Part of the work of the Center for Algebraic Thinking is to create modules of instruction for Mathematics Methods courses that make use of research on students’ thinking. The following modules integrate that research and formative assessments to facilitate preservice teachers learning about how students think when doing algebra. Each module consists of a set that introduces the concepts of the module, integrates video of students discussing their thinking regarding the concepts, formative assessments, teaching strategies, and discussion of research. These modules are in development, so any feedback is appreciated (algebrathinking”at”gmail.com). Each title is a link to a PDF. You can access the videos at VIMEO, but you will need permission. Here is the list of videos with a bit of information about each one (LINK).

#### ALGEBRAIC RELATIONS: Informal Procedures and Student Intuition

When solving equations, students often resort to applying taught algorithms to find the answer. However, when students use their intuition and informal procedures they can learn to move flexibly between solution strategies.

#### ALGEBRAIC RELATIONS: The Meaning of the Equal Sign

The goal of this module is to help pre-service teachers to develop their understanding of “=” and help them to understand the relational properties of “=” and highlight the difference between operators symbol such as “+, -, /,…”, and “=”

#### ANALYSIS OF CHANGE: Axes

Students struggle to understand the dynamic between the axes in a graph and the relationship between x and y.

#### ANALYSIS OF CHANGE: Connecting Graphs to Real World Data

Students frequently have difficulty creating graphs of real world situations. In this module candidates focus on connecting changing relationships to graphical representations.

#### ANALYSIS OF CHANGE: Connecting Graphs to Algebraic Relationships

It is important for students to be able to construct a graph that represents an algebraic equation. Equally important is the ability to form an algebraic equation that represents a simple graph (such as a linear graph). This module is designed to help candidates understand some of the important skills and common misconceptions associated with connecting algebraic expressions to graphical representations.

#### VARIABLES & EXPRESSIONS: Generalization

The transition from arithmetic to algebra is primarily one of moving from concrete to abstract thinking. Abstract thinking in algebra is often the generalization of specific instances in arithmetic to a rule or formula that captures the pattern algebraically.

#### VARIABLES & EXPRESSIONS: Representation

There is a developmental progression in understanding the meaning of variables. A key concept is understanding what variables can represent and how that is influenced by context.

#### MODELING: Translating Words into Equations--Letters

This module is focused on the teaching of algebra translation as it appears between words and equations. Almost all word problems begin with a problem statement in written or spoken language that requires a semantic understanding of the problem situation and task. In algebra, there is usually a translation of that understanding into the “language” and syntax of algebraic expressions and equations. There is a related, although cognitively different, skill that is required to understand the semantics of a quantitative model that is derived from the interpretation of an algebraic expression or equation. This module focuses specifically upon use of letters to signify quantities that may be:

a) letter as unknown as in 5n = 2(n-3) + 8

b) letter as variable as in 5n = y/8

#### MODELING: Translating Words into Equations--Numbers

This module is focused on the teaching of algebra translation as it appears between words and equations. Almost all word problems begin with a problem statement in written or spoken language that requires a semantic understanding of the problem situation and task. In algebra, there is usually a translation of that understanding into the “language” and syntax of algebraic expressions and equations. There is a related, although cognitively different, skill that is required to understand the semantics of a quantitative model that is derived from the interpretation of an algebraic expression or equation. This module focuses specifically upon use of numbers as constants as in ‘a’ above (the number 8) and numbers as factors that imply operations (as in the 5 in 5n).

#### MODELING: Translating Words into Equations--Relevant Variables

This module is focused on the teaching of algebra translation as it appears between words and equations. Almost all word problems begin with a problem statement in written or spoken language that requires a semantic understanding of the problem situation and task. In algebra, there is usually a translation of that understanding into the “language” and syntax of algebraic expressions and equations. There is a related, although cognitively different, skill that is required to understand the semantics of a quantitative model that is derived from the interpretation of an algebraic expression or equation. This module focuses specifically upon use of variables of interest --- the need for determination of which variables matter and the avoidance of irrelevant and extraneous variables and maintaining their definitions after algebraic transformations of equations

#### MODELING: Translating Equations into Words

This module is focused on the teaching of algebraic modeling of quantitative relationships observed in the translation of equations into verbal descriptions. The skills needed for students to make sense of equations that are used to describe quantitative relationships in the sciences are often related to their understanding of algebraic equations and expressions. This is especially critical to study in science, engineering, economics, social science, business, construction, etc.

#### MODELING: Physical Representation

This module is focused on the teaching of algebraic modeling of quantitative relationships observed in physical representations. By “physical representation”, we mean any physical event that may be observed and/or manipulated to represent quantitative relationships “in the world.” These events are “real world” in so far as they may be experienced in the world. Their relevance to student lives may occur only in the context of the experience generated to create learning in or out of the classroom that is motivated by the sense-making that comes from interpretation, problem-solving, or the engagement through active participation. The skills developed to model physical events are critical to study in science, engineering, economics, social science, business, construction, etc.

#### PATTERNS & FUNCTIONS: Connecting Informal Thinking and Observations to Algebraic Notation

Students need support with seeing how visual representations are constructed if they are to make generalizations about the representation. If they are given opportunities to construct or draw the pictorial growth pattern they are better able to “see” the important relationships (i.e., the number of chips from one figure to the next, the figure number and the number of chips).

#### PATTERNS & FUNCTIONS: What is a Function?

The concept of function is a complex mathematical idea whose formal definition has evolved over time. This module provides an opportunity for pre-service and in-service teachers to grapple with a number of key misconceptions that often hinder students’ deeper understanding of functions.

#### PATTERNS & FUNCTIONS: How do Examples Impact our Understanding of Functions?

Students understandings of key concepts and their ability to generalize about mathematical ideas are directly affected by the examples their teachers choose to use in the course of instruction. In this module, pre-service and in-service teachers have the opportunity to explore strategies for choosing algebraic examples that support deeper conceptual understanding for students.