Examining Two Middle School Mathematics Teachers’ Knowledge for Teaching Manipulation of Algebraic Expressions during Lesson Planning and Instruction

Year 2019, , 588 - 616, 13.12.2019

Dilek Girit Yıldız , Didem Akyüz

https://doi.org/10.16949/turkbilmat.487243

Abstract

Teachers use
their content and pedagogical content knowledge for teaching algebra. For this
reason, the examination of how teachers use this knowledge may help shed light
on how students learn algebra, especially in determining why they usually have
difficulties. The aim of the current study is to reveal what teachers know, and
propose what they actually need to know for teaching the simplification and
equivalence of algebraic expressions. The multiple-case study design was used
for this study to compare and contrast the two middle school teachers’ lesson
planning and instruction. The data corpus included lesson plans, actual
instruction records, and post-observation interviews. Data analysis was
conducted using the Mathematical Knowledge for Teaching (MKT) model. The
findings indicated that both teachers had a lack of specialized content
knowledge about mathematical representations such as algebra tiles. They did
not use algebra tiles effectively and could not link algebraic and geometric
representations that underlie the idea of multiplication. It was observed that
both teachers generally used unknowns and variables interchangeably indicating
the inadequacy of their common content knowledge. In the planning process, the
two teachers were able to state the common misconceptions that the students
generally had and the ways of addressing them. Through the cases of these two
teachers, it was observed that teachers need to have a good conceptual
mathematical understanding and also knowledge of students’ thinking in order to
design effective lessons. Based on the findings, the types of knowledge that
the teachers need to have are outlined and the theoretical and practical implications
of the study are discussed.

Keywords

Equivalence of algebraic expressions, mathematical knowledge for teaching, simplification of algebraic expressions, middle school mathematics teacher

