Approved by Faculty Senate

University Studies Course Approval Proposal

Basic Skills Mathematics

The Department of Mathematics and Statistics proposes the following
requirement for Basic

Skills in Mathematics at Winona State University. This was approved by the full department

on Thursday, September 21, 2000.

Basic Skills in Mathematics

Successful completion of any 100-level, 3-4 s.h., Mathematics (MATH) or Statistics

STAT) course satisfies the Basic Skills Mathematics requirement.

At the same time, we also propose that the University Studies
Subcommittee consider adopting a

change to the Basic Skills whereby students may use courses required for their major or
minor

program to satisfy Basic Skills requirements.

If the proposed requirement above is approved, any of the following
courses would be acceptable

as satisfying the Basic Skills in Mathematics Requirement:

Math 100 Survey of Mathematics (3 s.h.)
Math 110 Finite Mathematics (3 s.h.)

Math 120 Precalculus (4 s.h.)
Math 130 Matrix Algebra (3 s.h.)

Math 140 Applied Calculus (3 s.h.)
Math 150 Math Earth/Life Sciences I (3 s.h.)

Math 155 Math Earth/Life Sciences II (3 s.h.)
Math 160
Calculus I (4 s.h.)

Math 165 Calculus II (4 s.h.)
Stat 110 Fundamentals of Statistics (3 s.h.)

Adhering to the current University Studies policy, none of the above
courses would be allowed

to be listed in other categories of the General Education Program.

All of these are existing courses, previously approved by A2C2.

Department Contact Person for this entire proposal, including all courses contained herein:

Jeffrey R. Anderson, Mathematics and Statistics Department Chair

Email janderson@winona.edu

University Studies Program

Basic Skills in Mathematics

Description and Outcomes

3. Mathematics (3 S.H.)

The purpose of the Mathematics requirement in University Studies is to
help students develop an

appreciation of the uses and usefulness of mathematical models of our world, as applied in
a

variety of specific contexts. Students should complete the requirement as soon as
possible,

preferably in their first year and certainly no later than their third semester. Only
approved

courses offered by the Department of Mathematics and Statistics can be used to satisfy the

University Studies requirements for Basic Skills in Mathematics. Each of these courses
must

address at least four of the following outcomes.

These courses must include requirements and learning activities that promote students' abilities to...

a. use logical reasoning by studying mathematical patterns and
relationships;

b. use mathematical models to describe real-world phenomena and to solve real-world
problems -

as well as understand the limitations of models in making predictions
and drawing conclusions;

c. organize data, communicate the essential features of the data, and interpret the data
in a meaningful way;

d. do a critical analysis of scientific and other research;

e. extract correct information from tables and common graphical displays, such as line
graphs,

scatter plots, histograms, and frequency tables;

f. express the relationships illustrated in graphical displays and tables clearly and
correctly in

words; and/or

g. use appropriate technology to describe and solve quantitative problems.

**Discussion of Basic Skills Mathematics Course Work and the Required
Outcomes**

In general, every mathematics, mathematics education, and statistics
course addresses all of the

outcomes (a.- g.) to some extent. From a global point of view, mathematics may be viewed
as

the result of a human desire to describe, quantify, and predict and to be able to do so
with an

unlimited degree of accuracy and precision. Put in a slightly different manner,
mathematics is the

natural result of people wanting to have methods of communication which do not allow for

misinterpretation beyond a predetermined amount of allowable error (to include the
possibility

that no allowable error is desired). This is what drove ancient Greeks to begin the
development

of logic and geometry. It is what drove Galileo to the use of measurement and functions to

communicate relationships between physical quantities. It is similarly what drove Pascal
and

Descartes to the development of coordinate geometry as a graphical means for examining

functions and data. Newton and Leibniz, equally motivated by a need to predict, developed
the

calculus to predict the motion of planets and, by the same machinery, predict the motion
of

terrestrial objects due to gravity. The quest continues today as scientists in the field
grapple with

these issues of description, quantification, and prediction as applied to everyday
problems

concerned with the stock market, insurance and risk, the weather, quantum machines, and
the

teaching of mathematics, to name just a few. All of these objectives are therefore
inherent in the

program of mathematics itself.

