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Supporting prospective mathematics teachers in noticing students' proportional reasoning

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Abstract

This study investigates the effect of a professional development program designed to develop prospective mathematics teachers’ ability to notice students’ proportional reasoning. It examines how those teachers attended to student thinking, how they interpreted it, and how they decided to respond. Sixteen prospective teachers (PSTs) at a state university participated in a seven-week online professional development (PD) program. We provided participants with a research-based framework for proportional reasoning and engaged them in various ways via interviews with middle school students. They also wrote a reflection paper and received feedback. Combining multiple elements with interviewing was a unique feature of this program. The improvement in participants’ noticing skills was statistically significant. Specifically, in the post-assessment, the participants paid attention to most of the mathematical details of the students’ strategies, provided interpretations supported by conceptual underpinnings and evidence, and made research-based student-oriented decisions that focused on key issues. Implications for future teacher education practices and suggestions for further research are provided.

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Notes

  1. We are referring to the research underpinning the framework presented in the PD program.

References

  • Ball, D. L., & Cohen, D. K. (1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education. In L. Darling-Hammond & G. Sykes (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 3–32). Jossey-Bass Publishers.

    Google Scholar 

  • Buforn, À., Llinares, S., Fernández, C., Coles, A., & Brown, L. (2022). Pre-service teachers’ knowledge of the unitizing process in recognizing students’ reasoning to propose teaching decisions. International Journal of Mathematical Education in Science and Technology, 53(2), 425–443. https://doi.org/10.1080/0020739X.2020.1777333

    Article  Google Scholar 

  • Burgos, M., & Godino, J. D. (2022). Prospective primary school teachers’ competence for the cognitive analysis of students’ solutions to proportionality tasks. Journal Für Mathematik-Didaktik, 43(2), 347–376. https://doi.org/10.1007/s13138-021-00193-4

    Article  Google Scholar 

  • Casey, S., Lesseig, K., Monson, D., & Krupa, E. E. (2018). Examining preservice secondary mathematics teachers’ responses to student work to solve linear equations. Mathematics Teacher Education and Development, 20(1), 132–153.

    Google Scholar 

  • Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20(1), 37–44. https://doi.org/10.1177/001316446002000104

    Article  Google Scholar 

  • Cramer, K., Post, T., & Graeber, A. O. (1993). Connecting research to teaching: Proportional reasoning. Mathematics Teacher, 86(5), 404–407. https://doi.org/10.5951/MT.86.5.0404

    Article  Google Scholar 

  • Ellemor-Collins, D. L., & Wright, R. J. (2008). Assessing student thinking about arithmetic: Videotaped interviews. Teaching Children Mathematics, 15(2), 106–111.

    Article  Google Scholar 

  • Fernandes, A. (2012). Mathematics preservice teachers learning about English Language Learners through task-based interviews and noticing. Mathematics Teacher Educator, 1(1), 10–22. https://doi.org/10.5951/mathteaceduc.1.1.0010

    Article  Google Scholar 

  • Fernández, C., Llinares, S., & Rojas, Y. (2020). Prospective mathematics teachers’ development of noticing in an online teacher education program. ZDM-Mathematics Education, 52(5), 959–972. https://doi.org/10.1007/s11858-020-01149-7

    Article  Google Scholar 

  • Fernández, C., Llinares, S., & Valls, J. (2012). Learning to notice students’ mathematical thinking through on-line discussions. ZDM-Mathematics Education, 44(6), 747–759. https://doi.org/10.1007/s11858-012-0425-y

    Article  Google Scholar 

  • Field, A. (2012). Discovering statistics using IBM SPSS statistics. Berlin: Sage Publications.

    Google Scholar 

  • Ginsburg, H. (1981). The clinical interview in psychological research on mathematical thinking: Aims, rationales, techniques. For the Learning of Mathematics, 1(3), 4–11.

    Google Scholar 

  • Guner, P., & Akyuz, D. (2020). Noticing students’ mathematical thinking within the context of lesson study. Journal of Teacher Education, 71(5), 568–583. https://doi.org/10.1177/0022487119892964

    Article  Google Scholar 

  • Hunting, R. P. (1997). Clinical interview methods in mathematics education research and practice. The Journal of Mathematical Behavior, 16(2), 145–165. https://doi.org/10.1016/S0732-3123(97)90023-7

    Article  Google Scholar 

  • Ivars, P., Fernández, C., & Llinares, S. (2020). A learning trajectory as a scaffold for pre-service teachers’ noticing of students’ mathematical understanding. International Journal of Science and Mathematics Education, 18(3), 529–548. https://doi.org/10.1007/s10763-019-09973-4

    Article  Google Scholar 

  • Jacobs, V. R., & Empson, S. B. (2016). Responding to children’s mathematical thinking in the moment: An emerging framework of teaching moves. ZDM-Mathematics Education, 48(1–2), 185–197. https://doi.org/10.1007/s11858-015-0717-0

