Diverge Then Converge: A Strategy for Deepening Understanding Through Analyzing and Reconciling Contrasting Patterns of Reasoning

One of the challenges of teaching content courses for prospective elementary teachers (PTs) is engaging PTs in deepening their conceptual understanding of mathematics they feel they already know (Thanheiser, Philipp, Fasteen, Strand, & Mills, 2013). We introduce the Diverge then Converge strategy for orchestrating mathematical discussions that we claim (1) engenders sustained engagement with a central conceptual issue and (2) supports a deeper understanding of the issue by engaging PTs in considering both correct and incorrect reasoning. We describe a recent implementation of the strategy and present an analysis of students’ written responses that are coordinated with the phases of the discussion. We close by considering conditions under which the strategy appears particularly relevant, factors that appear to influence its effectiveness, and questions for future research.

Contributor Notes

Theresa J. Grant, Department of Mathematics, Western Michigan University, 4427 Everett Tower, Kalamazoo, Michigan 49008-5152; terry.grant@wmich.edu

Mariana Levin, Department of Mathematics, Western Michigan University, 4421 Everett Tower, Kalamazoo, Michigan 49008-5152; mariana.levin@wmich.edu

(Corresponding author is Grant terry.grant@wmich.edu)(Corresponding author is Levin mariana.levin@wmich.edu)
Mathematics Teacher Educator
  • 1.

    BallD. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal93(4) 373397.

    • Search Google Scholar
    • Export Citation
  • 2.

    BallD. L. ThamesM. H. & PhelpsG. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education59(5) 389407.

    • Search Google Scholar
    • Export Citation
  • 3.

    BeckerJ. (2004). Reconsidering the role of overcoming perturbations in cognitive development: Constructivism and consciousness. Human Development47(2) 7793.

    • Search Google Scholar
    • Export Citation
  • 4.

    BorasiR. (1994). Capitalizing on errors as “springboards for inquiry”: A teaching experiment. Journal for Research in Mathematics Education25(2) 166208.

    • Search Google Scholar
    • Export Citation
  • 5.

    ChamberlinM. T. (2005). Teachers’ discussions of students’ thinking: Meeting the challenge of attending to students’ thinking. Journal for Mathematics Teacher Education8141170.

    • Search Google Scholar
    • Export Citation
  • 6.

    DurkinK. & Rittle-JohnsonB. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction22(3) 206214.

    • Search Google Scholar
    • Export Citation
  • 7.

    FrankeM. L. KazemiE. & BatteyD. (2007). Mathematics teaching and classroom practice. Second Handbook of Research on Mathematics Teaching and Learning1(1) 225256.

    • Search Google Scholar
    • Export Citation
  • 8.

    GrantT. J. (2015). Reflecting on a decade of curriculum design: The importance of setting the tone . In T. G. Bartell K. N. Bieda R. T. Putnam K. Bradfield & H. Dominguez (Eds.) Proceedings of the thirty-seventh Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Columbus, OH: ERIC Clearinghouse on Science, Mathematics, and Environmental Education.

    • Search Google Scholar
    • Export Citation
  • 9.

    GrantT. J. & LoJ. (2009). Reflecting on the process of task adaptation and extension: The case of computational starters. In B. Clarke R. Millman & B. Grevholm (Eds.) Effective tasks in primary mathematics teacher education (pp. 2334). Springer, New York.

    • Search Google Scholar
    • Export Citation
  • 10.

    HiebertJ. CarpenterT. FennemaE. FusonK. HumanP. MurrayH. OliverA. & WearneD. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher251221.

    • Search Google Scholar
    • Export Citation
  • 11.

    Hufferd-AcklesK. FusonK. C. & SherinM. G. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education35(2) 81116.

    • Search Google Scholar
    • Export Citation
  • 12.

    KazemiE. & HintzA. (2014). Intentional talk: How to structure and lead productive mathematical discussions. Stenhouse Publishers, Portland, ME.

    • Search Google Scholar
    • Export Citation
  • 13.

    KazemiE. & StipekD. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms. Elementary School Journal1025980.

    • Search Google Scholar
    • Export Citation
  • 14.

