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This article reports on elementary students' understanding of time in the context of common classroom manipulatives and notational systems. Students in Grades 2 (n = 72) and 4 (n = 72) participated in problem-solving interviews involving different clocks. Quantitative results revealed that students' performances were significantly different as a function of the tool available. Descriptive case studies of 3 Grade 4 students are presented in which students demonstrated competencies in conventions related to benchmark numeric conversions between hours and minutes and counting by 5s around the clock, yet only partial competencies related to the integral relationship between hours and minutes. Implications for theory and the treatment of time in curriculum and instruction are discussed.

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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 areas 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, such knowledge could lead to recommendations for how to support the curricular work of teachers and novice teachers in particular.

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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.

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Fourth graders engage in mathematical reasoning and problem solving that leads to insights about the relationship between hour and minute units.