RESEARCH PAPER
Pre-service Teachers’ Perceptions of the Use of Representations and Suggestions for Students’ Incorrect Use
Mi Yeon Lee 1  
,  
 
 
More details
Hide details
1
Arizona State University, USA
2
Oakland University, USA
CORRESPONDING AUTHOR
Mi Yeon Lee   

Arizona State University
Online publish date: 2019-04-12
Publish date: 2019-04-12
 
EURASIA J. Math., Sci Tech. Ed 2019;15(9):em1745
KEYWORDS
TOPICS
ABSTRACT
In this study, we investigated how elementary pre-service teachers (PSTs) perceive using representations in teaching mathematics and what fractional representations (e.g., manipulatives or models) they suggest to guide students’ incorrect use of representations in learning fractions. A written questionnaire was administrated to 151 PSTs at a large Southwestern university in the US. An inductive content analysis approach including both qualitative and quantitative analyses was used to analyze the data. Findings suggested that fraction-related topics were the PSTs’ main choices for using representations, and they valued understanding concepts and making connections between representations and concepts. Also, the findings showed the PSTs’ tendency to use models procedurally and their predominant dependency on a few types of models (e.g., wedged circular models) in guiding students who use representations incorrectly. Implications for designing mathematics methods courses that support effective use of representations are discussed.
 
REFERENCES (75)
1.
Ball, D. L. (1992). Magical hopes: manipulatives and the reform of math education. American Educator, 16(2), 14–18.
 
2.
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407. https://doi.org/10.1177/002248....
 
3.
Behr, M. J., & Post, T. R. (1992). Teaching rational number and decimal concepts. In T. R. Post (Ed.), Teaching mathematics in grades K-8: Research-based methods (2nd ed.) (pp. 201–248). Boston: Allyn and Bacon.
 
4.
Behr, M. J., Wachsmuth, I., & Post, T. (1988). Rational number learning aids: Transfer from continuous models to discrete models. Focus on Learning Problems in Mathematics, 10(4), 1–17.
 
5.
Berenson, S. B., Valk, T. V. D., Oldham, E., Runesson, U., Moreira, C. Q., & Broekman, H. (1997). An international study to investigate prospective teachers’ content knowledge of the area concept. European Journal of Teacher Education, 20(2), 137–150. https://doi.org/10.1080/026197....
 
6.
Bezuk, N., & Cramer, K. (1989). Teaching about fractions: What, when, and how? In P. Trafton (Ed.), National Council of Teachers of Mathematics 1989 yearbook: New directions for elementary school mathematics (pp. 156–167). Reston, VA: National Council of Teachers of Mathematics.
 
7.
Borko, H., Eisenhart, M., Brown, C., Underhill, R., Jones, D., & Agard, P. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education, 23(3), 194–222. https://doi.org/10.2307/749118.
 
8.
Bosse, M. J., Lynch-Davis, K., Adu-Gyamfi, K., & Chandler, K. (2016). Using integer manipulatives: Representational determinism. International Journal for Mathematics Teaching and Learning, 17(3). Retrieved from http://www.cimt.org.uk/ijmtl/i....
 
9.
Bray, W. S., & Abreu-Sanchez, L. (2010). Using number sense to compare fractions: Reflect and discuss. Teaching Children Mathematics, 17(2), 90–97.
 
10.
Cai, J. (2005). U.S. and Chinese teachers’ constructing, knowing, and evaluating representations to teach mathematics. Mathematical Thinking and Learning, 7(2), 135–169. https://doi.org/10.1207/s15327....
 
11.
Charles, K., & Nason, R. (2000). Young children’s partitioning strategies. Educational Studies in Mathematics, 43, 191–221. https://doi.org/10.1023/A:1017....
 
12.
Clarke, D., & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72(1), 127–138. https://doi.org/10.1007/s10649....
 
13.
Clarke, D., Roche, A., & Mitchell, A. (2008). 10 practical tips for making fractions come alive and make sense. Mathematics Teaching in the Middle School, 13(7), 373–380.
 
14.
Collins, A. (2011). Representational competence: A commentary on the Greeno analysis. In T. Koschmann (Ed.), Theories of learning and research into instructional practice (pp. 105–112). New York, NY: Springer. https://doi.org/10.1007/978-1-....
 
15.
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), 111–144. https://doi.org/10.2307/749646.
 
16.
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, ratios, and proportions (pp. 41–48). Reston, VA: National Council of Teachers of Mathematics.
 
17.
Cramer, K., & Whitney, S. (2010). Learning rational number concepts and skills in elementary school classrooms. In D.V. Lambdin & F.K. Lester, Jr. (Eds.), Teaching and learning mathematics: Translating research for elementary school teachers (pp. 15–22). Reston, VA: NCTM.
 
18.
Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School, 13(8), 490–496.
 
19.
Cross, D. F., Lee, M. Y., Zeybek. Z., & Adefope, O. (2015). Delving into the pieces: Drawing connections between different domains of teacher knowledge. Paper presented at the Annual Conference of American Educational Research Association (AERA), Chicago, Illinois, USA.
 
