Approved by University Studies Sub-Committee. A2C2 action pending
University Studies Course Approval Proposal
Flag Requirements – Writing Flag
The Department of Mathematics and Statistics proposes the following course for inclusion in
University Studies as a course satisfying the requirements for a Writing Flag. This was approved
by the full department on Thursday, November 7, 2002.
Course: Abstract Algebra (Math 440), 4 s.h.
Catalog Description: Axiomatic development of groups, rings, fields. This is a University
Studies course satisfying the Writing Flag. Prerequisite: Math 210
This is an existing course, previously approved by A2C2.
Department Contact Person for this Course:
Name: Steven D. Leonhardi, Department of Mathematics and Statistics
Title: Associate Professor of Mathematics and Statistics
Email: [email protected]
Discussion of University Studies:
The Writing Flag in relation to Abstract Algebra (MATH 440)
University Studies: Writing Flag
Flagged courses will normally be in the student’s major or minor program. Departments will
need to demonstrate to the University Studies Subcommittee that the courses in question
merit the flags. All flagged courses must require the relevant basic skills course(s) as
prerequisites (e.g., the “College Reading and Writing” Basic Skill course is a prerequisite for
Writing Flag courses), although departments and programs may require additional
prerequisites for flagged courses. The University Studies Subcommittee recognizes that it
cannot veto department designation of flagged courses.
The purpose of the Writing Flag requirement is to reinforce the outcomes specified for the
basic skills area of writing. These courses are intended to provide contexts, opportunities,
and feedback for students writing with discipline-specific texts, tools, and strategies. These
courses should emphasize writing as essential to academic learning and intellectual
development.
Courses can merit the Writing Flag by demonstrating that section enrollment will allow for
clear guidance, criteria, and feedback for the writing assignments; that the course will require
a significant amount of writing to be distributed throughout the semester; that writing will
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comprise a significant portion of the student’s final course grade; and that students will have
opportunities to incorporate readers’ critiques of their writing.
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How Abstract Algebra (MATH 440) Satisfies the General Writing Flag Requirements:
The purpose of a writing flag is twofold: (1) to reinforce the outcomes specified for the basic
skills area of writing, and (2) to provide contexts, opportunities, and feedback for students
writing with discipline-specific texts, tools, and strategies.
This course addresses Criterion (2) in the following manner. The unique discipline-specific
writing in which students of mathematics engage is that of writing proofs. Since our majors
have little opportunity to practice this skill in classes other than Advanced Calculus and
Abstract Algebra, the main content of these courses centers on writing proofs, with feedback
and revision.
This course addresses Criterion (1) in the following manner. Another equally important
writing skill for a student of mathematics is that of summarizing the general strategy
underlying a proof in a clear and thorough, but non-rigorous, fashion. That is, a student of
mathematics should be able to write a clear abstract of a proof that would be readable by
others who are in the field of mathematics but are not necessarily familiar with the specific
proof being described. To develop this skill, students in this course will be required to write
short abstracts of several of their proofs, with the opportunity for feedback and revision on
each.
Together, the above two types of writing provide students with opportunities for feedback on
their writing in discipline-specific contexts. Additionally, both types of writing, since they
involve the usual paragraph and grammatical structures learned in an introductory course on
English composition, will reinforce the outcomes specified for the basic skills area of writing.
How Abstract Algebra (MATH 440) Satisfies the Detailed Writing Flag Requirements:
Writing Flag courses must include requirements and learning activities that promote students’
abilities to:
a. practice the processes and procedures for creating and completing successful
writing in their fields;
This course is a rigorous introduction to the concepts of Abstract Algebra. To
successfully complete the course, the student is required to demonstrate not only an
understanding of the mathematical concepts involved in Abstract Algebra, but also an
ability to convey those concepts in concise written form, both formally (proof) and
informally (abstract). Mathematical proof represents a very precise writing style that has
developed over two thousand years, and writing an abstract of a proof requires an
understanding of, and an ability to articulate, the methods and strategies used to construct
the proof. Proper use of this writing style requires a knowledge of the relevant
terminology and a facility with the grammar and sentence structure that is germane to
good expository writing. The student receives feedback on his or her written presentation
of logical arguments throughout the semester, with the opportunity to refine both the
proofs and abstracts.
