** **

**Math 3390**

**Spring 2005**

** **

**I.
****Course: Math 3390 Introduction to Mathematical Systems**

Department
of Mathematics, College of Science and Mathematics

Kennesaw
State University

**II. Instructor:** Dr. Mary Garner

** **Office: Science 522

Phone: 770-423-6664

e-mail: mgarner@kennesaw.edu

Web Page: http://ksuweb.kennesaw.edu/~mgarner/

WEBCT: http://courses.kennesaw.edu

Office Hours: 2 PM – 3 PM Tues and Thurs

10 AM – Noon Wed

Other times by appointment

**III. Class Sessions:**
Mondays and Wednesdays 6:30 PM –
7:45 PM. CL 1007

**IV.
Texts (required): ***Sets,
Functions, and Logic: An Introduction to Abstract*

* Mathematics*,
third edition, by Devlin, 2004.

You
must access WEBCT to obtain the syllabus, assignments, announcements, important
dates, and any other course materials. You also must access WEBCT to complete
course requirements. Through WEBCT students can ask questions or engage in
discussions with the class.

**V. Catalog
Description: **

Introduction to Mathematical Systems is a course
specifically designed to introduce students to the study of mathematics from a
mathematical systems approach. A mathematical system consists of undefined
terms, axioms, and theorems. Mathematical systems studied will vary according
to the instructor and may be chosen from sets, number systems, and/or geometry.
The major emphasis of this class will be on the development of skills in
communicating and justifying mathematical ideas and conclusions.

** **

**VI.
****Purpose/Rationale:
**

** **

The faculty of Kennesaw
State University endorses the standards for the preparation of teachers of
mathematics proposed by the Mathematical Association of America (MAA) in *A
Call for Change: Recommendations for the Mathematical Preparation of Teachers
of Mathematics* and by the National Council of Teachers of Mathematics
(NCTM) in the *Curriculum and Evaluation Standards for School Mathematics*
and the *Professional Standards for Teaching Mathematics* and subscribed
to by the National Council for Accreditation of Teacher Education. Thus, the
mathematics education courses at Kennesaw State University are designed so that
teachers will

1.
View
mathematics as a system of interrelated principles.

2.
Communicate
mathematics accurately, both orally and in writing.

3.
Understand
the elements of mathematical modeling.

4.
Understand
the use of calculators and computers appropriately in the teaching and learning
of mathematics.

5.
Appreciate
the development of mathematics both historically and culturally.

6.
Understand
the mathematics content that is necessary to teach the grades 6-12 mathematics
envisioned by the MAA and the NCTM

In
preparing teachers, this course emphasizes mathematics as a model of inquiry,
not just a collection of facts and techniques to be learned. In addition, the
principles advocated in the NCTM *Standards* are woven throughout the
course, so that the preservice teacher will have knowledge of the kind of
pedagogy that is being prescribed and will be able to serve as an agent of
change. This course will require the student to solve problems, think
critically, and reflect.

COLLABORATIVE DEVELOPMENT OF
EXPERTISE IN

*TEACHING AND LEARNING*

* *

The Professional Teacher Education Unit (PTEU)
at Kennesaw State University is committed to developing expertise among
candidates in initial and advanced programs as teachers and leaders who possess
the capability, intent and expertise to facilitate high levels of learning in
all of their students through effective, research-based practices in classroom
instruction, and who enhance the structures that support all learning. To that
end, the PTEU fosters the development of candidates as they progress through
stages of growth from novice to proficient to expert and leader. Within the
PTEU conceptual framework, expertise is viewed as a process of continued
development, not an end-state. To be effective, teachers and educational
leaders must embrace the notion that teaching and learning are entwined and
that only through the implementation of validated practices can all students
construct meaning and reach high levels of learning. In that way, candidates
are facilitators of the teaching and learning process. Finally, the PTEU
recognizes, values and demonstrates collaborative practices across the college
and university and extends collaboration to the community-at-large. Through
this collaboration with professionals in the university, the public and private
schools, parents and other professional partners, the PTEU meets the ultimate
goal of assisting Georgia schools in bringing all students to high levels of
learning.

__ __

__Knowledge
Base: __

Teacher
development is generally recognized as a continuum that includes four phases:
preservice, induction, in-service, renewal (Odell, Huling, and Sweeny, 2000).
Just as Sternberg (1996) believes that the concept of expertise is central to
analyzing the teaching-learning process, the teacher education faculty at KSU
believes that the concept of expertise is central to preparing effective
classroom teachers and teacher leaders. Researchers describe how during the
continuum phases teachers progress from being Novices learning to survive in
classrooms toward becoming Experts who have achieved elegance in their
teaching. We, like Sternberg (1998), believe that expertise is not an end-state
but a process of continued development.

