Investigating Mathematics Pre-service Teachers’ Knowledge for Teaching: Focus on Quadratic Equations
Keywords:
Mathematics teacher education research, Algebra, Quadratics, Mathematics content knowledge, Mathematics pedagogical content knowledgeAbstract
A substantial body of research has documented that the types of knowledge that mathematics teachers draw upon during their practice often differs from the knowledge of individuals working in other fields. Drawing on the Mathematical Knowledge for Teaching (MKT) Framework and the School Mathematics Teaching Pedagogical Content Knowledge (SMTPCK) Framework, we investigated the knowledge that mathematics preservice secondary teachers (M-PSTs) used when solving quadratic equations and talking about teaching this topic during task-based interviews. Most of the M-PSTs were able to draw on their Common Content Knowledge (in MKT) and Content Knowledge in a Pedagogical Context (in SMTPCK) for procedures such as using the quadratic formula and completing the square. the M-PSTs, however, more often struggled and expressed uncertainty when asked to draw on their Specialised Content Knowledge (in MKT) and Clearly Pedagogical Content Knowledge (in SMTPCK) to, for example, provide multiple representations to support student learning. Our findings support persistent calls from professional organisations for a series of courses in secondary mathematics teacher education programs that provide opportunities for M-PSTs to engage with and investigate secondary mathematics content from an advanced perspective. Such experiences have the potential to enhance development of several domains of M-PSTs’ MKT and SMTPCK. Similarities, differences, affordances, and limitations of MKT and SMTPCK frameworks are discussed.
References
Allaire, P. R., & Bradley, R. E. (2001). Geometric approaches to quadratic equations from other times and places. Mathematics Teacher, 94(4), 308–315. https://www.jstor.org/stable/20870675
Alvey, C., Hudson, R. A., Newton, J. & Males, L. M. (2016). Secondary pre-service teachers’ algebraic reasoning about linear equation solving. Issues in the Undergraduate Mathematics Preparation of School Teachers: The Journal, 1. https://files.eric.ed.gov/fulltext/EJ1115162.pdf
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215–241. https://doi.org/10.1023/A:1024312321077
Asquith, P., Stephens, A. C., & Knuth, E. J., & Alibali, M. W. (2007). Middle school mathematics teachers’ knowledge of students’ understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning, 9(3), 249–272. https://doi.org/10.1007/BF02655899
Association of Mathematics Teacher Educators. (2017). Standards for preparing teachers of mathematics. https://amte.net/standards
Ball, D. L., Sleep, L., Boerst, T. A., & Bass, H. (2009). Combining the development of practice and the practice of development in teacher education. The Elementary School Journal, 109(5), 458–474. https://www.jstor.org/stable/10.1086/596996
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389–407. https://doi.org/10.1177/0022487108324554
Blanton, M. (2022). The role of learning progressions in “democratizing” students’ access to algebra. In A. E. Lischka, E. B. Dyer, R. S. Jones, J. N. Lovett, J. Strayer, & S. Drown (Eds.), Proceedings of the forty-fourth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Middle Tennessee State University (pp. 30-42).
Carrillo-Yañez, J., Climent, N., Montes, M., Contreras, L. C., Flores-Medrano, E., Escudero-Ávila, D., Vasco, D., Rojas, N., Flores, P., Aguilar-González, A., Ribeiro, M., & Muñoz-Catalán, M. C. (2018). The mathematics teacher’s specialised knowledge (MTSK) model. Research in Mathematics Education, 20(3), 236–253. https://doi.org/10.1080/14794802.2018.1479981
Chick, H. L. (2007). Teaching and learning by example. In J. Watson & K. Beswick (Eds.), Mathematics: Essential research, essential practice (Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 3–21). MERGA. https://files.eric.ed.gov/fulltext/ED503746.pdf
Chick, H., & Baker, M., Pham, T., & Cheng, H. (2006, July). Aspects of teachers’ pedagogical content knowledge for decimals. In J. Novotná, H. Moraová, M. Krátká, & N. Stehliková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (pp. 297–304). PME. http://www.igpme.org/publications/current-proceedings/
Chick, H., & Beswick, K. (2018). Teaching teachers to teach Boris: A framework for mathematics teacher educator pedagogical content knowledge. Journal of Mathematics Teacher Education, 21(5), 475–499. https://doi.org/10.1007/s10857-016-9362-y
Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. American Mathematical Society and Mathematical Association of America. https://www.cbmsweb.org/archive/MET2/met2.pdf
Copur-Gencturk, Y., Tolar, T., Jacobson, E., & Fan, W. (2019). An empirical study of the dimensionality of the mathematical knowledge for teaching construct. Journal of Teacher Education, 70(5), 485–497. https://doi.org/10.1177/0022487118761860
Cuoco, A. & Rotman, J. J. (2013). Learning modern algebra: From early attempts to prove Fermat's last theorem. Mathematical Association of America.
