# Algebra

## A Student Asks About (-5)!

A first-year algebra student’s curiosity about factorials of negative numbers became a starting point for an extended discovery lesson into territory not usually explored in secondary school mathematics.

## Problematizing and Assessing Secondary Mathematics Teachers’ Ways of Thinking

Engage with presenters as they discuss assessment and rubrics designed to measure secondary teachers’ mathematical habits of mind.

Work in secondary mathematics education takes many approaches to content, pedagogy, professional development and assessment. This session aims to illuminate the richness of hte content of secondary mathematics and the field of secondary mathematics education by sharing two such approaches and reflecting on the differences and commonalities between the two.

## Constructing and Role-Playing Student Avatars in a Simulation of Teaching Algebra for Diverse Learners

From the perspectives of Graduate Research Assistants (GRAs), this study examines the design and implementation of a simulated teaching environment in Second Life (SL) for prospective teachers to teach algebra for diverse learners. Drawing upon the Learning-for-Use framework, the analyses provide evidence on the development of student avatars in construction and role-playing activities. The study reveals challenges, procedures, and suggestions for future simulations. This study also calls for research efforts toward preparing mathematics teachers for cultural diversity.

This study examines the design and implementation of a simulated teaching environment in Second Life for prospective teachers.

## Illuminating Coordinate Geometry with Algebraic Symmetry

A symmetric polynomial is a polynomial in one or more variables in which swapping any pair of variables leaves the polynomial unchanged. For example, f(x, y, z) = xy +xz + yz is a symmetric polynomial.

A symmetric polynomial is a polynomial in one or more variables in which swapping any pair of variables leaves the polynomial unchanged. For example, f(x, y, z) = xy +xz + yz is a symmetric polynomial. If we interchange the variables x and y, we obtain yx + yz + xz, which is the same as f(x, y, z); likewise, swapping x and z (or y and z) returns the original polynomial. These polynomials arise in many areas of mathematics, including Galois theory and combinatorics, but they are rarely taught in a high school curriculum. In this article, we describe an application of symmetric polynomials to a familiar problem in coordinate geometry, thus introducing this powerful tool in a context that is accessible to high school students.

## SmartGraphs: Algebra

The Concord Consortium has developed 19 activities for teaching and learning algebra that are available online or as an app for iPad or Android tablet computers. These activities—which cover a variety of algebra topics, from linear equations to transformations of functions—help students develop skills creating and using algebraic functions and graphs to solve problems.

The Concord Consortium has developed 19 activities for teaching and learning algebra that are available online or as an app for iPad or Android tablet computers. These activities—which cover a variety of algebra topics, from linear equations to transformations of functions—help students develop skills creating and using algebraic functions and graphs to solve problems. Hints and scaffolds support learners who need help.

## Teaching Viable Argumentation and Measuring the Effects

How do we encourage referent-based mathematical argumentation without encouraging students to request that examples accompany otherwise viable arguments? Assessment concerns are explored and discussed.

The LAMP project has developed a sequence of lessons in a hypothetical learning trajectory that targets students’ ability to write viable arguments in algebraic contexts. Most of the lessons encourage students to produce a referent (e.g., variable expression or equation, generic example, diagram) as the foundation of their argument. Students come to the lessons with a predisposition for example production in support of their claims and to augment arguments.

## Student Materials, Professional Development, and Assessment Organized Around Habits of Mind in the CCSSM

Learn about three projects centered on algebraic habits of mind: a puzzle-centric curriculum for middle school and at-risk algebra students, professional development on the Standards for Mathematical Practice, and an assessment for teachers.

Algebraic habits of mind, at the core of five of the Standards for Mathematical Practice, become both a potent and appealing intervention for at-risk algebra students and a solid prevention-model middle-school course either to accelerate algebra or to ensure success in a later algebra course. The session focuses on the habits of mind in that context, in related professional development work that addresses the Standards for Mathematical Practices, and on assessment of algebraic habits of mind in teachers.

## Discussion of Promising Scale-up Strategies for Reaching Classrooms

Participants and the presenters will discuss their experiences—including releasing free and paid apps—and provide suggestions to others for successfully reaching many users.

Over a period of five years the SmartGraphs project developed HTML5 software for teaching and learning STEM subjects that make use of line graphs and scatter plots. SmartGraphs activities help students understand the “story” represented by a graph. The project created dozens of activities for algebra, physical science, and other STEM subjects, as well as an authoring system allowing non-computer-programmers to create and disseminate free online activities.

## Assessing Secondary Teachers’ Algebraic Habits of Mind

Participants provide feedback on a preliminary paper-and-pencil assessment of secondary teachers’ mathematical habits of mind (MHoM) and use classroom video to examine MHoM in practice.

In Assessing Secondary Teachers’ Algebraic Habits of Mind, the project team is developing tools to study the following questions: What are the mathematical habits of mind (MHoM) that secondary teachers use, how do they use them, and how can we measure them?

In this session, presenters share a paper-and-pencil assessment being developed to measure how teachers use MHoM when they do mathematics for themselves. The presenters also share classroom video and a preliminary framework for examining MHoM in teaching practice.