Skip to main content
SpringerLink
Log in

Balancing on the edge of competency-oriented versus procedural-oriented practices: orchestrating whole-class discussions of complex mathematical problems

  • Original Article
  • Published:
Mathematics Education Research Journal Aims and scope Submit manuscript

Abstract

Establishing and maintaining key mathematical practices in whole-class discussions, such as justifying claims, representing mathematical objects and making connections between the representations, is crucial to the development of students' mathematical competencies. The aim of this article is to investigate how and why the establishment of key mathematical practices in whole-class discussions varies in a teaching mathematics through problem-solving project. Analyses of transcribed video-recorded whole-class discussions result in the suggestion that the complexity of the mathematical problem itself as well as the complexity related to teaching the problem may contribute to more procedure-oriented practices instead of competency-oriented practices that create opportunities for the students to develop their mathematical competencies on a broad front. However, the results also suggest that the teaching of complex mathematical problems might develop the teacher's establishment and maintenance of key mathematical practices. Researchers initiating an intervention project hence have to consider both the students' and the teacher's learning trajectories, which might not always coincide. Other important aspects for the researcher to consider are discussed and pointed out as important areas for future research, such as how explicit introduction of appropriate frameworks may support teachers in learning to teach mathematics through problem solving.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. For example, the formula 4n + 1 for the number of black tiles in the Mosaic problem was classified to be partly justified since it was explicitly stated where the term 1 originate from (the black tile in the middle of the pattern), but it was not stated where the term 4n originate from.

  2. Each classroom interaction was partitioned into several episodes in which the interaction focused on a specific topic.

References

  • Adler, J., & Davis, Z. (2006). Opening another black box: researching mathematics for teaching in mathematics teacher education. Journal for Research in Mathematics Education, 37, 270–296.

    Google Scholar 

  • Adler, J., & Davis, Z. (2011). Modelling teaching in mathematics teacher education and the constitution of mathematics for teaching. In K. Ruthven & T. Rowland (Eds.), Mathematical knowledge in teaching (pp. 139–160). London: Springer.

    Chapter  Google Scholar 

  • Askew, M., Brown, M., Rhodes, V., Johnson, D., & Wiliam, D. (1997). Effective teachers of numeracy. London: School of Education, King’s College.

    Google Scholar 

  • Ball, D. L. (1993). With an eye on the mathematical horizon: dilemmas of teaching elementary school mathematics. The Elementary School Journal, 94, 373–397.

    Article  Google Scholar 

  • Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: what makes it special? Journal of Teacher Education, 59, 389–407.

    Article  Google Scholar 

  • Bednarz, N., Kieran, C., & Lee, L. (1996). Approaches to algebra: Perspectives for research and teaching. Dordrecht: Kluwer.

    Google Scholar 

  • Bergqvist, E., Bergqvist, T., Boesen, J., Helenius, O., Lithner, J., Palm, T., & Palmberg, B. (2009). Matematikutbildningens mål och undervisningens ändamålsenlighet [The goals of mathematics education and the adequateness of instruction; in Swedish]. Gothenburg: National Center for Mathematics Education.

    Google Scholar 

  • Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex, and setting. Philadelphia: Open University Press.

    Google Scholar 

  • Boaler, J. (2002). Exploring the nature of mathematical activity: using theory, research and ‘working hypothesis’ to broaden conceptions of mathematics knowing. Educational Studies in Mathematics, 51, 3–21.

    Article  Google Scholar 

  • Choppin, J. (2011). The role of ‘local theories’: teacher knowledge and its impact on engaging students with challenging tasks. Mathematics Education Research Journal, 23, 5–25.

    Article  Google Scholar 

  • Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32, 9–13.

    Article  Google Scholar 

  • Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10, 113–163.

    Article  Google Scholar 

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.

    Article  Google Scholar 

  • Emanuelsson, J., & Sahlström, F. (2008). The price of participation: teacher control versus student participation in classroom interaction. Scandinavian Journal of Educational Research, 52, 205–223.

    Article  Google Scholar 

  • Franke, L. M., Kazemi, E., & Battey, D. (2007). Understanding teaching and classroom practice in mathematics. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 225–257). Charlotte: Information Age Publishing.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven: Yale University Press.

    Google Scholar 

  • Larsson, M. (2007). 32 rika problem i matematik [32 rich mathemacal problems; in Swedish]. Stockholm: Liber.

  • Larsson, M., & Ryve, A. (2011). Effective teaching through problem‐solving by sequencing and connecting student solutions. In G. H. Gunnarsdóttir, F. Hreinsdóttir, G. Pálsdóttir, M. Hannula, M. Hannula‐Sormunen, E. Jablonka, U. T. Jankvist, A. Ryve, P. Valero, & K. Waege (Eds.), Proceedings of NORMA11: The sixth Nordic conference on mathematics education in Reykjavik, May 11–14 2011 (pp. 425–434). Reykjavik: University of Iceland Press.

  • Leikin, R., & Rota, S. (2006). A case study on the development of teacher’s proficiency in managing an inquiry-based classroom. Mathematics Education Research Journal, 18, 44–68.

