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An Abstract Algebra Class Designed for Secondary Mathematics Teachers by Justin R. Hill and Dr. Christopher Thron Texas A&M UniversityCentral Texas

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Obstacles to Abstract Algebra For a 8- 12 Certifying Student: Abstract Algebra requires: 1)An ease in generalization of patterns, concepts, and definitions. 2)Constant but unprompted mental play with concrete examples. 3)High familiarity with algebraic structures such as complex numbers, modular arithmetic, and matrices. 4)An ease in thinking in terms of sets and functions. 5)Previous experience with proofs.

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Textbook Methodology: Based on material from Abstract Algebra: Theory and Applications, by Thomas W. Judson; and Proofs and Concepts, by Dave Witte Morris and Joy Morris. (both available on the web; modified with permission) We revised and amplified that material, focusing on concrete examples and exercises to develop and practice each concept, skill, and proof technique. Classroom Methodology: Scaled back lectures in favor of group work on exercises that elucidated skills and concepts being taught. Lecture was used instead to: (1) review students comprehension of the reading; and (2) fill in gaps of understanding and skill apparent from group work.

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Skills in Generalization and Mental Play with Concrete Examples – Work with concrete examples. – They see and extend patterns through discussion/examples/ exercises. – Use directed exercises to generalize the use of a concept or definition. All generalizing and mental work is modeled.

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Familiarity with Complex Numbers, Modular Arithmetic, and Matrices Chapter 1: Complex Numbers – The Origin of Complex Numbers – Arithmetic with… – Alternative Representations of… – Applications of… Chapter 2: Modular Arithmetic – The Basics – The Integers Mod n – Modular Equations

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Thinking in Terms of Sets and Functions Chapter 3: Sets – Set Basics – Set Relations – Set Operations – Properties of Set Operations – Do the subsets of a set form a group? Chapter 4: Functions – Cartesian Product – Introduction to Functions – One-to-One Functions – Onto Functions – Bijections – Composition of Functions – Inverse Functions v

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Proofs Direct Proofs Contradiction Induction Element-by-Element One-to-One Onto Reflexive Transitive Symmetric Algebraic Proofs iff Proofs Uniqueness Proofs

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Proofs (cont.)

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Did it Work? Survey: All but one student said the same two things: What they Appreciated: The abundance and use of examples and exercises in the textbook and in the classroom What it lacked: More examples. Grades:

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Algebraic Proofs. 1. Transitive property of equality 2. Symmetric property of equality 3. Reflexive property of equality 4. Substitution 5. Addition property.

Algebraic Proofs. 1. Transitive property of equality 2. Symmetric property of equality 3. Reflexive property of equality 4. Substitution 5. Addition property.

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