# Algebra

## Using Schemas to Develop Algebraic Thinking

This article describes how students generalize and formalize patterns using student developed schemas including subtracting out and building up. The article includes samples of student work including students explanations for each schema. This article could supplement readings from section II and IV.

## Using Data-Collection Devices to Enhance Students' Understanding

This article examines four areas of difficulty students have with graphing and modeling. The four areas are identified as connecting graphs with physical concepts, connecting graphs with the real world, transitioning between graphs and physical events, and building graphical concepts through student discourse. The article looks at studentsí misconceptions and the role of technology in helping to form accurate graphical concepts. This article could be used in conjunction with the readings from section IV.

## Teaching Realistic Mathematics Modeling: A Teaching Experiment with Fifth Graders

This article can be used to supplement CTS Sections III and IV. The article describes the design and the results of an exploratory teaching experiment carried out to test the hypothesis that it is feasible to develop in pupils a disposition toward (more) realistic mathematical modeling. This goal is achieved by immersing them in a classroom culture in which word problems are conceived as exercises in mathematical modeling, with a focus on the assumptions and the appropriateness of the model underlying any proposed solution.

## Strategies Used by Second-Year Algebra Students to Solve Problems

This article is a brief overview of the results of a research project focusing on solution strategies algebra students used to solve non-linear function problems. The article focuses on the variety of strategies, the relationship to achievement and using multiple representations. Examples of the constructed response items and student solutions are given. This reading could be used to supplement readings from section II and IV.

## Some Misconceptions Concerning the Concept of the Variable

The selected reading, pages 313-315, can be used for addtional readings in Section IV to examine research on student learning.

## Middle School Students' Understanding of Core Algebraic Concepts: Equivalence + Variable

This article describes the results of a multi-year research project on algebraic reasoning in middle school students. The article describes middle school studentsí understandings and/or misunderstandings of two core algebraic ideas ñ equivalence and variables. The article gives student response examples and discusses implications for instruction. This article could be used to supplement the readings from sections II and IV.

## How Students Learn: Mathematics in the Classroom

How Students learn uses the principles and findings from How People Learn within the context of the mathematics classroom. An introduction to the principles as they apply to mathematics is included in chapter 2, whole numbers at the elementary level is the focus of chapter 3, rational numbers at the middle level is the focus of chapter 4 and chapter 5 focuses on functions at the high school level. This would be an additional reference to sections II, III and IV.

## Developing Students' Understanding of Variable

This article describes the implementation of an activity designed to confront a common misconception about variable. The article describes common misconceptions held by middle level students and describes a problem situation designed to uncover studentsí misconceptions. This article could be used to supplement the readings from sections II and IV.

## Building a Foundation for Learning Algebra in the Elementary Grades

The research summarized in this Volume of In Brief focuses on the ability of elementary students to reason algebraically. The articles can be used with CTS Section IV.

## Balancing Act: The Truth behind the Equals Sign

This article illustrates the misconceptions that students have when using the equals sign and describes an activity used with students to develop the foundation for an accurate conception of equivalency. The author states that developing an accurate understanding of the equal sign is the basis for comprehending equations and inequalities. This article could be used to supplement the readings from sections II and IV.