The use of a variety of tasks that encourage students to examine the advantages and limitations of recursive and explicit reasoning. The authors discuss the reasoning for employing different types of reasoning and how to encourage students to justify their thinking.

# Search Results

### 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.

### John K. Lannin and Kathryn B. Chval

Use these specific strategies to confront assumptions about teaching and learning mathematics.

### John K. Lannin, Delinda van Garderen and Jessica Kamuru

This manuscript discusses two important ideas for developing student foundational understanding of the number line: (a) student views of the number sequence, and (b) recognizing units on the number line. Various student strategies and activities are included.

### Brian E. Townsend, John K. Lannin and David D. Barker

Helping students determine different strategies to explore algebraic reasoning tasks can help them draw accurate conclusions.

### Kathryn B. Chval, John K. Lannin, Fran Arbaugh and Angela D. Bowzer

Educators who can elicit preservice teachers' beliefs about teaching mathematics can effectively challenge and change unrealistic expectations.

### John K. Lannin, Fran Arbaugh, David D. Barker and Brian E. Townsend

Give me a productive error over a boring, mundane, and unproductive fact any day.

### John K. Lannin, Brian E. Townsend, Nathan Armer, Savanna Green and Jessica Schneider

An important goal of school mathematics involves helping students use the powerful forms of representation that have been developed over the centuries through the work of mathematicians throughout the world. However, challenges exist in encouraging students to develop meaning for the mathematical symbols used in formal algebra. Research has demonstrated that students often fail to develop a deep understanding of the meaning of symbolic representations of variables (e.g., Booth 1984; Clement 1982), so much so that Thompson (1994) found that a limited understanding of the meaning of variables negatively impacts students who later take college calculus. The question arises as to how we can develop meaning for formal algebraic symbols in the middle grades so that instruction can build on this meaning throughout students' high school and college experiences.