The Department of Mathematics and Statistics proposes the following course for inclusion in University Studies courses satisfying the Oral Communication Flag requirement at Winona State University. This was approved by the full department on Thursday, January 18, 2001.

Course: Abstract Algebra (MATH 440), 4 s.h.

Catalog Description: Axiomatic development of groups, rings, and fields. This is a University Studies course satisfying the Oral Communication Flag requirement. Prerequisite: MATH 210.

This is an existing course, previously approved by A2C2.

Department Contact Person for this course:

Steven D. Leonhardi, Department of Mathematics and Statistics

Email leonhardi@winona.edu

General Discussion of University Studies – Oral Communication Flag in relation to MATH 440:

The purpose of the Oral Communication Flag requirement is to complete the process of providing graduates of Winona State University with the knowledge and experience required to enable them to become highly competent communicators by the time they graduate.

Courses can merit the Oral Communication Flag by demonstrating that they allow for clear guidance, criteria, and feedback for the speaking assignments; that the course requires a significant amount of speaking; that speaking assignments comprise a significant portion of the final course grade; and that students will have opportunities to obtain student and faculty critiques of their speaking.

These courses must include requirements and learning activities that promote students’ abilities to…

1. Earn significant course credit through extemporaneous oral presentations;

 

Students in this course are required to learn and perform three different types of speaking in this course: (1) the discussion necessary for a group of 2 to 4 students to construct and verify a proof or disproof of a mathematical claim; (2) presenting completed mathematical proofs to the class; and (3) presenting the results of an expository research project to the class. Typically, the expository project and presentation is worth about 10% of the student’s final grade, and homework problems that are worked on in groups are worth about 40% of the student’s final grade, although this varies somewhat depending upon the instructor and year. A full week of the semester is used for oral presentations of student projects.

 

 

2. Understand the features and types of speaking in their disciplines;

 

The three types of speaking about mathematics listed above reflect the three major types of speaking needed by mathematicians and mathematics teachers. The first type, discussion among peers about mathematical concepts, examples, and arguments, is the primary means of progress in mathematical research, and is also a highly effective means of helping students at any level construct their own mathematical knowledge. Even to mathematics majors, mathematics is essentially a foreign language. This course is intended to help students learn to use and communicate mathematical terminology and arguments correctly and at a level of rigor and following the stylistic standards appropriate to the discipline.

 

The second type of speaking, presenting a complete proof to an audience, corresponds to the type of presentation that a mathematician might give at a specialized conference, or a presentation that a teacher would give to a class after they have had time to work on a problem. The audience is assumed to already know a significant amount of background and terminology, but a complete, step-by-step explanation of the specific claim must be given. This type of speaking requires the highest level of rigor out of the three types.

 

The third type of speaking, presenting results of an expository research project, corresponds to the type of talk one might give at a general mathematics conference, in which the audience is presumed to have only a minimal amount of background knowledge, and in which a large amount of information is condensed and summarized to the main themes and introduction.

 

3. Adapt their speaking to field-specific audiences;

 

Students in Abstract Algebra learn to adapt their speaking to communicate effectively with (1) students working together on the same problems, (2) "experts" (i.e., students in the same class working on different problems) who know the background info but still need to hear details of the speaker’s specific work, and (3) "non-experts" with only minimal background in the topic about which they are speaking, as described in item b. above.

 

4. Receive appropriate feedback from teachers and peers, including suggestions for improvement;

 

Students receive immediate feedback from peers during discussion done in groups. The instructor circulates throughout the class during this group work, confirming when terminology and arguments are appropriate, and offering corrections, hints, and suggestions for improvement. The instructor also comments orally after problem solutions and proofs are presented to the class.

 

For the expository project and presentation, the instructor offers comments on a preliminary outline, then comments on a first draft, with suggestions for improvement and suggestions on the oral presentation, and comments on the final presentation.

 

 

5. Make use of the technologies used for research and speaking in the fields; and

 

Students typically use the blackboard and chalk to present solutions to homework problems; some use overhead slides or printed handouts to supplement their explanations. For their project presentations, many students use Power Point (sometimes with audio supplements), some students use overhead slides, some use printed handouts, and some use the blackboard-- in approximately the same proportions as would be represented at a professional conference for mathematicians or mathematics teachers.

 

6. Learn the conventions of evidence, format, usage, and documentation in their fields.

This is a major focus of the course: for students to learn how to evaluate and present evidence, correct usage, and what exactly constitutes a "proof" in mathematics, and for students to learn how to communicate their ideas, conjectures, and conclusions. In particular, students must learn how to move through the process of communicating their informal intuitions based on concrete examples, to developing a formal, rigorous, general proof, and then finally to explaining their proof in a way that others can understand. That is, the type of speaking that is most effective for "discovering" a theorem and/or its proof is very different from the type of speaking that is required for presenting a formal statement of a theorem and its proof. Students must learn how to carry out both types of speaking, and learn to recognize which type of speaking (namely, what level of formality in terms of evidence and usage) is appropriate in different situations.

