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ZDM

, Volume 49, Issue 4, pp 585–597 | Cite as

Using comparison of multiple strategies in the mathematics classroom: lessons learned and next steps

  • Kelley Durkin
  • Jon R. Star
  • Bethany Rittle-Johnson
Original Article
First Online:

Abstract

Comparison is a fundamental cognitive process that can support learning in a variety of domains, including mathematics. The current paper aims to summarize empirical findings that support recommendations on using comparison of multiple strategies in mathematics classrooms. We report the results of our classroom-based research on using comparison of multiple strategies to help students learn mathematics, which includes short-term experimental research and a year-long randomized controlled trial using a researcher-designed supplemental Algebra I curriculum. Findings indicated that comparing different solution methods for solving the same problem was particularly effective for supporting procedural flexibility across students and for supporting conceptual and procedural knowledge among students with some prior knowledge of one of the methods, but that teachers may need additional support in deciding what to compare and when to use comparison. Drawing from this research, we offer instructional recommendations for the effective use of comparison of multiple strategies for improving mathematics learning, including (a) regular and frequent comparison of alternative strategies, particularly after students have developed some fluency with one initial strategy; (b) judicious selection of strategies and problems to compare; (c) carefully-designed visual presentation of the multiple strategies; and (d) use of small group and whole class discussions around the comparison of multiple strategies, focusing particularly on the similarities, differences, affordances, and constraints of the different approaches. We conclude with suggestions for future work on comparing multiple strategies, including the continuing need for the development of, and rigorous evaluation of, curriculum materials and specific instructional techniques that effectively promote comparison.

Keywords

Comparison Multiple strategies Explanation Mathematics learning 
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Notes

Acknowledgements

Much of the research reported in this article was supported by grants from the National Science Foundation (DRL0814571) and the Institute of Education Sciences (R305H050179); the ideas in this paper are those of the authors and do not represent official positions of NSF or IES.

