# Snapshots of mathematics teacher noticing during task design

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## Abstract

Designing a mathematically worthwhile task is critical for promoting students’ reasoning. To improve task design skills, teachers often engage in collaborative lesson planning activities such as lesson study. However, to learn from the process of lesson study, it is important for teachers to notice productively the concepts, students’ confusion and the design of the task. But what researchers mean by productive noticing varies. In this article, I present the FOCUS Framework which highlights two characteristics of productive noticing: having an explicit focus for noticing and focusing noticing through pedagogical reasoning. Using these two characteristics, I develop snapshots of noticing as a representation of practice to present a fine-grained analysis of teacher noticing. Through vignettes of teachers discussing the design of a task to teach fractions, I illustrate how two teachers’ noticing can be analysed and represented using snapshots of noticing. To conclude, I highlight what snapshots of noticing tell us about a teacher’s noticing and suggest ways to use these snapshots in future studies of noticing.

## Keywords

Mathematics teacher noticing Productive teaching practices Task design Representations of practice Fractions## Notes

### Acknowledgments

I would like to express my heartfelt thanks to my PhD supervisors, Professors Mike Thomas and Caroline Yoon, for their advice and suggestions regarding the initial drafts. I would also like to thank the reviewers for their insightful comments which were helpful for my revision of the draft. Last but not least, I would like to thank Ms. Amy Soh for helping me with the proofreading. This article is an extended version of an earlier paper presented at the Psychology of Mathematics Education Conference in 2014.