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b. understand the main features and uses of writing in their fields;
It is in rigorous courses such as Abstract Algebra that the student’s conceptual
understanding of mathematics is expanded into a rigorous understanding of the logical
underpinnings of mathematical abstraction. This logical foundation, by its very nature, is
inextricably interwoven with the precise writing that is used to express it. It is here that
the student gains an awareness that proofs of mathematical theorems and propositions lie
not in convincing pictures or clever examples, but in very precise and carefully applied
logical analysis. Such analysis is only as clear as its exposition. A proof is not clear
unless the reader has a prior organizational structure within which to interpret the proof.
An abstract serves the purpose of providing the reader with this necessary tool.
c. adapt their writing to the general expectations of readers in their fields;
Writing a mathematical proof is a very different type of writing compared to most other
exposition. In this course, the successful student must learn to weave good sentence
structure with mathematical formulae and symbolism in a way that brings clarity to the
subject of the exposition. Particularly close attention must be paid to the implications of
uni- and bi-conditional statements and the differences among theorems, conjectures,
lemmas, and definitions. On the other hand, to write an abstract of a proof, the student
must have a facility with these ideas that runs deeply enough to allow him/her to
accurately present the essence of the thinking behind a proof without becoming
excessively technical.
d. make use of the technologies commonly used for research and writing in their fields;
and
When attempting to uncover patterns in analysis and abstraction, the student routinely
makes use of various graphical and algebraic computer aids, such as graphing calculators
or computer algebra systems. Additionally, there are special scripting languages for
typesetting mathematical exposition, such as TeX and LaTeX. Either these or the
equation editors in most popular word processors may be used to render proofs in this
course. The use of typesetting tools is not a requirement of the course, but instead an
option whose implementation is left to the discretion of the instructor.
e. learn the conventions of evidence, format, usage, and documentation in their fields.
As discussed above, the student encounters heavy use of mathematical terminology,
mathematical reasoning, and the expository strategies unique to the field of mathematical
analysis as well as the conventions for citing previously proven results, such as lemmas or
theorems, in a mathematical proof.
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Winona State University
Department of Mathematics and Statistics
Course Outline—M4401
Course Title: Abstract Algebra
Number of Credits: 4 S.H.
Prerequisite: Discrete Mathematics and Foundations (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, fields. This is a University
Studies course satisfying the Writing Flag.
Statement of Major Focus and Objectives of the Course: The major focus of this course is to
introduce students to the logical underpinnings of mathematical analysis and abstraction
and to provide students with the ability to demonstrate a rigorous understanding of
analysis and abstraction by writing clear, accurate, and concise proofs.
Possible Texts:
 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
Methods of Instruction: Lecture, Discussion, Problem Sets (possibly cooperative), computer lab
projects..
Course Requirements: None other than the text.
Evaluation Process: Hour exams and/or quizzes, homework, expository research projects, and a
final exam.
University Studies: Writing Flag
Flagged courses will normally be in the student’s major or minor program. Departments will
need to demonstrate to the University Studies Subcommittee that the courses in question
merit the flags. All flagged courses must require the relevant basic skills course(s) as
prerequisites (e.g., the “College Reading and Writing” Basic Skill course is a prerequisite for
Writing Flag courses), although departments and programs may require additional
prerequisites for flagged courses. The University Studies Subcommittee recognizes that it
cannot veto department designation of flagged courses.
The purpose of the Writing Flag requirement is to reinforce the outcomes specified for the
basic skills area of writing. These courses are intended to provide contexts, opportunities, and
1
Prepared by Felino G. Pascual on November 17, 2002; revised by Steven D. Leonhardi on
March 31, 2003.
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feedback for students writing with discipline-specific texts, tools, and strategies. These
courses should emphasize writing as essential to academic learning and intellectual
development.
Courses can merit the Writing Flag by demonstrating that section enrollment will allow for
clear guidance, criteria, and feedback for the writing assignments; that the course will require
a significant amount of writing to be distributed throughout the semester; that writing will
comprise a significant portion of the student’s final course grade; and that students will have
opportunities to incorporate readers’ critiques of their writing.
These courses must include requirements and learning activities that promote students’
abilities to:
a) practice the processes and procedures for creating and completing successful writing
in their fields;
b) understand the main features and uses of writing in their fields;
c) adapt their writing to the general expectations of readers in their fields;
d) make use of the technologies commonly used for research and writing in their fields;
and
e) learn the conventions of evidence, format, usage, and documentation in their fields.
Topics below which include such requirements and learning activities are indicated below using
lowercase, boldface letters a-d corresponding to these requirements.
