APPENDIX B: TEMPLATE FOR NEW AND REVISED COURSE PROPOSALS (2002)

Prefix, Number, and Name of Course: MAT 501 Math for Teachers: Algebra

Credit Hours: 3

In-Class Instructional Hours: 3 Labs: 0 Field Work: 0

Catalog Description:

Prerequisite: 24 credit hours of undergraduate mathematics

Operational systems, number systems, groups, rings, fields, ordered fields, functions over fields, algebraic properties of the trigonometric functions.

Reasons for Revision:

To provide a graduate level course that provides an in-depth study of the Algebra topics that are covered in the 7 – 12 bands of the newly revised (2005) New York State Mathematics Standards.
To provide an advanced, in-depth study of the connections between school mathematics and abstract algebra.
To provide an opportunity for students to examine the mathematical theory behind the mathematics that they teach.

Student Learning Outcomes:	Course Content References:	Assessment:
1. Students will investigate non-standard operational systems and discover their various properties.	I - IV	1. Group work in class, individual homework assignments, exams
2. Students will explore and prove theorems concerning divisibility, algebraic numbers and decimal representations.	II	2. Group work in class, individual homework assignments, exams
3. Students will examine a variety of algebraic structures such as groups, rings and fields.	I, III-V	3. Group work in class, individual homework assignments, exams, group projects.
4. Students will create examples, compose proofs and solve problems on relations, functions and polynomials over a field.	III -V	4. Group work in class, individual homework assignments, exams.
5. Students will use geometric techniques to develop the trigonometric functions and their properties.	V	5. Group work in class, individual homework assignments, exams.
6. Students will observe and demonstrate direct relationships between school mathematics and modern abstract algebra and trigonometry.	I - V	6. Group work in class, individual homework assignments, exam, individual project.

Student Learning Outcomes:

Course Content References:

Assessment:

1. Students will investigate non-standard operational systems and discover their various properties.

I - IV

1. Group work in class, individual homework assignments, exams

2. Students will explore and prove theorems concerning divisibility, algebraic numbers and decimal representations.

2. Group work in class, individual homework assignments, exams

3. Students will examine a variety of algebraic structures such as groups, rings and fields.

I, III-V

3. Group work in class, individual homework assignments, exams, group projects.

4. Students will create examples, compose proofs and solve problems on relations, functions and polynomials over a field.

III -V

4. Group work in class, individual homework assignments, exams.

5. Students will use geometric techniques to develop the trigonometric functions and their properties.

5. Group work in class, individual homework assignments, exams.

6. Students will observe and demonstrate direct relationships between school mathematics and modern abstract algebra and trigonometry.

I - V

6. Group work in class, individual homework assignments, exam, individual project.

Course Content:

I. Algebra in Number Systems

A. Decimal Representations of Real Numbers (terminating, simple-periodic, delayed periodic)

B. Algebraic Numbers

C. Divisibility Properties of Integers and Polynomials

II. Operational Systems

A. Modular Systems, Integer Congruence, Applications to Cryptology

B. Other Non-standard Operational Systems

C. Integral Domains

III. Algebraic Structures

A. Groups

1. Definition and examples

2. Commutative and non-commutative

3. Solving equations in groups

4. Opposite operations

B. Rings

1. Definitions and examples

2. Properties of rings

3. Solving equations in rings

4. Reiterated sums and products

C. Fields

1. Field as an extension of rings

2. Properties

3. Solving equations and proofs

a) Quadratic formula

D. Algebraic Extensions of Fields, i.e. Complex Numbers

E. Ordered Fields

IV. Algebra in Functions

A. Functions Over Fields

1. Relations and mappings

2. 1-1, onto and bijections

3. Functions as operations of an algebraic system

4. Polynomial equations over fields

5. Equivalence relations

V. The Relationship between Geometry and Algebra: Trigonometry

A. Trigonometric Ratios as Functions

B. Pythagorean Theorem and the Law of Sines, Law of Cosines

C. Ptolemy’s Theorem

D. Derivation of Trigonometric Identities from Algebraic and Geometric Proofs

E. The Properties of Operations on Trigonometric Functions

Resources:

Classic Scholarship in the Field:

Apostol, T. (1989). Project Mathematics: Polynomials. Pasadena, CA: California Institute of Technology.

