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.

# Search Results

### Darrell Earnest

### Darrell Earnest and Julie M. Amador

Share news about happenings in the field of elementary school mathematics education, views on matters pertaining to teaching and learning mathematics in the early childhood or elementary school years, and reactions to previously published opinion pieces or articles. Find detailed department submission guidelines at http://www.nctm.org/WriteForTCM.

### Darrell Earnest and Aadina A. Balti

Incorporating algebra into the elementary grades has become a focus for teachers, principals, and administrators around the country. Algebra is commonly regarded as a gateway to future opportunity (e.g., Moses and Cobb 2001), and elementary mathematics standards at both the state and national levels now reflect this effort to provide students with opportunities to learn critical concepts before middle and high school (California Department of Education 1997; NCTM 2000). However, implementing algebra standards at the elementary level is challenging—how do mathematics educators effectively and meaningfully incorporate algebraic ideas into K–5 curriculum? When elementary teachers are unfamiliar with early algebra, lessons designed and labeled as algebraic may become arithmetic exercises; the algebra then remains hidden from both the teacher and students in the implementation. The result is that the algebra standard is only superficially addressed.

### Darrell Earnest, Susan Radtke and Siri Scott

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

### Jennifer Pfotenhauer, Rick Kleine, Yasmin Sitabkhan and Darrell Earnest

Students had been learning about integers and fractions on the number line. For a lesson on mixed numbers, they solved an assessment problem at the beginning of the lesson. After the lesson, the authors interviewed two students individually and asked each girl to solve the same problem again.

### Leslie Dietiker, Lorraine M. Males, Julie M. Amador and Darrell Earnest

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.

### David W. Carraher, Analúcia D. Schliemann, Bárbara M. Brizuela and Darrell Earnest

Algebra instruction has traditionally been postponed until adolescence because of historical reasons (algebra emerged relatively recently), assumptions about psychological development (“developmental constraints” and “developmental readiness”), and data documenting the difficulties that adolescents have with algebra. Here we provide evidence that young students, aged 9–10 years, can make use of algebraic ideas and representations typically absent from the early mathematics curriculum and thought to be beyond students' reach. The data come from a 30-month longitudinal classroom study of four classrooms in a public school in Massachusetts, with students between Grades 2–4. The data help clarify the conditions under which young students can integrate algebraic concepts and representations into their thinking. It is hoped that the present findings, along with those emerging from other research groups, will provide a research basis for integrating algebra into early mathematics education.