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, e
b) skills in carrying out the process of experimentation, conjecture, and verification. a, d,
e
c) skills in creating, critiquing, and communicating proofs in writing. a, b, c, d, e
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
a, d, e
b) the ability to communicate mathematical ideas in writing, using everyday and
mathematical language, including symbols.
a, b, c, d, e
c) the ability to communicate mathematical ideas orally, using both everyday and
mathematical language
d) the ability to make and evaluate mathematical conjectures and arguments and validate
their own mathematical thinking
a, c, d, e
e) an understanding of the interrelationships within mathematics
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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
d, e
h) the ability to use algebra to describe patterns, relations and functions and to model and
solve problems
d, e
i) an understanding of the role of axiomatic systems in different branches of mathematics,
such as algebra and geometry
d, e
j) an understanding of the major concepts of abstract algebra
k) the ability to use calculators in computational and problem-solving situations d
l) the ability to use computer software to explore and solve mathematical problems d
m) a knowledge of the historical development of mathematics that includes the
contributions of underrepresented groups and diverse cultures e
Course Outline of the Major Topics and Subtopics:
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
Additional Information about Writing Assignments: In accordance with criteria a, b, c, d,
and e, this course provides the rigorous underpinnings of proof construction and writing
that are expected of students planning to attend graduate school in mathematics. The
abstracts and proofs that students write in this course constitute the vast majority of their
grade. One such abstract/proof pair is given below as an example of the type of writing
required in this course:
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Abstract: In the following proof, we show that if a non-empty subset H of a group G satisfies
the property that ab 1 is in H whenever a and b are in H , then H is a subgroup of G. To
accomplish this, we begin by assuming the hypotheses and then show that this assumption leads
necessarily to the conclusion that H is a subgroup of G. Hence, we begin with the assumption
that H is a non-empty subset of group G and that ab 1 is in H whenever a and b are in H .
We must then prove that subset H satisfies the following four properties:
1. The operation of G is associative on H , that is, (ab)c  a(bc) for all
a , b, c  H ;
2. H contains an identity, that is, an element e such that ae  ea  a for all a  H ;
3. Every element a  H has an inverse in H , that is, some element b  H such that
ab  ba  e; and
4. H is closed under the operation of G , that is, ab  H whenever a  H and
b  H.
Showing these four properties is sufficient to show that H is a group under the same
operation as G , and therefore H is a subgroup of G.
Proof: Prove that if H is a non-empty subset of group G and if ab 1 is in H whenever a and
b are in H , then H is a subgroup of G.
Proof: Assume that H is a non-empty subset of group G and that ab 1 is in H
whenever a and b are in H . We must show that H is a subgroup of G.
Firstly, the operation of G is associative on H because H is a subset of G.
Secondly, we must show that H contains an identity element e. Since H is given to be
non-empty, we can fix some element a  H . Applying the hypothesis to a and b  a gives us
that ab 1  aa 1  e  H , where e acts as the identity on all elements of G , and therefore e acts
as the identity on all elements of H , since H is a subset of G.
Thirdly, let c  H be chosen arbitrarily; we must show that its inverse c 1  H . Apply
the hypothesis with a  e (where we know by the previous paragraph that this is an element of
H ) and with b  c (where we know by the choice of c that this is in H ) ; the hypothesis ensures
that ab 1  H . However, ab 1  ec 1  c 1 , so c 1  H , as was to be shown.
Finally, to show closure, let c, d  H be chosen arbitrarily; we must show that cd  H .
Let a  c and let b  d 1 . By the choice of c, d  H and by the existence of inverses in H as
shown above, we must have a, b  H ; therefore the hypothesis ensures that ab 1  H . However,
 
ab 1  c(d 1 ) 1  cd , using the fact that d 1
1
 d , so cd  H , as was to be shown. QED
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Approval/Disapproval Recommendations
Department Recommendation:
Approved
Chairperson Signature
Dean’s Recommendation:
Disapproved
Date
Date
Approved
Dean’s Signature
Disapproved
Date
Date
*In the case of a Dean’s recommendation to disapprove a proposal, a written rationale for the recommendation
to disapprove shall be provided to USS.
USS Recommendation:
Approved
University Studies Director’s Signature
A2C2 Recommendation:
Approved
Approved
Date
Disapproved
Date
Date
Approved
VP’s Signature
President’s Decision:
Disapproved
Date
FA President’s Signature
Academic VP’s Recommendation:
Date
Date
A2C2 Chairperson Signature
Faculty Senate Recommendation:
Disapproved
Disapproved
Date
Date
Approved
President’s Signature
Disapproved
Date
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Date