Department of Mathematics

Syllabus

Math 01.115-Contemporary Mathematics

**CATALOG DESCRIPTION:**

Math 01.115 – Contemporary Mathematics 3 S.H.

Prerequisites: Basic Algebra II

This course is designed to develop an appreciation of what mathematics

is and how it is used today. Topics covered include: statistics and

probability; graphs, trees and algorithms; geometrical perspectives

including transformations, symmetry, and similarity; and the mathematics

of social choice. Students are expected to have completed equivalents of

Basic Algebra and Basic Skills Reading.

**OBJECTIVES:**

This course will help students to:

**· ** develop their problem solving and critical
thinking skills

**· ** expand their understanding of and appreciation
for modern mathematics and its applications

**· ** understand both continuous and discrete applications
of mathematics, highlighting some of the more
recent developments in mathematics

**· ** improve their mathematical and computer skills,
through the use

of computational and computer-related algorithms

**CONTENT:**

I. Statistics (4 weeks)

A. Elementary Sampling Theory and Experimental Design

1. Random sampling and bias

2. Experimental design

B. Descriptive Statistics

1. Graphical descriptions and exploratory data analysis

2. Measures of location and variability with a discussion
of computer

algorithms and computational efficiency

3. Regression line - graphical description, with little
emphasis on

computation

C. Probability

1. The frequency concept of probability

2. Mathematical description of probability and expectation
- Students

should appreciate how these are used in gambling,
lotteries, and insurance

3. Sampling distributions with an emphasis on the difference
between

discrete and continuous distributions

4. Central limit theorem

D. Inferential Statistics

1. Confidence intervals

2. The effect of hidden variables on data analysis

II. DISCRETE MATHEMATICAL MODELS (3 weeks)

A. Euler Circuits

1. Graphs as mathematical models

2. Graphs, edges, and vertices and their applications

3. Valence and the existence of Euler circuits

B. Hamiltonian Circuits

1. Algorithms for finding a minimum-cost Hamiltonian circuit

2. Trees, sets, and counting techniques

3. Traveling Salesman Problem (TSP) and the need for computationally

efficient algorithms

C. Directed Graphs and Scheduling

1. Directed Graphs

2. Critical Paths

3. Priority list scheduling

III. TOPICS IN THE MATHEMATICS OF SOCIAL CHOICE (2 weeks)

(Choose from the following.)

A. Discrete and Continuous Versions of Fair Division Problems (optional)

1. Formulations of fair division problems used to
illustrate intuitive

and precise meaning of "continuous"
and "discrete"

2. The procedures used to solve these problems can
be thought of as

algorithms

3. The assumptions needed in order that the procedures
achieve fair

division can be thought of as axioms

B. The Mathematics of Voting (optional)

1. Plurality Method and Condorcet Criterion

2. Borda Count Method

3. Sequential Pairwise Voting and the Pareto Condition

4. Arrow’s Impossibility Theorem

5. Weighted Voting and the Banzaf Power Index

C. Discrete Models of Continuous Data (optional)

1. Relationship between Integers and Rational Numbers with
respect to apportionment problems

2. Examples of apportionment problems:

a. Electoral College

b. House of Representatives

c. College class scheduling

3. Undesirable outcomes in apportionment schemes turn out
to be a

feature of any reasonable apportionment scheme
(Balinsky-Young Theorem)

IV. GEOMETRY (4 weeks)

A. Symmetry, Patterns, Tilings

1. Symmetry

a. As an aesthetic or non-mathematical
idea

b. Isometry of the plane
(this mathematical concept gives precision

to what should be meant by symmetry in a two-dimensional context)

c. Using the classification
of all isometries of the plane, a

complete classification of all 1- and 2- dimensional patterns can be given

d. Concept of a Group

2. Tilings

a. Regular, periodic, and
nonperiodic tilings of the plane

b. Plane geometry, algebra,
and deduction are used to show that only

the equilateral
triangle, rectangle, and hexagon can tile the plane

edge-to-edge

B. Mensuration, Growth, and Form

1. Review of mensuration formulae for areas and volumes
of common

geometrical shapes

2. Geometric similarity and the scaling of real objects

3. How surface area and volumes increase as dimensions
increase; the

implications for the growth of
animate and inanimate objects

C. Fractal Geometry (optional)

TEXTS:

** Excursions in Modern Mathematics, Tannenbaum,P., Prentice-Hall, 7th
edition, 2010.

** For All Practical Purposes: Introduction to Contemporary Mathematics.

W.H. Freeman and Co., New York, 8th edition, 2009.

Mathematics Beyond the Numbers, Gilbert, G. & Hatcher, R. John Wiley &

Sons, Inc., 2000.

Mathematics All Around, Pirnot, T., Addison-Wesley, Inc., 2nd Edition,
2003.

Mathematics and the Modern World, Hathaway, P., Addison Wesley, 2000