Course Title: Abstract Algebra MATH 440

Number of Credits: 4 S.H. Frequency of Offering: offered fall semester

Prerequisite(s): MATH 210

Grading: Grade only for all majors, minors, options, concentrations and licensures within the Department of Mathematics and Statistics. The P/NC option is available to others.

Course Description: Axiomatic development of groups, rings, and fields. This is a University Studies course satisfying the Oral Communication Flag requirement.

The purpose of the Oral Communication Flag requirement is to complete the process of providing graduates of Winona State University with the knowledge and experience required to enable them to become highly competent communicators by the time they graduate.

Courses can merit the Oral Communication Flag by demonstrating that they allow for clear guidance, criteria, and feedback for the speaking assignments; that the course requires a significant amount of speaking; that speaking assignments comprise a significant portion of the final course grade; and that students will have opportunities to obtain student and faculty critiques of their speaking.

These courses must include requirements and learning activities that promote students’ abilities to…

a. Earn significant course credit through extemporaneous oral presentations;

b. Understand the features and types of speaking in their disciplines;

c. Adapt their speaking to field-specific audiences;

d. Receive appropriate feedback from teachers and peers, including suggestions for improvement;

e. Make use of the technologies used for research and speaking in the fields; and

f. Learn the conventions of evidence, format, usage, and documentation in their fields.

Course objectives that include such requirements and learning activities are indicated below using lowercase, boldface letters (a-f) corresponding to these.

Statement of major focus and objectives of the course:

The major focus of this course is to provide students with

a) knowledge of the content of abstract algebra. a, b, c, d, f

b) skills in carrying out the process of experimentation, conjecture, and verification. b, d, f

c) skills in creating, critiquing, and communicating proofs, both orally and in writing.

a, b, c, d, e, f

Note that a focus of the course will be to prepare students to develop the competencies outlined in the following Minnesota Standards of Effective Teaching Practice for Beginning Teachers: Standard 1 -- Subject Matter;

Objectives: To develop within the future teacher ...

a) the ability to use a problem-solving approach to investigate and understand mathematical content b, d, f

b) the ability to communicate mathematical ideas in writing, using everyday and mathematical language, including symbols

c) the ability to communicate mathematical ideas orally, using both everyday and mathematical language a, b, c, d, e, f

d) the ability to make and evaluate mathematical conjectures and arguments and validate their own mathematical thinking b, d, f

e) an understanding of the interrelationships within mathematics

f) an understanding of and the ability to apply concepts of number, number theory and number systems

g) an understanding of and the ability to apply numerical computational and estimation techniques and the ability to extend them to algebraic expressions

h) the ability to use algebra to describe patterns, relations and functions and to model and solve problems b, d, f

i) an understanding of the role of axiomatic systems in different branches of mathematics, such as algebra and geometry a, b, d, f

j) an understanding of the major concepts of abstract algebra a, b, c, d, e, f

k) the ability to use calculators in computational and problem-solving situations e

l) the ability to use computer software to explore and solve mathematical problems e

m) a knowledge of the historical development of mathematics that includes the contributions of underrepresented groups and diverse cultures a, b, c, d, e, f

 

Course Outline of the Major Topics and Subtopics (ordering at the discretion of the instructor):

I. Preliminaries

A. Historical origins of abstract algebra

B. Basic set theory

C. Review of methods of proof and the axiomatic method as applied to:

1. The integers and the Greatest Common Divisor Identity

2. Matrix algebra

3. Complex numbers

4. Functions and compositions

5. Relations and equivalence relations

II. Groups

A. Permutations, symmetries of a polygon

B. Groups and subgroups

C. Cyclic groups

D. Permutation groups

III. Rings

A. Rings and subrings

B. Factorization, uniqueness of factorization, units, and associates

C. Integral domains and fields

IV. Homomorphisms and Quotient Structures

A. Homomorphisms

B. Isomorphisms

C. Normal subgroups

D. Quotient subgroups

E. Ideals

F. Quotient rings

Method of Instruction: Lecture, discussion, group work, student presentations, computer lab projects.

Evaluation Procedure: Hour exams and/or quizzes, homework, student presentation of solved homework, expository research project and presentation, and a final exam.

Textbooks or Alternatives:

A First Course in Abstract Algebra, Anderson and Feil

A First Course in Abstract Algebra, Fraleigh

Contemporary Abstract Algebra, Gallian

Abstract Algebra, Herstein

Abstract Algebra: An Introduction, Hungerford

A Book of Abstract Algebra, Pinter

List of References and Bibliography:

Laboratory Experiences in Group Theory: A Manual to be used with Exploring Small Groups, by Ellen Maycock Parker

 

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