Preservice Teachers’ Algebraic Reasoning and Symbol Use on a Multistep Fraction Word Problem
Previous research on preservice teachers’ understanding of fractions and algebra has focused on one or the other. To extend this research, we examined 85 undergraduate elementary education majors and middle school mathematics education majors’ solutions and solution paths (i.e., the ways or methods in which preservice teachers solve word problems) when combining fractions with algebra on a multistep word problem. In this article, we identify and describe common strategy clusters and approaches present in the preservice teachers’ written work. Our results indicate that preservice teachers’ understanding of algebra include arithmetic methods, proportions, and is related to their understanding of a whole.
Behr, M., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–126). New York, NY: Academic Press.
Berk, D., Tabor, S. B., Gorowara, C. C., & Poetzel, C. (2009). Developing prospective elementary teachers’ flexibility in the domain of proportional reasoning. Mathematical Thinking and Learning, 11(3), 113–135.
Çağlayan, G., & Olive, J. (2011). Referential commutativity: Preservice K–8 teachers’ visualization of fraction operations using pattern blocks. In L. R. Wiest & T. Lamberg (Eds.), Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 303–311). Reno, NV.
Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2015). Children’s mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann.
Cechlárová, K., Furcoňová, K., & Harminc, M. (2014). Strategies used for the solution of a nonroutine word problem: A comparison of secondary school pupils and pre-service mathematics teachers. IM Preprint, series A, No. 4/2014. Retrieved from http://umv.science.upjs.sk/phocadownload/userupload/gabriela.novakova/A4-2014.pdf
Friedlander, A., & Michal T. (2001). Promoting multiple representations in algebra. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics: 2001 yearbook (pp. 173–185). Reston, VA: National Council of Teachers of Mathematics.
Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. New York, NY: Aldine.
Hallagan, J. E., Rule, A. C., & Carlson, L. F. (2009) Elementary school preservice teachers’ understandings of algebraic generalizations. Montana Mathematics Enthusiast, 6(1–2), 201–206.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406. doi:10.3102/00028312042002371
Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers’ mathematical knowledge: What knowledge matters and what evidence counts? In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 111–156). Charlotte, NC: Information Age.
Hunter, J. (2010). “You might say you’re 9 years old but you’re actually B years old because you’re always getting older”: Facilitating young students’ understanding of variables. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematics education: Proceedings of the 33rd Annual Conference of the Mathematics Education Research Group of Australasia (pp. 256–263). Fremantle, Australia: Mathematics Education Research Group of Australasia. Retrieved from https://www.merga.net.au/documents/MERGA33_HunterJ.pdf
Kaput, J. J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K–12 curriculum. In National Council of Teachers of Mathematics, Mathematical Sciences Education Board, & National Research Council (Eds.), The nature and role of algebra in the K–14 curriculum (pp. 25–26). Washington, DC: National Academy Press.
Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). New York, NY: Macmillan.
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 707–762). Charlotte, NC: Information Age.
Kieren, T. E. (1976). On the mathematical, cognitive and instructional foundations of rational numbers. In R. Lesh & D. Bradbard (Eds.), Number and measurement: Papers from a research workshop (pp. 101–144). Columbus, OH.
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). Mahwah, NJ: Lawrence Erlbaum.
Knuth, E. J., Stephens, A. C., McNeil, M. N., & Alibali, M.W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312.
Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. Journal of the Learning Sciences, 13(2), 129–164.
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 (Vol. 1, pp. 629–667). Charlotte, NC: Information Age.
Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30(1), 39–65.
Linsell, C., & Anakin, M. (2013). Foundation content knowledge: What do pre-service teachers need to know? In V. Steinle, L. Ball, & C. Bardini (Eds.), Mathematics education: Yesterday, today and tomorrow (Proceedings of the 36th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 442–449). Melbourne, Australia: Mathematics Education Research Group of Australasia. Retrieved from https://www.merga.net.au/documents/Linsell_et_al_MERGA36-2013.pdf
Luo, F., Lo, J.-J., & Leu, Y.-C. (2011). Fundamental fraction knowledge of preservice elementary teachers: A cross-national study in the United States and Taiwan. School Science and Mathematics, 111(4), 164–177.
MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33, 1–19.
Miles, M. B., Huberman, A. M., & Saldaña, J. (2014). Qualitative data analysis: A methods sourcebook (3rd ed.). Thousand Oaks, CA: SAGE.
Mills, B. (2012). The equal sign: An operational tendency does not mean an operational view. In L. R. Van Zoest, J.-J. Lo, & J. L. Kratky (Eds.), Proceedings of the 34th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 745–748). Kalamazoo, MI.
National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel (ED 00424P). Washington, DC: U.S. Department of Education.
Pomerantsev, L., & Korosteleva, O. (2003). Do prospective elementary and middle school teachers understand the structure of algebraic expressions? Issues in the Undergraduate Mathematics Preparation of School Teachers: The Journal, Vol. 1: Content Knowledge. Retrieved from http://www.k-12prep.math.ttu.edu/journal/1.contentknowledge/pomer01/article.pdf
Poon, K.-K., & Leung, C.-K. (2010). Pilot study on algebra learning among junior secondary students. International Journal of Mathematical Education in Science and Technology 41(1), 49–62. doi:10.1080/00207390903236434
Prediger, S. (2010). How to develop mathematics-for-teaching and for understanding: The case of meanings of the equal sign. Journal of Mathematics Teacher Education, 13(1), 73–93.
Richardson, K., Berenson, S., & Staley, K. (2009). Prospective elementary teachers use of representation to reason algebraically. Journal of Mathematical Behavior, 28(2), 188–199.
Stacey, K., & MacGregor, M. (1999). Learning the algebraic method of solving problems. Journal of Mathematical Behavior, 18(2), 149–167. doi:10.1016/S0732-3123(99)00026-7
Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). Implementing standards-based mathematics instruction: A casebook for professional development (2nd ed.). New York, NY: Teachers College Press.
Stump, S., & Bishop, J. (2002). Preservice elementary and middle school teachers’ conceptions of algebra revealed through the use of exemplary curriculum materials. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Heide, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 1903–1910). Athens, GA. Retrieved from http://files.eric.ed.gov/fulltext/ED471781.pdf
Swafford, J. O., & Langrall, C. W. (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31, 89–112.
Thanheiser, E., Browning, C., Edson, A. J., Kastberg, S., and Lo, J.-J. (2013, September 26). Building a knowledge base: Understanding prospective elementary school teachers’ mathematical content knowledge. International Journal for Mathematics Teaching and Learning. Retrieved from http://www.cimt.plymouth.ac.uk/journal/thanheiser.pdf
Tobias, J. M. (2009). Preservice elementary teachers' development of rational number understanding through the social perspective and the relationship among social and individual environments (Doctoral dissertation). Available from ProQuest Dissertations and Theses database. (UMI No. 3383698)
Tobias, J. M. (2013). Prospective elementary teachers' development of fraction language for defining the whole. Journal of Mathematics Teacher Education, 16(2), 85–103.
Van Dooren, W., Verschaffel, L., & Onghena, P. (2002). The impact of preservice teachers’ content knowledge on their evaluation of students’ strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33(5), 319–351.
Walkington, C., Sherman, M., & Petrosino, A. (2012). “Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation. Journal of Mathematical Behavior, 31(2), 174¬–195. doi:10.1016/j.jmathb.2011.12.009
Wieman, R., & Arbaugh, F. (2013). Success from the start: Your first years teaching secondary mathematics. Reston, VA: National Council of Teachers of Mathematics.
Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379–402. doi:10.1023/A:1020291317178