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 i

dicated. 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 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 l

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

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 155 – Mathematics for the Earth and Life Sciences II.

**Mathematics 155 – Mathematics for the Earth and Life Sciences II
– 3 s.h.**

Course Description: This course was developed in conjunction with the earth and life science departments. Its main focus is a study of the mathematics of change as it applies to the earth and life sciences. Emphasis is on the creation and analysis of mathematical models of natural phenomena. Modeling begins with finite difference equations and quickly progresses into continuous models, leading to a study of derivatives. A qualitative analysis of differential equations and systems follows the study of derivatives, and the course ends with a brief discussion of numerical schemes for integration and the exploration of more in-depth models. This is a University Studies course which satisfies the Basic Skills in Mathematics.

**Text:** Contemporary Calculus Through Applications by the North
Carolina School of Science and Mathematics.

**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 155 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;
- extract correct information from tables and common graphical displays, such as line graphs, scatter plots, histograms, and frequency tables;
- express the relationships illustrated in graphical displays and tables clearly and correctly in words.

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

Topics Covered:

- Models of Change – discrete versus continuous phenomena and the mathematical
expressions involved. Average rates of change as approximations of instantaneous rates of
change. Applications to population models.
**(I, II, IV, V)** - The mathematics of instantaneous rates of change – the derivative. Calculation of
the derivative from a function. Rules for calculation of the derivative of a product and
of a quotient of two functions. Estimation of the derivative from graphically or
numerically presented data.
**(I, III, IV, V)** - Applications of the derivative in optimization, numerical root finding schemes, and
quantification of related rates. Applications to population models (eg., determination of
maximal population increase), disease models (eg., determination of relationship between
inoculation rate and rate of increase in infected individuals), geological models
(quantification of maximal stress and rate of change of stresses in shifting plates), and
determination of an "optimal" linear model given a data set.
**(II, III)** - Physical models involving differential equations. Examples include population models
incorporating exponential growth laws, logistic growth laws, and more general polynomial
growth laws. Determination of physical parameters in these laws from given data (according
to a specificed optimal fit criterion). Graphical, numerical, and exact solution methods.
**(I, II, III, IV, V)** - Applied projects including topics from prediction of the spread of infectious diseases,
models of population growth, models of the permeability of red blood cells, and
predator-prey models.
**(II, IV, V)**

**Methods of Instruction:** Lecture, discussion, projects (possibly
cooperative).

**Course Requirements:** Use of a graphing calculator is strongly
recommended. A computer aided mathematics system such as Mathematica or Matlab can also be
helpful.

**Evaluation Process:** Tests, quizzes, projects.