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
Web Page: http://ksuweb.kennesaw.edu/~mgarner/
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.
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.
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:
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.
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.
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.
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.
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.
Principles And Standards For School Mathematics, 2000.
Historical Topics For The Mathematics Classroom,
edited by Baumgart, 1969.
Mathematics Teaching in the Middle School
The syllabus is subject to change with notice.