    Article  Google Scholar 

  • Leinhardt, G., & Steele, M. D. (2005). Seeing the complexity of standing to the side: instructional dialogues. Cognition and Instruction, 23(1), 87–163.

    Article  Google Scholar 

  • Lester, F. K., & Lambdin, D. V. (2004). Teaching mathematics through problem solving. In B. Clarke, D. M. Clarke, G. Emanuelsson, B. Johansson, D. V. Lambdin, F. K. Lester, A. Wallby, & K. Wallby (Eds.), Proceedings of the Midsummer World Mathematics Education Conference, Göteborg, Sweden: National Center for Mathematics Education (NCM): International Perspectives on Learning and Teaching Mathematics (pp. 189–204). Gothenburg: National Center for Mathematics Education.

    Google Scholar 

  • Linell, P. (1998). Approaching dialogue: Talk and interaction in dialogical perspectives. Amsterdam: John Benjamins Publishing.

    Google Scholar 

  • Lithner, J., Bergqvist, E., Bergqvist, T., Boesen, J., Palm, T., & Palmberg, B. (2010). Mathematical competencies: A research framework. In C. Bergsten, E. Jablonka, & T. Wedege (Eds.), Mathematics and mathematics education: Cultural and social dimensions: Proceedings from the seventh mathematics education research seminar held in Stockholm (pp. 157–167). Linköping: SMDF.

    Google Scholar 

  • Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65–86). Dordrecht: Kluwer.

    Google Scholar 

  • Mason, J. (1998). Enabling teachers to be real teachers: necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1, 243–267.

    Article  Google Scholar 

  • Mercer, N., & Littleton, K. (2007). Dialogue and the development of children’s thinking: A sociocultural approach. London: Routledge.

    Google Scholar 

  • Muir, T. (2008). Principles of practice and teacher actions: influences on effective teaching of numeracy. Mathematics Education Research Journal, 20, 78–101.

    Article  Google Scholar 

  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics.

    Google Scholar 

  • National Research Council (2001). Adding it up: Helping children learn mathematics. In J. Kilpatrick, J. Swafford, & B. Findell (Eds.), Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academic Press.

    Google Scholar 

  • Nilsson, P., & Ryve, A. (2010). Focal event, contextualization, and effective communication in the mathematics classroom. Educational Studies in Mathematics, 74, 241–258.

    Google Scholar 

  • Niss, M., & Jensen, T. H. (Eds.). (2002). Kompetencer og matematiklæring - Ideer og inspiration til udvikling af matematikundervisningen i Danmark [Competencies and the learning of mathematics - Ideas and inspiration for the development of mathematics education in Denmark; in Danish]. Copenhagen: Undervisningsministeriets forlag.

    Google Scholar 

  • Rowland, T. (2008). The purpose, design and use of examples in the teaching of elementary mathematics. Educational Studies in Mathematics, 69, 149–163.

    Article  Google Scholar 

  • Ryve, A., Larsson, M., & Nilsson, P. (2011). Analyzing content and participation in classroom discourse: dimensions of variation, mediating tools, and conceptual accountability. Scandinavian Journal of Educational Research. Advance online publication. doi:10.1080/00313831.2011.628689.

  • Ryve, A., Nilsson, P., & Mason, J. (2012). Establishing mathematics for teaching within classroom interactions in teacher education. Educational Studies in Mathematics, 81, 1–14.

    Google Scholar 

  • Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Skovsmose, O., & Borba, M. (2004). Research methodology and critical mathematics education. In P. Valero & R. Zevenbergen (Eds.), Researching the socio-political dimensions of mathematics education: Issues of power in theory and methodology (pp. 207–226). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Stein, M. K., Boaler, J., & Silver, E. A. (2003). Teaching mathematics through problem solving: Research perspectives. In H. L. Schoen & R. I. Charles (Eds.), Teaching mathematics through problem solving: Grades 6–12 (pp. 245–256). Reston: National Council of Teachers of Mathematics.

    Google Scholar 

  • 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, 313–340.

    Article  Google Scholar 

  • 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 

  • Weber, K., Maher, C., Powell, A., & Lee, H. S. (2008). Learning opportunities from group discussions: warrants become the objects of debate. Educational Studies in Mathematics, 68, 247–261.

    Article  Google Scholar 

  • Wertsch, J. V. (2007). Mediation. In H. Daniels, M. Cole, & J. V. Wertsch (Eds.), The Cambridge Companion to Vygotsky (pp. 178–192). Cambridge: Cambridge University Press.

    Google Scholar 

  • Wertsch, J. V., & Kazak, S. (2011). Saying more than you know in instructional settings. In T. Koschmann (Ed.), Theorizing learning practice (pp. 134–167). Mahwah: Erlbaum.

    Google Scholar 

  • Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Education, Culture and Communication, Mälardalen University, Box 883, 721 23, Västerås, Sweden

    Maria Larsson & Andreas Ryve

Authors
  1. Maria Larsson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Ryve
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Larsson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Larsson, M., Ryve, A. Balancing on the edge of competency-oriented versus procedural-oriented practices: orchestrating whole-class discussions of complex mathematical problems. Math Ed Res J 24, 447–465 (2012). https://doi.org/10.1007/s13394-012-0049-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13394-012-0049-0

Keywords

Search

Navigation