20.
Duval, R. (2006). A cognitive analysis of problems of comprehension in the learning of mathematics. Educational Studies in Mathematics, 61, 103–131. https://doi.org/10.1007/s10649....
 
21.
Grbich, C. (2007). Qualitative data analysis: An introduction. Thousand Oaks, CA: Sage.
 
22.
Greeno, J. G., & Hall, R. P. (1997). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 78, 361–367.
 
23.
Greer, B. (2009). Representational flexibility and mathematical expertise. ZDM, 41(5), 697–702. https://doi.org/10.1007/s11858....
 
24.
Grow-Maienza, J., & Beal, S. (2005, April). What we can learn from Asian mathematics textbooks. Paper presented at the Research Presession of the National Council of Teachers of Mathematics, Anaheim, CA.
 
25.
Hackenberg, A. (2013). The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. Journal of Mathematical Behavior, 32, 538–563. https://doi.org/10.1016/j.jmat....
 
26.
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.
 
27.
Hackenberg, A. J., & Lee, M. Y. (2016). Students' distributive reasoning with fractions and unknowns. Educational Studies in Mathematics, 93(2), 245-263. https://doi.org/10.1007/s10649....
 
28.
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.
 
29.
Hodges, T. E., Cady, J., & Collins, R. L. (2008). Fraction representation: The not-so-common denominator among textbooks. Mathematics Teaching in the Middle School, 14(2), 78–84.
 
30.
Huinker, D. (2015). Representational competence: A renewed focus for classroom practice in mathematics. Wisconsin Teachers of Mathematics, 67(2), 4–8.
 
31.
Hunting, R. P. (1986). Rachel’s schemes for constructing fraction knowledge. Educational Studies in Mathematics, 17(1), 49–66. https://doi.org/10.1007/BF0030....
 
32.
Izsák, A. (2008). Mathematical knowledge for teaching fraction multiplication. Cognition and Instruction, 26(1), 95–143. https://doi.org/10.1080/073700....
 
33.
Kamii, C., Lewis, B. A., & Kirkland, L. (2001). Manipulatives: When are they useful? Journal of Mathematical Behavior, 20(1), 21–31. https://doi.org/10.1016/S0732-....
 
34.
Kawanaka, T., Stigler, J. W., & Hiebert, J. (1999). Studying mathematics classrooms in Germany, Japan and the United States: Lessons from the TIMSS videotape study. International comparisons in mathematics education, 11, 86.
 
35.
Lamon, S. J. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (2nd ed.). Mahwah, NJ: Erlbaum.
 
36.
Lamon, S. J. (2007). Rational numbers and proportional reasoning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Charlotte, NC: Information Age.
 
37.
Lee, M. Y. (2017). Pre-service teachers' flexibility with referent units in solving a fraction division problem. Educational Studies in Mathematics, 96(3), 327-348. https://doi.org/10.1007/s10649....
 
38.
Lee, M. Y., & Hackenberg, A. J. (2014). Relationships between Fractional Knowledge and Algebraic Reasoning: The case of Willa. International Journal of Science and Mathematics Education, 12(4), 975-1000. https://doi.org/10.1007/s10763....
 
39.
Lee, S. J., Brown, R. E., & Orrill, C. H. (2011). Mathematics teachers’ reasoning about fractions and decimals using drawn representations. Mathematical Thinking and Learning, 13(3), 198–220. https://doi.org/10.1080/109860....
 
40.
Lesh, R., English, L. D., Sevis, S., & Riggs, C. (2013) Modeling as a means for making powerful ideas accessible to children at an early age. In S. Hegedus, & J. Roschelle, (Eds.) The SimCalc Vision and Contributions: Democratizing Access to Important Mathematics (419–436). Springer Science+Business Media Dordrecht. https://doi.org/10.1007/978-94....
 
41.
Lesh, R., Post, T., & Behr, M. (1987). Representations and Translations among Representations in Mathematics Learning and Problem Solving. In C. Janvier, (Ed.), Problems of Representations in the Teaching and Learning of Mathematics (pp. 33–40). Hillsdale, NJ: Lawrence Erlbaum.
 
42.
Lo, J. J., & Luo, F. (2012). Prospective elementary teachers’ knowledge of fraction division. Journal of Mathematics Teacher Education, 15(6), 481–500. https://doi.org/10.1007/s10857....
 
43.
Mack, N. K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26(5), 422–441. https://doi.org/10.2307/749431.
 
44.
Martin, T., & Schwartz, D. L. (2005). Physically distributed learning: adapting and reinterpreting physical environments in the development of fraction concepts. Cognitive Science, 29(4), 587–625. https://doi.org/10.1207/s15516....
 
45.
McKendree, J., Small, C., & Stenning, K. (2002). The role of representation in teaching and learning critical thinking. Educational Review, 54, 57–67. https://doi.org/10.1080/001319....
 