    KelemanikG. LucentaA. & CreightonS. J. (2016). Routines for reasoning: Fostering the mathematical practices in all students. Portsmouth, NH: Heinemann.

    • Search Google Scholar
    • Export Citation
  • 15.

    LampertM. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal27(1) 2963.

    • Search Google Scholar
    • Export Citation
  • 16.

    LeathamK. R. PetersonB. E. StockeroS. L. & Van ZoestL. R. (2015). Conceptualizing mathematically significant pedagogical opportunities to build on student thinking. Journal for Research in Mathematics Education46(1) 88124.

    • Search Google Scholar
    • Export Citation
  • 17.

    LeinhardtG. & SteeleM. D. (2005). Seeing the complexity of standing to the side: Instructional dialogues. Cognition and Instruction23(1) 87163.

    • Search Google Scholar
    • Export Citation
  • 18.

    National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for mathematics. Retrieved from http://www.corestandards.org/Math.

  • 19.

    Rittle-JohnsonB. & StarJ. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology99(3) 561.

    • Search Google Scholar
    • Export Citation
  • 20.

    SantagataR. (2005). Practices and beliefs in mistake-handling activities: A video study of Italian and US mathematics lessons. Teaching and Teacher Education21(5) 491508.

    • Search Google Scholar
    • Export Citation
  • 21.

    SatyamV. R. LevinM. SmithJ. GrantT. J. VoogtK. & BaeY. (2018). Graphing as a tool for exploring students’ affective experience as mathematics learners. In A. Weinberg C. Rasmussen J. Rabin M. Wawro & S. Brown (Eds.) Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics EducationSan Diego, CA.

    • Search Google Scholar
    • Export Citation
  • 22.

    SaxeG. B. ShaughnessyM. M. ShannonA. Langer-OsunaJ. M. ChinnR. & GearhartM. (2007). Learning about fractions as points on a number line. The learning of mathematics: sixty-ninth yearbook (pp. 221237). Reston VA.

    • Search Google Scholar
    • Export Citation
  • 23.

    SchenkeK. & RichlandL. E. (2017). Preservice teachers’ use of contrasting cases in mathematics instruction. Instructional Science45(3) 311329.

    • Search Google Scholar
    • Export Citation
  • 24.

    SherinM. G. (2002). When teaching becomes learning. Cognition and Instruction20(2) 119150.

  • 25.

    SilverE. A. GhousseiniH. GosenD. CharalambousC. & StrawhunB. T. F. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. The Journal of Mathematical Behavior24(3–4) 287301.

    • Search Google Scholar
    • Export Citation
  • 26.

    SteinM. K. EngleR. A. SmithM. S. & HughesE. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning10(4) 313340.

    • Search Google Scholar
    • Export Citation
  • 27.

    SteinM. K. & SmithM. P. (2011). Five practices for orchestrating productive mathematics discussions (2nd ed.). Thousand Oaks, CA: Corw (Corwin Press).

    • Search Google Scholar
    • Export Citation
  • 28.

    Smith IIIJ. P. & StarJ. R. (2007). Expanding the notion of impact of K-12 standards-based mathematics and reform calculus programs. Journal for Research in Mathematics Education38(1) 334.

    • Search Google Scholar
    • Export Citation
  • 29.

    Smith IIIJ. P. LevinM. BaeY. SatyamV. R. & VoogtK. (2017). Exploring undergraduates’ experience of the transition to proof. In A. Weinberg C. Rasmussen J. Rabin M. Wawro & S. Brown (Eds.) Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics EducationSan Diego, CA.

    • Search Google Scholar
    • Export Citation
  • 30.

    ThanheiserE. PhilippR. FasteenF. StrandK. & MillsB. (2013). Preservice-teacher interviews: A tool for motivating mathematics learning. Mathematics Teacher Educator1(2) 137147.

    • Search Google Scholar
    • Export Citation
  • 31.

    Van ZoestL. R. PetersonB. E. LeathamK. R. & StockeroS. L. (2016). Conceptualizing the teaching practice of building on student mathematical thinking. North American Chapter of the International Group for the Psychology of Mathematics Education, Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (38th, Tucson, AZNov 3–6, 2016).

    • Search Google Scholar
    • Export Citation

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