## References

- Alston, A. S., Pedrick, L., Morris, K. P., & Basu, R. (2011). Lesson study as a tool for developing teachers’ close attention to students’ mathematical thinking. In L. C. Hart, A. S. Alston, & A. Murata (Eds.),
*Lesson study research and practice in mathematics education: learning together*(pp. 135–151). Netherlands: Springer.CrossRefGoogle Scholar - Anthony, G., & Walshaw, M. (2009).
*Effective pedagogy in mathematics*. Geneva: International Bureau of Education.Google Scholar - Ball, D. L. (2011). Foreword. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: seeing through teachers’ eyes*(pp. xx–xxiv). New York: Routledge.Google Scholar - Ball, D. L., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.),
*A research companion to principles and standards for school mathematics*(pp. 27–44). Reston: National Council of Teachers of Mathematics.Google Scholar - Ball, D. L., & Forzani, F. M. (2009). The work of teaching and the challenge for teacher education.
*Journal of Teacher Education, 60*(5), 497–511. doi: 10.1177/0022487109348479.CrossRefGoogle Scholar - 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. doi: 10.1177/0022487108324554.CrossRefGoogle Scholar - Barnhart, T., & van Es, E. (2015). Studying teacher noticing: examining the relationship among pre-service science teachers’ ability to attend, analyze and respond to student thinking.
*Teaching and Teacher Education, 45*, 83–93. doi: 10.1016/j.tate.2014.09.005.CrossRefGoogle Scholar - Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology.
*Qualitative Research in Psychology, 3*, 77–101.CrossRefGoogle Scholar - Breen, S., & O’Shea, A. (2010). Mathematical thinking and task design.
*Irish Mathematics Society Bulletin, 66*, 39–49.Google Scholar - Brodie, K. (2010). Mathematical reasoning through tasks: learners’ responses. In K. Brodie (Ed.),
*Teaching mathematical reasoning in secondary school classrooms*(pp. 43–56). New York: Springer.CrossRefGoogle Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics*(N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Trans. N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield Eds.). Dordrecht, The Netherlands: Kluwer Academic.Google Scholar - Bryman, A. (2012).
*Social research methods*(4th ed.). New York: Oxford University Press.Google Scholar - Choy, B. H. (2013). Productive mathematical noticing: what it is and why it matters. In V. Steinle, L. Ball, & C. Bardini (Eds.),
*Proceedings of 36th annual conference of Mathematics Education Research Group of Australasia*(pp. 186–193). Melbourne: MERGA.Google Scholar - 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.Google Scholar - Choy, B. H. (2015).
*The FOCUS framework: snapshots of mathematics teacher noticing*(Unpublished doctoral dissertation). New Zealand: University of Auckland.Google Scholar - Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. In A. Kelly, R. Lesh, & J. Baek (Eds.),
*Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching*(pp. 68–95). New York: Routledge.Google Scholar - Cobb, P., & Whitenack, J. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcripts.
*Educational Studies in Mathematics, 30*(3), 213–228.CrossRefGoogle Scholar - Cohen, D. K., Raudenbush, S. W., & Ball, D. L. (2003). Resources, instruction, and research.
*Educational Evaluation and Policy Analysis, 25*(2), 119–142.CrossRefGoogle Scholar - Doerr, H. M. (2006). Examining the tasks of teaching when using students’ mathematical thinking.
*Educational Studies in Mathematics, 62*(1), 3–24. doi: 10.1007/s10649-006-4437-9.CrossRefGoogle Scholar - Fernandez, C., Cannon, J., & Chokshi, S. (2003). A US–Japan lesson study collaboration reveals critical lenses for examining practice.
*Teaching and Teacher Education, 19*(2), 171–185. doi: 10.1016/s0742-051x(02)00102-6.CrossRefGoogle Scholar - Fernandez, C., Llinares, S., & Valls, J. (2012). Learning to notice students’ mathematical thinking through on-line discussions.
*ZDM Mathematics Education, 44*, 747–759. doi: 10.1007/s11858-012-0425-y.CrossRefGoogle Scholar - Franke, M. L., Kazemi, E., & Battey, D. (2007). Understanding teaching and classroom practice in mathematics. In J. Frank & K. Lester (Eds.),
*Second handbook of research on mathematics teaching and learning*(pp. 225–256). Charlotte, NC: Information Age Publishing.Google Scholar - Glaser, B. G. (1965). The constant comparative method of qualitative analysis.
*Social Problems, 12*(4), 436–445.CrossRefGoogle Scholar - 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). New York: Routledge.Google Scholar - Grossman, P., & McDonald, M. (2008). Back to the future: directions for research in teaching and teacher education.
*American Educational Research Journal, 45*(1), 184–205. doi: 10.3102/0002831207312906.CrossRefGoogle Scholar - Henningsen, M., & Stein, M. K. (1997). Mathematics tasks and student cognition: classroom-based factors that support and inhibit high-level mathematical thinking and reasoning.
*Journal for Research in Mathematics Education, 28*(5), 524–549.CrossRefGoogle 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.Google Scholar - Jacobs, V. R., Lamb, L. L. C., Philipp, R. A., & Schappelle, B. P. (2011a). Deciding how to respond on the basis of children’s understandings. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: seeing through teachers’ eyes*(pp. 97–116). New York: Routledge.Google Scholar - Jacobs, V. R., Philipp, R. A., & Sherin, M. G. (2011). Preface. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: seeing through teachers’ eyes*(pp. xxv - xxix). New York: Routledge.Google Scholar - Lampert, M. (2001).
