*[*

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 Stat 110 – Fundamentals of Statistics.

**Statistics 110 – Fundamentals of Statistics – 3 s.h.**

Course Description: Introductory statistics with emphasis on applications. NOTE: Students who have completed MATH 140 or MATH 160 should take STAT 210 instead of STAT 110. Prerequisite: qualifying score on the mathematics placement exam or MATH 050. This is a University Studies course which satisfies the Basic Skills in Mathematics.

**Possible Textbooks:** Blaisdell, E. A. Statistics in Practice
Saunders College Publishing, 1993. McClave, J. T. & Sincich, T. Statistics (8th Ed.)
Prentice Hall, 2000. Moore, D.S. & McCabe, G.P. Introduction to the Practice of
Statistics (3^{rd} Ed) 1999.**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. Statistics
110 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;
- do a critical analysis of scientific and other research;
- extract correct information from tables and common graphical displays, such as line graphs, scatter plots, histograms, and frequency tables;
- 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-VI.

Topics Covered:

- Introductory terms and methods of data collection including a discussion of validity.
**(III)** - Descriptive statistics: Stem-and-leaf displays; Frequency distributions and histograms;
Measures of central tendency; Measures of variation; Measures of position.
**(III), (V)** - Scatterplots and correlation, including a discussion of reliability.
**(V)** - Introduction to probability concepts.
- The Normal distribution.
**(I)** - Sampling distributions and the central limit theorem.
**(II)** - Confidence interval estimation: One sample; Independent samples; Dependent samples.
- The logic of hypothesis testing: Statements of hypotheses; Type I and Type II errors;
Probability values.
**(IV)** - Hypothesis testing involving one sample: One sample z- and t-tests; Wilcoxon Signed Rank
test.
**(IV)** - Hypothesis testing involving two means/distributions: Independent samples; Dependent
samples; Wilcoxon Rank Sum test.
**(IV)** - Contingency table analysis: Chi-square test; Construction of three-way contingency tables.
- Introduction to Analysis of Variance.
- Additional topics as time permits.

**Method of Instruction:** Methods may include lecture, group work,
case studies, 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. **(VI)**