Exploring how understandings from abstract algebra can influence the teaching of structure in early algebra

Nicholas Wasserman


Empirical work connecting content knowledge to teaching practice has become increasingly important in discussions around a professional knowledge base for teaching. This paper aims to continue that work, reporting a single exploratory case study of one elementary mathematics teacher. The purpose of the study was to explore how introducing teachers to advanced mathematics, particularly ideas in abstract algebra, might inform and influence their instructional practices for teaching early algebra topics about arithmetic structure. The findings indicate that instructional practices did change, especially in ways that more explicitly incorporated structure in early algebra with students, and fostered increased reasoning and sense-making about these ideas. Implications for mathematics teacher education are discussed.


early algebra; mathematical knowledge for teaching; abstract algebra

Full Text:



Author(s) (2013)

Author(s) (2014)

Author(s) (2015)

Author(s) (2016a)

Author(s) (2016b)

Ball, D.L., Thames, M.H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), pp. 389-407.

Common Core State Standards in Mathematics (CCSS-M). (2010). Retrieved from: http://www.corestandards.org/the-standards/mathematics

Denzin, N.K. (1978). The research act: A theoretical introduction to sociological methods (2nd ed.). New York, NY: McGraw-Hill.

Fernández, S. & Figueiras, L. (2014). Horizon content knowledge: Shaping MKT for a continuous mathematical education. REDIMAT, 3(1), pp. 7-29.

Gibson, W.J. & Brown, A. (2009). Working with qualitative data. Thousand Oaks, CA: Sage.

Harry, B., Sturges, K., & Klingner, J.K. (2005). Mapping the process: An exemplar of process and challenge in grounded theory analysis. Educational Researcher, 34(2), pp. 3-13.

Hill, H.C., Sleep, L., Lewis, J.M., & Ball, D.L. (2007). Assessing teachers’ mathematical knowledge: What knowledge matters and what evidence counts. In F.K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 111-155). Charlotte, NC: Information Age.

Jakobsen, A., Thames, M.H., & Ribeiro, C.M. (2013). Delineating issues related to horizon content knowledge for mathematics teaching. Paper presented at the Eighth Congress of European Research in Mathematics Education (CERME-8). Retrieved from: http://cerme8.metu.edu.tr/wgpapers/WG17/WG17_Jakobsen_Thames_Ribeiro.pdf

Kaput, J.J. (2008). What is algebra? What is algebraic reasoning? In J.J. Kaput, D.W. Carraher, & M.L. Blanton (Eds.), Algebra in the Early Grades (pp. 5-17). New York, NY: Lawrence Erlbaum.

Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40, pp. 173-196.

Monk, D.H. (1994). Subject area preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), pp. 125-145.

Morris, A. (1999). Developing concepts of mathematical structure: Pre-arithmetic reasoning versus extended arithmetic reasoning. Focus on Learning Problems in Mathematics, 21(1), pp. 44-67.

Rowland, T. (2012). Introduction. Retrieved from: http://www.knowledgequartet.org/introduction

Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers' mathematics subject knowledge: the knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), pp. 255-281.

Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009). Developing Primary Mathematics Teaching: Reflecting on practice with the Knowledge Quartet. London: Sage.

Shulman, L.S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), pp. 4-14.

Silverman, J., & Thompson, P.W. (2008). Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Teacher Education, 11(6), pp. 499-511.

Simon, M.A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8(4), pp. 359-371.

Slavit, D. (1999). The role of operation sense in transitions from arithmetic to algebra thought. Educational Studies in Mathematics, 37(3), pp. 251-274.

Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage.

Warren, E. (2003). The role of arithmetic structure in the transition from arithmetic to algebra. Mathematics Education Research Journal, 15(2), pp. 122-137.

Yin, R.K. (2008). Case study research: Design and methods (4th ed.). Beverly Hills, CA: SAGE.

Zazkis, R., & Leikin, R. (2010). Advanced mathematical knowledge in teaching practice: Perceptions of secondary mathematics teachers. Mathematical Thinking and Learning, 12(4), pp. 263-281.

Zazkis, R., & Mamolo, A. (2011). Reconceptualizing knowledge at the mathematical horizon. For the Learning of Mathematics, 31(2), pp. 8-13.


  • There are currently no refbacks.