References

• An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school teachers in China and the U.S. Journal of Mathematics Teacher Education, 7, 145-172.
• Artigue, M., Assude, T., Grugeon, B., & Lenfant, A. (2001). Teaching and learning algebra: Approaching complexity through complementary perspectives. In H. Chick, H. K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra (pp. 21-32). Australia: University of Melbourne.
• Ashlock, R. B. (2006). Error patterns in computation: Using error patterns to improve instruction. Colombus: OH: Merrill Prentice Hall.
• Aslan-Tutak, F., & Ertaş, F. G. (2013, February). Practices to enhance preservice secondary teachers’ specialized content knowledge. Paper presented at the Eighth Congress of European Research in Mathematics Education (CERME 8), Antalya, Turkey.
• Aydın, E., & Gündoğdu, L. (2016). Middle school 6th grade mathematics textbook. Ankara, Turkey: Sevgi Publications.
• Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.
• Bair, 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.
• Banerjee, R., & Subramaniam, K. (2012). Evolution of a teaching approach for beginning algebra. Educational Studies in Mathematics, 80(3), 351-367.
• Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412-446.
• Blömeke, S., & Delaney, S. (2012). Assessment of teacher knowledge across countries: A review of the state of research. ZDM Mathematics Education, 44, 223-247.
• Booth, L. R. (1988). Children’s difficulties in beginning algebra. In A. F. Coxford & A. P. Shulte (Eds.), The ideas of algebra, K-12 (pp. 299-306). Reston: National Council of Teachers of Mathematics.
• Caglayan, G. (2013). Prospective mathematics teachers' sense-making of polynomial multiplication and factorization modeled with algebra tiles. Journal of Mathematics Teacher Education, 16(5), 349-378.
• Cai, J., Ng, S. F., & Moyer, J. C. (2011). Developing students’ algebraic thinking in earlier grades: Lessons from China and Singapore. In J. Cai, & E., Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives. (pp. 25-41). Verlag Berlin Heidelberg: Springer.
• Capraro, M. M., & Joffrion, H. (2006). Algebraic equations: Can middle-school students meaningfully translate from words to mathematical symbols? Reading Psychology, 27(2-3), 147-164.
• Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87-115.
• Chick, H. L., Baker, M., Pham, T., & Cheng, H. (2006). Aspects of teachers’ pedagogical content knowledge for decimals. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 297-304). Prague: PME.
• Cochran, K. F., DeRuiter, J. A., & King, R. A. (1993). Pedagogical content knowing: An integrative model for teacher preparation. Journal of Teacher Education, 44(4), 263-272.
• Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five traditions (2nd ed.). Thousand Oaks, CA: Sage Publications.
• Ding, M., & Heffernan, K. (2018). Transferring specialized content knowledge to elementary classrooms: Preservice teachers’ learning to teach the associative property. International Journal of Mathematical Education in Science and Technology, 49(6), 899-921.
• Doerr, H. M. (2004). Teachers’ knowledge and the teaching of algebra. In K. Stacey, & H. Chick (Eds.), The Future of the Teaching and Learning of Algebra: The 12th ICMI Study (pp. 267-290). Dordrecht, The Netherlands: Kluwer.
• Dogbey, J. (2016). Using variables in school mathematics: Do school mathematics curricula provide support for teachers? International Journal of Science and Mathematics Education, 14(6), 1175-1196.
• El Mouhayar, R. R., & Jurdak, M. E. (2013). Teachers’ ability to identify and explain students’ actions in near and far figural pattern generalization tasks. Educational Studies in Mathematics, 82(3), 379-396.
• Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21(6), 521-544.
• Even, R., Tirosh, D., & Robinson, N. (1993). Connectedness in teaching equivalent algebraic expressions: Novice versus expert teachers. Mathematics Education Research Journal, 5(1), 50-59.
• Fennema, E., & Franke, M. L. (1992). Teachers' knowledge and its impact. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 147-164). NewYork: Macmillan.
• Ferrini-Mundy, J., McCrory, R., & Senk, S. (2006, April). Knowledge of algebra teaching: Framework, item development, and pilot results. Paper presented at the Research Presession of Annual Meeting of the National Council of Teachers of Mathematics. St. Louis: MO.
• Filloy, E., & Sutherland, R. (1996). Designing curricula for teaching and learning algebra. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (Vol. 1, pp. 139-160). Dordrecht, the Netherlands: Springer.
• Gallardo, A. (2000). Historical-epistemological analysis in mathematics education: Two works in didactics of algebra. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspective on school algebra (pp. 121-139). Dordrecht, The Netherlands: Kluwer Academic Publishers.
• Graeber, A., & Tirosh, D. (2008). Pedagogical content knowledge: Useful concept or elusive notion. In P. Sullivan, & T. Wood (Eds.), The international handbook of mathematics teacher education (pp. 117-132). Rotterdam: Sense Publishers.
• Grossman, P. L. (1990). The making of a teacher: Teacher knowledge and teacher education. New York: Teachers College.
• Gunnarsson, R., Sönnerhed, W. W., & Hernell, B. (2016). Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations? Educational Studies in Mathematics, 92(1), 91-105.
• Hallagan, J. E. (2004). A teacher’s model of students’ algebraic thinking about equivalent expressions. In M. J. Hoines, & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 1-8). Bergen, Norway: International Group for the Psychology of Mathematics Education.
• Hill, H. C., Ball, D. B., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Teacher Education, 39(4), 372-400.
• Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406.
• 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. 257-315). Reston, VA: National Council of Teachers of Mathematics.
• Hoch, M., & Dreyfus, T. (2004). Structure sense in high school algebra: The effect of brackets. In M. J. Høines, & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 49-56). Bergen: PME.
• Huang, R., & Kulm, G. (2012). Prospective middle-grade mathematics teachers' knowledge of algebra for teaching. The Journal of Mathematical Behavior, 31(4), 417-430.
