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http://dx.doi.org/10.7468/mathedu.2022.61.1.1

Exploring fraction knowledge of the stage 3 students in proportion problem solving  

Lee, Jin Ah (Gwanggyo Middle School)
Lee, Soo Jin (Department of Mathematics Education, Korea National University of Education)
Publication Information
The Mathematical Education / v.61, no.1, 2022 , pp. 1-28 More about this Journal
Abstract
The purpose of this study is to explore how students' fractional knowledge is related to their solving of proportion problems. To this end, 28 clinical interviews with four middle-grade students, each lasting about 30~50 minutes, were carried out from May 2021 to August 2021. The present study focuses on two 7th grade students who exhibited their ability to coordinate three levels of units prior to solving whole number problems. Although the students showed interiorization of three levels of units in solving whole number problems, how they coordinated three levels of units were different in solving proportion problems depending on whether the problems required reasoning with whole numbers or fractions. The students could coordinate three levels of units prior to solving the problems involving whole numbers, they coordinated three levels of units in activity for the problems involving fractions. In particular, the ways the two students employed partitioning operations and how they coordinated quantitative unit structures were different in solving proportion problems involving improper fractions. The study contributes to the field by adding empirical data corroborating the hypotheses that students' ability to transform one three levels of units structure into another one may not only be related to their interiorization of recursive partitioning operations, but it is an important foundation for their construction of splitting operations for composite units.
Keywords
improper fractions; splitting operation for composite units; interiorization of recursive partitioning; proportion;
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Times Cited By KSCI : 3  (Citation Analysis)
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1 Lee, S. J., & Shin, J. H. (2020). Students' proportion problem solving with different units coordination stages. Journal of Educational Research in Mathematics. 30(2), 245-279. https://doi.org/10.29275/jerm.2020.05.30.2.245.   DOI
2 Steffe, L. P., & Olive, J. (2010). Children's fractional knowledge. Springer.
3 Johnson, H. L. (2015). Secondary students' quantification of ratio and rate: A framework for reasoning about change in covarying quantities. Mathematical Thinking and Learning, 17(1), 64-90. https://doi.org/10.1080/10986065.2015.981946   DOI
4 Ulrich, C. (2012). The addition and subtraction of signed quantities. Invited chapter in R. Mayes, L. Hatfield, & M. Mackritis (Eds), Quantitative reasoning and mathematical modeling: A drive for STEM integrated education and teaching in context (pp. 127-141). University of Wyoming.
5 Lee, J. Y. (2019). A study on introducing fractions in mathematics textbooks: Focused on stages of units coordination. Journal of Elementary Mathematics Education in Korea, 23(3), 323-345.
6 Lobato, J., Ellis, A. B., Charles, R. I., & Zbiek, R. M. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematicsin grades 6-8. The National Council of Teachers of Mathematics.
7 Suh, B. (2020). A study on restructuring of 'Number and operations area' in middle school mathematics curriculum. The Mathematical Education, 59(2), 167-183. https://doi.org/10.7468/mathedu.2020.59.2.167   DOI
8 Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel, & J. Confrey (Eds.), The Development of multiplicative reasoning in the learning of mathematics (pp. 181-234). SUNY Press.
9 Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education, WISDOMe Monographs (Vol. 1, pp. 33-57). University of Wyoming.
10 Yoo, J. Y., & Shin, J. H. (2020). A fourth grade student's units coordination for fractions. Education of Primary School Mathematics, 23(2), 87-116. https://doi.org/10.7468/jksmec.2020.23.2.87   DOI
11 Park, J. S. (2008). Characteristics of students' problem solving using additive strategy in ratio and proportion tasks. School Mathematics, 10(4), 605-626.
12 Behr, M., Harel, G., Post, T., & Lesh, R. (1993). Rational numbers: towards 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. 49-84). Lawrence Erlbaum.
13 Clement, J. (2000). Analysis of clinical interview: Foundations and model viability. In Lesh, R. and Kelly, A., Handbook of research methodologies of science and mathematics education (pp. 341-385). Lawrence Erlbaum.
14 Ezzy, D. (2002). Qualitative analysis: Practice and innovation. Routledge.
15 Yoo, J. Y., & Shin, J. H. (2021). Construction of recursive partitioning operation and fraction schemes with interiorized two levels of units. School Mathematics, 23(1), 1-31. https://doi.org/10.29275/sm.2021.03.23.1.1   DOI
16 Park, K. S. (2010). A critical analysis on definitions of biyoul and value of bi in elementary mathematics in Korea. Journal of Educational Research in Mathematics, 20(3). 397-411.
