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Preservice teachers’ use of contrasting cases in mathematics instruction

Abstract

Drawing comparisons between students’ alternative solution strategies to a single mathematics problem is a powerful yet challenging instructional practice. We examined 80 preservice teachers’ when asked to design a short lesson when given a problem and two student solutions—one correct and one incorrect. These micro-teaching events were videotaped and coded, revealing that fewer than half of participants (43%) made any explicit comparison or contrasts between the two solution strategies. Those who did were still not likely to use additional support strategies to draw students’ attention to key elements of the comparison. Further, correlations suggest that participants’ mathematical content knowledge may be related to whether participants’ showed contrasting cases but not to whether they used specific pedagogical cues to support those comparisons. While these micro-teaching events differ from the interactive constraints of a classroom, they reveal that participants did not immediately orient toward differing student solutions as a discussion opportunity, and that future instruction on contrasting cases must highlight the utility of this practice.

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Notes

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    Please note that because the prompt was originally an item on the CKT-M assessment form b, the student solutions cannot be provided publicly. For more information see Hill et al. (2005, 2008).

References

  1. Alfieri, L., Nokes-Malach, T. J., & Schunn, C. D. (2013). Learning through case comparisons: a meta-analytic review. Educational Psychologist, 48(2), 87–113.

  2. Alibali, M. W., Nathan, M. J., Wolfgram, M. S., Church, R. B., Jacobs, S. A., Johnson Martinez, C., et al. (2014). How teachers link ideas in mathematics instruction using speech and gesture: A corpus analysis. Cognition and Instruction, 32(1), 65–100.

  3. 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.

  4. Baroody, A. J., & Ginsburg, H. P. (1983). The effects of instruction on children’s understanding of the” equals” sign. The Elementary School Journal, 84(2), 199–212.

  5. Bartell, T. G., Webel, C., Bowen, B., & Dyson, N. (2013). Prospective teacher learning: recognizing evidence of conceptual understanding. Journal of Mathematics Teacher Education, 16(1), 57–79.

  6. Begolli, K., & Richland, L. E. (2016). Teaching mathematics by comparison: Analog visibility as a double-edged sword. Journal of Educational Psychology, 108(2), 194–213. doi:10.1037/edu0000056.

  7. Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34.

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

  9. Common Core State Standards Initiative (2010). Common Core State Standards for mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

  10. Dalehefte, I. M., Prenzel, M., & Seidel, T. (2012). Reflecting on learning from errors in school instruction: Findings and suggestions from a Swiss-German video study. In J. Bauer & C. Harteis (Eds.), Human fallibility. The ambiguity of errors for work and learning. Dordrecht: Springer.

  11. Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22(3), 206–214.

  12. English, L. D. (Ed.). (1997). Mathematical reasoning: analogies, metaphors, and images. Mahwah, NJ: Erlbaum.

  13. English, L. D. (2004). Promoting the development of young children’s mathematical and analogical reasoning. In L. D. English (Ed.), Mathematical and analogical reasoning of young learners. Mahwah, NJ: Lawrence Erlbaum.

  14. Fang, Z. (1996). A review of research on teacher beliefs and practices. Educational Research, 38(1), 47–65.

  15. Gick, M. L., & Holyoak, K. L. (1980). Analogical problem solving. Cognitive Psychology, 15, 306–355.

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

  17. Große, C. S., & Renkl, A. (2006). Effects of multiple solution methods in mathematics learning. Learning and Instruction, 16(2), 122–138.

  18. Große, C. S., & Renkl, A. (2007). Finding and fixing errors in worked examples: Can this foster learning outcomes? Learning and Instruction, 17(6), 612–634.

  19. Grossman, P. L., Valencia, S. W., Evans, K., Thompson, C., Martin, S., & Place, N. (2000). Transitions into teaching: Learning to teach writing in teacher education and beyond. Journal of Literacy Research, 32(4), 631–662.

  20. Hadfield, O. D., Littleton, C. E., Steiner, R. L., & Woods, E. S. (1998). Predictors of preservice elementary teacher effectiveness in the micro-teaching of mathematics lessons. Journal of Instructional Psychology, 25(1), 34–48.

  21. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28, 524–549.

  22. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., et al. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 Video Study., NCES 2003-013 Washington, DC: U.S. Department of Education, National Center for Education Statistics.

  23. Hill, H., Ball, D. L., & Schilling, S. (2008). Unpacking “pedagogical content knowledge”: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.

  24. Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42, 371–406.

  25. Kajander, A. (2007). Unpacking mathematics for teaching: A study of preservice elementary teachers’ evolving mathematical understandings and beliefs. Journal of Teaching and Learning, 5(1), 22–54.

  26. Kieran, C. (1981). Concepts associated with the equality symbol. Educational studies in Mathematics, 12(3), 317–326.

  27. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

  28. Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for research in Mathematics Education, 37(4), 297–312.