References

  1. Common Core State Standards in Mathematics. (2010). Washington D.C.: National Governors Association Center for Best Practices, Council of Chief State School Officers.Google Scholar
  2. Aleven, V. A. W. M. M., & Koedinger, K. R. (2002). An effective metacognitive strategy: Learning by doing and explaining with a computer-based Cognitive Tutor. Cognitive Science, 26(2), 147–179. doi: 10.1207/s15516709cog2602_1.CrossRefGoogle Scholar
  3. Alfieri, L., Nokes-Malach, T. J., & Schunn, C. D. (2013). Learning Through Case Comparisons: A Meta-Analytic Review. Educational Psychologist, 48(2), 87–113. doi: 10.1080/00461520.2013.775712.CrossRefGoogle Scholar
  4. Australian Education Ministers (2006). Statements of learning for mathematics. Carlton South Victoria. Australia: Curriculum Corporations.Google Scholar
  5. Ball, D. L. (1993). With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics. The Elementary School Journal, 93, 373–397.CrossRefGoogle Scholar
  6. Catrambone, R., & Holyoak, K. J. (1989). Overcoming contextual limitations on problem-solving transfer. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15, 1147–1156. doi: 10.1037/0278-7393.15.6.1147.Google Scholar
  7. Chi, M. T. H. (2000). Self-explaining: The dual processes of generating inference and repairing mental models. In R. Glaser (Ed.), Advances in instructional psychology: Educational design and cognitive science (pp. 161–238). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  8. Cho, S., Holyoak, K. J., & Cannon, T. D. (2007). Analogical reasoning in working memory: Resources shared among relational integration, interference resolution, and maintenance. Memory & Cognition, 35(6), 1445–1455. doi: 10.3758/BF03193614.CrossRefGoogle Scholar
  9. Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Teacher support for collective argumentation: A framework for examining how teachers support students’ engagement in mathematical activities. Educational Studies in Mathematics, 86(3), 401–429. doi: 10.1007/s10649-014-9532-8.CrossRefGoogle Scholar
  10. Cummins, D. (1992). Role of analogical reasoning in the induction of problem categories. Journal of Experimental Psychology: Learning, Memory, and Cognition, 18(5), 1103–1124. doi: 10.1037/0278-7393.18.5.1103.Google Scholar
  11. Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22, 206–214. doi: 10.1016/j.learninstruc.2011.11.001.CrossRefGoogle Scholar
  12. Fraivillig, J. L., Murphy, L. A., & Fuson, K. (1999). Advancing children’s mathematical thinking in everyday mathematics classrooms. Journal for Research in Mathematics Education, 30, 148–170. doi: 10.2307/749608.CrossRefGoogle Scholar
  13. Franke, M. L., Turrou, A. C., Webb, N. M., Ing, M., Wong, J., Shin, N., & Fernandez, C. (2015). Student engagement with others’ mathematical ideas: The role of teacher invitation and support moves. The Elementary School Journal, 116(1), 126–148. doi: 10.1086/683174.CrossRefGoogle Scholar
  14. Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy. Cognitive Science, 7(2), 155–170. doi: 10.1207/s15516709cog0702_3.CrossRefGoogle Scholar
  15. Gentner, D. (1989). The mechanisms of analogical learning. In A. Ortony & S. Vosniadou (Eds.), Similarity and analogical reasoning (pp. 199–241). New York, NY: Cambridge University Press.CrossRefGoogle Scholar
  16. Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology, 95, 393–405. doi: 10.1037/0022-0663.95.2.393.CrossRefGoogle Scholar
  17. Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1–38. doi: 10.1016/0010-0285(83)90002-6.CrossRefGoogle Scholar
  18. Guo, J., & Pang, M. F. (2011). Learning a mathematical concept from comparing examples: The importance of variation and prior knowledge. European Journal of Psychology of Education, 26(4), 495–525. doi: 10.1007/s10212-011-0060-y.CrossRefGoogle Scholar
  19. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., … Kersting, N. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study (NCES 2003-013). US Department of Education. Washington, DC: National Center for Education Statistics.Google Scholar
  20. Hodds, M., Alcock, L., & Inglis, M. (2014). Self-explanation training improves proof comprehension. Journal for Research in Mathematics Education, 45(1), 62–101. doi: 10.5951/jresematheduc.45.1.0062.CrossRefGoogle Scholar
  21. Johnson, D. W., & Johnson, R. T. (1994). Learning together and alone: Cooperative, competitive and individualistic learning (Vol. 4th). Boston, MA: Allyn and Bacon.Google Scholar
  22. Kilpatrick, J., Swafford, J. O., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington DC: National Academy Press.Google Scholar
  23. Kotovsky, L., & Gentner, D. (1996). Comparison and categorization in the development of relational similarity. Child Development, 67, 2797–2822. doi: 10.1111/j.1467-8624.1996.tb01889.x.CrossRefGoogle Scholar
