MATHS 250: Pre-college Mathematics From an Advanced Viewpoint (3)
Syllabus
1.
Prerequisite: Maths 150, 166, 215. Open only to Mathematics
Teaching majors.
2.
Course Description: In depth treatment of concepts underlying common
topics in the middle and high school mathematics curriculum. Topics include
number systems, polynomial and transcendental functions, analytic geometry,
theory of equations, and measurement.
3.
Course Objectives: Students completing the course will have a deeper
understanding of the mathematical content that they will be expected to teach
in middle school and high school. In addition, students will learn about
connections among topics as well as multiple representations of the content.
4.
Course Rationale: Several influential reports[1]
on the state of education of mathematics teachers indicate that mathematics
teachers are most effective when they have a deep understanding of, and
knowledge of the connections among, the topics that they are teaching. This
course is specifically designed to meet this need.
5.
Course Content: Instructors will discuss most of the following
topics:
·
Number Systems
Natural
numbers: Place value; primes and
factorization; computations in bases other than ten.
Rational
numbers: Fraction concepts; repeating
and terminating decimal expansions; countability.
Real
numbers: Description of the real
number system; uncountability; non-repeating decimal expansions; algebraic and
transcendental numbers; rationalizing the denominator;
Complex
numbers: Polar representation (need
trigonometry first); roots of unity; algebraic properties; properties of
complex analytic functions as compared to real differentiable functions.
·
Functions
Polynomials: Fundamental theorem of algebra; graphing; factoring;
multiple roots and real versus complex roots; quadratic formula; cubic formula.
Natural
exponential and logarithmic functions:
Origins and applications; laws of exponents.
Trigonometric
functions: Unit circle point of view;
trigonometric identities beyond sin2(t)+cos2(t)=1.
Sequences
and series: Arithmetic and geometric
sequences; power series representations of transcendental functions; complex
power series and eit=cos(t)+isin(t).
·
Measurement
Areas
and volumes of standard geometric objects: Circles, balls, cones, pyramids, et cetera.
Germs
of integral calculus: Method of
exhaustion and Archimedes’ determination of the area of the circle.
·
Analytic Geometry
Graphs
and properties of familiar equations:
Lines, planes, and conic sections, including parametric representations;
graphing transformations.
Solving
systems of equations: Intersection of
conics; intersection of a conic and a line; linear systems of equations.
·
Pathology/Oddities in
Mathematics
Nowhere
differentiable continuous functions; space filling curves; non-analytic
function, et cetera.
6.
Course Format: Lecture/discussion.
7. Method of Evaluating Student
Performance: Possible methods include in-class exams, homework, group
projects, take-home exams, and presentations.
8.
Evaluation of the Course: The instruction of the course is evaluated by
departmental student evaluations and peer evaluation. The course is reviewed
and revised periodically by the Departmental Undergraduate Programs Committee.
9.
Addendum: Course materials will be prepared by the instructor.
R.
Bremigan, J. Lorch, 12/11/00
Revised 2/04 M. Karls