Depaepea, F., Verschaffel, L., & Kelchtermans, G. (2013). Pedagogical content knowledge: A systematic review of the way in which the concept has pervaded mathematics educational research. Teaching and Teacher Education, 34, 12-25. https://doi.org/10.1016/j.tate.2013.03.001
Eraslan, A., & Aspinwall, L. (2007). Quadratic functions: Students' graphic and analytic representations. Mathematics Teacher, 101(3), 233–237. https://www.jstor.org/stable/20876091
Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. Aldine de Gruyter.
Gunter, D. (2016). Jumping to quadratic models. Mathematics Teacher, 110(3), 174–180. https://www.jstor.org/stable/10.5951/mathteacher.110.3.0174
Heid, M. K., Wilson, P. S., & Blume, G. W. (Ed.). (2015). Mathematical understanding for secondary teaching: A framework and classroom-based situations. Information Age.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406. https://doi.org/10.3102/00028312042002371
Hoffer, T. B., Venkataraman, L., Hedberg, E. C., & Shagle, S. (2007). Final report on the National Survey of Algebra Teachers for the National Math Panel. http://www2.ed.gov/about/bdscomm/list/mathpanel/final-report-algebra-teachers.pdf
Huang, R., & Kulm, G. (2012). Prospective middle grade mathematics teachers’ knowledge of algebra for teaching. Journal of Mathematical Behavior, 31(4), 417–430. https://doi.org/10.1016/j.jmathb.2012.06.001
International Association for the Evaluation of Educational Achievement. (2013). TIMSS 2015 assessment frameworks. TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College.
Kazemi, E., & Franke, M. L. (2004). Teacher learning in mathematics: Using student work to promote collective inquiry. Journal of Mathematics Teacher Education, 7(3), 203–235. https://doi.org/10.1023/B:JMTE.0000033084.26326.19
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Information Age.
Kind, V. (2009). Pedagogical content knowledge in science education: Perspectives and potential for progress. Studies in Science Education, 45(2), 169-204. https://doi.org/10.1080/03057260903142285
Koehler, M. J., Mishra, P., Kereluik, K., Seob Shin, T., & Graham, C. R. (2014). The Technological Pedagogical Content Knowledge Framework. In J. M. Spector, M. D. Merrill, J. Elan, & M. J. Bishop (Eds.), Handbook of research on educational communications and technology (pp. 101-111). Springer.
Kunter, M., Klusmann, U., Baumert, J., Richter, D., Voss, T., & Hachfeld, A. (2013). Professional competence of teachers: Effects on instructional quality and student development. Journal of Educational Psychology, 105(3), 805. https://doi.org/10.1037/a0032583
Lai, M. Y., & Clark, J. (2018). Extending the notion of Specialized Content Knowledge: Proposing constructs for SCK. Mathematics Teacher Education and Development, 20(2), 75–95. https://mted.merga.net.au/index.php/mted/article/view/403
Litke, E. (2020). The nature and quality of algebra instruction: Using a content-focused observation tool as a lens for understanding and improving instructional practice. Cognition and Instruction, 38(1), 57-86. https://doi.org/10.1080/07370008.2019.1616740
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum.