  24. Kullberg, A., Runesson, U., & Marton, F. (2017). What is made possible to learn when using the variation theory of learning in teaching mathematics? ZDM Mathematics Education, 49(4). doi: 10.1007/s11858-017-0858-4
  25. Kultusministerkonferenz. (2004). Bildungsstandards im Fach Mathematik für den Primarbereich [Educational Standards in Mathematics for Primary Schools]. Neuwied, Germany: Luchterhand.Google Scholar
  26. 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, 29–63. doi: 10.3102/00028312027001029.CrossRefGoogle Scholar
  27. Loewenstein, J., Thompson, L., & Gentner, D. (1999). Analogical encoding facilitates knowledge transfer in negotiation. Psychonomic Bulletin and Review, 6(4), 586–597. doi: 10.3758/BF03212967.CrossRefGoogle Scholar
  28. Lynch, K., & Star, J. R. (2014a). Exploring teachers’ implementation of comparison in Algebra I. The Journal of Mathematical Behavior, 35, 144–163. doi: 10.1016/j.jmathb.2014.07.003.CrossRefGoogle Scholar
  29. Lynch, K., & Star, J. R. (2014b). 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. doi: 10.5951/jresematheduc.45.1.0006.CrossRefGoogle Scholar
  30. Morrison, R. G., Krawczyk, D. C., Holyoak, K. J., Hummel, J. E., Chow, T. W., Miller, B. L., & Knowlton, B. J. (2004). A neurocomputational model of analogical reasoning and its breakdown in frontotemporal lobar degeneration. Journal of Cognitive Neuroscience, 16(2), 260–271. doi: 10.1162/089892904322984553.CrossRefGoogle Scholar
  31. Namy, L. L., & Gentner, D. (2002). Making a silk purse out of two sow’s ears: Young children’s use of comparison in category learning. Journal of Experimental Psychology: General, 131(1), 5–15. doi: 10.1037/0096-3445.131.1.5.CrossRefGoogle Scholar
  32. Newton, K. J., & Star, J. R. (2013). Exploring the Nature and Impact of Model Teaching With Worked Example Pairs. Mathematics Teacher Educator, 2(1), 86–102. doi: 10.5951/mathteaceduc.2.1.0086.CrossRefGoogle Scholar
  33. Newton, K. J., Star, J. R., & Lynch, K. (2010). Understanding the development of flexibility in struggling algebra students. Mathematical Thinking and Learning, 12(4), 282–305. doi: 10.1080/10986065.2010.482150.CrossRefGoogle Scholar
  34. Oakes, L. M., & Ribar, R. J. (2005). A Comparison of infants’ categorization in paired and successive presentation familiarization tasks. Infancy, 7(1), 85–98. doi: 10.1207/s15327078in0701_7.CrossRefGoogle Scholar
  35. Oxford English Dictionary. (2016, August 25). Compare [v.2]. Retrieved from http://www.oed.com/view/Entry/37441?rskey=moZ2mC&result=3&isAdvanced=false.
  36. Richland, L. E., Holyoak, K. J., & Stigler, J. W. (2004). Analogy use in eighth-grade mathematics classrooms. Cognition and Instruction, 22, 37–60. doi: 10.1207/s1532690Xci2201_2.CrossRefGoogle Scholar
  37. 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, 249–273. doi: 10.1016/j.jecp.2006.02.002.CrossRefGoogle Scholar
  38. Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics. Educational Psychologist, 47(3), 189–203. doi: 10.1080/00461520.2012.667065.CrossRefGoogle Scholar
  39. Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316(5828), 1128–1129. doi: 10.1126/science.1142103.CrossRefGoogle Scholar
  40. Rittle-Johnson, B. (2006). Promoting transfer: Effects of self-explanation and direct instruction. Child Development, 77(1), 1–15. doi: 10.1111/j.1467-8624.2006.00852.x.CrossRefGoogle Scholar
  41. Rittle-Johnson, B., Loehr, A. M., & Durkin, K. (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM Mathematics Education. doi: 10.1007/s11858-017-0834-z.
  42. Rittle-Johnson, B., Saylor, M., & Swygert, K. E. (2008). Learning from explaining: Does it matter if mom is listening? Journal of Experimental Child Psychology, 100(3), 215–224. doi: 10.1016/j.jecp.2007.10.002.CrossRefGoogle Scholar
  43. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346–362. doi: 10.1037/0022-0663.93.2.346.CrossRefGoogle Scholar
  44. 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(3), 561–574. doi: 10.1037/0022-0663.99.3.561.CrossRefGoogle Scholar
  45. Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101, 529–544. doi: 10.1037/a0014224.CrossRefGoogle Scholar
  46. Rittle-Johnson, B., & Star, J. R. (2011). The power of comparison in learning and instruction: learning outcomes supported by different types of comparisons. Psychology of Learning and Motivation-Advances in Research and Theory, 55, 199.CrossRefGoogle Scholar
  47. Rittle-Johnson, B., Star, J. R., & 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. doi: 10.1037/a0016026.CrossRefGoogle Scholar
