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.

  1. use logical reasoning by studying mathematical patterns and relationships
  2. 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 110 – Finite Mathematics.

Mathematics 110 – Finite Mathematics – 3 s.h.

Course Description: Applications of elementary mathematics on matrices, linear programming, probability, and statistics to
real-life problems. This course provides the non-calculus mathematics background necessary for students in business,
management, and social sciences. This course satisfies Basic Skills in Mathematics. 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 Texts: Finite Mathematics by Mizrahi and Sullivan. Other references the instructor may want to use:
Finite Mathematics by Smith

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
that promote students' abilities to...

  1. use logical reasoning by studying mathematical patterns and relationships;
  2. 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;
  3. extract correct information from tables and common graphical displays, such as line graphs, scatter plots, histograms, and frequency tables;
  4. 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: Specific topics will be selected by the instructor as indicated in the course description. Possible topics may include:

  1. Linear equations: Rectangular coordinates and lines; Parallel and intersecting lines; Applications. (I), (II)
  2. Systems of Linear Equations: Substitution and elimination; Matrix reduction; Matrix algebra, matrix multiplication, inverse of a matrix; Using technology to solve large systems; Applications. (II), (IV)
  3. Linear Programming - a geometric approach: Linear inequalities; Linear Programming problems. (III)
  4. Finance: Interest; Compound interest; Annuities; Amortization; Applications. (II)
  5. Counting techniques: Sets; Multiplication principle; Permutations; Combinations; Applications. (I)
  6. Probability: Sample spaces and probability models; Properties of the probability of an event; Probability based on counting techniques; Conditional probability; Independent events. (III)

Method of Instruction: Determined by the instructor. Typically lecture with class discussion.

Course Requirements: A scientific calculator. Access to Microsoft EXCEL. (IV)

Evaluation Process: Determined by the instructor. Typically a combination of semester exams and quizzes, homework, group projects, journals, and a final exam.