The mathematics and statistics courses proposed herein are those which
a student may complete

to satisfy the Basic Skills in Mathematics requirement, and these courses contain a
portion of this

legacy. Additionally, they are all reasonably accessible to new, incoming students at WSU.
On the

other hand, students come to WSU with a wide range of backgrounds in mathematics. It is to
be

expected, then, that Basic Skills in Mathematics may be more meaningful to students if
their course

begins at a level of development that is appropriate to their existing mathematical
background and

abilities. Such is the rationale behind providing a choice of courses, as opposed to
selecting one

course, for Basic Skills in Mathematics.

In each of the course outlines, the outcomes which are predominantly
addressed in that particular

course are indicated. A general discussion of each objective, including examples of
student activities

which address these, is now presented.

- use logical reasoning by studying mathematical patterns and relationships

Students will gain experience with valid reasoning while investigating
and applying patterns,

sequences, and other relationships. For example, in attempting to conclude that a sequence

of annual population data follows a logistic equation, students are faced with the
difference

between inductive reasoning and deductive reasoning. They learn that while inductive
arguments,

i.e., based upon experiment, are all that is possible given a finite amount of data,
deductive

arguments, i.e., based upon logical reasoning, are very powerful in examining questions o

of long-term growth or decline of that population. The idea here is that once a
mathematical

relationship or equation has been decided upon, perhaps as a model for some physical
situation,

then the full power of logical reasoning may be applied to discover what additional
information

must necessarily follow from adoption of that equation.

b. use mathematical models to describe
real-world phenomena and to solve real-world problems

-as well as understand the limitations of
models in making predictions and drawing conclusions;

Students will begin with a mathematical model, suggested by the results of experiment or resulting

from processes such as linear/nonlinear regression applied to data, and use that model to study

the predictions of the model. For example, data regarding the height versus weight of a group of

people may be gathered. Linear regression may be applied to conclude a mathematical equation

(linear) which relates height to weight. This may then be used to quantify what the "average"

(in some sense) weight of a person is given their height. As the linear equation will return a weight

for any height, no matter how large, students see that there is necessarily a limit to how the model

may be applied in order to describe real people. They will also see the limitations in a calculation

such as "average" via comparison of results from the model with actual data.

c. organize data, communicate the essential features of the data, and interpret the data in a meaningful way;

Given a collection of data, students will investigate several methods of organization and graphical display.

They will examine methods of isolating only that data which is necessary for the solution of a certain problem. Finally, they will learn that there are usually reasons to be cautious in the interpretation of data due to possible contamination of the data itself, e.g., flawed data collection methodologies used, and possible confounding

factors not reflected in the data.

d. do a critical analysis of scientific and other research;

Students will examine the methodologies, such as data collection and analysis of this data, and results

reported due to the analyses used in the research literature which is accessible to them (taking into

account background material necessary to understand the topics of such literature). They will critique

these works for the validity of reasoning used, for the soundness of the mathematics applied, and for the appropriateness of the conclusions given the data collected.

e. extract correct information from
tables and common graphical displays, such as line graphs,

scatter plots, histograms, and
frequency tables;

It is a simple newspaper editor's trick to change the scale on a vertical or horizontal axis of a graph to

fool the eye into thinking that something is increasing at an alarming rate (such as the price of oil) or is

decreasing very slowly (such as unemployment). Also, there is another type of graph, called an area

graph, which is sometimes used to communicate, e.g., that one magazine's readership is double that of

its competitor, but really shows a symbol which is four times that of a smaller symbol representing the

competitor's readership. Graphical displays communicate information very quickly, but they can also

communicate incorrect information based upon shape, size, or some other factor. Students will learn

to carefully examine all parts of graphically presented data in order to draw correct conclusions from these.

f. express the relationships
illustrated in graphical displays and tables clearly and correctly in

words;

Once a student has gleaned the correct information from graphical or tabular presented data, there

is the issue of communicating this in English. Converting from a very precise language, Mathematics,

to a very imprecise language, English, is generally a difficult problem. Students must learn to take care

in communicating the information accurately while not introducing ambiguities due to use of the English

language.

g. use appropriate technology to describe and solve quantitative problems.