  29. Korthagen, F. A., & Kessels, J. P. (1999). Linking theory and practice: Changing the pedagogy of teacher education. Educational Researcher, 28(4), 4–17.

  30. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29–63.

  31. Lynch, K., & Star, J. R. (2014). Views of struggling students on instruction incorporating multiple strategies in Algebra I: An exploratory study. Journal for Research in Mathematics Education, 45(1), 6–18.

  32. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers understanding of fundamental mathematics in China and the United States. Hillsdale, NJ: Erlbaum.

  33. McDonald, M., Kazemi, E., Kelley-Petersen, M., Mikolasy, K., Thompson, J., Valencia, S. W., et al. (2014). Practice makes practice: Learning to teach in teacher education. Peabody Journal of Education, 89(4), 500–515.

  34. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.

  35. Nespor, J. (1987). The role of beliefs in the practice of teaching. Journal of Curriculum Studies, 19(4), 317–328.

  36. Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110.

  37. Philipp, R. A. (2007). Mathematics teachers. Beliefs and affect. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning. Reston, VA: National Council of Teachers of Mathematics.

  38. Richland, L. E., & McDonough, I. (2010). Learning by analogy: Discriminating between potential analogs. Contemporary Educational Psychology, 35(1), 28–43.

  39. Richland, L. E., Holyoak, K. J., & Stigler, J. W. (2004). Analogy generation in eighth grade mathematics classrooms. Cognition and Instruction, 22(1), 37–60.

  40. Richland, L. E., Morrison, R. G., & Holyoak, K. J. (2006). Children’s development of analogical reasoning: Insights from scene analogy problems. Journal of Experimental Child Psychology, 94(3), 249–273.

  41. Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316, 1128–1129.

  42. Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99, 561–574.

  43. Rittle-Johnson, B., Star, J., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836–852.

  44. Ross, B. H. (1987). This is like that: The use of earlier problems and the separation of similarity effects. Journal of Experimental Psychology, 13(4), 629–639.

  45. Santagata, R. (2004). “Are you joking or are you sleeping”. Cultural beliefs and practices in Italian and U.S. teachers’ mistake-handling strategies. Linguistics and Education, 15, 141–164.

  46. Santagata, R. (2005). Practice and beliefs in mistake-handling activities: A video study of Italian and US mathematics lessons. Teaching and Teacher Education, 21, 491–508.

  47. Santagata, R., & Yeh, C. (2014). Learning to teach mathematics and to analyze teaching effectiveness: Evidence from a video-and practice-based approach. Journal of Mathematics Teacher Education, 17(6), 491–514.

  48. Schwartz, D. L., Chase, C. C., Oppezzo, M. A., & Chin, D. B. (2011). Practicing versus inventing with contrasting cases: The effects of telling first on learning and transfer. Journal of Educational Psychology, 103(4), 759–775.

  49. Seaman, C. E., & Szydlik, J. E. (2007). Mathematical sophistication among preservice elementary teachers. Journal of Mathematics Teacher Education, 10(3), 167–182.

  50. Siegler, R. S. (2002). Microgenetic studies of self-explanation. In N. Garnott & J. Parziale (Eds.), Microdevelopment: A process-oriented perspective for studying development and learning (pp. 31–58). Cambridge: Cambridge University Press.

  51. Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287–301.

  52. Smith, M., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. Reston: National Council of Teachers of Mathematics.

  53. Star, J. R., Caronongan, P., Foegen, A., Furgeson, J., Keating, B., Larson, M. R., Lyskawa, J., McCallum, W. G., Porath, J., & Zbiek, R. M. (2015). Teaching strategies for improving algebra knowledge in middle and high school students (NCEE 2014-4333). Washington, DC: National Center for Education Evalua- tion and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education. Retrieved from the NCEE website: http://whatworks.ed.gov.

  54. Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Helping teachers learn to better incorporate student thinking. Mathematical Thinking and Learning, 10(4), 313–340.

  55. Stigler, J. W., & Hiebert, J. (2004). Improving mathematics teaching. Educational Leadership, 61(5), 12–17.

  56. Stockero, S. L. (2008). Using a video-based curriculum to develop a reflective stance in prospective mathematics teachers. Journal of Mathematics Teacher Education, 11(5), 373–394.

  57. Van Es, E. A., & Sherin, M. G. (2002). Learning to notice: Scaffolding new teachers’ interpretations of classroom interactions. Journal of Technology and Teacher Education, 10(4), 571–595.

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Acknowledgements

This work was supported by a National Science Foundation CAREER Award to the second author, NSF#0954222, and an NSF Science of Learning Center, SPE 0541957. We would like to thank the three anonymous reviewers for their feedback on previous versions on this paper.

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Correspondence to Katerina Schenke.

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Schenke, K., Richland, L.E. Preservice teachers’ use of contrasting cases in mathematics instruction. Instr Sci 45, 311–329 (2017). https://doi.org/10.1007/s11251-017-9408-2

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Keywords

  • Teacher cognition and practices
  • Professional development
  • Mathematics education