Teacher Noticing: Attending To and Interpreting Students’ Mathematical Fluency Through Written and Oral Representations

Authors

  • Katherin Cartwright The University of Sydney

Keywords:

teacher notcing, mathematical fluency, student mathematical thinking

Abstract

Teacher noticing of students’ thinking is a key component in the development of teaching expertise. The central focus of the study was on investigating teachers’ noticing–attending and interpreting–of mathematical fluency. This article reports on findings from nine primary teachers’ observations of upper primary students during mathematics lessons and their reported noticings in post-lesson group discussions. Teachers’ noticings focused on attending to mathematically important details of students’ strategies which aligns with previous teacher noticing research. The study contributes to the field of teacher noticing by illustrating how the use of a detailed observational tool provided specificity of the mathematical features to attend to, and a common language for discussion. The tool’s emphasis on observing students’ oral and written representations widens the scope of what evidence teachers used to interpret mathematical fluency (differing from how fluency is generally observed via numerical/written responses alone). In this study’s case oral responses were most important; teachers relied heavily on restating student oral responses to interpret their noticings.

References

Australian Curriculum, Assessment and Reporting Authority. (2025). Australian Curriculum: Mathematics: Understanding this learning area. https://www.australiancurriculum.edu.au/curriculum-information/understand-this-learning-area/mathematics

Anthony, G., Hunter, J., & Hunter, R. (2015). Supporting prospective teachers to notice students’ mathematical thinking through rehearsal activities. Mathematics Teacher Education and Development, 17(2), 7–24.

Amador, J. M., & Weston, T. L. (2024). A review of analytic frameworks for noticing in mathematics and science: Comparing noticing frameworks across disciplines and over time. International Journal of Science and Mathematics Education, 22(8), 1739–1760.

Bandura, A. (1977). Self-efficacy: Toward a unifying theory of behavioral change. Psychological Review, 84(2), 191–215.

Baroody, A. J. (2006). Why children have difficulties mastering the basic number combinations and how to help them. Teaching Children Mathematics 13(8), 22–31.

Bay-Williams, J. M., & Kling, G. (2019). 5 fundamentals for developing basic fact fluency. Prime Number, 34(2), Article 4.

Beisiegel, M., Mitchell, R., & Hill, H. C. (2018). The design of video-based professional development: An exploratory experiment intended to identify effective features. Journal of Teacher Education, 69(1), 69–89.

Bragg, L. A., Herbert, S., Loong, E. Y. K., Vale, C., & Widjaja, W. (2016). Primary teachers notice the impact of language on children’s mathematical reasoning. Mathematics Education Research Journal, 28(4), 523–544.

Brantley-Dias, L., Dias, M., Frisch, J., & Rushton, G. (2008, March). The role of digital video and critical incident analysis in learning to teach science. In W. F. Tate & J. D. Anderson (Eds.), Research on schools, neighborhoods, and communities: Toward civic responsibility Paper presented at the American Educational Research Association Annual Meeting, New York.

Cai, J., & Lester, F. K. (2005). Solution representations and pedagogical representations in Chinese and US classrooms. The Journal of Mathematical Behavior, 24(3–4), 221–237.

Cartwright, K. (2018). Exploring mathematical fluency: teachers' conceptions and descriptions of students. In Hunter, J., Perger, P., & Darragh, L. (Eds.). Making waves, opening spaces (Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia, pp. 202–209). Auckland: MERGA.

Cartwright, K. (2019). “Because 7 and 8 are always in all of them”: What do Students Write and Say to Demonstrate their Mathematical Fluency. In G. Hine, S. Blackley, & A. Cooke (Eds.). Mathematics education research: impacting practice (Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia, pp. 156–163). Perth: MERGA.

Cartwright, K. (2020). Analyzing students' communication and representation of mathematical fluency during group tasks. The Journal of Mathematical Behavior, 60, Article 100821. https://doi.org/10.1016/j.jmathb.2020.100821

Choy, B. H. (2014). Teachers’ productive mathematical noticing during lesson preparation. In C. Nicol, P. Liljedahl, S. Oesterle, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 2, pp. 297–304), Vancouver, Canada. PME.

Choy, B. H. (2016). Snapshots of mathematics teacher noticing during task design. Mathematics Education Research Journal, 28(3), 421–440.

Cochran-Smith, M., & Lytle, S. L. (2015). Inquiry as stance: Practitioner research for the next generation. Teachers College Press.

Dole, S., Carmichael, P., Thiele, C., Simpson, J., & O'Toole, C. (2018). Fluency with number facts: Responding to the Australian Curriculum: Mathematics. In J. Hunter, P. Perger, & L. Darragh (Eds.), Making waves, opening spaces (Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia, pp. 266–273). Auckland: MERGA.

