The authors used two technology frameworks to design three seventh-grade mathematics lessons, focusing on opportunities to address collaboration, authenticity, or personalization.

# Browse

## Two Technology Frameworks: Their Use in Lesson Design

### Elizabeth B. Harkey, Angela T. Barlow, and Victoria Groves-Scott

## Understanding Algebraic Expressions Through Figural Patterns

### Casey Hawthorne and John Gruver

This instructional sequence develops your students’ meaningful understanding of algebraic expressions.

## Happy Numbers: A Platform for Conjecture and Exploration

### James A. Gerald and David Jay Hebert

Happy Numbers offer a fun platform for exploring numerical patterns, making conjectures, and investigating mathematical thought in recreational mathematics.

## Using Series to Construct Pythagorean Triples

### Darien DeWolf and Balakrishnan Viswanathan

This article provides a series-focused approach to computing Pythagorean triples.

## GPS: Replicating Patterns

### Monica G. McLeod and Daniel K. Siebert

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.

## Computer-Supported Assessment of Geometric Exploration Using Variation Theory

### Yael Luz and Michal Yerushalmy

We report on an innovative design of algorithmic analysis that supports automatic online assessment of students’ exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic activity format for if–then propositions and implemented the activity on a platform that collects and analyzes students’ work. Finally, we searched for ways to use variation theory to analyze ninth-grade students’ recorded work. We scored and classified data and found correlation between patterns in exploration stages and the conjectures students generated. We demonstrate how automatic identification of mistakes in the early stages is later reflected in the quality of conjectures.

## What Can the Realization Tree Assessment Tool Reveal About Explorative Classroom Discussions?

### Merav Weingarden and Einat Heyd-Metzuyanim

One of the challenges of understanding the complexity of so-called reform mathematics instruction lies in the observational tools used to capture it. This article introduces a unique tool, drawing from commognitive theory, for describing classroom discussions. The Realization Tree Assessment tool provides an image of a classroom discussion, depicting the realizations of the mathematical object manifested during the discussion and the narratives that articulate the links between these realizations. We applied the tool to 34 classroom discussions about a growing-pattern algebraic task and, through cluster analysis, found three types of whole-class discussion. Associations with classroom-level variables (track, but not grade level or teacher seniority) were also found. Implications with respect to applications and usefulness of the tool are discussed.

##
Variations on a Theme of Fibonacci^{
1
}

### Sheldon P. Gordon and Michael B. Burns

We introduce variations on the Fibonacci sequence such as the sequences where each term is the sum of the previous three terms, the difference of the previous two, or the product of the previous two. We consider the issue of the ratio of the successive terms in ways that reinforce key behavioral concepts of polynomials.

## Delving Deeper: What Else Comes after 1 + 2 = 3?

### Charles F. Marion

The simplest of prekindergarten equations, 1 + 2 = 3, is the basis for an investigation involving much of high school mathematics, including triangular numbers, arithmetic sequences, and algebraic proofs.

## The Importance of Play in Middle School Mathematics

### Kate Degner

Using question 28 from the May Problems to Ponder in volume 114, the author and her seventh- and eighth-grade students launched into a discussion of creativity, linearity, piecewise, and recursive definitions of functions. This pattern to ponder provided rich mathematical opportunities for all students in my middle school classroom.