    Article  Google Scholar 

  • 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://doi.org/10.5951/jresematheduc.41.2.0169

    Article  Google Scholar 

  • Kastberg, S. E., Lischka, A. E., & Hillman, S. L. (2020). Characterizing mathematics teacher educators’ written feedback to prospective teachers. Journal of Mathematics Teacher Education, 23(2), 131–152. https://doi.org/10.1007/s10857-018-9414-6

    Article  Google Scholar 

  • Krupa, E. E., Huey, M., Lesseig, K., Casey, S., & Monson, D. (2017). Investigating secondary preservice teacher noticing of students’ mathematical thinking. In E. O. Schack, M. H. Fisher, & J. A. Wilhelm (Eds.), Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks (pp. 49–72). Springer.

    Chapter  Google Scholar 

  • Lamon, S. J. (1995). Ratio and proportion: Elementary didactical phenomenology. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 167–198). State University of New York Press.

    Google Scholar 

  • Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Information Age Publishing.

    Google Scholar 

  • Lamon, S. J. (2012). Teaching fractions and ratios for understanding (3rd ed.). Routledge.

    Book  Google Scholar 

  • LaRochelle, R., Nickerson, S. D., Lamb, L. C., Hawthorne, C., Philipp, R. A., & Ross, D. L. (2019). Secondary practicing teachers’ professional noticing of students’ thinking about pattern generalization. Mathematics Teacher Education and Development, 21(1), 4–27.

    Google Scholar 

  • Lee, M. Y. (2018). Further investigation into the quality of teachers’ noticing expertise: A proposed framework for evaluating teachers’ models of students’ mathematical thinking. EURASIA Journal of Mathematics, Science and Technology Education. https://doi.org/10.29333/ejmste/92019

    Article  Google Scholar 

  • Lee, M. Y. (2019). The development of elementary pre-service teachers’ professional noticing of students’ thinking through adapted lesson study. Asia-Pacific Journal of Teacher Education, 47(4), 383–398. https://doi.org/10.1080/1359866X.2019.1607253

    Article  Google Scholar 

  • Lesseig, K., Casey, S., Monson, D., Krupa, E. E., & Huey, M. (2016). Developing an interview module to support secondary PSTs’ noticing of student thinking. Mathematics Teacher Educator, 5(1), 29–46. https://doi.org/10.5951/mathteaceduc.5.1.0029

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Monson, D., Krupa, E., Lesseig, K., & Casey, S. (2020). Developing secondary prospective teachers’ ability to respond to student work. Journal of Mathematics Teacher Education, 23(2), 209–232. https://doi.org/10.1007/s10857-018-9420-8

    Article  Google Scholar 

  • Moyer, P. S., & Milewicz, E. (2002). Learning to question: Categories of questioning used by preservice teachers during diagnostic mathematics interviews. Journal of Mathematics Teacher Education, 5(4), 293–315. https://doi.org/10.1023/A:1021251912775

    Article  Google Scholar 

  • National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematics success for all. Author.

  • Nickerson, S. D., Lamb, L., & LaRochelle, R. (2017). Challenges in measuring secondary mathematics teachers’ professional noticing of students’ mathematical thinking. In E. O. Schack, M. H. Fisher, & J. A. Wilhelm (Eds.), Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks (pp. 381–398). Springer.

    Chapter  Google Scholar 

  • Pişkin Tunç, M., & Çakıroğlu, E. (2022). Fostering prospective mathematics teachers’ proportional reasoning through a practice-based instruction. International Journal of Mathematical Education in Science and Technology, 53(2), 269–288. https://doi.org/10.1080/0020739x.2020.1844909

    Article  Google Scholar 

  • Rodgers, C. (2002). Seeing student learning: Teacher change and the role of reflection. Harvard Educational Review, 72(2), 230–253. https://doi.org/10.17763/haer.72.2.5631743606m15751

    Article  Google Scholar 

  • Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2015). Developing pre-service teachers’ noticing of students’ understanding of the derivative concept. International Journal of Science and Mathematics Education, 13(6), 1305–1329. https://doi.org/10.1007/s10763-014-9544-y

    Article  Google Scholar 

  • Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2019). Relationships among prospective secondary mathematics teachers’ skills of attending, interpreting and responding to students’ understanding. Educational Studies in Mathematics, 100(1), 83–99. https://doi.org/10.1007/s10649-018-9855-y

    Article  Google Scholar 

  • Schack, E. O., Fisher, M. H., Thomas, J. N., Eisenhardt, S., Tassell, J., & Yoder, M. (2013). Prospective elementary school teachers’ professional noticing of children’s early numeracy. Journal of Mathematics Teacher Education, 16(5), 379–397. https://doi.org/10.1007/s10857-013-9240-9

    Article  Google Scholar 

  • Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (2011). Situating the study of teacher noticing. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 3–13). Routledge.

