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Conceptual Model-Based Problem Solving

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Posing and Solving Mathematical Problems

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

Abstract

While mathematics problem-solving skills are well recognized as critical for virtually all areas of daily life and successful functioning on the job, many students with learning disabilities or difficulties in mathematics (LDM) fail to acquire these skills during their early school studies, thereby subjecting themselves to lifelong challenges with mathematical problem solving. This chapter will introduce a conceptual model-based problem-solving (COMPS) approach that aims to promote elementary students’ generalized word problem-solving skills. With the emphasis on algebraic representation of mathematical relations in cohesive mathematical models, the COMPS program makes connections among mathematical ideas; it offers elementary school teachers a way to bridge the gap between algebraic and arithmetic teaching and learning. The COMPS program may be especially helpful for students with LDM who are likely to experience disadvantages in working memory and information organization. Findings from a series of empirical research studies will be presented, and implications for elementary mathematics education will be discussed pertinent to all students meeting the new Common Core State Standards for Mathematics (CCSSM, 2012).

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References

  • Baxter, J., Woodward, J., & Olson, D. (2001). Effects of reform-based mathematics instruction in five third grade classrooms. Elementary School Journal, 101(5), 529–548.

    Article  Google Scholar 

  • Bloom, B. S. (1976). Human characteristics and school learning. New York: McGraw-Hill.

    Google Scholar 

  • Blomhøj, M. (2004). Mathematical Modelling – A Theory for Practice. In B. Clarke et al. (Eds.), International Perspectives on Learning and Teaching Mathematics (pp. 145–159). Göteborg: National Center for Mathematics Education.

    Google Scholar 

  • Blum, W., & Leiss, D. (2005). Modellieren mit der “Tanken”-Aufgaben. Mathematic Lehren, 128, 18–21.

    Google Scholar 

  • Boulineau, T., Fore, C., Hagan-Burke, S., & Burke, M. D. (2004). Use of story-mapping to increase the story-grammar text comprehension of elementary students with learning disabilities. Learning Disability Quarterly, 27, 105–121.

    Article  Google Scholar 

  • Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Duran, R., Reed, B. S., et al. (1997). Learning by understanding: The role of multiple representations in learning algebra. American Educational Research Journal, 34(4), 663–689.

    Article  Google Scholar 

  • Bryant, D. P., Bryant, B. R., Roberts, G., Vaughn, S., Pfannenstiel, K. H., Porterfield, J., et al. (2011). Early numeracy intervention program for first-grade students with mathematics difficulties. Council for Exceptional Children, 78(1), 7–23.

    Google Scholar 

  • Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.

    Google Scholar 

  • Cathcart, W. G., Pothier, Y. M., Vance, J. H., & Bezuk, N. S. (2006). Learning mathematics in elementary and middle schools: A learner-centered approach (4th ed.). Upper Saddle River, NJ: Pearson Merrill Prentice Hall.

    Google Scholar 

  • Cawley, J., Parmar, R., Foley, T. E., Salmon, S., & Roy, S. (2001). Arithmetic performance of students: Implications for standards and programming. Exceptional Children, 67, 311–328.

    Google Scholar 

  • Chen, Z. (1999). Schema induction in children’s analogical problem solving. Journal of Educational Psychology, 91, 703–715.

    Article  Google Scholar 

  • Common Core State Standards Initiative. (2012). Introduction: Standards for mathematical practice. Retrieved from http://www.corestandards.org/the-standards/mathematics

  • Ding, M., & Li, X. (2014). Transition from concrete to abstract representations: The distributive property in a Chinese textbook series. Educational Studies in Mathematics, 87(1), 103–121.

    Article  Google Scholar 

  • Fuchs, D., & Deshler, D. D. (2007). What we need to know about responsiveness to intervention (and shouldn’t be afraid to ask). Learning Disabilities Research and Practice, 22, 129–136.

    Article  Google Scholar 

  • Fuchs, L. S., Fuchs, D., & Hollenbeck, K. N. (2007). Expanding responsiveness to intervention to mathematics at first and third grade. Learning Disabilities Research and Practice, 22, 13–24.

    Google Scholar 

  • Gick, M., & Holyoak, K. H. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1–38.

    Article  Google Scholar 

  • 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). New York: Macmillan.

    Google Scholar 

  • Horner, R. D., & Baer, D. M. (1978). Multiple-probe technique: A variation of the multiple baseline. Journal of Applied Behavior Analysis, 11, 189–196.

    Google Scholar 

  • Hamson, M. J. (2003). The place of mathematical modeling in mathematics education. In S. J. Lamon, W. A. Parker, & K. Houston (Eds.), Mathematical modeling: A way of life. Chichester, England: Horwood.

    Google Scholar 

  • Hutchison, L. F. (2012). Addressing the STEM teacher shortage in American schools: Ways to recruit and retain effective STEM teachers. Action in Teacher Education, 34, 541–550.

    Article  Google Scholar 

  • Jonassen, D. H. (2003). Designing research-based instruction for story problems. Educational Psychology Review, 15, 267–296.

    Article  Google Scholar 

  • Kazdin, A. E. (1982). Single-case research designs: Methods for clinical and applied settings. New York: Oxford.

    Google Scholar 

  • Lesh, R., Doerr, H. M., Carmona, G., & Hjalmarson, M. (2003). Beyond constructivism. Mathematical Thinking and Learning, 5(2&3), 211–233.

    Article  Google Scholar 

  • Lesh, R., Landau, M., & Hamilton, E. (1983). Conceptual models in applied mathematical problem solving research. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts & processes (pp. 263–343). New York: Academic Press.