Maher, N., Muir, T., & Chick, H. (2015). Examining PCK in a senior secondary mathematics lesson. In M. Marshman, V. Geiger, & A. Bennison (Eds.), Mathematics Education in the Margins (Proceedings of the 38th Annual Conference of the Mathematics Education Research Group of Australasia) (pp. 389–396). MERGA. https://files.eric.ed.gov/fulltext/ED572479.pdf
McCrory, R., Floden, R., Ferrini-Mundy, J., Reckase, M. D., & Senk, S. L. (2012). Knowledge of algebra for teaching: A framework of knowledge and practices. Journal for Research in Mathematics Education, 43(5), 584–615. https://www.jstor.org/stable/10.5951/jresematheduc.43.5.0584
Merriam, S. B., & Tisdell, E. J. (2016). Qualitative research: A guide to design and implementation. Jossey-Bass.
Moses, R. P., & Cobb, C. E. (2001). Radical equations. Beacon.
Murray, E., Durkin, K., Chao, T., Star, J. R., & Vig, R. (2018). Exploring connections between content knowledge, pedagogical content knowledge, and the opportunities to learn mathematics: Findings from the TEDS-M dataset. Mathematics Teacher Education and Development, 20(1), 4–22. https://mted.merga.net.au/index.php/mted/article/view/310
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Author.
Newton, J., Maeda, Y., Alexander, V., & Senk, S. (2014). How well are secondary mathematics teacher education programs aligned with the recommendations made in MET II? Notices of the American Mathematical Society, 61(3), 292-295. https://www.ams.org/journals/notices/201403/201403-full-issue.pdf
Organisation for Economic Co-operation and Development. (2019). PISA 2018 assessment and analytical framework. OECD. https://doi.org/10.1787/19963777
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), 255–281. https://doi.org/10.1007/s10857-005-0853-5
Schifter, D. (2001). Learning to see the invisible: What skills and knowledge are needed to engage with students’ mathematical ideas? In T. Wood, B. S. Nelson, & J. Warfield (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics (pp. 109–134). Erlbaum.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. https://www.jstor.org/stable/24636916
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1–22. https://doi.org/10.17763/haer.57.1.j463w79r56455411
Speer, N. M., King, K. D., & Howell, H. (2015). Definitions of mathematical knowledge for teaching: Using these constructs in research on secondary and college mathematics teachers. Journal of Mathematics Teacher Education, 18, 105–122. https://doi.org/10.1007/s10857-014-9277-4
Stacey, K., & McGregor, M. (1999). Learning the algebraic method of solving problems. Journal of Mathematical Behavior, 18, 149–167. https://doi.org/10.1016/S0732-3123(99)00026-7
Stein, M. K., Kaufman, J. H., Sherman, M., & Hillen, A. F. (2011). Algebra: A challenge at the
crossroads of policy and practice. Review of Educational Research, 81(4), 453–492. https://doi.org/10.3102/0034654311423025
Strauss, A. L. (1987). Qualitative analysis for social scientists. Cambridge University.
Sultan, A., & Artzt, A. F. (2010). The mathematics that every secondary school math teacher needs to know. Routledge.
Teppo, A. R. (2015). Grounded theory methods. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education: Examples of methodology and methods (pp. 3-21). Springer.
Teuscher, D., Dingman, S. W., Nevels, N. N., & Reys, B. J. (2008). Curriculum standards, course requirements, and mandated assessments for high school mathematics. Journal of Mathematics Education Leadership, 10(2), 49–55.
Vaiyavutjamai, P., & Clements, M. A. (2006). Effects of classroom instruction on students' understanding of quadratic equations. Mathematics Education Research Journal, 18(1), 47–77. https://doi.org/10.1007/BF03217429
Zakaria, E., & Maat, S. M. (2010). Analysis of students’ error in learning of quadratic equations. International Education Studies, 3(3), 105–110. https://doi.org/10.5539/ies.v3n3p105