Math 3495 develops a strong conceptual understanding of the algebraic concepts necessary for 5-8 and 7-12 teachers of mathematics, it also integrates various technologies that are used in the classrooms of today. The instruction in MATH 3495 also models the “teacher as facilitator” as various instructional strategies are employed, such as cooperative groups and joint projects. Students are actively engaged in all phases of their learning.

__Use
of Technology: __

The use of calculators and computers is an encouraged
and accepted practice to enable students to discover mathematical relationships
and approach real world applications.
Familiarizing the pre-service teacher with a variety of technological
tools is an integral part of the math sequence for teachers. Students in Math 3495 routinely use computer
software and graphing calculators.

A variety of materials and instructional strategies
will be employed to meet the needs of the different learning styles of diverse
learners in class. Candidates will gain knowledge as well as an
understanding of differentiated strategies and curricula for providing
effective instruction and assessment within multicultural classrooms. One
element of course work is raising candidate awareness of critical multicultural
issues. A second element is to cause candidates to explore how multiple
attributes of multicultural populations influence decisions in employing
specific methods and materials for every student. Among these attributes
are** age, disability, ethnicity, family structure, gender, geographic region,
giftedness, language, race, religion, sexual orientation, and socioeconomic
status**. An emphasis on cognitive style differences provides a
background for the consideration of cultural context.

Kennesaw State University provides
program accessibility and accommodations for persons defined as disabled under
Section 504 of the Rehabilitation Act of 1973 or the Americans with
Disabilities Act of 1990. A number of
services are available to support students with disabilities within their
academic program. In order to make
arrangements for special services, students must visit the Office of Disabled
Student Support Services (ext. 6443) and develop an individual assistance
plan. In some cases, certification of
disability is required.

Please be aware there are other support/mentor groups on the campus of Kennesaw State University that address each of the multicultural variables outlined above.

**VII.
****Goals
and Objectives: **

** **

Students completing
this course will be able to:

·
Communicate mathematical concepts using the
language of sets, logic, and mathematical systems.

·
Construct a logically valid argument.

·
Write precise mathematical definitions.

·
Demonstrate understanding of basic concepts of
orders, relations, and functions.

·
Construct an elementary mathematical system from
a set of axioms and undefined terms.

·
Discuss and give examples of the role that
mathematical systems have played in the development of modern mathematics.

·
Use truth tables to determine the truth value of
statements forms for statements.

·
Use quantifiers with logical connectives and
negate quantified statements.

·
Write precise definitions and apply definitions
for the fundamental set operations.

·
Produce accurate proofs for statements involving
sets.

·
Write precise definitions and apply definitions
for functions and special types of functions.

·
Write precise definitions and apply definitions
for special mathematical relations.

·
Demonstrate an understanding of equivalence
relations.

·
Demonstrate understanding of the role of axioms,
definitions, and theorems in mathematical systems.

** **

**VIII.
****Course
Requirements/Assignments:**

** **

·
Throughout the course, students will be asked
to present orally and submit written explanations of concepts or proofs of
theorems germane to the topics

·
A final group
project is required in which the students will develop a mathematical system
and present their results orally and in writing. Students will be given part of
a mathematical system, the undefined terms, and some axioms. They will then be
asked to develop definitions, propose other axioms, propose and prove theorems
valid for the system.

·
Each student
will select from a list of topics and prepare a written report on the
historical and philosophical development of mathematical systems.

·
Homework will
be assigned after topics are discussed. Students are expected to work on
class-related preparation and homework approximately two hours outside of class
for each hour in class.

·
There will be
opportunities for students to work in cooperative learning groups during class
and outside class. There will be regular group assignments; sample work will be
evaluated either individually or as group assignments.

·
There will be
two brief 20 minute quizzes during the semester.

·
There will be
two exams (full class period) during the semester and a comprehensive final
exam. There will be no make-up tests.

** **

__Schedule and Important Dates:__

(These are tentative dates and subject to
change with notice.)

Jan
17 Holiday, no classes

Feb 7 Quiz 1 (logic)

Feb
23 Test 1 (logic and sets)

March
4 Last day to withdraw without
academic penalty

March 5-11 Spring break, no classes

March
21 Quiz 2 (functions)

April
11 Test 2 (relations and
functions)

April
18 History reports due.

