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Nuances of Prospective Teachers’ Interpretations of Integer Word Problems

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Exploring the Integer Addition and Subtraction Landscape

Part of the book series: Research in Mathematics Education ((RME))

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Abstract

This chapter identifies the ways in which 15 prospective teachers engage the strands of mathematical proficiency as they solve word problems involving integer addition and subtraction. The prospective teachers, through think-aloud interviews, demonstrated a strong focus on solving problems using procedures, which some did not explain and others explained in detail. Number line representations were popular ways to illustrate solution methods, especially to highlight distances to and from zero. Further, some problems elicited a variety of strategies, while others mainly elicited procedures. The collective think-aloud data reveal strong, interconnected strands that could help individuals reflect on procedural versus conceptual knowledge and how best to explain and make connections among the ideas involved in the problems.

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References

  • Almeida, R., & Bruno, A. (2014). Strategies of pre-service primary school teachers for solving addition problems with negative numbers. International Journal of Mathematical Education in Science and Technology, 45(5), 719–737. https://doi.org/10.1080/0020739X.2013.877605

    Article  Google Scholar 

  • Bishop, J. P., Lamb, L. L., Philipp, R. A., Whitacre, I., & Schappelle, B. P. (2016). Leveraging structure: Logical necessity in the context of integer arithmetic. Mathematical Thinking and Learning, 18(3), 209–232. https://doi.org/10.1080/10986065.2016.1183091

    Article  Google Scholar 

  • Bofferding, L. (2010). Addition and subtraction with negatives: Acknowledging the multiple meanings of the minus sign. In P. Brosnan, D. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (pp. 703–710). Columbus, OH: The Ohio State University.

    Google Scholar 

  • Bofferding, L. (2014). Negative integer understanding: Characterizing first graders’ mental models. Journal for Research in Mathematics Education, 45(2), 194–245. https://doi.org/10.5951/jresematheduc.45.2.0194

    Article  Google Scholar 

  • Bofferding, L., & Richardson, S. E. (2013). Investigating integer addition and subtraction: A task analysis. In M. Martinez & A. Superfine (Eds.), Proceedings of the 35th annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (pp. 111–118). Chicago, IL: University of Illinois at Chicago.

    Google Scholar 

  • Bofferding, L., & Wessman-Enzinger, N. M. (2017). Subtraction involving negative numbers: Connecting to whole number reasoning. The Mathematics Enthusiast, 14, 241–262. https://scholarworks.umt.edu/tme/vol14/iss1/14

    Google Scholar 

  • Bursal, M., & Paznokas, L. (2006). Mathematics anxiety and preservice elementary teachers’ confidence to teach mathematics and science. School Science and Mathematics, 106(4), 173–180. https://doi.org/10.1111/j.1949-8594.2006.tb18073.x

    Article  Google Scholar 

  • Ericsson, K. A. (2006). Protocol analysis and expert thought: Concurrent verbalizations of thinking during experts’ performance on representative task. In K. A. Ericsson, N. Charness, P. Feltovich, & R. R. Hoffman (Eds.), Cambridge handbook of expertise and expert performance (pp. 223–242). Cambridge, UK: Cambridge University Press.

    Chapter  Google Scholar 

  • Ericsson, K. A., & Simon, H. A. (1993). Protocol analysis; Verbal reports as data (Rev. Ed.). Cambridge, MA: Bradford Books/MIT Press.

    Google Scholar 

  • Fagnant, A., Vlassis, J., & Crahay, M. (2005). Using mathematical symbols at the beginning of the arithmetical and algebraic learning. In L. Verschaffel, E. De Corte, G. Kanselaar, & M. Valcke (Eds.), Powerful environments for promoting deep conceptual and strategic learning (pp. 81–95). Leuven, Belgium: Leuven University Press.

    Google Scholar 

  • Fuson, K. (1992). Research on whole number addition and subtraction. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243–295). New York, NY: Macmillan.

    Google Scholar 

  • Gallardo, A. (1994). Negative numbers in algebra. In J. de Ponte & J. Matos (Eds.), Proceedings for the 18th international conference for the Psychology of Mathematics Education (pp. 376–383). Lisbon, Portugal: PME.

    Google Scholar 

  • Guerrero, A., & Martinez, E. D. (1982). Additive and subtractive aspects of the comparison relationship. In A. Vermandel (Ed.), Proceedings of the 6th conference of the International Group for the Psychology of Mathematics Education (pp. 150–155). Antwerp, Belgium: PME.

    Google Scholar 

  • Hertel, J., & Wessman-Enzinger, N. M. (2017). Examining Pinterest as a curriculum resource for negative integers: An initial investigation. Education Sciences, 7, 45. https://doi.org/10.3390/educsci7020045

    Article  Google Scholar 

  • Holm, J., & Kajander, A. (2011). “I finally get it”: Developing mathematical understanding during teacher education. International Journal of Mathematical Education in Science and Technology, 43(5), 563–574. https://doi.org/10.1080/0020739X.2011.622804

    Article  Google Scholar 

  • Kajander, A., & Holm, J. (2013). Pre-service teachers’ mathematical understanding: Searching for differences based on school curriculum background. Fields Mathematics Education Journal, 1, 3–20.