Fereday, J., & Muir-Cochrane, E. (2006). Demonstrating rigor using thematic analysis: A hybrid approach of inductive and deductive coding and theme development. International Journal of Qualitative Methods, 5(1), 80–92.

Fisher, M. H., Thomas, J., Jong, C., Schack, E. O., & Dueber, D. (2019). Comparing preservice teachers’ professional noticing skills in elementary mathematics classrooms. School Science and Mathematics, 119(3), 142–149.

Foster, C. (2016). Confidence and competence with mathematical procedures. Educational Studies in Mathematics, 91(2), 271–288.

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), 380–392.

Goldsmith, L. T., & Seago, N. (2011). Using classroom artifacts to focus teachers’ noticing. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 169–187). Routledge.

Jacobs, V. R., Empson, S. B., Jessup, N. A., Dunning, A., Pynes, D. A., Krause, G., & Franke, T. M. (2024). Profiles of teachers’ expertise in professional noticing of children’s mathematical thinking. Journal of Mathematics Teacher Education, 27(3), 295–324.

Jacobs, V. R., Lamb, L. L., & Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.

Jacobs, V. R., & Philipp, R. A. (2004). Mathematical thinking: Helping prospective and practicing teachers focus. Teaching Children Mathematics, 11(4), 194–201.

Jazby, D. (2016). An ecological analysis of mathematics teachers' noticing. In B. White, M. Chinnappan, & S. Trenholm (Eds.), Opening up mathematics education research (Proceedings of the 39th annual conference of the Mathematics Education Research Group of Australasia, pp. 360–367). Adelaide: MERGA.

Johnston-Wilder, S., & Lee, C. (2010). Mathematical resilience. Mathematics Teaching, 218, 38–41.

Kalinec Craig, C. A., Bannister, N., Bowen, D., Jacques, L. A., & Crespo, S. (2021). “It was smart when:” Supporting prospective teachers’ noticing of students’ mathematical strengths. Journal of Mathematics Teacher Education, 24(4), 375–398. https://doi.org/10.1007/s10857-020-09464-2

Kamii, C., Lewis, B. A., & Kirkland, L. D. (2001). Fluency in subtraction compared with addition. The Journal of Mathematical Behavior, 20(1), 33–42.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. National Academy Press.

König, J., Santagata, R., Scheiner, T., Adleff, A. K., Yang, X., & Kaiser, G. (2022). Teacher noticing: A systematic literature review of conceptualizations, research designs, and findings on learning to notice. Educational Research Review, 36, Article 100453. https://doi.org/10.1016/j.edurev.2022.100453

Kooloos, C., Oolbekkink-Marchand, H., van Boven, S., Kaenders, R., & Heckman, G. (2022). Making sense of student mathematical thinking: The role of teacher mathematical thinking. Educational Studies in Mathematics, 110(3), 503–524.

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.

Lampert, M. (2001). Teaching problems and the problems of teaching. Yale University Press.

Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. SAGE Publications.

Louie, N., Adiredja, A. P., & Jessup, N. (2021). Teacher noticing from a sociopolitical perspective: The FAIR framework for anti-deficit noticing. ZDM-Mathematics Education, 53(1), 95–107. https://doi.org/10.1007/s11858-021-01229-2

Mason, J. (2002). Researching your own practice: The discipline of noticing. Routledge.

Melhuish, K., Thanheiser, E., & Guyot, L. (2020). Elementary school teachers’ noticing of essential mathematical reasoning forms: Justification and generalization. Journal of Mathematics Teacher Education, 23(1), 35–67.

Mitchell, R. N., & Marin, K. A. (2015). Examining the use of a structured analysis framework to support prospective teacher noticing. Journal of Mathematics Teacher Education, 18(6), 551–575.

Morano, S., Randolph, K., Markelz, A. M., & Church, N. (2020). Combining explicit strategy instruction and mastery practice to build arithmetic fact fluency. Teaching Exceptional Children, 53(1), 60–69.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. NCTM Ed. Officers.

Patton, M. Q. (2015). Qualitative research and evaluation methods (4th ed.). SAGE Publications.

Pierce, R., & Stacey, K. (2004). A framework for monitoring progress and planning teaching towards the effective use of computer algebra systems. International Journal of Computers for Mathematical Learning, 9(1), 59–93.

Pimm, D. (2019). Routledge revivals: Speaking mathematically (1987): Communication in mathematics classrooms. Routledge.

Planas, N., & Schütte, M. (2018). Research frameworks for the study of language in mathematics education. ZDM–Mathematics Education, 50(6), 965-974.

Russo, J., & Hopkins, S. (2018). Measuring mental computational fluency with addition: A novel approach. In J. Hunter, P. Perger, & L. Darragh (Eds.), Making waves, opening spaces (Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia, pp. 661–668). Auckland: MERGA.

Santagata, R., Zannoni, C., & Stigler, J. W. (2007). The role of lesson analysis in pre-service teacher education: An empirical investigation of teacher learning from a virtual video-based field experience. Journal of Mathematics Teacher Education, 10(2), 123–140.

