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http://dx.doi.org/10.7468/mathedu.2012.51.1.021

An Analysis of Students' Understanding of Operations with Whole Numbers and Fractions  

Kim, Kyung-Mi (Center for Curriculum and Instruction studies, Korea University)
Whang, Woo-Hyung (Dept. of Math. Education, Korea University)
Publication Information
The Mathematical Education / v.51, no.1, 2012 , pp. 21-45 More about this Journal
Abstract
The purpose of the study was to investigate how students understand each operations with whole numbers and fractions, and the relationship between their knowledge of operations with whole numbers and conceptual understanding of operations on fractions. Researchers categorized students' understanding of operations with whole numbers and fractions based on their semantic structure of these operations, and analyzed the relationship between students' understanding of operations with whole numbers and fractions. As the results, some students who understood multiplications with whole numbers as only situations of "equal groups" did not properly conceptualize multiplications of fractions as they interpreted wrongly multiplying two fractions as adding two fractions. On the other hand, some students who understood multiplications with whole numbers as situations of "multiplicative comparison" appropriately conceptualize multiplications of fractions. They naturally constructed knowledge of fractions as they build on their prior knowledge of whole numbers compared to other students. In the case of division, we found that some students who understood divisions with whole numbers as only situations of "sharing" had difficulty in constructing division knowledge of fractions from previous division knowledge of whole numbers.
Keywords
Whole Numbers; fractions; Addition; Subtraction; Multiplication; Division; Understanding;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
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1 Schmittau, J. (1991). A theory-driven inquiry into the conceptual structure of multiplication. Focus on Learning Problems in Mathematics, 13(4), 50-64.
2 Sharp, J., & Adams, B. (2002). Children' constructions of knowledge for fraction division after solving realistic problems. The Journal of Educational Research, 95(6), 333-347.   DOI
3 Siebert, I. (2002). Connecting informal thinking and algorithms: The case of division of fraction. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions (pp.247-256). Reston, VA: NCTM.
4 Sierpinska, A. (1994). Understanding in mathematics. London, NY: Routledge.
5 Sinicrope, R., Mick, H. W., & Kolb, J. R. (2002). Interpretations of fraction division. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions (pp.153-161). Reston, VA: NCTM.
6 Skemp, R. (1987). The psychology of learning mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.
7 Slavit, D. (1999). The role of operation sense in transitions from arithmetic to algebraic thought. Educational Studies in Mathematics, 37, 251-274.
8 Steffe, L. P. (2004). On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning, 6(2), 129-162.   DOI
9 Steffe, L. P., & Olive, J. (1993). Children's mathematical learning in computer microworlds. Paper presented at the International Conference on Rational Number Learning, University of Georgia, Athens, GA.
10 Streefland, L. (1984). Unmasking N-Distracters as a source of failures in learning fractions. In B. Southwell (Ed.), Proceedings of the 8th International Conference for the Psychology of Mathematics Education. Sydney: Mathematical Association of New South Wales.
11 Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm ofdevelopmental research. Dordrecht, The Netherlands: Kluwer Academic Publications.
12 Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25.   DOI
13 Tirosh, D., Wilson, J., Graeber, A., & Fischbein, E.(1993). Conceptual adjustments in progressing from whole to rational numbers. Paper presented at the International Conference on Rational Number Learning, University of Georgia, Athens, GA.
14 Toluk, Z., & Middleton, J. A. (2004). The development of students'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.
15 Peled, I., & Segalis, B. (2007). It''s Not Too Late to Conceptualize: Constructing a Generalized Subtraction Schema by Abstracting and Connecting Procedures. Mathematical Thinking and Learning, 7(3), 207-230.
16 Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26, 165-190.   DOI
17 Riley, M., Greeno, J. G., & Heller, J. I. (1983). Development of students' problem-solving ability in arithmetic. In H. P. Ginsberg (Ed.), The development of mathematical thinking (pp.53-196). NY: Academic Press.
18 Sáenz-Ludlow, A. (2003). A collective chain of signification in conceptualizing fractions. Journal of Mathematical Behavior, 22, 181-211.   DOI
19 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.
20 Moss, J., & Case, R. (1999). Developing students'understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122-147.   DOI
21 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 (Vol. 2, pp.601-610). Hobart, Sydney: MERGA.
22 Ni, Y., & Zhou, Y. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27-52.   DOI
23 Nickerson, R. S. (1985). Understanding understanding. American Journal of Education, 93(2), 201-239.   DOI
24 Olive, J. (1999). From fractions to rational numbers of arithmetic: A reorganization hypothesis. Mathematical Thinking and Learning, 1, 279-314.   DOI
25 김경미․황우형 (2009). 분수의 덧셈, 뺄셈에 대한 아동의 이해 분석. 한국수학교육학회지 시리즈 E <수학교육논문집>, 23(3), 707-734.
26 김경미․황우형 (2011). 분수의 곱셈과 나눗셈에 대한 학생의 이해와 문장제 해결의 관련성 분석. 한국수학교육학회지 시리즈 A <수학교육>, 50(3), 337-356.
27 남승인․서찬숙 (2004). 문제 장면의 모델화를 통한 수업이 곱셈적 사고력과 곱셈 능력 신장에 미치는 영향. 한국수학교육학회지 시리즈 C <초등수학교육>, 8(1),33-50.
