1984). Search for the roots of ratio: Some thoughts on the long term learning process. Part I. Educational Studies in Mathematics 15, 327-347.   DOI
49 Streefland, L. (1985). Search for the roots of ratio: Some thoughts on the long term learning process. Part II. Educational Studies in Mathematics 16, 75-94.   DOI
50 Streefland, L., Simon, A., Burrill, G., & Middleton, J. A. (1997). Grasping Sizes. Teacher Guide. In T. A. Romberg (Ed.), Mathematics in Contexts: A Connected Curriculum for Grades 5-8. Chicago: Encyclopedia Britannica Educational Corporation.
51 Streefland, L., Simon, A., Burrill, G., & Middleton, J. A. (2004). 달은 얼마나 높이 있을까? Grasping Sizes. (나온교육연구소 역). 서울: 도서출판 이우. (영어원작은 1997년 출판).
52 Sylvana, N. S. (2014). Design Research On Mathematics Education: Ratio Table In Developing The Students' Proportional Reasoning. Unpublished master dissertation, Sriwijaya University(In Collaboration between Sriwijaya University and Utrecht University), Palembang.
53 The School Mathematics Project(2003a). SMP Interact Practice for Book 7T. Cambridge University Press.
54 The School Mathematics Project(2003b). SMP Interact Practice for Book 8T. Cambridge University Press.
55 The School Mathematics Project(2003c). SMP Interact Practice for Book 9T. Cambridge University Press.
56 The School Mathematics Project(2003d). SMP Interact Practice for Book 7S. Cambridge University Press.
57 The School Mathematics Project(2003e). SMP Interact Practice for Book 8S. Cambridge University Press.
58 The School Mathematics Project(2003f). SMP Interact Practice for Book 9S. Cambridge University Press.
59 The School Mathematics Project(2003g). SMP Interact Practice for Book 9C. Cambridge University Press.
60 Thompson, P. W., & Thompson, A. G. (1994). Talking about rates conceptually, Part I: A teacher's struggle. Journal for Research in Mathematics Education, 25(3), 279-303.   DOI
61 Van den Heuvel-Panhuizen, M., Streefland, L., Middleton, J. A., & Meyer, M. R. (2004). 백 분율은 백을 좋아해. Per Sense. (나온교육연구소 역). 서울: 도서출판 이우. (영어원작은 1997년 출판).
62 Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204.   DOI
63 Treffers, A. (1987). Three Dimensions. A Model of Goal and Theory Description in Mathematics Instruction-The Wiskobas Project. Dordrecht: D. Reidel Publishing Company.
64 Van den Heuvel Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics 54, 9-35.   DOI
65 Van de Walle, J. A. (2008). 수학을 어떻게 가르칠 것인가. (남승인 외 역). 서울: 경문사. (영어 원작은 2004년 출판).
66 Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on properties of overgeneralization. Cognition and Instruction, 23(1), 57-86.   DOI
67 Van Dooren, W., De Bock, D., Janssens, D., & Verschaffel, L. (2008). The linear imperative: An inventory and conceptual analysis of students' overuse of linearity. Journal for Research in Mathematics Education, 39(3), 311-342.
68 Van Galen, F., Wijers, M., Burrill, G., & Spence, M. S. (2004). 똑같이 나누어요. Some of the Parts. (나온교육연구소 역). 서울: 도서출판 이우. (영어원작은 1997년 출판).
69 Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert, & M. Behr (Eds.), Number Concepts and Operations in the Middle Grades, Vol. 2 (pp. 141-161). Reston, VA: National Council of Teachers of Mathematics.
70 Webb, D. C., Boswinkel, N., & Dekker, T. (2008). Beneath the tip of the iceberg: Using representations to support student understanding. Mathematics Teaching in the Middle School, 14(2), 110-113.
71 WSOY(2012a). 핀란드 초등 수학교과서 Laskutaito 4-2. 양재욱.도영 역. 서울: 솔빛길출판사.
72 WSOY(2012b). 핀란드 초등 수학교과서 Laskutaito 5-1. 문보람.도영 역. 서울: 솔빛길출판사.
73 WSOY(2012c). 핀란드 초등 수학교과서 Laskutaito 5-2. 문보람.도영 역. 서울: 솔빛길출판사.
74 WSOY(2012d). 핀란드 초등 수학교과서 Laskutaito 6-1. 오수현.도영 역. 서울: 솔빛길출판사.
75 WSOY(2012e). 핀란드 초등 수학교과서 Laskutaito 6-2. 이영석.도영 역. 서울: 솔빛길출판사.