How Middle Grade Teachers Think about Algebraic Reasoning


  • David Glassmeyer Kennesaw State University
  • Belinda Edwards Kennesaw State University


algebraic reasoning, in-service teachers, functional thinking, generalizations


Algebraic reasoning is an essential habit of mind for building conceptual knowledge in K-12 mathematics, yet little is known about how middle school mathematics teachers think about algebraic reasoning. In this article we describe a research project examining how algebraic reasoning was considered by grades 6, 7, or 8 mathematics teachers in a two-week professional development and over the following two months. We found these 21 teachers initially described algebraic reasoning in a way requiring only procedural knowledge to solve problems with a single solution, solution strategy, or representation. Teachers reported three activities influenced a shift in their thinking about algebraic reasoning, specifically by requiring conceptual knowledge to solve problems using multiple solutions, solution strategies, or representations. While some teachers also associated aspects of generalization and functional thinking as part of algebraic reasoning, two months after the professional development no teachers continued to associate these aspects as part of algebraic reasoning. These findings suggest the kinds of activities other teacher educators can use to develop teachers’ thinking about algebraic reasoning, and supports the need for additional research and interventions to support middle school teachers’ consideration of algebraic reasoning in more advanced ways.

Author Biographies

David Glassmeyer, Kennesaw State University

Assistant Professor of Mathematics Education

Mathematics Department

Kennesaw State University

Belinda Edwards, Kennesaw State University

Associate Professor of Mathematics Education

Mathematics Department

Kennesaw State University



Atweh, B. & Goos, M. (2011). The Australian mathematics curriculum: A move forward or back to the future? Australian Journal of Education, 55(3), 183–278.

Blanton, M. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. Portsmouth, NA: Heinemann.

Blanton, M., & Kaput, J. (2011). Functional Thinking as a Route Into Algebra in the Elementary Grades. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 5-23): Springer Berlin Heidelberg.

Blanton, M., & Kaput, J. (2004). Elementary grades students’ capacity for functional thinking. Proceedings of the International Group for the Psychology of Mathematics Education, 2, 135-142. Bergen, Norway: Bergen University College.

Blair, S. L., & Rich, B. S. (2011). Characterizing the development of specialized mathematical content knowledge for teaching in algebraic reasoning and number theory. Mathematical Thinking and Learning, 13(4), 292-321.

Borko, H. (2004). Professional development and teacher learning: Mapping the terrain. Educational Researcher, 33(8), 3 – 15.

Borko, H., Frykholm, J.A., Pittman, M., Eiteljor, E., Nelson, M., Jacobs, J., Clark, K., & Schneider, C. (2005). Preparing teachers to foster algebraic thinking. Zentralblatt fur Didaktik der Mathematik: International Reviews on Mathematical Education, 37(1), 43-52.

Bottoms, G. (2003). Getting students ready for Algebra I: What middle grades students need to know and be able to do. Atlanta, GA: Southern Regional Education Board.

Bush, S., & Karp, K. (2013). Prerequisite algebra skills and associated misconceptions of middle grade students: A review. The Journal of Mathematical Behavior, 32, 613-632.

Cai, J. & Knuth, E. (2011). Introduction: The development of students’ algebraic thinking in earlier grads from curricular, instructional and learning perspectives. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 1-5): Springer Berlin Heidelberg.

Carpenter, T., Franke, M., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann.

Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Ernest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87–115.

Carraher, D. W., Schliemann, A. D., & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the Early Grades (pp. 235–272), New York, NY: Routledge.

Ellis, A. B. (2011). Generalizing-promoting actions: How classroom collaborations can support students’ mathematical generalizations. Journal for Research in Mathematics Education, 42(4), 308-345.

Georgia Department of Education. (2015). Accelerated pre-calculus, Georgia Standards of Excellence Frameworks. Retrieved 27 July 2015 from

Greenes, C. E., & Findell, C., (1999). Developing students’ algebraic reasoning abilities. In L.V. Stiff (Ed), Developing Mathematical Reasoning in Grades K–12, 127-37.

Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades. Mahwah, NJ: Lawrence Erlbaum/Taylor & Francis Group & National Council of Teachers of Mathematics.

Kaput, J. & Blanton, M. (2005). Algebrafying the elementary mathematics experience in a teacher-centered, systemic way. In T.A. Romber, T.P. Carpenter, & F. Dremock (Eds.), Understanding Mathematics and Science Matters. Mahwah, NJ: Lawrence Erlbaum Associates.

Kaput, J. & Blanton, M. (2008). Algebra from a symbolization point of view. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 19-55). New York: Lawrence Erlbaum/NCTM.

Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Educator, Singapore, 8(1), 139-151.

Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lister (ed.). Second Handbook of Research on Mathematics Teaching and Learning (pp. 707-762). Reston, VA: NCTM.

Knuth, E. J., Alibali, M. W., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13(9), 514-519.

Lee, L. (2001). Early algebra: But which algebra? In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra. Proceedings of the 12th ICMI Study Conference, (pp. 392-399).

Lesh, R., Landau, M., & Hamilton, E. (1983). Conceptual models and applied mathematical problem-solving research. Acquisition of mathematics concepts and processes, 263-343.

National Council of Teachers of Mathematics (2001). Principles and standards for school mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA.

National Curriculum Board. (2011). Shape of the Australian curriculum: Mathematics. Retrieved 29 May 2015 from Curriculum_-_Maths.pdf

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC.

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

Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. 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 Behavior, 24(3), 287-301.

Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 95-132). New York: Erlbaum.

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.