28 이종욱 (2007). 한 초등학교 2학년 아동의 곱셈과 나눗셈 해결 전략에 관한 사례 연구. 한국수학교육학회지 시리즈 A <수학교육>, 46(2), 155-171.
29 황우형․김경미 (2008). 자연수의 사칙연산에 대한 아동의 이해 분석, 한국수학교육학회지 시리즈 A <수학교육>, 47(4), 519-543.
30 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 (Vol. 2, pp.49-56). Melbourne: PME.
31 Carraher, D. W. (1993). Building rational number concepts upon students' prior understanding. Paper presented at the International Conference on Rational Number Learning, University of Georgia, Athens, GA.
32 Barmby, P., Harries, T., & Higgins, S. (2009). The array representation and primary children's understanding and reasoning in multiplication. Educational Studies in Mathematics, 70, 217-241.   DOI
33 Behr, M. J., Wachsmuth, I., Post, T., & Lesh, R. (1984). Order and equivalence of rational numbers. Journal for Research in Mathematics Education, 15, 323-341.   DOI
34 Biddlecomb, B. D. (2002). Numerical knowledge as enabling and constraining fraction knowledge: an example of the reorganization hypothesis. Journal of Mathematics Behavior, 21, 167-190.   DOI
35 Cramer, K., & Henry, A. (2002). Using manipulative models to build number sense for addition of fractions. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, rations, and proportions (pp. 41-48). Reston, Virginia: NCTM.
36 Davis, G., Hunting, R., & Pearn, D. (1993). What might a fraction mean to a child and would a teacher know? Journal of Mathematical Behavior, 12, 63-76.
37 Doerr, H., & Bowers, J. (1999). Revealing pre-service teachers' thinking about functions through concept mapping. In F. Hitt & M. Santos (Eds.), Proceedings of the 21st Annual Meeting of PME-NA (Vol. 1, pp.364-369). Columbus, OH:ERIC.
38 English, L. D., & Halford, G. S. (1995). Mathematics education models and processes. NY: Lawrence Erlbaum Associates.
39 Fischbein, E., Deri, M., Nello, M. S., & Merino, M. S.(1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16, 3-17.   DOI
40 Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht : Reidel Publishing Company.
41 Hackengerg, A. J. & Tillema, E. S. (2009). Students' whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. The Journal of Mathematical Behavior, 28, 1-18.   DOI
42 Fuson, K. (1992). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp.243-275). NY: Macmillan.
43 Gray, E. M. (1993). The transition from whole number to fraction: The need for flexible thinking. Paper presented at the International Conference on Rational Number Learning, University of Georgia, Athens, GA.
44 Greer, B. (1992). Multiplication and division as models of situations. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp.276-295). NY: Macmillan.
45 Herscovics, N. (1996). The construction of conceptual schemes in mathematics. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin & B. Greer (Eds.), Theories of mathematics learning (pp.351-379). NJ: Lawrence Erlbaum Associates.
46 Hiebert, J., & Carpenter, T. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics research and teaching (pp.65-100). NY: MacMillan.
47 Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
48 Huinker, D. (2002). Examining dimensions of fraction operation sense. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions (pp.72-78). Reston, VA: NCTM.
49 Hunting, R. P. (1986). Rachel's schemes for constructing fraction knowledge. Educational Studies in Mathematics, 17, 49-66.   DOI
50 Hunting, R. P., Davis, G., & Pearn, C. A. (1997). The role of whole number knowledge in rational number learning. In F. Biddulph & K. Carr (Eds.), People in mathematics education. Proceedings of the 20th Annual Conference of the Mathematics Education Research Group of Australasia (pp.239-246). Rotorua, New Zealand: MERGA.
51 Leikin, R., Chazan, D., & Yerushalmy, M. (2001). Understanding teachers' changing approaches to school algebra: Contributions of concept maps as part of clinical interviews. Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp.289-296). Utrecht, Netherlands: Freudenthal Institute.
52 Lesh, R., & Landau, M. (1983). Acquisition of mathematics concepts and processes. NY: Academic Press.
53 Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp.33-40). Hillsdale, NJ: Lawrence Erlbaum Associates.
54 Mack, N. K. (1990). Learning fractions with understanding: building on informal knowledge. Journal for Research in Mathematics Education, 21(1), 16-32.   DOI
55 Vai senstein, A. (2006) . A Look at a Chi ld's Understanding of Mathematical Ideas through His Representations. In Smith, S. Z., & Smith, M. E. (Eds.), Teachers engaged in research: Inquiry into mathematics classrooms, grades pre-K-2 (pp. 95-108). Greenwich, CT: Information Age Publishing.
56 Van de Walle, J. (2006). Elementary and middle school mathematics: Teaching developmentally (6thed.). Boston, MA: Allyn & Bacon.
57 Vinner, S., & Dreyfus, T. (1989). Image and definition for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-366.   DOI
58 Wu, Z. (2001). Multiplying fractions. Teaching Children Mathematics, 8(3), 174-177.
59 Wilcox, S. K., & Sahloff, M. (1998). Another perspective on concept maps: Empowering students. Mathematics Teaching in the Middle School, 3, 464-469.
60 Williams, C. (1998). Using concept maps to access conceptual knowledge of function. Journal for Research in Mathematics Education, 29(4), 414-421.   DOI
61 Yanik, H. B., Helding, B., & Flores, A. (2008). Teaching the concept of unit in measurement interpretation of rational numbers. Elementary Education Online, 7(3), 693-705.