April
27 Last class, projects due

May
2 Comprehensive final exam
6:30-8:30

**IX. Evaluation and Grading:**

** **

2 quizzes 60 points each 120

2 exams 100 points each 200

Comprehensive final exam 160

Group projects and assignments 100

(includes final project
80 points)

Individual projects and assignments 120

(history project 60
points,

graded homework 20
points,

class participation 20
points

WEBCT participation 20
points)

Total Possible Points 700

Grades will be assigned as follows:

630-700 A

560-629 B

490-559 C

420-489 D

Below
420 F

**X. Academic Honesty Statement:**

** **

Every KSU student is responsible for upholding the provisions of the Student Code of Conduct, as published in the Undergraduate and Graduate Catalogs. Section II of the Student Code of Conduct addresses the University’s policy on academic honesty, including provisions regarding plagiarism and cheating, unauthorized access to University materials, misrepresentation/falsification of University records or academic work, malicious removal, retention, or destruction of library materials, malicious/intentional misuse of computer facilities and/or services, and misuse of student identification cards. Incidents of alleged academic misconduct will be handled through the established procedures of the University Judiciary Program, which includes either an “informal” resolution by a faculty member, resulting in a grade adjustment, or a formal hearing procedure, which may subject a student to the Code of Conduct’s minimum one semester suspension requirement.

**XI.
****Class
Attendance Policy: **

** **

Regular
attendance is assumed and will be monitored.
Although it is impossible to reconstruct classroom lectures,
discussions, and activities, in the event of unavoidable absence, the student
will assume full responsibility for any material and/or announcement
missed. Daily assignments and quizzes
that are missed cannot be made up. The
instructor strongly urges each student to form a study team with one or two
other students. Points will be deducted from any late assignments.

** **

**Withdrawal
Policy:**

Students choosing to withdraw from
this course without academic penalty must do so by **March 4, 2005**.
Withdrawal forms can be obtained from the Office of the Registrar. The completed form must be approved by the
Registrar. A student ceasing to attend
class and completing course requirements will be assigned a failing grade if
official withdrawal has not been completed.
There is a new University policy on the total number of withdrawals a
student may have (see 2004-2005 Undergraduate Catalogue p. 42).

** **

**XII.
****Course
Outline:**

A.
Introduction to
Mathematical Systems

i.
Definition and
examples

ii.
Historical perspective
(included throughout the course)

B.
The Logic and Language
of Proofs – In mathematics, logical arguments are used to deduce implications
(theorems) from assumptions (axioms). This segment of the course provides the
common language and rules of logic necessary to make mathematically meaningful
statements and construct valid arguments (proofs).

i.
Statements,
predicates, and quantifiers.

ii.
Mathematical
implications.

iii.
Proofs (direct,
indirect, contrapositive) and logical equivalence.

C.
Sets – Sets are the
building blocks of mathematical structures. This section of the course takes
advantage of the student’s informal understanding of sets to develop the
important ideas and theorems of set theory from the undefined notions of set,
element, and belonging.

i.
Introduction.

ii.
Operations on sets.

iii.
Indexed families.

iv.
The set of natural
numbers, including inductive reasoning and the axiom of induction.

D.
Relations and orders –
Relations are encountered in mathematical and non-mathematical settings and
provide the language to describe basic concepts of size, order and equivalence.

i.
Relations.

ii.
Equivalence relations,
partitions and identifications.

E.
Functions – In this
section, a student’s informal understanding of functions is put on a more
rigorous foundation.

i.
Functions as
relations, composition and inverses.

ii.
Functions viewed
globally.

iii.
Binary relations.

iv.
Definition and
examples.

F.
Mathematical systems
revisited.

** **

**XIII.
****References/Bibliography:**

Mathematical
Association of America & American Mathematical Society (2001*). The *

* Conference
Board of the Mathematical Sciences: Mathematical Education of *

* Teachers
Part I.* Washington, DC: MAA.

National
Council of Teachers of Mathematics (NCTM), Reston, VA.

Publications:

*Principles And Standards For School
Mathematics, 2000.*

* **Historical
Topics For The Mathematics Classroom*,

* *edited by Baumgart, 1969.

* *Periodicals:

* Mathematics
Teacher*

* Mathematics
Teaching in the Middle School*

*The
syllabus is subject to change with notice.*

* *