17 Hackenberg, A. J. (2013). The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. Journal of Mathematical Behavior, 32(3), 538-563. https://doi.org/10.1016/j.jmathb.2013.06.007   DOI
18 Hackenberg, A. J., & Tillema, E. S. (2009). Students' whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. Journal of Mathematical Behavior, 28(1), 1-18. https://doi.org/10.1016/j.jmathb.2009.04.004   DOI
19 Hong, G. J. (2013). A discussion on the terms related to ratio and rate from the revised 2007 curriculum textbook. Journal of Educational Research in Mathematics, 23(20), 285-295.
20 Hong, J. Y., & Kim, M. K. (2013). A study on children's proportional reasoning based on an ill-structured problem. School Mathematics, 15(4), 723-742.
21 Steffe, L. P. (2003). Fractional commensurate, composition, and adding scheme. Learning trajectories of Jason and Laura: Grade 5. Journal of Mathematical Behavior, 22(3), 237-295. https://doi.org/10.1016/S0732-3123(03)00022-1   DOI
22 Steffe, L. P., Liss II, D. R., & Lee, H. Y. (2014). On the operations that generate intensive quantity. In K. C. Moore, L. P. Steffe, & L. L. Hatfield (Eds.), Epistemic algebra students: Emerging models of students' algebraic knowing (Vol. 4, pp. 49-80). WISDOMe Monographs, University of Wyoming.
23 Kim, S. H., & Na, G. S. (2008). Teaching the concept of rate and ratio: Focused on using the reconstructed textbook. Journal of Educational Research in Mathematics, 18(3), 309-333.
24 Eom, S. Y., & Kwean, H. J. (2011). The analysis of 6th-grade elementary school student's proportional reasoning ability and strategy according to academic achievement. Communications of Mathematical Education, 25(3), 537-556. https://doi.org/10.7468/jksmee.2011.25.3.537   DOI
25 Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (part 1). For the Learning of Mathematics, 35(3), 2-7.
26 Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to the principles and standards for school mathematics (pp. 95-113). National Council of Teachers of Mathematics.
27 Ulrich, C. (2016). Stages in constructing and coordinating units additively and multiplicatively (part 2). For the Learning of Mathematics, 36(1), 34-39.
28 Yoo, K. W., Jong, J. W., Kim, Y. S., & Kim, H. B. (2018). Understanding qualitative research methods (pp. 75-110). Park Young Story.
29 Lee, J. A., & Lee S. J. (2020). Secondary students' reasoning about covarying quantities in quadratic function problems. Journal of Educational Research in Mathematics, 30(4), 705-732. https://doi.org/10.29275/jerm.2020.11.30.4.705   DOI
30 Liss, D. R., II (2019). The development of distributive partitioning operations. The Journal of Mathematical Behavior, 56, Article 100775 https://doi.org/10.1016/j.jmathb.2019.04.004   DOI
31 Kaput, J., & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 235-287). State University of New York Press.
32 Ahn, S. H., & Pang, J. S. (2008). A survey on the proportional reasoning ability of fifth, sixth, and seventh graders. Journal of Educational Research in Mathematics, 18(1), 103-121.
33 Carney, M. B., Smith, E., Hughes, G. R., Brendefur, J. L., & Crawford, A. (2016). Influence of proportional number relationships on item accessibility and students' strategies. Mathematics Education Research Journal, 28, 503-522. https://doi.org/10.1007/s13394-016-0177-z   DOI
34 Chong, Y. O. (2015). Teaching proportional reasoning in elementary school mathematics. Journal of Educational Research in Mathematics, 25(1), 21-58.
35 Hackenberg, A. J., & Lee, M. Y. (2015). Relationships between students' fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46(2), 196-243. https://doi.org/10.5951/jresematheduc.46.2.0196   DOI
36 Jeong, E. S. (2003a). An educational analysis on ratio concept. Journal of Educational Research in Mathematics, 13(3), 247-265.
37 Jeong, E. S. (2003b). A historical. mathematical, psychological analysis on ratio concept. School Mathematics, 5(4), 421-440.
38 Jeong, E. S. (2013). Study on proportional reasoning in elementary school mathematics. Journal of Educational Research in Mathematics, 23(4), 505-516.
39 Corbin, J., & Strauss, A. (2015). Basics of qualitative research (4th ed.). Sage Publications.
40 Hackenberg, A. J. (2010). Students' reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383-432. https://doi.org/10.1080/07370008.2010.511565   DOI
41 Kim, K. H., & Paik, H. S. (2010). Comparison of the curricula and the textbooks concerning the proportion and ratio area between Korea and Singapore. School Mathematics, 12(4), 473-491.
