Mathematics Teacher Noticing and Opportunities to Notice Student Thinking April 11, 2019 Darl Rassi

Background The mathematics classroom.

Background Mathematical Thinking Area? Area?! Oh, no! I hate math…

Background Mathematical Thinking Area? 𝐴= 1 2 ×𝑏𝑎𝑠𝑒×ℎ𝑒𝑖𝑔ℎ𝑡 = 1 2 ×4×3 = 6 square inches

Background Mathematical Thinking Area? There are 12 square inches in the rectangle. Area? Area is the number of square inches. Mathematical Thinking I could make a rectangle that is two of the triangles. There must be half that many in the triangle. Area is 6 square inches! 5 in 3 in 4 in

Rationale for Study Mathematics Standards in the U.S. and National Council of Teachers of Mathematics recommend that teachers elicit and use student mathematical thinking (SMT) (NCTM, 2000; 2014; NGA & CCSSO, 2010). Teachers who elicit and use SMT continue to improve in their ability to teach mathematics throughout their careers (Frank, Carpenter, Levi, & Fennema, 2001). Must notice SMT to use it in instruction. SMT must be present in the classroom (Opportunities to Notice)

How do teachers notice SMT? Professional Noticing of Children’s Mathematical Thinking (Jacobs, Lamb, & Philipp, 2010). Attention Interpretation Response Most teacher noticing research Describe what teachers notice Changes in noticing Ways to improve noticing Descriptions of Attention/Interpretation/Response

How do teachers notice SMT? Perceptual Cycle Model (Neisser, 1976) Environment Knowledge and Expectations Physiological Perceptual Processes (Opportunities to Notice) Impact perception Current Presentation Describe what teachers notice Changes in noticing Ways to improve noticing Descriptions of Attention/Interpretation/Response

How do teachers notice SMT? Perceptual Cycle Model (Neisser, 1976) Environment Knowledge and Expectations Physiological Perceptual Processes (Opportunities to Notice) Impact perception Complex relationships. Case study of 4 university students in secondary mathematics education. (Preservice Teachers; PSTs). Describe what teachers notice Changes in noticing Ways to improve noticing Descriptions of Attention/Interpretation/Response

Methods PSTs enrolled in a secondary mathematics methods course. Field experience in local high schools with a mentor teacher. Taught and video-recorded 3 lessons Were interviewed after teaching the three lessons about what they noticed about SMT while teaching.

Analysis What SMT is present in the classroom (opportunities to notice)? Identified opportunities to notice using a framework for student mathematics and then SMT (Leatham, Peterson, Stockero, & Van Zoest, 2015).

Results – Opportunities to Notice Qualitative Data Analysis Software Timeline of class period Bradley’s first lesson. Episodes (Topics) SM SMT

Results – Opportunities to Notice Bradley’s Lessons Sadie’s Lessons Randy’s Lessons Jamie’s Lessons

Results – Opportunities to Notice

Results – Opportunities to Notice Overall: Few opportunities to notice Eliciting and using SMT is difficult (Fraivillig, Murphy, & Fuson, 1999) Randy: Lesson 1 Low-stakes Student collaboration/ problem-solving Jamie: Minimal Focuses on telling students correct way to do the problems.

Implications Shift focus from “Can teachers notice SMT?” to “Can teachers create opportunities to notice SMT?” Why are teachers (not) creating opportunities to notice SMT? Allow PSTs to teach in low-stakes environment. Have students collaborate on open-ended problem-solving tasks.

Next Steps What SMT was noticed? Analyze interview transcripts.

Next Steps Connect opportunities to notice (environment) with what was noticed (perception)?

Opportunities to Notice & Noticing Interesting example Logarithm log⁡( ) has a base of 10 Natural log is abbreviated ln⁡( ) and has a base of 𝑒.

Opportunities to Notice & Noticing Researcher: She [Ann] asked a question. Do you remember that? Sadie: Vaguely yeah. She hasn't been there for a lot of my lessons, so I totally forgot she was in this one. [Conversation about other students]. Sadie : I think that's where Ann was coming from was she had that concrete thinking that it's [the base of the natural logarithm] always 10. Then we were able to reason together the difference, like if you see an L-N then it's an e and if it's a log then it's a 10. I remember she asked, "How do you know when to use which one?" I think that's what she asked, so then I was like, ""What's the difference?" From there I think she would be thinking I either use an e or a 10, but I don't know where, and I don't know when.

Opportunities to Notice & Noticing From classroom video: Sadie : X is by itself. So, we're done. Okay? So, the trick in this one was just using that natural log to get that base E. Ann (Student): Wait, so you know how the e equals 10 to the 10 there instead of the e there. So, you wouldn't solve that out though 10 to the third equals e to the third equals X. Sadie : The 10 is a different kind of log. When we have that L and that N, that should trigger you to think of e. So, we'll go through your, that scenario with the other log. Actually, right now. So, if we go through this one right now, Ann brings up a good point. How do I know if there's nothing there to put 10 or an e. I know those are my two options, but how do I know which one do I do?

Conclusion PSTs create few opportunities to notice SMT The lesson that had the most opportunities to notice SMT was low-stakes and had students collaborating with problem-solving tasks. Further work needs to be done to connect opportunities to notice to what teachers notice about SMT.

Thank you! Questions? Frank, M. L., Carpenter, T. P., Levi, L., & Fennema, E. (2001). Capturing teachers’ generative change: A follow-up study of professional development in mathematics. American Educational Research Journal, 38(3), 653689. Fraivillig, J. L., Murphy, L. A., & Fuson, K. C. (1999). Advancing children’s mathematical thinking in everyday mathematics classrooms. Journal for Research in Mathematics Education, 30(2), 148170. doi.org/10.2307/749608 Jacobs, V. R., Lamb, L. C., & Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169202. Leatham, K. R., Peterson, B. E., Stockero, S. L., & Van Zoest, L. R. (2015). Conceptualizing mathematically significant pedagogical opportunities to build on student thinking. Journal for Research in Mathematics Education, 46(1), 88124. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010a). Common core state standards for mathematics. Washington, DC: Authors. Neisser, U. (1976). Cognition and reality. San Francisco, CA: W.H. Freeman.