    Google Scholar 

  • Lewis, A. B. (1989). Training students to represent arithmetic word problems. Journal of Educational Psychology, 81, 521–531.

    Article  Google Scholar 

  • Maletsky, E. M., et al. (2004). Harcourt math (Indianath ed.). Chicago: Harcourt.

    Google Scholar 

  • Mevarech, Z. R., & Kramarski, B. (2008). Mathematical modeling and meta-cognitive instruction. Retrieved November 10, 2008, from http://www.icme-organisers.dk/tsg18/S32MevarechKramarski.pdf

  • National Assessment of Educational Progress Result. (NAEP, 2013). Are higher and lower performing students making gains? Retrieved November 12, 2013, from http://nationsreportcard.gov/reading_math_2013/#/gains-percentiles

  • Pressley, M. (1986). The relevance of the good strategy user model to the teaching of mathematics. Educational Psychologist, 21(1–2), 139–161.

    Article  Google Scholar 

  • Schmidt, S., & Weiser, W. (1995). Semantic structures of one-step word problems involving multiplication or division. Educational Studies in Mathematics, 28, 55–72.

    Article  Google Scholar 

  • Schwartz, J. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 41–52). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Sherin, B. (2001). How students understand physics equations. Cognition and Instruction, 19(4), 479–541.

    Article  Google Scholar 

  • Sowder, L. (1988). Children’s solutions of story problems. Journal of Mathematical Behavior, 7, 227–238.

    Google Scholar 

  • Thompson, P. (1989). Artificial intelligence, advanced technology, and learning and teaching algebra. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 135–161). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Van de Walle, J. A. (2004). Elementary and middle school mathematics: Teaching developmentally (5th ed.). Boston: Allyn and Bacon.

    Google Scholar 

  • Xin, Y. P. (2007). Word-problem-solving tasks presented in textbooks and their relation to student performance: A cross-curriculum comparison case Study. Journal of Educational Research, 100, 347–359.

    Article  Google Scholar 

  • Xin, Y. P. (2008). The effect of schema-based instruction in solving word problems: An emphasis on pre-algebraic conceptualization of multiplicative relations. Journal for Research in Mathematics Education, 39, 526–551.

    Google Scholar 

  • Xin, Y. P. (2012). Conceptual model-based problem-solving (COMPS) approach: Teach students with learning difficulties to solve math problems. Boston: Sense.

    Book  Google Scholar 

  • Xin, Y. P., Kastberg, S., & Chen, V. (2015). Conceptual Model-based Problem Solving (COMPS): A Response to Intervention Program for Students with LDM. National Science Foundation funded project.

    Google Scholar 

  • Xin, Y. P., Liu, J., & Zheng, X. (2011). A cross-cultural lesson comparison on teaching the connection between multiplication and division. School Science and Mathematics, 111(7), 354–367.

    Article  Google Scholar 

  • Xin, Y. P., Tzur, R., & Si, L. (2008). Nurturing multiplicative reasoning in students with learning disabilities in a computerized conceptual-modeling environment. Grant received from the National Science Foundation.

    Google Scholar 

  • Xin, Y. P., Wiles, B., & Lin, Y. (2008). Teaching conceptual model-based word-problem story grammar to enhance mathematics problem solving. Journal of Special Education, 42, 163–178.

    Article  Google Scholar 

  • Xin, Y. P., & Zhang, D. (2009). Exploring a conceptual model-based approach to teaching situated word problems. Journal of Educational Research, 102(6), 427–441.

    Article  Google Scholar 

  • Xin, Y. P., Tzur, R., Si, L., Hord, C., Liu, J., Park, J. Y., Cordova, M., & Ruan, L. Y. (2013, April). A comparison of teacher-delivered instruction and an intelligent tutor-assisted math problem-solving intervention program. Paper presented at the 2013 American Educational Research Association (AERA) Annual Meeting, San Francisco.

    Google Scholar 

  • Xin, Y. P., Si, L., Hord, C., Zhang, D., Cetintas, S., & Park, J. Y. (2012). The effects of computer-assisted instruction in teaching conceptual model-based problem solving. Learning Disabilities: A Multidisciplinary Journal, 18(2), 71–85.

    Google Scholar 

  • Xin, Y. P., Zhang, D., Park, J. Y., Tom, K., Whipple, A., & Si, L. (2011). A comparison of two mathematics problem-solving strategies: Facilitate algebra-readiness. Journal of Educational Research, 104, 381–395.

    Article  Google Scholar 

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Authors and Affiliations

  1. Purdue University, 100 North University Street, West Lafayette, IN, 47907-2098, USA

    Yan Ping Xin Ph.D.

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  1. Yan Ping Xin Ph.D.
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Correspondence to Yan Ping Xin Ph.D. .

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Editors and Affiliations

  1. University of Chile, Santiago, Chile

    Patricio Felmer

  2. University of Helskini, Helsinki, Finland

    Erkki Pehkonen

  3. University of Georgia, Athens, USA

    Jeremy Kilpatrick

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Xin, Y.P. (2016). Conceptual Model-Based Problem Solving. In: Felmer, P., Pehkonen, E., Kilpatrick, J. (eds) Posing and Solving Mathematical Problems. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-28023-3_14

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  • DOI: https://doi.org/10.1007/978-3-319-28023-3_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28021-9

  • Online ISBN: 978-3-319-28023-3

  • eBook Packages: EducationEducation (R0)

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