42 Kim, S. H., Shin, J. H., & Lee, S. J. (2018). A theoretical analysis of students' solving equal sharing problems with the framework of Steffe's partitioning operations and its application. School Mathematics, 20(1), 17-42. https://doi.org/10.29275/sm.2018.03.20.1.17   DOI
43 Ko, E. S., & Lee, K. H. (2007). Analysis on elementary students' proportional thinking: A case study with two 6-graders. Journal of Educational Research in Mathematics, 17(4), 359-380.
44 Kwon, O. N., Park, J. S., & Park, J. H. (2007). An analysis on mathematical for proportional reasoning in the middle school mathematics curriculum. The Mathematical Education, 46(3), 315-329.
45 Nabors, W. K. (2003). From fractions to proportional reasoning: A cognitive schemes of operation approach. Journal of Mathematical Behavior, 22(2), 133-179. https://doi.org/10.1016/S0732-3123(03)00018-X   DOI
46 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). National Council of Teachers of Mathematics.
47 Lobato, J., Hohensee, C., Rhodehamel, B., & Diamond, J. (2012). Using student reasoning to inform the development of conceptual learning goals: The Case of Quadratic Functions. Mathematical Thinking and Learning, 14, 85-119. https://doi.org/10.1080/10986065.2012.656362   DOI
48 Ministry of Education (2015). Mathematics curriculum. Proclamation of the Ministry of Education No. 2015-74 [Vol. 8].
49 Norton, A., & Boyce, S. (2013). A cognitive core for common state standards. Journal of Mathematical Behavior, 32(2), 266-279. https://doi.org/10.1016/j.jmathb.2013.01.001   DOI
50 Norton, A., & Boyce, S. (2015). Provoking the construction of a structure for coordinating n+1 levels of units. Journal of Mathematical Behavior, 40, 211-232. https://doi.org/10.1016/j.jmathb.2015.10.006   DOI
51 Norton, A., Boyce, S., Ulrich, C., & Phillips, N. (2015). Students' units coordination activity: A cross-sectional analysis. Journal of Mathematical Behavior, 39, 51-66. https://doi.org/10.1016/j.jmathb.2015.05.001   DOI
52 Olive, J., & Steffe, L. P. (2002). The construction of an iterative fractional scheme: The case of Joe. Journal of Mathematical Behavior, 20, 413-437. https://doi.org/10.1016/S0732-3123(02)00086-X   DOI
53 Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259-309. https://doi.org/10.1016/1041-6080(92)90005-Y   DOI
54 Shin, J. H., Lee, S. J., & Steffe, L. P. (2020). Problem solving activities of two middle school students with distinct levels of units coordination. Journal of Mathematical Behavior, 59, 1-19. https://doi.org/10.1016/j.jmathb.2020.100793   DOI
55 Shin, J. H., & Lee, S. J. (2019). Conceptual analysis for solving a missing value problem using a proportional relationship. Journal of Educational Research in Mathematics, 29(2), 227-250. https://doi.org/10.29275/jerm.2019.5.29.2.227   DOI
56 Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95-132). Lawrence Erlbaum.
57 Steffe, L. P. (2001). A new hypothesis concerning children's fractional knowledge. Journal of Mathematical Behavior, 20(3), 267-307. https://doi.org/10.1016/S0732-3123(02)00075-5   DOI
58 Steffe, L. P. (2010). Operations that produce numerical counting schemes. In L. P. Steffe & J. Olive (Eds.), Children's fractional knowledge (pp. 27-47). Springer.
59 Steffe, L. P. (2014). On children's construction of quantification. In R. Mayes, & L. Hatfield (Eds), Quantitative reasoning and mathematics and science education: Papers from an international STEM research symposium (Vol. 3, pp. 13-41). WISDOMe Monographs, University of Wyoming.
60 Lamon, S. J. (1993). Ratio and proportion: Connecting content and children's thinking. Journal for Research in Mathematics Education, 24(1), 41-61. https://doi.org/10.5951/jresematheduc.24.1.0041   DOI
61 Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629-667). Information Age Publishing and the National Council of Teachers of Mathematics.
62 Lee, J. Y., & Pang, J. S. (2016). Reconsideration of teaching addition and subtraction of fractions with different denominators: Focused on quantitative reasoning with unit and recursive partitioning. School Mathematics, 18(3), 625-645.