algebraic notation and procedures but instead provide a basis for using algebraic reasoning by building on students' understanding of number and operations. As such, the goal of the task was to leverage students' understanding of positive integer factors and

### George J. Roy, Jessica S. Allen and Kelly Thacker

### Joyce W. Bishop, Albert D. Otto and Cheryl A. Lubinski

The changing applications of mathematics have contributed to a shift from the perception that mathematics is a fixed body of arbitrary rules to the realization that the discipline is “a vigorous active science of patterns” (National Research Council 1989, p. 13). NCTM's *Curriculum and Evaluation Standards for School Mathematics* (1989) recommends using patterns to promote mathematical understanding and, in particular, algebraic reasoning. A number of other mathematics education reform documents make similar recommendations (e.g., AAAS [1989]; National Research Council [1990]; Steen [1990]; NCTM [2000]). Researchers have begun to identify different approaches that students use to reason about patterns (Bishop 1997; MacGregor and Stacey 1993; Orton and Orton 1996; Stacey 1989). Research also shows that using students' thinking about patterns can help them develop a better understanding of mathematical concepts and the representations that reflect those concepts (Carey 1992; Fennema, Carpenter, and Peterson 1989). This article illustrates how students' thinking about geometric patterns can be used to help them develop algebraic reasoning and to make sense of mathematical notation and symbols.

### Maria L. Blanton and James J. Kaput

We present here results of a case study examining the classroom practice of one thirdgrade teacher as she participated in a long-term professional development project led by the authors. Our goal was to explore in what ways and to what extent the teacher was able to build a classroom that supported the development of students' algebraic reasoning skills. We analyzed 1 year of her classroom instruction to determine the robustness with which she integrated algebraic reasoning into the regular course of daily instruction and its subsequent impact on students' ability to reason algebraically. We took the diversity of types of algebraic reasoning, their frequency and form of integration, and techniques of instructional practice that supported students' algebraic reasoning as a measure of the robustness of her capacity to build algebraic reasoning. Results indicate that the teacher was able to integrate algebraic reasoning into instruction in planned and spontaneous ways that led to positive shifts in students' algebraic reasoning skills.

### Mitchell J. Nathan and Kenneth R. Koedinger

Mathematics teachers and educational researchers ordered arithmetic and algebra problems according to their predicted problem-solving difficulty for students. Predictions deviated systematically from algebra students' performances but closely matched a view implicit in textbooks. Analysis of students' problem-solving strategies indicates specific ways that students' algebraic reasoning differs from that predicted by most teachers and researchers in the sample and portrayed in common textbooks. The Symbol Precedence Model of development of algebraic reasoning, in which symbolic problem solving precedes verbal problem solving and arithmetic skills strictly precede algebraic skills, was contrasted with the Verbal Precedence Model of development, which provided a better quantitative fit of students' performance data. Implications of the findings for student and teacher cognition and for algebra instruction are discussed.

### Victoria R. Jacobs, Megan Loef Franke, Thomas P. Carpenter, Linda Levi and Dan Battey

A yearlong experimental study showed positive effects of a professional development project that involved 19 urban elementary schools, 180 teachers, and 3735 students from one of the lowest performing school districts in California. Algebraic reasoning as generalized arithmetic and the study of relations was used as the centerpiece for work with teachers in Grades 1–5. Participating teachers generated a wider variety of student strategies, including more strategies that reflected the use of relational thinking, than did nonparticipating teachers. Students in participating classes showed significantly better understanding of the equal sign and used significantly more strategies reflecting relational thinking during interviews than did students in classes of nonparticipating teachers.

### F. D. Rivera and Joanne Rossi Becker

Findings, insights, and issues drawn from a three-year study on patterns are intended to help teach prealgebra and algebra.

### Temple A. Walkowiak

A talkative second grader helps her teacher describe, extend, and generalize about growing patterns.

### Sheryl L. Stump

Such classic problems as the Painted cube help teachers and students deepen their understanding and expand their algebraic thinking.

### Drew Polly

Try these approaches to boost your students' understanding and higher-order thinking skills.

### John K. Lannin

NCTM's (2000) recommendations for algebra in the middle grades strive to assist students' transition to formal algebra by developing meaning for the algebraic symbols that students use. Further, students are expected to have opportunities to develop understanding of patterns and functions, represent and analyze mathematical situations, develop mathematical models, and analyze change. By helping students move from specific numeric situations to develop general rules that model all situations of that type, teachers in fact begin to address the NCTM's recommendations for algebra. Generalizing numeric situations can create strong connections between the mathematical content strands of number and operation and algebra (as well as with other content strands). In addition, these generalizing activities build on what students already know about number and operation and can help students develop a deeper understanding of formal algebraic symbols.