• Kahan, J. A., Cooper, D. A., & Bethea, K. A. (2003). The role of mathematics teachers' content knowledge in their teaching: A framework for research applied to a study of student teachers. Journal of Mathematics Teacher Education, 6(3), 223-252.
• Kaput, J. J. (2000). Teaching and learning a new algebra with understanding. U.S. Department of Education Office of Educational Research and Improvement (OERI) Educational Resources Information Center. (ERIC Document Reproduction Service No. ED441 662).
• Kieran, C. (2007). Learning and teaching algebra at the middle school from college levels: Building meaning for symbols and their manipulation. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp.707-762). Charlotte, NC: Information Age.
• Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. Wagner, & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 33-56), Reston, Virginia: NCTM, and Hillsdale, N.J.: Erlbaum.
• Lepak, J. R., Wernet, J. L., & Ayieko, R. A. (2018). Capturing and characterizing students’ strategic algebraic reasoning through cognitively demanding tasks with focus on representations. The Journal of Mathematical Behavior, 50, 57-73.
• Livneh, D., & Linchevski, L. (2007). Algebrification of arithmetic: Developing algebraic structure sense in the context of arithmetic. In J. W. Woo, H. C., Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the Psychology of Mathematics Education (Vol. 3, pp. 217-225). Seoul, Korea: International Group for the Psychology of Mathematics Education.
• Livy, S., & Downton, A. (2018). Exploring experiences for assisting primary pre-service teachers to extend their knowledge of student strategies and reasoning. The Journal of Mathematical Behavior, 51, 150-160.
• Lucariello, J., Tine, M. T., & Ganley, C. M. (2014). A formative assessment of students’ algebraic variable misconceptions. The Journal of Mathematical Behavior, 33, 30-41.
• MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1-19.
• Malara, N. A., & Navarra, G. (2009). The analysis of classroom-based processes as a key task in teacher training for the approach to early algebra. In B. Clarke, B. Grevholm, & R. Millman (Eds.), Tasks in primary mathematics teacher education (pp. 235-262). Berlin: Springer.
• Marchini, C., & Papadopoulos, I. (2011). Are useless brackets useful for teaching? In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 185-192). Ankara: PME.
• Merriam, S. B. (2009). Qualitative research: A guide to design and implementation: Revised and expanded from qualitative research and case study applications in education. San Francisco: Jossey-Bass.
• Molina, M., Rodríguez-Domingo, S., Cañadas, M. C., & Castro, E. (2017). Secondary school students’ errors in the translation of algebraic statements. International Journal of Science and Mathematics Education, 15(6), 1137-1156.
• Nathan, M. J., & Koedinger, K. R. (2000). An investigation of teachers’ beliefs of students’ algebra development. Cognition and Instruction, 18(2), 209-237.
• National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Author.
• Novotná, J., & Hoch, M. (2008). How structure sense for algebraic expressions or equations is related to structure sense for abstract algebra. Mathematics Education Research Journal, 20(2), 93-104.
• Ojose, B. (2015). Common misconceptions in mathematics: Strategies to correct them. America: University Press of America.
• Prendergast, M., & O’Donoghue, J. (2014). ‘Students enjoyed and talked about the classes in the corridors’: Pedagogical framework promoting interest in algebra. International Journal of Mathematical Education in Science and Technology, 45(6), 795-812.
• 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.
• Seng, L. K. (2010). An error analysis of form 2 (grade 7) students in simplifying algebraic expressions: A descriptive study. Electronic Journal of Research in Educational Psychology, 8(1), 139-162.
• Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
• Stephan, M., Pugalee, D., Cline, J., & Cline, C. (2017). Lesson imagining in math and science: Anticipating student ideas and questions for deeper STEM learning. Alexandria, VA: ASCD.
• Stephens, A. C. (2008). What “counts” as algebra in the eyes of preservice elementary teachers? The Journal of Mathematical Behavior, 27(1), 33-47.
• Stump, S. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124-144.
• Subramaniam, K., & Banerjee, R. (2004). Teaching arithmetic and algebraic expressions. In M. J. Hoines, & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 121-128). Bergen, Norway: International Group for the Psychology of Mathematics Education. Sullivan, P. (2018). Supporting teachers in improving their knowledge of mathematics. The Journal of Mathematical Behavior, 51, 161-166.
• Sullivan, P., Askew, M., Cheeseman, J., Clarke, D., Mornane, A., Roche, A., & Walker, N. (2015). Supporting teachers in structuring mathematics lessons involving challenging tasks. Journal of Mathematics Teacher Education, 18(2), 123-140.
• Tchoshanov, M., Quinones, M. C., Shakirova, K. B., Ibragimova, E. N., & Shakirova, L. R. (2017). Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics. The Journal of Mathematical Behavior, 47, 54-69.
• Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying algebraic expressions: Teacher awareness and teaching approaches. Educational Studies in Mathematics, 35, 51-64.
• Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford (Ed.), The ideas of algebra, K-12 (pp. 8-19). Reston, VA; National Council of Teachers of Mathematics.
• Van Dooren, W., Verschaffel, L., & Onghena, P. (2002). The impact of preservice teachers’ content knowledge on their evaluation of students’ strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33(5), 319-351.
• Weinberg, A., Dresen, J., & Slater, T. (2016). Students’ understanding of algebraic notation: A semiotic systems perspective. The Journal of Mathematical Behavior, 43, 70-88.
• Wilkie, K. J. (2014). Upper primary school teachers’ mathematical knowledge for teaching functional thinking in algebra. Journal of Mathematics Teacher Education, 17(5), 397-428. Yin, R. K. (2003). Case study research: Design and methods (3rd ed.). Thousand Oaks, California: Sage Publications.