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.

Schack, E. O., Fisher, M. H., & Wilhelm, J. A. (Eds.). (2017). Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks. Springer International Publishing.

Sevinc, S., & Galindo, E. (2022). Noticing student mathematical thinking: Self-contemplation of a pre-service teacher. European Journal of Science and Mathematics Education, 10(2), 154–169.

Sherin, M., Jacobs, V., & Philipp, R. (Eds.). (2011a). Mathematics teacher noticing: Seeing through teachers' eyes. Routledge.

Sherin, M. G., Linsenmeier, K. A., & van Es, E. A. (2009). Selecting video clips to promote mathematics teachers' discussion of student thinking. Journal of Teacher Education, 60(3), 213–230.

Sherin, M. G., Russ, R. S., & Colestock, A. A. (2011b). Accessing mathematics teachers’ in-the-moment noticing. In M. G. Sherin, V. R. Jacobs, & R.A. Phillipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 79–94). Routledge.

Sherin, M. G., & van Es, E. A. (2009). Effects of video club participation on teachers’ professional vision. Journal of Teacher Education, 60(1), 20–37. https://doi.org/10.1177/0022487108328155

Stockero, S. (2021). Transferability of teacher noticing. ZDM–Mathematics Education, 53(1), 73–84. https://doi.org/10.1007/s11858-020-01198-y

Sullivan, P., & Lilburn, P. (1997). Open-ended maths activities: Using "good" questions to enhance learning. Oxford University Press.

Sullivan, P., & Mornane, A. (2014). Exploring teachers’ use of, and students’ reactions to, challenging mathematics tasks. Mathematics Education Research Journal, 26(2), 193–214. https://doi.org/10.1007/s13394-013-0089-0

Sullivan, P., Borcek, C., Walker, N., & Rennie, M. (2016). Exploring a structure for mathematics lessons that initiate learning by activating cognition on challenging tasks. The Journal of Mathematical Behavior, 41, 159–170.

Tekin-Sitrava, R., Kaiser, G., & Işıksal-Bostan, M. (2022). Development of prospective teachers’ noticing skills within initial teacher education. International Journal of Science and Mathematics Education, 20(7), 1611–1634.

Teuscher, D., Leatham, K. R., & Peterson, B. E. (2017). From a framework to a lens: Learning to notice student mathematical thinking. In E. O. Schack, M. H. Fisher, & J. A. Wilhelm (Eds.), Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks (pp. 31–48). Springer.

Thomas, D. R. (2006). A general inductive approach for analyzing qualitative evaluation data. American Journal of Evaluation, 27(2), 237–246.

Tripp, T., & Rich, P. (2012). Using video to analyze one’s own teaching. British Journal of Educational Technology, 43(4), 678–704.

van den Kieboom, L. A., Magiera, M. T., & Moyer, J. C. (2017). Learning to notice student thinking about the equal sign: K-8 preservice teachers’ experiences in a teacher preparation program. In E. O. Schack, M. H. Fisher, & J. A. Wilhelm (Eds.), Teacher noticing: Bridging and broadening perspectives, contexts, and frameworks (pp. 141–160). Springer.

van Es, E. A. (2011). A framework for learning to notice student thinking. In M. G. Sherin, V. R. Jacobs, & R.A. Phillipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 164–181). Routledge.

van Es, E.A., & Sherin, M.G. (2002). Learning to notice: Scaffolding new teachers’ interpretations of classroom interactions. Journal of Technology and Teacher Education, 10(4), 571–596.

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, 244–276.

van Es, E. A., & Sherin, M. G. (2021). Expanding on prior conceptualizations of teacher noticing. ZDM–Mathematics Education, 53(1), 17–27. https://doi.org/10.1007/s11858-020-01211-4

Watson, A., & Sullivan, P. (2008). Teachers learning about tasks and lessons. In D. Tirosh & T. Wood (Eds.), Tools and processes in mathematics teacher education (pp. 109–135). Sense Publishers.

Wei, Y., Zhang, Q., Guo, J., & Chen, M. (2023). Learning to teach through noticing: A bibliometric review of teacher noticing research in mathematics education during 2006–2021. Humanities and Social Sciences Communications, 10(1), 1–15.

Weyers, J., Koenig, J., Scheiner, T., Santagata, R., & Kaiser, G. (2024). Teacher noticing in mathematics education: A review of recent developments. ZDM–Mathematics Education, 56(1), 249–264. https://doi.org/10.1007/s11858-023-01527-x

Zamarian, L., Lopez-Rolon, A., & Delazer, M. (2007). Neuropsychological case studies on arithmetic processing. In D. B. Berch & M. M. M. Mazzocco (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 245–264). Paul Brookes.

Downloads

Published

2026-10-14

Issue

Section

Articles