There are many different forms of hardware and software currently in use by those who apply mathematics

to the solution of real-world problems. As the amount of realism in a mathematical model increases, so

does the difficulty of calculations necessary to deduce the predictions of that model. Therefore, it is useful

for students to gain experience with technology such as graphing calculators, MatLab and Mathematica

software packages, and JMP statistical analysis packages as they apply mathematics to the analysis needed

in studying applications motivated problems that include a greater level of realism.

The following is an example syllabus for Math 100 – Survey of
Mathematics. Although course syllabi will vary

from instructor to instructor, the common elements of all syllabi will include (1) the
course description designating

the course as a general studies courses, (2) the University Studies Program outcomes for
Basic Skills in Mathematics

and where these are addressed in the course, and the (3) topics/instructional
methodologies as delineated in the

Mathematics and Statistics Department course outline.

The following is an example syllabus for Math 130 – Matrix Algebra.

**Mathematics 130 – Matrix Algebra – 3 s.h.**

Course Description: An introduction to matrix algebra and elementary vector spaces with emphasis on computation. Prerequisite: MATH 120. This is a University Studies course which satisfies the Basic Skills in Mathematics.

**Possible Textbooks:** Anton, Howard. Elementary Linear Algebra with
Applications, (8^{th} ed) John Wiley & Sons, 2000. Hill, Richard. Elementary
Linear Algebra with Applications, (2^{nd} ed) Saunders College Publishing, 1991.
Johnson L., Riess, R. D., Arnold, J. Introduction to Linear Algebra, (4^{th} ed)
Addison Wesley, 1997. Lay, David C. Linear Algebra and its Applications, (2^{nd}
ed) Addison Wesley, 1998.

**Basic Skills in Mathematics: **The purpose of the Mathematics
requirement in University Studies is to help students develop an appreciation of the uses
and usefulness of mathematical models of our world, as applied in a variety of specific
contexts. Students should complete the requirement as soon as possible, preferably in
their first year and certainly no later than their third semester. Only approved courses
offered by the Department of Mathematics and Statistics can be used to satisfy the
University Studies requirements for Basic Skills in Mathematics. Mathematics 130 contains
requirements and learning activities that promote students' abilities to...

- use logical reasoning by studying mathematical patterns and relationships;
- use mathematical models to describe real-world phenomena and to solve real-world problems - as well as understand the limitations of models in making predictions and drawing conclusions;
- organize data, communicate the essential features of the data, and interpret the data in a meaningful way;
- use appropriate technology to describe and solve quantitative problems.

In the description of class topics and requirements below, these objectives in this list are referred to by I-IV.

Topics covered:

- Systems of Linear Equations: Gaussian Elimination; Existence and Uniqueness; Vector and
Matrix Equations.
**(I), (III)** - Linear Transformations: Matrices: Operations; Diagonal, Upper and Lower Triangular
Matrices; Inverses; Rank.
**(I)** - Determinants: Computing Determinants, Cofactors, 2X2 and 3X3 Cases; Cramer's Rule.
**(II)** - Vector Spaces: Definition; R
^{n}and Subspaces; Linear Independence and Span; Bases and Dimension. - Eigenvectors and Eigenvalues: Computing Eigenvectors and Eigenvalues; Characteristic
Equations.
**(II)**

Optional Topics: Partitioned Matrices; LU Decomposition; Change of Basis.

**Method of Instruction:** Methods may include lecture, group work,
discussion of examples, and discussion of computer output.

**Evaluation Procedures:** Possible methods include examinations,
quizzes, computer assignments, homework problems, and a final examination.

**Note:** It is required that computer assignments using an appropriate
computer package be included in the course. **(IV)**