Learning Math: Patterns, Functions, and Algebra
Algebraic Thinking Part A: A Framework for Algebraic Thinking (15 minutes)
Session 1, Part A
What does algebra mean to you? For many of us, the word “algebra” conjures classroom memories of xs and ys, manipulating numbers and symbols according to prescribed rules, and solving for the unknown in an equation. It may not have been clear where the rules came from, or why x could be different in every problem.
In recent years, the vision of how algebra is taught has been changing. Algebraic thinking begins as a study of generalized arithmetic. The focus is on operations and processes rather than numbers and computations. When algebra is studied this way, the rules for manipulating letters and numbers in equations don’t seem arbitrary, but instead are a natural extension of what we know about computation.
In this video segment, participants discuss their initial impressions of algebraic thinking.
Does thinking algebraically require formal “algebra?” Think of some situations in everyday life in which a person might think algebraically but would not consider themselves to be performing “algebra.”
You can find this segment on the session video, approximately 2 minutes and 19 seconds after the Annenberg Media logo.
Read the following description of algebraic thinking:
What does algebraic thinking really mean? Two components of algebraic thinking, the development of mathematical thinking tools and the study of fundamental algebraic ideas, have been discussed by mathematics educators and within policy documents (e.g., NCTM, 1989, 1993, 2000; Driscoll, 1999). Mathematical thinking tools are analytical habits of mind. They include problem solving skills, representation skills, and reasoning skills. Fundamental algebraic ideas represent the content domain in which mathematical thinking tools develop. Within this framework, it is understandable why conversations and debates occur within the mathematics community regarding what mathematics should be taught and how mathematics should be taught. In reality, both components are important. One can hardly imagine thinking logically (mathematical thinking tools) with nothing to think about (algebraic ideas). On the other hand, algebra skills that are not understood or connected in logical ways by the learner remain “factoids” of information that are unlikely to increase true mathematical understanding and competence.
From Shelley Kriegler’s project “Mathematics Content Programs for Teachers,” UCLA Department of Mathematics, January 2000.
This passage points out two components of algebraic thinking: mathematical thinking tools and algebraic ideas. In this session, and in the sessions that follow, we will immerse ourselves in these two components of algebraic thinking. We’ll use mathematical thinking tools like problem-solving, reasoning, and representation skills to help us make sense of situations. We’ll also take a look at algebraic ideas, including patterns, variables, and functions. Note 2
Problem A1: Write and Reflect
What would you define as algebraic thinking?
Problem A2: Write and Reflect
When do you think students begin to think algebraically?
Write down your answers to Problems A1 and A2, because we will return to them at the end of the course.
There’s no clear line separating formal algebra from informal algebraic ideas. Though you may not realize it, the kind of logical thinking required in reasoning about real-life situations and reasoning about mathematics is often very similar.
As you work on the rest of the problems in this session, focus on the kind of thinking that’s required to solve them and the kinds of representations that are most helpful in your reasoning. These problems may or may not be considered “formal algebra,” but they will hopefully reinforce the notion that “making sense” is a big part of what mathematics is all about.
The goal of this session, as well as many that follow, is to immerse ourselves in mathematics that illustrates two components of algebraic thinking: mathematical thinking tools (problem solving, representation, and reasoning skills) and algebraic ideas (functions, patterns, variables, generalized arithmetic, and symbolic manipulation).
Groups: Discuss how these two components relate to the current debate in mathematics reform. Then, divide into small groups to answer Problems A1 and A2. When finished, everyone should share their answers with the whole group.
At this point, the idea of what algebraic thinking is may not be clear. The purpose here is not to construct a precise definition, but to consider initial ideas without judgment as a way of beginning a longer conversation. In fact, in Session 9 we will consider the same questions again, in order to evaluate whether our conception of algebraic thinking has broadened and strengthened.
Session 1 Algebraic Thinking
In this initial session, we will explore algebraic thinking first by developing a definition of what it means to think algebraically, then by using algebraic thinking skills to make sense of different situations.
Session 2 Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is.
Session 3 Functions and Algorithms
In Session 1, we looked at patterns in pictures, charts, and graphs to determine how different quantities are related. In Session 2, we used patterns and variables to describe relationships in tables and in situations like toothpicks and triangles. This session extends the exploration of relationships to include the concepts of algorithm and function. Note1
Session 4 Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.
Session 5 Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.
Session 6 Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.
Session 7 Nonlinear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations.
Session 8 More Nonlinear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.
Session 9 Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the K-2 grade band.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 3-5 grade band.
Session 12 Classroom Case Studies, Grades 6-8
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 6-8 grade band.