Abstract

There is a strong relationship between the quality of education, teachers’ decisions and instructional actions, and knowledge. All of them have an important role outcome of the lesson and students’ learning. The focus of this study was to discover how the process of students’ learning or incapability of learning was affected by teaching practices. Thus, the process of teaching and pre and post-lesson interviews conducted with a middle school mathematics teacher were analyzed. The data were collected through reflective interviews that were conducted with the teacher before and after the lesson as well as the video recording of the teacher’s two-hour lesson. In light of the findings, it can possibly be argued that the teacher had actions that caused incorrect learning and misconceptions. This result cannot be explained by a single factor related to the teacher. It would not be wrong to claim that the instructional decisions (decision-making processes) in the teaching and teacher’s content knowledge had decisive roles in the formation of this result, too. Therefore, the findings were discussed within the framework of instructional decisions and teachers’ content knowledge.  

Keywords

References

• Arcavi, A., & Schoenfeld, A. H. (2008). Using the unfamiliar to problematize the familiar: The case of mathematics teacher in-service education. Canadian Journal of Science, Mathematics and Technology Education, 8, 280-295. https://doi.org/10.1080/14926150802315122
• 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. https://doi.org/10.1177/0022487108324554
• Barnett, J., & Hodson, D. (2001). Pedagogical context knowledge: Toward a fuller understanding of what good science teachers know. Science Education, 85, 426–453. https://doi.org/10.1002/sce.1017
• Barton B., & Sheryn, L. (2009). The mathematical needs of secondary teachers: data from three countries. International Journal of Mathematics Education, Science and Technology, 40, 101–108. https://doi.org/10.1080/00207390802576807
• Berg, D. (2012). Algebraic thinking in the elementary classroom. Simon Fraser University.
• Booth, L. (1988). Children’s difficulties in beginning algebra. In A. F. Coxford, & A. P. Shulte (Eds.). The ideas of algebra, K-12 (pp. 20-32). National Council of Teachers of Mathematics.
• Caram, C. A., & Davis, P. B. (2005). Inviting student engagement with questioning. Kappa Delta Pi Record, 42(1), 19-23. https://doi.org/10.1080/00228958.2005.10532080
• Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Heinemann.
• Chalouh, L., & Herscovics, N. (1988). Teaching algebraic expressions in a meaningful way. In a.f. Coxford Shulte (Eds.). The ideas of algebra K-12. NCTM.
• Clarkson, P., & Presmeg, N. (2008). Critical issues in mathematics education. Springer.
• Confrey, J. (1993). Learning to see children's mathematics: Crucial challenges in constructivist reform. In K. Tobin (Eds.), Constructivist perspectives in science and mathematics, (pp. 299-321). American Association for the Advancement of Science.
• Crespo, S. (2000). Seeing more than right and wrong answers: Prospective teachers’ interpretations of students’ mathematical work. Journal of Mathematics Teacher Education, 3(2), 155-181. https://doi.org/10.1023/A:1009999016764
• Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five approaches (2nd ed.). London: Sage.
• Doerr, H. M. (2006). Examining the tasks of teaching when using students’ mathematical thinking. Educational Studies in Mathematics, 62(1), 3-24. https://doi.org/10.1007/s10649-006-4437-9
• Dreyfus, H. & Dreyfus, S. (1987). Mind over machine: The power of human intuition. The Free Press. https://doi.org/10.1109/MEX.1987.4307079
• El Mouhayar, R., & Jurdak, M. (2015a). Variation in strategy use across grade levels by pattern generalization types. International Journal of Mathematical Education in Science and Technology, 46(4), 553–569. https://doi.org/10.1080/0020739X.2014.985272
• Empson, S. B., & Jacobs, V. R. (2008). Learning to listen to children’s mathematics. In D. Tirosh & T. Wood (Eds.), Tools and processes in mathematics teacher education (pp. 257-281). Sense.
• Fennema, E., Franke, M. L., Carpenter, T. P., & Carey, D. A. (1993). Using children’s mathematical knowledge in instruction. American Educational Research Journal, 30(3), 555-583. https://doi.org/10.3102/00028312030003555
• Franke, M. L., & Kazemi, E. (2001). Learning to teach mathematics: Focus on student thinking. Theory into Practice, 40(2), 102-109. https://doi.org/10.1207/s15430421tip4002_4
• Franke, M. L., Webb, N. M., Chan, A. G., Ing, M., Freund, D., & Battey, D. (2009). Teacher questioning to elicit students’ mathematical thinking in elementary school classrooms. Journal of Teacher Education, 60(4), 380–39. https://doi.org/10.1177/0022487109339906
• Franke, M. L., Webb, N. M., Chan, A., & Battey, D. (2007). Eliciting student thinking in elementary school mathematics classrooms (ED498545). ERIC.
• Franke, M.L., 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 (pp. 225–256). Information Age Publishing.
• Goodwin, C. (1994). Professional vision. American Anthropologist, 96, 606–633.
• Goos, M. (2013). Knowledge for teaching secondary school mathematics: what counts? The effects of teacher mathematics knowledge and pedagogy on student achievement in Rural Guatemala. International Journal of Mathematical Education in Science and Technology, 44(7), 972-983. https://doi.org/10.1080/0020739X.2013.826387
• Hawkins, S. R. (2016). Scaffolding preservice teachers’ noticing of elementary students’ scientific thinking [Unpublished doctoral dissertation]. Indiana University, Indiana.
• Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59-78. https://doi.org/10.1007/BF01284528
• Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Eds.), Second handbook of research on mathematics teaching and learning (pp. 371-404). Information Age.
