*[*

Approved by Faculty Senate.

**University Studies Course Approval Proposal
Unity and Diversity – Critical Analysis**

The Department of Mathematics and Statistics proposes the following course for inclusion in University Studies, Unity and Diversity, Critical Analysis at Winona State University. This was approved by the full department on Thursday, January 4, 2001.

Course: Discrete Mathematics and Foundations (MATH 210), 4 s.h.

Catalog Description: Introductory discrete mathematics including symbolic logic,
elementary number theory, sequences,

sets and combinatorics. Valid and invalid argument forms are studied, and direct and
indirect methods of proof are

introduced. This is a University Studies course satisfying requirements in Critical
Analysis. Prerequisite: MATH 110

or MATH 120 or MATH 150.

This is an existing course, previously approved by A2C2.

Department Contact Person for this course:

Jeffrey R. Anderson, Mathematics and Statistics Department Chair

Email janderson@winona.edu

General Discussion of University Studies – Critical Analysis in relation to MATH 210:

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**University Studies: Critical Analysis**

Critical Analysis courses in the University Studies program are devoted to teaching critical thinking or analytic problem-solving skills. These skills include the ability to identify sound arguments and distinguish them from fallacious ones. The objective of these courses is to develop students abilities to effectively use the process of critical analysis. Disciplinary examples should be selected to support the development of critical analysis skills. These courses must include requirements and learning activities that promote students abilities to…

a.

evaluate the validity and reliability of information;b.

Information often comes in the form of a claim or theory. It is unfortunately all too common that when such information appears in print, the public reaction is to take the truth of such on the faith that those things which are typeset in the newspaper or in textbooks must be true (especially if the information involved some sort of mathematics and/or statistics). In MATH 210, students develop the ability to examine arguments first on the method of proof used – either deductive or inductive (i.e., based upon experiment) – and then on the symbolic form of that method. Students study the reliability of inductive reasoning versus deductive reasoning. Further, through symbolic logic, they learn to systemize their evaluative methods of examining the validity of deductive arguments. As such, students learn a very powerful mathematical technique towards evaluating the validity and reliability of claims or theories.

c.

analyze modes of thought, expressive works, arguments, explanations, or theories;

One example of the form a theory might take is "If something happens, then something will also happen". Although a seemingly clear and simple promise of when one thing will follow from another, there are several ways such a theory may be stated equivalently yet appear to say something quite different. Yet another possibility is that an "If-then" sort of theory may be twisted around a bit, appear to be equivalent to the original theory, and, in fact, be completely different. Terminologies associated with such an example are conditional, contrapositive, converse, and inverse. Students develop the ability to see which of these are equivalent, which are not, and why. They also learn to recognize the many different forms, involving such terms as "necessary" and "sufficient", that these may take in everyday print (with or without mathematics). Students develop the ability to understand the ways in which these statements may be validly deduced, and they learn the ways in which a faulty line of reasoning may be turned around to look like it is valid.

d.

recognize possible inadequacies or biases in the evidence given to support arguments or conclusions; ande.

The way in which evidence is assembled and used to argue the truth of a point may result in either a solid, deductive argument or one of several classic errors in reasoning. For example, a popular movie some years back included a court case where a key witness, a colonel as it turns out, claims to have had nothing to do with a certain soldier's death. Upon cross-examination, he offers three key pieces of testimony. First, all of his soldiers follow orders. No exceptions. Second, he ordered his soldiers not to harm another soldier on his base in any way. Third, before the soldier who died met with his untimely death, this colonel issued him a transfer to another base. The reason given was that this soldier was substandard, and other soldier may harm the man for not being up to standards. Given some thought, it is possible to see that the colonel has lied. Students in MATH 210 develop the ability to apply symbolic logic to clearly recognize the contradiction in the evidence provided. They then go on to other, more elaborate examples, to see that such contradictions are quite common occurrences and that, turning the example around a bit, "proof by contradiction" becomes a very powerful method of deductive reasoning.

f.

advance or support claims.

One of the key goals of MATH 210 is that students learn to develop the truth of mathematical claims or theories through valid methods of deductive reasoning. Further, they learn to apply the meaning of quantified statements toward proper reasoning and also toward discovery of those claims which turn out to be false. Open-ended problems challenge students to either develop the truth of a claim or show that it is false, and, ultimately, students develop their own "theorems" as extensions of this work.

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Discrete Mathematics and Foundations (MATH 210) 4 s.h.

**Course Syllabus/Outline**

Course Title: Discrete Mathematics and Foundations MATH 210

Number of Credits: 4 S.H. Frequency of Offering: Every Semester

Prerequisite(s): MATH 110 or MATH 120 or MATH 150

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: An introduction to symbolic logic, quantifiers, arguments and their use in proving theorems. Examples are taken from elementary number theory, sequences, sets, and everyday situations. This is a University Studies course satisfying requirements in Critical Analysis.

Statement of major focus and objectives of the course: The major focus of this course is to provide students with the ability to write proofs, the ability to read proofs and decide what method is being used to prove the statement, the ability to apply symbolic logic and methods of reasoning to everyday situations.

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 ...

� the ability to use a problem-solving approach to investigate and understand mathematical content

� the ability to formulate and solve problems from both mathematical and everyday situations

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

� the ability to communicate mathematical ideas orally, using both everyday and mathematical language

� the ability to make and evaluate mathematical conjectures and arguments and validate their own mathematical thinking

� the ability to connect mathematics to other disciplines and real-world situations

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

� the ability to use algebra to describe patterns, relations and functions and to model and solve problems

__University Studies: Critical Analysis__

Critical Analysis courses in the University Studies program are devoted to teaching critical thinking or analytic problem-solving skills. These skills include the ability to identify sound arguments and distinguish them from fallacious ones. The objective of these courses is to develop students abilities to effectively use the process of critical analysis. Disciplinary examples should be selected to support the development of critical analysis skills. These courses must include requirements and learning activities that promote students abilities to…

g. evaluate the validity and reliability of information;

h. analyze modes of thought, expressive works, arguments, explanations, or theories;

i. recognize possible inadequacies or biases in the evidence given to support arguments or conclusions; and

j. advance or support claims.

Topics below which include such requirements and learning activities are indicated below using lowercase, boldface letters a.-d. corresponding to these.

Course Outline of the Major Topics and Subtopics:

I. The Logic of Compound Statements: Logical Form and Logical Equivalence; Logical Implication; Valid and Invalid Arguments.

a., b., c.II. The Logic of Quantified Statements: Predicates and Quantified Statements; Arguments with Quantified Statements.

a., b., c.III. Methods of Proof: Direct Proof and Counterexample; Argument by Contradiction; More on Indirect Argument.

a.-d.IV. Mathematical Induction: Sequences; Mathematical Induction; Strong Mathematical Induction and the Well-Ordering Principle.

a.-d.V. More Applications of Proof Techniques: Sets and Set theory (Unions and Intersections, "Element Chasing" Proofs, Cartesian Products and Relations, Equivalence Relations, Partitions); Functions (Construction of Functions, One-to-one and Onto Functions); Cardinality (Cardinality and Cardinal Numbers).

b., d.

Method of Instruction: Lecture, Discussion, Group work

Evaluation Procedure:__ __Hour exams, Quizzes, Final exam, Group
work

Textbooks or Alternatives:

__Discrete Mathematics with Applications__, Epp

__Discrete Mathematics and its Applications__, Rosen

List of References and Bibliography:

__A Transition to Advanced Mathematics__, Smith, Egen, & St.
Andre

__Numbers and Mathematics__, Dodge

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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