Apostol, T. (1989). Project Mathematics: Sines and Cosines, Part I. Pasadena, CA: California Institute of Technology.

Apostol, T. (1989). Project Mathematics: Sines and Cosines, Part II. Pasadena, CA: California Institute of Technology.

Apostol, T. (1989). Project Mathematics: Sines and Cosines, Part III. Pasadena, CA: California Institute of Technology.

Apostol, T. (1989). Project Mathematics: The Story of Pi. Pasadena, CA: California Institute of Technology.

Beaumont, R. A. & Pierce, R. E. (1963). The Algebraic Foundations of Mathematics. Reading, MA: Addison-Wesley.

Comprehensive School Mathematics Program (1973). Elements of Mathematics - Book 0 Intuitive Background Chapter 10: Algebra in Operational Systems. St Louis, MO: CEMREL.

Comprehensive School Mathematics Program (1973). Elements of Mathematics - Book 0 Intuitive Background Chapter 13: Algebra Of Real Functions. St Louis, MO: CEMREL.

Comprehensive School Mathematics Program (1973). Elements of Mathematics - Book 0 Intuitive Background Chapter 1 Operational Systems. St Louis, MO: CEMREL.

Driscoll, M. (1999). Algebraic Thinking: A Guide for Teachers. Portsmouth, NH: Heinemann.

Herstein, I. N. (1996). Abstract Algebra, 3^rd ed. New York: John Wiley & Sons.

Hobson, E. (1921). A Treatise on Plane Trigonometry. New York: Cambridge University Press.

Moses, B., ed. (1999). Algebraic Thinking. Reston, VA: National Council of Teachers of Mathematics.

Niven, I. (1956). Irrational Numbers. Washington, D.C.: Mathematical Association of America.

Rademacher, H & Toeplitz, O. (1957). The Enjoyment of Mathematics. Princeton, NJ: Princeton University Press.

Rotman, J. (1999). An Introduction to the Theory of Groups, 4^th ed. New York: Springer-Verlag, Inc.

Current Scholarship in the Field:

Burke, M., Erickson, D., Lott, J. & Obert, M. (2001). Navigating through Algebra in Grades 9 - 12. Reston, VA: National Council of Teachers of Mathematics.

Dossey, J., McCrone, S., Giordano, F. & Weir, M. (2001). Methods and Modeling for Today's Mathematics Classroom. Pacific Grove, CA: Brooks Cole.

Friel, S., Rachlin, S. & Doyle, D. (2001). Navigating through Algebra in Grades 6 - 8. Reston, VA: National Council of Teachers of Mathematics.

Gelfand, I. M. & Saul, M. (2001). Trigonometry. Boston: Birkhauser.

National Research Council (2005). How Students Learn: Mathematics in the Classroom. Washington, DC: The National Academies Press.

Papick, I. (2006). Algebra Connections. Upper Saddle River, NJ: Pearson Prentice Hall.

Usiskin, Z., Peressini, A., Marchisotto, E. A. & Stanley, D. (2003). Mathematics for High School Teachers: An Advanced Perspective. Upper Saddle River, NJ: Prentice Hall.

Periodicals:

College Mathematics Journal

Educational Studies in Mathematics

Mathematics Magazine

Teaching Mathematics in the Middle School

The Mathematics Teacher

Electronic Resources:

Committee on the Undergraduate Program in Mathematics, Mathematics and the Mathematical Sciences in 2010: What Should Students Know? http://www.maa.org/news/cupm_front.pdf

Committee on the Undergraduate Program in Mathematics, Undergraduate Programs and Courses
in the Mathematical Sciences: CUPM Curriculum Guide 2004, http://www.maa.org/cupm/curr_guide.html

National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, http://standards.nctm.org/