Styles and Strategies for Teaching High School Mathematics

21 Techniques for Differentiating Instruction and Assessment

Edward J. Thomas, John R. Brunsting, Pam L. Warrick

• Publisher: Corwin
• Year: 2010
• ISBN:
  9781452295060
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

This book offers effective, research-based strategies that can be mixed and matched to differentiate mathematics instruction for high school students through four different learning styles.

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1

Mastery Strategies

OVERVIEW

Mastery strategies help students remember mathematical content and procedures and practice their computational skills. They are especially engaging to Mastery math students.

Mastery math students …

• Want to learn practical information and set procedures.

• Like math problems that are like problems they have solved before and that use algorithms to produce a single solution.

• Approach problem solving in a step-by-step manner.

• Experience difficulty when mathematics becomes too abstract or when faced with nonroutine problems.

• Want a math teacher who models new skills, allows time for practice, and builds in feedback and coaching sessions.

The six Mastery strategies in this chapter can help you meet these NCTM Process Standards (see Figure 1.0).

FIGURE 1.0 Correlation of Mastery Strategies to NCTM Process Standards

*For more information on the National Council of Teachers of Mathematics (NCTM) Process Standards, please consult their Principles and Standards for School Mathematics (2000), or visit their website at www.nctm.org.

Convergence Mastery

Strategy Overview

In every math classroom, after every quiz or test, a wonderful instructional opportunity presents itself: the opportunity to help students learn from their mistakes. But the truth is, in most math classrooms, tests are returned with red marks on them, and the next unit begins. This means that the great majority of math students are missing the chance to root out errors, clarify confusions, and grow as learners.

The Convergence Mastery strategy is a simple but powerful way to provide students with multiple opportunities to learn from their mistakes and achieve mastery of important math procedures and skills. At the heart of the strategy is a series of short quizzes focused on a single core skill (e.g., factoring polynomials). Before taking the first quiz, students practice the skill and review in pairs or small groups. Students take the first quiz individually, return to their groups, and grade one another’s quizzes as the teacher provides the correct answers. Only two grades are possible:

1. Students with one or more incorrect answers receive an Incomplete

2. Students who answer every question correctly receive an A

Students who receive an A are not required to take additional quizzes. Instead, they help group members who received an incomplete to review, make corrections, and prepare for the next quiz. The strategy continues until all students have received an A.

How to Use the Strategy

1. Select a math procedure or skill you want all students to master.

2. Develop three to five short quizzes that contain problems representative of the skill.

3. Explain the quiz process and grading procedures to students. Be sure they understand how the process is designed to help them.

4. Provide a few minutes for students to review the skill in small groups.

5. Administer a quiz to all students. Quizzes are timed (usually 5 minutes per quiz).

6. Share the correct answers, and have students grade other group member’s quizzes.

7. Excuse any students who received an A from further quizzes. Have these students help their group members correct errors and prepare for the next quiz. If necessary, provide coaching sessions to struggling students.

8. Continue the process (steps 5–7) until virtually all students have received an A.

The Strategy in Action: Examples

Figure 1.1 shows a sample set of quizzes for finding the roots of functions.

FIGURE 1.1 Sample Convergence Mastery Quizzes: Roots of Functions

Figure 1.2 shows a sample set of quizzes for finding trigonometric identities.

FIGURE 1.2 Sample Convergence Mastery Quizzes: Trigonometric Identities

Source: Silver, H. F., Brunsting, J. R., & Walsh, T. (2008). Math Tools, Grades 3–12: 64 Ways to Differentiate Instruction and Increase Student Engagement. (p. 44)

Curriculum Connections

Convergence Mastery provides students opportunities to practice and increase their proficiency in important mathematical procedures and skills. Consider using this strategy in courses and for topics such as

Pre-Algebra

• Simplifying numeric expression using order of operations

• Simplifying numeric expressions that require operations with fractions

Algebra I

• Factoring polynomials

• Graphing linear equations

Geometry

• Writing a two-column proof

• Performing constructions with a compass and straightedge

Algebra II

• Finding the roots of a quadratic equation

• Cramer’s rule

Precalculus/Calculus

• Proving trigonometric identities

• Graphing trigonometric functions

Why the Strategy Works

Sometimes, teaching strategies come from real-life experiences. The initial seed for the Convergence Mastery strategy was planted at Boy Scouts camp where one of the authors of this book (Ed Thomas) spent his summers as a young boy. At summer camp, scouts had opportunities to earn awards and merit badges for meeting various challenges. One challenge was to make fire without matches. On certain nights, scouts were given a rock, a piece of steel, and 30 minutes to produce a fire. Scouts who succeeded were given a “Singed Eyebrows” certificate. Scouts who failed were invited to try again the next time the Singed Eyebrows station was open. Whenever a scout produced fire, whether it was on the first try or the fourth, he received his Singed Eyebrows certificate. This open-door policy on success motivated scouts to keep trying, learn from their mistakes, and achieve mastery in the skill of making fire.

Convergence Mastery takes the wisdom of Boy Scouts camp and puts it to work in the mathematics classroom. The strategy provides students with repeated and controlled practice opportunities, which help build students’ debugging skills and maximize skill acquisition. Convergence Mastery also provides teachers with an easy way to differentiate instruction according to students’ readiness levels. Students who need more practice opportunities and more coaching receive both. At the same time, students who have already mastered the skill do not sit around idly; instead, they become part of the teaching and learning process. What’s more, by having students help other students who have yet to receive an A on a quiz, Convergence Mastery capitalizes on the power of peer-coaching partnerships, which have been shown to increase students’ academic intensity (Fuchs, Fuchs, Mathes, & Simmons, 1997) and lead to academic gains and more positive attitudes toward subject matter (King-Sears & Bradley, 1995).

Planning Considerations

The idea behind Convergence Mastery is that all students converge toward mastery of the highlighted skill or procedure by achievin...