    Chapter  Google Scholar 

  • Simpson, A., & Haltiwanger, L. (2017). “This is the first time I’ve done this”: Exploring secondary prospective mathematics teachers’ noticing of students’ mathematical thinking. Journal of Mathematics Teacher Education, 20(4), 335–355. https://doi.org/10.1007/s10857-016-9352-0

    Article  Google Scholar 

  • So, W.W.-M. (2012). Quality of learning outcomes in an online video-based learning community: Potential and challenges for student teachers. Asia-Pacific Journal of Teacher Education, 40(2), 143–158. https://doi.org/10.1080/1359866X.2012.669828

    Article  Google Scholar 

  • Son, J.-W. (2013). How preservice teachers interpret and respond to student errors: Ratio and proportion in similar rectangles. Educational Studies in Mathematics, 84(1), 49–70. https://doi.org/10.1007/s10649-013-9475-5

    Article  Google Scholar 

  • Stockero, S. L., Rupnow, R. L., & Pascoe, A. E. (2017). Learning to notice important student mathematical thinking in complex classroom interactions. Teaching and Teacher Education, 63, 384–395. https://doi.org/10.1016/j.tate.2017.01.006

    Article  Google Scholar 

  • Tabak, I. (2004). Synergy: A complement to emerging patterns of distributed scaffolding. Journal of the Learning Sciences, 13(3), 305–335. https://doi.org/10.1207/s15327809jls1303_3

    Article  Google Scholar 

  • Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 179–234). SUNY Press.

    Google Scholar 

  • Ulusoy, F. (2020). Prospective teachers’ skills of attending, interpreting and responding to content-specific characteristics of mathematics instruction in classroom videos. Teaching and Teacher Education, 94, 103103. https://doi.org/10.1016/j.tate.2020.103103

    Article  Google Scholar 

  • Ulusoy, F., & Çakıroğlu, E. (2021). Exploring prospective teachers’ noticing of students’ understanding through micro-case videos. Journal of Mathematics Teacher Education, 24(3), 253–282. https://doi.org/10.1007/s10857-020-09457-1

    Article  Google Scholar 

  • Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th ed.). Berlin: Pearson.

    Google Scholar 

  • Van Dooren, W., De Bock, D., & Verschaffel, L. (2010a). From addition to multiplication…and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360–381. https://doi.org/10.1080/07370008.2010.488306

    Article  Google Scholar 

  • Van Dooren, W., De Bock, D., Vleugels, K., & Verschaffel, L. (2010b). Just answering … or thinking? Contrasting pupils’ solutions and classifications of missing-value word problems. Mathematical Thinking and Learning, 12(1), 20–35. https://doi.org/10.1080/10986060903465806

    Article  Google Scholar 

  • van Es, E. A. (2011). A framework for learning to notice student thinking. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 134–151). Routledge.

    Google Scholar 

  • Walkoe, J. (2015). Exploring teacher noticing of student algebraic thinking in a video club. Journal of Mathematics Teacher Education, 18(6), 523–550. https://doi.org/10.1007/s10857-014-9289-0

    Article  Google Scholar 

  • Weiland, I. S., Hudson, R. A., & Amador, J. M. (2014). Preservice formative assessment interviews: The development of competent questioning. International Journal of Science and Mathematics Education, 12(2), 329–352. https://doi.org/10.1007/s10763-013-9402-3

    Article  Google Scholar 

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Acknowledgements

We would like to thank Aybüke Karagözler and Ayşe Akkır for their contributions in the data collection and part of the coding within this study.

Funding

This research was partly supported by İstanbul Aydın University Scientific Research Projects funding under Grant Number 2021/09.

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Correspondence to Sinem Bas-Ader.

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All procedures performed in studies involving human participants were in accordance with the comparable ethical standards and approved by İstanbul Aydın University Research Ethics Committee.

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Appendix

Appendix

Tables 4, 5, 6, 7.

Table 4 The problems used in video clip #1, video clip #2, and video clip #3, student strategies involved and characteristics of students’ proportional reasoning
Table 5 A sample coding scheme for attending (Elif’s strategies in video clip #2)
Table 6 The coding scheme for interpreting along with examples of coding for each point from PSTs' comments on the student’s strategies in Video clip #2
Table 7 The coding scheme for deciding along with examples of coding for each point from PSTs' comments on the student’s strategies in Video clip #3

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Bas-Ader, S., Ader, E. & Taylan, R.D. Supporting prospective mathematics teachers in noticing students' proportional reasoning. J Math Teacher Educ (2024). https://doi.org/10.1007/s10857-024-09621-x

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