    Google Scholar 

  • Kilhamn, C. (2009). Making sense of negative numbers through metaphorical reasoning. In C. Bergsten, B. Grevholm, & T. Lingefjärd (Eds.), Perspectives on mathematical knowledge. Proceedings of madif6 (pp. 30–35). Linköping, Sweden: SMDF.

    Google Scholar 

  • Murray, J. C. (1985). Children’s informal conceptions of integer arithmetic. In L. Streefland (Ed.), Proceedings of the Ninth Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 147–153). Noordwijkerhout, the Netherlands: International Group for the Psychology of Mathematics Education.

    Google Scholar 

  • National Council of Teachers of Mathematics. (2014). Principles to action. Reston, VA: Author.

    Google Scholar 

  • National Research Council. (2001). In J. Kilpatrick, J. Swafford, & B. Findell (Eds.), Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press. https://doi.org/10.17226/9822

    Chapter  Google Scholar 

  • National Research Council. (1999). Serving the needs of pre-college science and mathematics education: Impact of a digital national library on teacher education and practice. Washington, DC: Author.

    Google Scholar 

  • National Research Council/National Science Foundation. (1996). From analysis to action: Undergraduate education in science, mathematics, engineering, and technology: Report of a convocation. Washington, DC: Author.

    Google Scholar 

  • Peled, I., Mukhopadhyay, S., & Resnick, L. (1989). Formal and informal sources of mental models for negative numbers. In G. Vergnaud, J. Rogalski, & M. Artique (Eds.), Proceedings of the thirteenth international conference for the Psychology of Mathematics Education (Vol. 3, pp. 106–110). Paris, France: PME.

    Google Scholar 

  • Ryan, J. T., Williams, J. S., & Doig, B. A. (1998). National tests: Educating teachers about their children’s mathematical thinking. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 81–88). Stellenbosch, South Africa: University of Stellenbosch.

    Google Scholar 

  • Schwarz, B. B., Kohn, A. S., & Resnick, L. B. (1993–1994). Positives about negatives: A case study of an intermediate model for signed numbers. Journal of the Learning Sciences, 3(1), 37–92. doi: https://doi.org/10.1207/s15327809jls0301_2.

  • Sfard, A. (2001). Balancing the unbalanceable: The NCTM Standards in light of theories of learning mathematics. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), A research companion for NCTM Standards (pp. 353–392). Reston, VA: National Council for Teachers of Mathematics.

    Google Scholar 

  • Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–401. http://www.jstor.org/stable/30034943

    Google Scholar 

  • Steiner, C. J. (2009). A study of pre-service elementary teachers’ conceptual understanding of integers. Unpublished doctoral dissertation, Kent, OH: Kent State University.

    Google Scholar 

  • Tesch, R. (1990). Qualitative research: Analysis types and software tools. New York, NY: Falmer.

    Google Scholar 

  • The Math Forum. (2016). Beginning to problem solve with “I Notice, I Wonder.” Handout. Retrieved from http://mathforum.org/pow/noticewonder/

  • Wessman-Enzinger, N. M. (2015). Developing and describing the use and learning of conceptual models for integer addition and subtraction of grade 5 students. Doctoral dissertation. Retrieved from Proquest. (Order No. 3725577).

    Google Scholar 

  • Wessman-Enzginer, N. M. (in press). Consistency of integer number sentences to temperature problems. Teaching Mathematics in the Middle School.

    Google Scholar 

  • Wessman-Enzinger, N. M., & Mooney, E. S. (2014). Making sense of integers through storytelling. Mathematics Teaching in the Middle School, 20(4), 202–205.

    Article  Google Scholar 

  • Whitacre, I., Bishop, J. P., Lamb, L. L. C., Philipp, R. A., Bagley, S., & Schappelle, B. P. (2015). ‘Negative of my money, positive of her money’: Secondary students’ ways of relating equations to a debt context. International Journal of Mathematical Education in Science and Technology, 46(2), 234–249. https://doi.org/10.1080/0020739X.2014.956822

    Article  Google Scholar 

  • Widjaja, W., Stacey, K., & Steinle, V. (2011). Locating negative decimals on the number line: Insights into the thinking of pre-service primary teachers. Journal of Mathematical Behavior, 30, 80–91. https://doi.org/10.1016/j.jmathb.2010.11.004

    Article  Google Scholar 

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Acknowledgment

Data collection was supported by a Purdue Research Foundation Grant.

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Correspondence to Laura Bofferding .

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Bofferding, L., Wessman-Enzinger, N.M. (2018). Nuances of Prospective Teachers’ Interpretations of Integer Word Problems. In: Bofferding, L., Wessman-Enzinger, N. (eds) Exploring the Integer Addition and Subtraction Landscape. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-90692-8_8

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  • DOI: https://doi.org/10.1007/978-3-319-90692-8_8

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