The Missing Link: Essential Concepts for Middle School Math Teachers

The Missing Link: Essential Concepts for Middle School Math Teachers

Recent studies show that once American students reach middle school, they begin to fall behind the students of many other countries in mathematical understanding and achievement. This video workshop familiarizes you with four concepts that have been identified by TIMSS (the Third International Mathematics and Science Study) as crucial to your students' future success. You’ll come to better understand the content of these math topics through demonstration of instructional techniques that show you how to involve your students in their own learning. In the “Discovery” session of each topic-pair, Master Teacher Jan Robinson (from the First in the World Consortium) leads the on-camera learner-teachers as they investigate a series of problem-based activities. The learner-teachers customize and expand upon these lessons in their own classrooms, and return with samples of their students’ work. In the “In Practice” sessions, they report on their experiences, evaluate the student work, and develop new instructional and assessment techniques.

    ISBN: 1-57680-290-6

Individual Program Descriptions

1. Workshop 1. Proportionality and Similar Figures: Discovery
The teachers discover what makes similar figures similar and how a scale factor affects side lengths, angles, perimeters, and areas when figures are enlarged or reduced.

2. Workshop 2. Proportionality and Similar Figures: In Practice
In this follow-up to Workshop 1, the teachers discuss how their students approached the proportionality lessons. They evaluate their students’ work as a way to strengthen their instructional practice, and create a new lesson based on their assessments.

3. Workshop 3. Patterns and Functions: Discovery
The teachers use real-life problems to display experimental data in graphs and tables, and analyze the resulting patterns to make predictions and develop algebraic equations.

4. Workshop 4. Patterns and Functions: In Practice
In this follow-up to Workshop 3, the teachers discuss how they taught the patterns lessons in their classrooms, learn to evaluate student work, and design new lessons.

5. Workshop 5. Polygons and Angles: Discovery
The teachers tackle hands-on activities to investigate angle measurements and their relationships in triangles, quadrilaterals, pentagons, and other polygons.

6. Workshop 6. Polygons and Angles: In Practice
In this follow-up to Workshop 5, the teachers discuss how they taught the polygon lessons in their classrooms, learn to evaluate student work, and design new lessons.

7. Workshop 7. Sampling and Probability: Discovery
The teachers collect data and determine the probability of an event, use probability to make predictions, and learn how to conduct random sampling.

8. Workshop 8. Sampling and Probability: In Practice
In this follow-up to Workshop 7, the teachers discuss how they taught the sampling lessons in their classrooms, learn to evaluate whether or not student work meets standards.

Learning Math: Patterns, Functions, and Algebra

A video- and Web-based course on basic concepts of algebra for K-8 teachers; 9 half-hour and 1 one-hour video programs, course guide, and Web site; graduate credit available

Learning Math: Patterns, Functions, and Algebra is the first of five video- and Web-based mathematics courses for elementary and middle school teachers. These courses, organized around the content standards of the National Council of Teachers of Mathematics (NCTM), will help you better understand the mathematics concepts underlying the content that you teach.

Patterns, Functions, and Algebra explores the “big ideas” in algebraic thinking, such as finding, describing, and using patterns; using functions to make predictions; understanding linearity and proportional reasoning; understanding non-linear functions; and understanding and exploring algebraic structure. The concluding case studies show you how to apply what you have learned in your own classroom. The course consists of 10 two-and-a-half hour sessions with a half-hour of video programming each, problem-solving activities available in print and on the Web, and class discussions.

Produced by WGBH Educational Foundation. 2002.

ISBN: 1-57680-469-0

Individual Program Descriptions

1. Video 1. Algebraic Thinking
Begin to explore what it means to think algebraically and learn to use algebraic thinking skills to make sense of different situations. This session covers describing situations through pictures, charts, graphs, and words; interpreting and drawing conclusions from graphs; and creating graphs to match written descriptions of real-life situations.

2. Video 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.

3. Video 3. Functions and Algorithms
Investigate algorithms and functions. Topics covered include the importance of doing and undoing in mathematics, determining when a process can or cannot be undone, using function machines to picture and undo algorithms, and the unique outputs produced by functions.

4. Video 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.

5. Video 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.

6. Video 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.

7. Video 7. Non-Linear 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.

8. Video 8. More Non-Linear 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.

9. Video 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.

10. Video 10. Classroom Case Studies (60 min.)
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 session is divided into three grade bands: K-2, 3-5, and 6-8.

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