  48. Rittle-Johnson, B., Star, J. R., & Durkin, K. (2012). Developing procedural flexibility: Are novices prepared to learn from comparing procedures? British Journal of Educational Psychology, 82(3), 436–455. doi: 10.1111/j.2044-8279.2011.02037.x.CrossRefGoogle Scholar
  49. Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16, 475–522. doi: 10.1207/s1532690xci1604_4.CrossRefGoogle Scholar
  50. Siegler, R. S., & Chen, Z. (2008). Differentiation and integration: Guiding principles for analyzing cognitive change. Developmental Science, 11(4), 433–448. doi: 10.1111/j.1467-7687.2008.00689.x.CrossRefGoogle Scholar
  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. doi: 10.1016/j.jmathb.2005.09.009.CrossRefGoogle Scholar
  52. Singapore Ministry of Education. (2006). Secondary mathematics syllabuses. Curriculum planning and development division. Retrieved from http://www.moe.gov.sg/education/syllabuses/sciences/files/maths-secondary.pdf.
  53. Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411.Google Scholar
  54. Star, J. R., Newton, K., Pollack, C., Kokka, K., Rittle-Johnson, B., & Durkin, K. (2015). Student, teacher, and instructional characteristics related to students’ gains in flexibility. Contemporary Educational Psychology, 41, 198–208. doi: 10.1016/j.cedpsych.2015.03.001.CrossRefGoogle Scholar
  55. Star, J. R., Pollack, C., Durkin, K., Rittle-Johnson, B., Lynch, K., Newton, K., & Gogolen, C. (2015). Learning from comparison in algebra. Contemporary Educational Psychology, 40, 41–54. doi: 10.1016/j.cedpsych.2014.05.005.CrossRefGoogle Scholar
  56. Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology, 102, 408–426. doi: 10.1016/j.jecp.2008.11.004.CrossRefGoogle Scholar
  57. Star, J. R., Rittle-Johnson, B., & Durkin, K. (2016). Comparison and explanation of multiple strategies: One example of a small step forward for improving mathematics education. Policy Insights from the Behavioral and Brain Sciences. doi: 10.1177/2372732216655543.Google Scholar
  58. Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340. doi: 10.1080/10986060802229675.CrossRefGoogle Scholar
  59. Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: Free Press.Google Scholar
  60. Treffers, A. (1991). Realistic mathematics education in the Netherlands 1980–1990. In L. Streefland (Ed.), Realistic mathematics education in primary school (pp. 11–20). Utrecht, Netherlands: Freudenthal Institute.Google Scholar
  61. Tyminski, A. M., Zambak, V. S., Drake, C., & Land, T. J. (2014). Using representations, decomposition, and approximations of practices to support prospective elementary mathematics teachers’ practice of organizing discussions. Journal of Mathematics Teacher Education, 17(5), 463–487. doi: 10.1007/s10857-013-9261-4.CrossRefGoogle Scholar
  62. VanderStoep, S. W., & Seifert, C. M. (1993). Learning “how” versus learning “when”: Improving transfer of problem-solving principles. Journal of the Learning Sciences, 3(1), 93–111. doi: 10.1207/s15327809jls0301_3.CrossRefGoogle Scholar
  63. Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111.CrossRefGoogle Scholar
  64. Webb, N. M. (1991). Task-related verbal interaction and mathematics learning in small groups. Journal for Research in Mathematics Education, 22(5), 366–389. doi: 10.2307/749186.CrossRefGoogle Scholar
  65. Webb, N. M., Franke, M. L., Ing, M., Wong, J., Fernandez, C. H., Shin, N., & Turrou, A. C. (2014). Engaging with others’ mathematical ideas: Interrelationships among student participation, teachers’ instructional practices, and learning. International Journal of Educational Research, 63, 79–93. doi: 10.1016/j.ijer.2013.02.001.CrossRefGoogle Scholar
  66. Woodward, J., Beckmann, S., Driscoll, M., Franke, M. L., Herzig, P., Jitendra, A. K., … Ogbuehi, P. (2012). Improving mathematical problem solving in grades 4 through 8: A practice guide. Washington, D. C.: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U. S. Department of Education.Google Scholar
  67. Ziegler, E., & Stern, E. (2014). Delayed benefits of learning elementary algebraic transformations through contrasted comparisons. Learning and Instruction, 33, 131–146. doi: 10.1016/j.learninstruc.2014.04.006.CrossRefGoogle Scholar
  68. Ziegler, E., & Stern, E. (2016). Consistent advantages of contrasted comparisons: Algebra learning under direct instruction. Learning and Instruction, 41, 41–51. doi: 10.1016/j.learninstruc.2015.09.006.CrossRefGoogle Scholar

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© FIZ Karlsruhe 2017

Authors and Affiliations

  1. 1.Peabody Research InstituteVanderbilt UniversityNashvilleUSA
  2. 2.Graduate School of EducationHarvard UniversityCambridgeUSA
  3. 3.Department of Psychology and Human Development, Peabody CollegeVanderbilt UniversityNashvilleUSA

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