Syllabus Foundations of Mathematical Structures Fall 2003


MATH 505 / Sec 01 Foundations of Mathematical Structures


5:00-8:00 PM / Wednesday / NSM A115


MAT 543 or concurrent enrollment. Students must have graduate standing and must have completed one year of full time secondary mathematics teaching.

Course Description

This course is designed for current middle school mathematics teachers. Mathematical topics include the basic algebraic properties of sets and operations applied both to the various classical number systems and to other contexts as well, the study of equivalence relations and equivalence classes, modular arithmetic and Diophantine equations, the decomposition of natural numbers and special families of natural numbers. The course also includes exposure to current research on the understanding of these topics by secondary students and their teachers.

Required Materials

Students need a course reader (available from our bookstore), a calculator, and various low-budget items (i.e., graph paper, ruler, colored pencils) to be specified throughout the semester.


Jackie Barab Office: NSM A121

Office Hours: 3:00 – 5:00 PM Monday

9:00 – 10:00 AM Wednesday, Friday

also, by appointment

Office Phone: (310) 243-3580

Office email:

Office FAX: (310) 516-3627 (Write: Attention J. Barab.)


After completing MAT 505 the student will be able to do each of the following and explain and justify the results:

* given a binary operation defined on a set, determine whether these properties hold: closure, associative, commutative, identity element, inverse elements.

* determine whether a set is a group under a given operation.

* determine whether or not various distributive properties hold on a given set with two specified operations.

* determine whether a relation is an equivalence relation on a given set; if so, identify the equivalence classes.

* describe and apply the structure of natural numbers and of special families of natural numbers.

* use modular arithmetic to analyze and solve linear Diophantine equations.

* demonstrate knowledge of current research on teaching and learning of mathematical structure in the school curriculum.


Syllabus Foundations of Mathematical Structures Fall 2003


Course Requirements

Students are expected to attend all classes, to do all assigned homework, and to take the scheduled exams. In addition, it is important for students to participate fully in class, which may include small group work, whole group discussions, presentations, and individual written work. In order to participate fully, students need to bring all materials requested by the instructor.

Grading Policy

Grades will be based on a midterm exam, a comprehensive final exam, and daily work consisting of homework, attendance, class participation, and classwork. Attendance of all classes is expected.

Task Points

Daily Work Composite (including attendance) 100

Midterm Exam [Wednesday, October 15] 100

Cumulative Final Exam [Wednesday, December 10]______ 150

Total Possible Points 350

Grades are based on a total of 350 points. They will be no lower than:

93% + A 87% + B+ 77% + C+ 67% + D+

90% + A- 83% + B 73% + C 63% + D

80% + B- 70% + C- < 60% F

Policy on Late Homework, Missed Classwork and Exams

Late homework and missed classwork will be accepted within a week of the due date for half-credit provided this work has not yet been discussed in class. If a student notifies the instructor on the day of an exam via phone or email and provides evidence of an appropriate excuse within the week, then make-up exam arrangements will be determined on an individual basis.

Academic Integrity

Maintaining academic integrity is of the utmost importance. The mathematics department does not tolerate academic dishonesty. For more details see the university catalogue under “Academic Integrity.”

Schedule Foundations of Mathematical Structures Fall 2003

Week Topic

1 Introduction to sets, operations, the closure property, the associative property

2 Identity elements and inverse properties

3 Group structure; commutative property

4 Well-defined operations; modular arithmetic

5 Distributive properties

6 Ring structure

7 Diophantine equations

8 Midterm

9 Relation properties: reflexive, symmetric, transitive

10 Equivalence relations and equivalence classes

11 Structure of natural numbers: primes and composites

12 Structure of triangular and trapezoidal numbers

13 Order relations

14 Introduction to fields

15 Finale: Identifying structures in a new context

Dec. 10 Final Exam