Approved by Faculty Senate February 10, 2003

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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, January 18, 2001.

**Course: **Advanced Calculus I (MATH 330), 4 S.H.

**Catalog Description: **A systematic approach to the theory of
differential and integral calculus

for functions and transformations in several variables. This is a University Studies course

satisfying requirements for a Writing Flag. *Prerequisite: *MATH 210,
and MATH 260.

This is an existing course, previously approved by A2C2.

**Department Contact Person for this Course:**

*Name: *Barry A. Peratt

*Title: *Associate Professor of Mathematics and Statistics

*Email: *bperatt@winona.edu

Discussion of University Studies:

The Writing Flag in relation to Advanced Calculus I (MATH 330)

**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

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 Advanced Calculus I (MATH 330) 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 Advanced Calculus I (MATH 330) 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 higher-level mathematical analysis. To

successfully complete the course, the student is required to demonstrate not only an

understanding of the mathematical concepts involved in analysis, 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 several hundred 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 expositional

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 analysis 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, 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 expositional 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—M3301

**Course Title: **Advanced Calculus I

**Number of Credits: **4 S.H.

**Prerequisite: **Discrete Mathematics and Foundations (M210) &
Multivariable Calculus (M260).

**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: **A systematic approach to the theory of
differential and integral calculus for

functions and transformations in one and several variables.

**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 to provide

students with the ability to demonstrate a rigorous understanding of analysis by writing

clear, accurate, and concise proofs.

**Possible Texts:**

· *Fundamental Ideas of Analysis *by
Reed.

· *Advanced Calculus *by Buck.

· *Advanced Calculus *by Fulks.

· *Advanced Calculus *by Widder.

· *Advanced Calculus, a Friendly
Approach *by Kosmala.

· *An Introduction to Analysis, 2nd edition *by
Bartle and Sherbert.

· *Introduction to Real Analysis *by
Gaughan.

· *Introduction to Real Analysis *by
Schramm.

· *Principles of Mathematical
Analysis *by Rudin.

· *Understanding Analysis *by
Abbott.

**Methods of Instruction: **Lecture, Discussion, Problem Sets (possibly
cooperative).

**Course Requirements: **None other than the text.

**Evaluation Process: **Tests, Quizzes, Problem Sets.

**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

1 Prepared by Barry A. Peratt on October 4, 2002.

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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.-e. **corresponding to these
requirements.

**Course Outline of the Major Topics and Subtopics:**

· The real number system and an
introduction to proof. **a., b., c., d., e.**

· Elementary
Topology—open/closed sets, countability, boundedness, compactness. **a., b.,**

**c., d., e.**

· Functions, Sequences, and Limits. **a.,
b., c., d., e.**

· Continuity. **a., b., c., d., e.**

· Differentiation. **a., b., c., d.,
e.**

· Integration. **a., b., c., d., e.**

· Vectors and Curves. **a., b., c.,
d., e.**

· Infinite Series. **a., b., c., d.,
e.**

**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 function *f *,
from set *S *to *T*, is a bijection,

then its inverse must also be a bijection. To accomplish this, we begin by
assuming that *f*

is a bijection and then show that this assumption leads necessarily to the conclusion that

1 - *f *is a bijection. Hence, we begin with the knowledge that *f
*has the following four

properties:

1. It is well-defined,

2. Its domain is all of the set *S*,

3. It is injective, and

4. It is surjective.

We must then prove that its inverse has the following four qualities:

1. It is well-defined,

2. Its domain is all of the set *T*,

3. It is injective, and

4. It is surjective.

We note that since *f *is injective, this will lead us to the
conclusion that its inverse is welldefined,

and the fact that *f *is well-defined will lead us to the conclusion
that 1 - *f *is

injective. Likewise, the fact that *f *is surjective leads to the
conclusion that its inverse has

*T *as its domain, and the fact that the domain of *f *is the set *S
*leads to the conclusion that

1 - *f *is surjective.

**Proof:**

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**Approval/Disapproval Recommendations**

Department Recommendation: Approved Disapproved Date

Chairperson Signature Date

Dean’s Recommendation: Approved Disapproved Date

Dean’s Signature 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 Disapproved Date

University Studies Director’s Signature Date

A2C2 Recommendation: Approved Disapproved Date

A2C2 Chairperson Signature Date

Faculty Senate Recommendation: Approved Disapproved Date

FA President’s Signature Date

Academic VP’s Recommendation: Approved Disapproved Date

VP’s Signature Date

President’s Decision: Approved Disapproved Date

President’s Signature Date