Curricular noticing: A comprehensive framework to describe teachers’ interactions with curriculum materials

Building on the work of Professional Noticing of Children’s Mathematical Thinking, we introduce the Curricular Noticing Framework to describe how teachers recognize opportunities within curriculum materials, understand their affordances and limitations, and use strategies to act on them. This framework builds on Remillard’s (2005) notion of participation with curriculum materials, connects with and broadens existing research on the relationship between teachers and written curriculum, and highlights new are as for research. We argue that once mathematics educators better understand the strategic curricular practices that support ambitious teaching, which we refer to as professional curricular noticing, then this knowledge can lead to recommendations for how to support the curricular work of teachers, particularly novice teachers

Supporting Generative Thinking about Number Lines, the Cartesian Plane, and Graphs of Linear Functions

This dissertation explores fifth and eighth grade students' interpretations of three kinds of mathematical representations: number lines, the Cartesian plane, and graphs of linear functions. Two studies were conducted. In Study 1, I administered the paper-and-pencil Linear Representations Assessment (LRA) to examine students' understanding of the three representations. The LRA had an experimental component that compared performance on routine problems to non-routine problems (problems not amenable to routine solution procedures). I administered the assessment to Grade 5 students (n=126) who had no formal instruction involving function graphs, and I compared their performances with those of Grade 8 students (n=131) enrolled in Algebra 1. A repeated measures ANOVA revealed students in each grade performed better on routine problems compared to non-routine problems, suggesting that routine problems may falsely indicate greater competence. Paired samples t-tests indicated no differences in performance between Grades 5 and 8 students on number line items, though Grade 8 students outperformed fifth graders on Cartesian plane and function graph items. Videotaped interviews with a subset of Grades 5 and 8 students revealed that students in each grade approached tasks across representations in similar ways, suggesting persisting misconceptions. Interviews also revealed patterns unique to each grade.In Study 2, I examined the efficacy of a tutorial intervention. The intervention introduced written definitions to support principled understandings of the number line, the Cartesian plane, and function graphs. A repeated measures ANOVA that compared pre/posttest scores of Grade 5 students (n=20) to a matched control group (n=20) revealed significant gains from pre- to posttest in the experimental group, with no detectable gains in a control. At posttest, Grade 5 tutorial students performed significantly better on non-routine LRA problems than Grade 8 students who did not receive the tutorial. Video analysis revealed a correlation between tutorial students' appropriate uptake of definitions and gains from pretest to posttest.Analyses across the two studies indicate that instruction that supports students' coordination of linear and numerical units can support students' learning with understanding. Potential applications include the development of curricula to support students' learning with understanding related to these representations and teacher professional development interventions

Time as a Measure: Elementary Students Positioning the Hands of an Analog Clock

Elementary students have difficulty with the topic of time. The present study investigated students’ actions to position hour and minute hands on an analog clock to indicate particular times of the day. Using one-on-one interviews with students in Grades 2 and 4 (n = 48), we analyzed whether students were more accurate for one hand indicator (hour or minute) versus the other as well as their solution approaches as they positioned each hand. We first present a quantitative analysis of student performance to document whether hour and minute hands posed differential challenges for students as they positioned hands to indicate particular times. Results indicate the hour hand is significantly more challenging to position accurately than the minute hand. Students’ solutions reflected varied approaches, including consideration of the quantitative hour-minute multiplicative relationship, attention to part-whole relations, and matching numbers from the provided time to numerals on the clock. We discuss implications for theory and instruction, including the relationship of time to length measure learning trajectories and the current treatment of time in K-12 mathematics standards for the United States