• Hiebert, J., Morris, A. K., Berk, D., & Jansen, A. (2007). Preparing teachers to learn from teaching. Journal of Teacher Education, 58(1), 47-61. https://doi.org/10.1177/0022487106295726
• International Association for the Evaluation of Educational Achievement (IEA). (2021). TIMSS 2019 International Results in Mathematics and Science https://timss2019.org/reports/achievement/index.html
• Jacobs, V. R., Empson, S. B., Krause, G. H., & Pynes, D. (2015, April). Responsive teaching with fractions [Paper presentation]. Annual Meeting of The Research Conference of the National Council of Teachers of Mathematics, Boston.
• Jacobs, V. R., Lamb, L. L. C. & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. https://www.jstor.org/stable/20720130
• Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema and T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133–155). Lawrence Erlbaum Associates, Inc., Publishers.
• Kieran, C. (1992). The learning and teaching of school algebra. In D.A. Grouws (Eds.). Handbook of research on mathematics teaching and learning (pp. 390‐419). Macmillan.
• Kieran, C. (2004). Algebraic thinking in the early grades: What is it. The Mathematics Educator, 8(1), 139-151.
• Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. Second Handbook of Research on Mathematics Teaching and Learning, 2, 707-762.
• Lee, M. Y., & Francis, D. C. (2018). Investigating the relationships among elementary teachers’ perceptions of the use of students’ thinking, their professional noticing skills, and their teaching practices. The Journal of Mathematical Behavior, 51, 118-128. https://doi.org/10.1016/j.jmathb.2017.11.007
• Lester, F. K. (2007). Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics. Information Age Publishing.
• Liu, Y. (2014). Teachers’ in-the-moment noticing of students’ mathematical thinking: a case study of two teachers [Unpublished doctoral dissertation]. The University of North Carolina, Carolina.
• MacGregor, M. E. (1986). A Fresh Look at Fruit Salad Algebra. Australian Mathematics Teacher, 42(3), 9-11.
• MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1-19.
• Martin, M. O., & Mullis, I. V. (2013). TIMSS and PIRLS 2011: Relationships among Reading, Mathematics, and Science Achievement at the Fourth Grade--Implications for Early Learning (ED545256). ERIC.
• Mason, J. (2011). Noticing: Roots and branches. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 35–50). Routledge.
• McGowen, M. A., & Tall, D. O. (2010). Metaphor or Met-Before? The effects of previous experience on practice and theory of learning mathematics. The Journal of Mathematical Behavior, 29(3), 169-179. https://doi.org/10.1016/j.jmathb.2010.08.002
• Meschede, N., Fiebranz, A., Möller, K., & Steffensky, M. (2017). Teachers’ professional vision, pedagogical content knowledge, and beliefs: On its relation and differences between pre-service and in-service teachers. Teaching and Teacher Education, 66, 158-170. https://doi.org/10.1016/j.tate.2017.04.010
• Perso, T. (1992). Overcoming misconceptions in algebra: using diagnostic (conflict) teaching. Mathematical Association of Western Australia.
• Pimm, D. (1987) Speaking mathematically: Communication in mathematics classrooms. Routledge and Kegan Paul.
• Pomerantsev, L., & Korosteleva, O. (2003). Do prospective elementary and middle school teachers understand the structure of algebraic expressions? Issues in the Undergraduate Mathematics Preparation of School Teachers: The Journal, 1(8), 1-10.
• Purdum-Cassidy, B., Nesmith, S., Meyer, R. D., & Cooper, S. (2015). What are they asking? An analysis of the questions planned by prospective teachers when integrating literature in mathematics. Journal of Mathematics Teacher Education, 18, 79–99. https://doi.org/10.1007/s10857-014-9274-7
• Schoenfeld, A. H. (2010). How we think. Routledge.
• Sherin, M. G., & van Es, E. A. (2005). Using video to support teachers’ ability to notice classroom interactions. Journal of Technology and Teacher Education, 13, 475-491.
• Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (2011). Situating the study of teacher noticing. In M.G. Sherin, V. R. Jacobs, and R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes, (pp. 3-13). Routledge.
• 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(4), 313-340. https://doi.org/10.1080/10986060802229675
• Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying algebraic expressions: Teacher awareness and teaching approaches. Educational Studies in Mathematics, 35(1), 51-64. https://doi.org/10.1023/A:1003011913153
• Toluk Uçar, Z. (2018). Approaches to algebraic thinking in terms of curriculum. Middle school mathematics curriculum: A historical review, 209-246.
• Usiskin, Z. (1997). Doing algebra in grades K‐4. In B. Moses (Eds.). Algebraic thinking, grades K‐12 (pp. 5‐7). NCTM. https://doi.org/10.5951/TCM.3.6.0346
• van Es, E. A. & Sherin, M. G. (2002). Learning to notice: Scaffolding new teacher's interpretations of classroom interactions. Journal of Technology and Teacher Education, 10(4), 571-595.
• van Es, E. A. & Sherin, M. G. (2008). Mathematics teachers ―learning to notice‖ in the context of a video club. Teaching and Teacher Education, 24(2), 244-276. https://doi.org/10.1016/j.tate.2006.11.005
• Van Zoest, L. R., Peterson, B. E., Leatham, K. R., & Stockero, S. L. (2016). Conceptualizing the teaching practice of building on student mathematical thinking. In M. B. Wood, E. E. Turner, M. Civil & J. A. Eli (Eds.), Proceedings of the 38th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1281–1288). University of Arizona.
• Watkins, J. D. (2018). Exploring the knowledge of algebra for teaching.
• Wilson, P. H., Mojica, G. F., & Confrey, J. (2013). Learning trajectories in teacher education: Supporting teachers’ understandings of students’ mathematical thinking. The Journal of Mathematical Behavior, 32(2), 103-121. https://doi.org/10.1016/j.jmathb.2012.12.003

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