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.

  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 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 165 – Calculus II.

Mathematics 165 – Calculus II – 4 s.h.

Catalog Description: Differential and integral calculus of functions of a single variable. Two semesters in sequence (Math 160, 165 - Calculus I, II). Prerequisite: Qualifying score on the mathematics placement exam or MATH 120. This is a University Studies course which satisfies the Basic Skills in Mathematics.

Text: Calculus, Single and Multivariable (Second Edition) by Hughes-Hallett, Gleason, McCallum, et al

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 165 contains requirements and learning activities 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. organize data, communicate the essential features of the data, and interpret the data in a meaningful way;
  4. extract correct information from tables and common graphical displays, such as line graphs, scatter plots, histograms, and frequency tables;
  5. express the relationships illustrated in graphical displays and tables clearly and correctly in words.

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

 

Topics Covered:

  1. "Real-world" problems and modeling. Examples: Determining volume, mass, and center of mass of a body; mathematics of sound waves; population growth; models with dependence on two or three variables. (II)
  2. Mathematical problems. Examples: Improper integrals; convergence of infinite series; approximation by Taylor and Fourier series; first order differential equations; vector manipulations; graphing and differential calculus for functions of two and three variables. (I), (V)
  3. Functions, graphs, and data. Examples:Slope fields for first order differential equations; graphing in three dimensions; visualization methods such as contours (i.e., level curves/surfaces). (III), (IV), (V)
  4. Limits. Examples: The limit of a function of two and three variables. (I)
  5. Differential calculus. Examples:Use of the derivative to study growth and decay models; partial derivatives, gradients, and directional derivatives for functions of two and three variables. (II), (V)
  6. Integral calculus. Examples: Use of integrals to determine area, volume, mass, center of mass; the definition and calculation of an improper integral; relationship of improper integrals to the issue of convergence of an infinite series; integration via Taylor series. (I), (II)

Method of Instruction: Classroom lecture/presentation/discussion, question-answer sessions, use of calculators/computers and student groups where appropriate.

Evaluation Procedure: Graded homework and/or projects, periodic quizzes and exams, comprehensive final exam.