Challenges in Measuring Secondary Mathematics Teachers’ Professional Noticing of Students’ Mathematical Thinking
Susan D. Nickerson, Lisa Lamb and Raymond LaRochelle
Abstract Our focus is on teachers’ professional noticing of children’s mathematical thinking which Jacobs et al. (Jacobs, Lamb, & Philipp, 2010) describe as a set of three interrelated skills: (1) attending to students’ strategies, (2) interpreting the students’ mathematical understandings, (3) deciding how to respond on the basis of students’ understandings. We focus on secondary teachers’ professional noticing of children’s mathematical thinking because we believe teachers with expertise in noticing children’s mathematical thinking are better poised to support their students’ mathematical performance and understanding. However, studying teacher noticing at the secondary level presents unique methodological challenges. We first consider methodological issues in measuring K-12 teachers’ professional noticing of children’s mathematical thinking, and then consider three methodological challenges that are particular to secondary mathematics classrooms. These issues center around the challenges of artifact selection, determining the relative sophistication of responses, and the lack of access to experts’ responses at the secondary level. Also, we consider the cultures of teaching in the elementary and secondary contexts.
Keywords Professional noticing Methodology Inservice Secondary teachers Mathematics
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E.O. Schack et al. (eds.), Teacher Noticing: Bridging and Broadening Perspectives, Contexts, and Frameworks, Research in Mathematics Education,
Imagine a teacher in a secondary mathematics classroom circulating while her 35 students work in small groups to solve an algebraic-generalization task. Perhaps she makes note of whether all students in a group are engaged and monitors students’ affect. She may wish that a particular student’s reasoning was visible or more understandable. She may or may not be looking for and may or may not be able to describe connections among the diverse mathematical responses. She likely observes many approaches taken to the task and critiques their sophistication, as well as their alignment with expected mathematical goals and the normative language, notation, and representations of mathematics. She wonders what statements, representations, or questions would support her students’ thinking. Furthermore, she must decide whose approaches to highlight: hers alone, one student’s approach, or several approaches. Suppose one approach to the task is representative of the majority of thinking. A different approach displays a misconception she feels should be discussed, while yet another approach is novel and unexpectedly touches upon a related mathematical concept. Two approaches may provide an opportunity to compare and contrast. In making the decision about what approach(es) to highlight, she perhaps considers students who need opportunities to share or who will model expertise. She reluctantly recalls the testing calendar. What pedagogical moves and subsequent tasks will best advance her mathematical agenda?
In a typical classroom, teachers must interact with and respond to an overwhelming amount of information. What to do with this information often involves making choices about to what to attend and how to make sense of that information. Before responding to an event in the classroom, a teacher must first notice that event. Now imagine measuring what the teacher noticed in the classroom scenario described above. We could observe the teacher’s practice and infer from the teacher’s actions what she is noticing. In the observation though, we cannot access what she noticed unless we see her act upon it. Or we may ask the teacher after class to reflect on what she noticed. Neither approach tells us directly about the in-the-moment, yet hidden, practice of professional noticing (Sherin, Russ, & Colestock, 2011). These are just some of the methodological challenges associated with assessing teacher noticing.
In this chapter, we argue that studying teacher noticing at the secondary level presents its own set of unique methodological challenges. Our goals for this chapter are to briefly discuss methodological issues for assessing teachers’ professional noticing of children’s mathematical thinking for all grade-level teachers (both in-service and preservice) and then to highlight methodological challenges that are particular to secondary mathematics classrooms. To underscore the challenges of studying the noticing of secondary mathematics teachers, we contrast the methods of Jacobs, Lamb, and Philipp (2010) in studying the professional noticing of K-3 teachers and our methods in our recent investigation with 16 secondary mathematics teachers. Because our work focused on secondary mathematics teachers, from this point forward we use the term students ’ mathematical thinking
(SMT) instead of children’s mathematical thinking. In the next section, we describe an overarching challenge related to data-collection and data-analysis methods for studying teachers’ in-the-moment noticing of students’ mathematical thinking.