*Teaching problems and the problems of teaching*. New Haven & London: Yale University Press.Google Scholar - Lampert, M., & Graziani, F. (2009). Instructional activities as a tool for teachers’ and teacher educators’ learning.
*The Elementary School Journal, 109*(5), 491–509.CrossRefGoogle Scholar - Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In A. Kelly & R. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 591–646). Mahwah: Lawrence Erlbaum Associates.Google Scholar - Lewis, C., Perry, R., & Hurd, J. (2009). Improving mathematics instruction through lesson study: a theoretical model and North American case.
*Journal of Mathematics Teacher Education, 12*(4), 285–304. doi: 10.1007/s10857-009-9102-7.CrossRefGoogle Scholar - Li, Y., Chen, X., & Kulm, G. (2009). Mathematics teachers’ practices and thinking in lesson plan development: a case of teaching fraction division.
*ZDM, 41*(6), 717–731. doi: 10.1007/s11858-009-0174-8.CrossRefGoogle Scholar - Mason, J. (2002).
*Researching your own practice: the discipline of noticing*. London: RoutledgeFalmer.Google Scholar - Mason, J. (2011). Noticing: roots and branches. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: aeeing through teachers’ eyes*(pp. 35–50). New York: Routledge.Google Scholar - Mason, J., & Johnston-Wilder, S. (2006).
*Designing and using mathematical tasks*. United Kingdom: Tarquin Publications.Google Scholar - 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. doi: 10.1016/j.jmathb.2010.08.002.CrossRefGoogle Scholar - Pegg, J. (1989). Analyzing a mathematics lesson to provide a vehicle for improving teaching practice.
*Mathematics Education Research Journal, 1*(2), 18-33.Google Scholar - Sherin, M. G., Jacobs, V. R., and Philipp, R. A. (2011a). 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. 1-13). New York: Routledge.Google Scholar - Sherin, M. G., Jacobs, V. R., and Philipp, R. A. (Eds.). (2011b).
*Mathematics teacher noticing: Seeing through teachers’ eyes*. New York: Routledge.Google Scholar - Sherin, M. G., Russ, R. S., & Colestock, A. A. (2011c). Accessing mathematics teachers’ in-the-moment noticing. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: seeing through teachers’ eyes*(pp. 79–94). New York: Routledge.Google Scholar - Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a lesson: successfully implementing high-level tasks.
*Mathematics teaching in the middle school, 14*(3), 132–138.Google Scholar - Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: from research to practice.
*Mathematics teaching in the middle school, 3*(5), 344–350.Google Scholar - Smith, M. S., & Stein, M. K. (2011).
*5 practices for orchestrating productive mathematics discussions*. Reston: National Council of Teachers of Mathematics Inc.Google Scholar - Star, J. R., Lynch, K., & Perova, N. (2011). Using video to improve preservice mathematics teachers’ abilities to attend to classroom features. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: seeing through teachers’ eyes*(pp. 117–133). New York: Routledge.Google Scholar - Star, J. R., & Strickland, S. K. (2008). Learning to observe: using video to improve preservice mathematics teachers’ ability to notice.
*Journal of Mathematics Teacher Education, 11*(2), 107–125. doi: 10.1007/s10857-007-9063-7.CrossRefGoogle Scholar - 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. doi: 10.1080/10986060802229675.CrossRefGoogle Scholar - Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: an analysis of mathematical tasks used in reform classrooms.
*American Educational Research Journal, 33*(2), 455–488.CrossRefGoogle Scholar - Stigler, J., & Hiebert, J. (1999).
*The teaching gap: best ideas from the world’s teachers for improving education in the classroom*. New York: The Free Press.Google Scholar - Stockero, S. L. (2014). Transitions in prospective mathematics teacher noticing. In J.-J. Lo, K. R. Leatham, & L. R. V. Zoest (Eds.),
*Research trends in mathematics teacher education*(pp. 239–259). Switzerland: Springer.CrossRefGoogle Scholar - Sullivan, P., Clarke, D., & Clarke, B. (2013).
*Teaching with tasks for effective mathematics learning*. New York: Springer.CrossRefGoogle Scholar - Takahashi, A., & McDougal, T. (2016). Collaborative lesson research: maximizing the impact of lesson study.
*ZDM*. doi: 10.1007/s11858-015-0752-x.Google Scholar - Thomas, M. O. J. (2006). Developing versatility in mathematical thinking. In A. Simpson (Ed.),
*Proceedings of Retirement as Process and Concept: a festshcrift for Eddie Gray and David Tall*(pp. 223–241). Prague: Charles University.Google Scholar - van Es, E. (2011). A framework for learning to notice students’ thinking. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.),
*Mathematics teacher noticing: seeing through teachers’ eyes*(pp. 134–151). New York: Routledge.Google Scholar - Vondrová, N., & Žalská, J. (2013).
*Paper presented at the 37th Conference of the International Group for the Psychology of Mathematics Education*(Mathematics for teaching and pre-service mathematics teachers’ ability to notice). Germany: Kiel.Google Scholar - Yang, Y., & Ricks, T. E. (2012). How crucial incidents analysis support Chinese lesson study.
*International Journal for Lesson and Learning Studies, 1*(1), 41–48. doi: 10.1108/20468251211179696.CrossRefGoogle Scholar - Zawojewski, J., Chamberlin, M. T., Hjalmarson, M. A., & Lewis, C. (2008). Developing design studies in mathematics education professional development: studying teachers’ interpretive systems. In A. Kelly, R. Lesh, & J. Baek (Eds.),
*Handbook of design research methods in education: innovations in science, technology, engineering, and mathematics learning and teaching*(pp. 219–245). New York: Routledge.Google Scholar