46.
McNamara, J., & Shaughnessy, M. M. (2010). Beyond pizzas & pies: 10 essential strategies for supporting fraction sense, grades 3-5. Sausalito, CA: Math Solutions Publications.
 
47.
Nathan, M. J., Alibali, M. W., Masarik, K., Stephens, A. C., & Koedinger, K. R. (2010). Enhancing middle school students’ representational fluency: A classroom-based study (WCER Working Paper No. 2010-9). Retrieved from University of Wisconsin–Madison, Wisconsin Center for Education Research website: http://wcer.wisc.edu/docs/work....
 
48.
National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston: Author.
 
49.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
 
50.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: NGA & CCSSO.
 
51.
Noyes, A. (2006). Using metaphor in mathematics teacher preparation. Teaching and Teacher Education, 22, 898–909. https://doi.org/10.1016/j.tate....
 
52.
Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of educational research, 62(3), 307–332. https://doi.org/10.3102/003465....
 
53.
Pape, S. J., & Tchoshanov, M. A. (2001). The role of representation(s) in developing mathematical understanding. Theory into Practice, 40, 118–127. https://doi.org/10.1207/s15430....
 
54.
Petit, M., Laird, R. E., & Marsden, E. L. (2010). A focus on fractions: Bringing research to the classroom. New York: Taylor & Francis. https://doi.org/10.4324/978020....
 
55.
Rosli, R., Han, S., Capraro, R., & Capraro, M. (2013). Exploring preservice teachers’ computational and representational knowledge of content and teaching fractions. Journal of Korean Society of Mathematics Education, 17(4), 221–141. https://doi.org/10.7468/jksmed....
 
56.
Saxe, G. B., Diakow, R., & Gearhart, M. (2012). Towards curricular coherence in integers and fractions: A study of the efficacy of a lesson sequence that uses the number line as the principal representational context. ZDM: The International Journal on Mathematics Education, 45(3), 343–364. https://doi.org/10.1007/s11858....
 
57.
Siebert, D., & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics, 12(8), 394–400.
 
58.
Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. https://doi.org/10.1016/j.cogp....
 
59.
Soto-Andrade, J. (2007). Metaphors and cognitive styles in the teaching-learning of mathematics. In Proceedings CERME (Vol. 5, pp. 191–200).
 
60.
Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. New York: Springer. https://doi.org/10.1007/978-1-....
 
61.
TeachingWorks. (n.d.). High-leverage content. Retrieved from http://www.teachingworks.org/w....
 
62.
Tunc-Pekkan, Z. (2015). An analysis of elementary school children’s fractional knowledge depicted with circle, rectangle, and number line representations. Educational Studies in Mathematics, 89, 419–441. https://doi.org/10.1007/s10649....
 
63.
Usiskin, Z. (2007). Some thoughts about fractions. Mathematics Teaching in the Middle School, 12(7), 370-373.
 
64.
van de Walle, J., Karp, K.S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th ed.). NJ: Pearson.
 
65.
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(1), 9–35. https://doi.org/10.1023/B:EDUC....
 
66.
Vig, R., Murray, E., & Star, J. R. (2014). Model breaking points conceptualized. Educational Psychology Review, 26, 73–90. https://doi.org/10.1007/s10648....
 
67.
von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning (Vol. 6). London: Falmer. https://doi.org/10.4324/978020....
 
68.
Watanabe, T. (2002). Representations in teaching and learning fractions. Teaching Children Mathematics, 8, 457–463.
 
69.
Watanabe, T. (2006). The teaching and learning of fractions: A Japanese perspective. Teaching Children Mathematics, 12(7), 368–374.
 
70.
Watanabe, T. (2007). Initial treatment of fractions in Japanese textbooks. Focus on Learning Problems in Mathematics, 29(2), 41–60.
 
71.
Woleck, K. R. (2001). Listen to their pictures; An investigation of children’s mathematical drawings. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles of Representation in School Mathematics, 2001 Yearbook (pp. 215–227). Reston, VA: National Council of Teachers of Mathematics.
 
72.
Wu, H. H. (2011). Understanding numbers in elementary school mathematics (Vol. 79). Providence, RI: American Mathematical Society. https://doi.org/10.1090/mbk/07....
 
73.
Zazkis, R., & Gadowsky, K. (2001). Attending to transparent features of opaque representations of natural numbers. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles of Representation in School Mathematics, 2001 Yearbook (pp. 44–52). Reston, VA: National Council of Teachers of Mathematics.
 
74.
Zhang, J. (1997). The nature of external representations in problem solving. Cognitive Science, 21(2), 179–217. https://doi.org/10.1207/s15516....
 
75.
Zhang, X., Clements, M. A., & Ellerton, N. F. (2015). Conceptual mis(understandings) of fractions: From area models to multiple embodiments. Mathematics Education Research Journal, 27, 233–261. https://doi.org/10.1007/s13394....
 
eISSN:1305-8223
ISSN:1305-8215