TEXAS A&M UNIVERSITY-Corpus Christi

Division of Computing and Mathematical Sciences

MATH 3313.001

Foundations of Higher Mathematics

Fall, 2005

I.  COURSE INFORMATION

Meeting Times & Places: TR 7-8:15 PM, CS-108

Professor:  Dr. Alex Sadovski

Office: CI 317

Office Hours

For more information about my schedule,

Phone:  (361) 825-2477

e-mailsadovski@falcon.tamucc.edu

II.  COURSE DESCRIPTION

This course assists students in the transition from lower-level courses such as calculus to higher-level courses such as advanced calculus and modern algebra. While lower-level mathematics courses emphasize skills and techniques needed for courses outside mathematics, higher-level mathematics courses require students to understand and write proofs and to think more abstractly. This course introduces students to fundamental ideas in logic and set theory needed for courses in higher mathematics and for secondary school and collegiate teaching. Techniques of proof, such as proof by contradiction and proof by induction, are used in various settings, such as analytic geometry and coordinate systems. The proper use of quantifiers, multiply quantified statements, properties of functions and relations on sets, modular arithmetic and equivalence relations, and partial orderings are emphasized. Examples used in this course will be taken from number theory, combinatorics, graph theory, modern algebra, and advanced calculus.
The following topics will be covered:

  • Set Theory, Logic and Propositional Calculus
  • Logic and Predicate Calculus (Quantifiers)
  • Applications to Proofs in Set Theory
  • Methods of Proof (Mathematical Induction, Indirect Proofs, Epsilon-Delta Proofs)
  • Relations (Equivalence Relations, Equivalence Classes, Partial Orderings)
  • Functions and Mappings (Injective, Surjective, Bijective, Image, Inverse Image)
  • Cardinality of Sets
  • Axioms of Real and Complex Number Systems
  • Geometry of Complex Arithmetic
  • Introduction to Advanced Calculus

Please note that this class is a prerequisite for MATH 4301, Advanced Calculus, and MATH 4306, Modern Algebra. This prerequisite will be enforced. If you do not get a grade of C or better in this course, you will need to repeat it before being permitted to take 4301 or 4306.

III. PREREQUISITES FOR THE COURSE

MATH 2414 Calculus II and MATH 2305, Discrete Math.

IV. TEXT AND OTHER SUPPLIES REQUIRED

A Transition to Advanced Mathematics, by Smith, Eggen and St. Andre, Brooks/Cole, 6th ed. is required. The book Mathematics for High School Teachers: An Advanced Perpective, by Usiskin, Peressini, Marchisotto and Stanley, Prentice Hall is recommended as a reference for future (and current) teachers.

V. COURSE OBJECTIVES

Students completing this course will learn to do the following:

  • Understand the structure and properties of the real and complex number systems
  • Read and understand arguments involving set theory and logic with minimal assistance from the instructor
  • Generalize mathematical observations of special cases
  • Write proofs of basic results in advanced calculus and set theory which include multiply quantified statements
  • Present mathematically precise arguments to peers, beginning college students, and secondary school students
  • Develop reasoning skills needed in higher mathematics course work and mathematics teaching

VI.  INSTRUCTIONAL METHODS AND ACTIVITIES

Class meetings will usually consist of a combination of small group work and whole-class discussion, with students presenting work at the board as well as a lecture over the material of the course. The focus both in class and outside will be on working problems and discussing solutions designed to lead students from an operational to a structural understanding of the course material. (Anna Sfard defines "operational" understanding to mean understanding at the level of process or computation , while "structural understanding" is defined as when students incorporate the ideas to create a new abstract mathematical object, which can in turn be the foundation of further mathematical objects. She has developed evidence to show that both historically and in individual students, operational understanding must come before structural.)

 

VII.  EVALUATION AND GRADE ASSIGNMENT

Type of assignment

Weight

Class participation/in-class work

10%

Paper-project

15%

Quizzes

25%

Midterm

25%

Final

25%

Letter grades will be assigned according to the table:

Grade

Range

A

86 to 100

B

76 to 85

C

66 to 75

D

50 to 65

F

below 50

Class participation/in-class work: As noted above, class meetings will consist of small-group work and whole-class discussion. You will self-assess your participation three times over the course of the semester using a rubric I will hand out. I reserve the right to alter your self-assessment if I feel it is much too high or too low.

Quizzes: No open books and notes. Quizzes are on understanding of the basic material of the course.

 Midterm and Final:  I will discuss these in more detail as the times for them approach.  The midterm will be given outside of class time so as to allow a longer period of time for you to take it. To compensate you for the time spent on the midterm, there will be no class meetings that week. Dates for the midterm and final are:

VIII.  TENTATIVE CLASS SCHEDULE

See attachment

IX.              OTHER CLASS POLICIES

IX.I Official Part

Attendance:  This is probably obvious, but since 10% of your grade is based on in-class work, unexcused absences will have a negative effect on your grade.

Missed midterm/final:  If you are unable to attend the midterm or the final and you wish to make it up, I need to hear from you no later than 24 hours after the missed test or final.  You should be able to provide adequate documentation of why your absence was necessary.  If you wait more than 24 hours to contact me, you will also need to provide adequate documentation of why you were unable to meet the 24-hour deadline.  As an example, "I was called out of town unexpectedly on business" might be a valid reason to miss a test, but it is not an adequate reason to miss the 24-hour notification requirement.

Students with disabilities: The Mathematics Program complies with the Americans with Disabilities Act in making reasonable accommodations for qualified students with disabilities.  If you need disability accommodations in this class, please see me as soon as possible.  Please have your accommodation letter from TAMU-CC Services for Students with Disabilities Office with you when you come see me.  If you suspect that you may have a disability (physical impairment, learning disability, psychiatric disability, etc.), please contact the Services for Students with Disabilities Office (located in Driftwood 101) at 825-5816.  It is important that you contact them in a timely fashion as it may take several days to review requests and prepare accommodations.

 

1          Attendance required, exceptions are sickness, job and family emergencies, but I will not use class roll at any time, because it is your responsibility to be in class and attend to the process of learning (see also II.2.).

2          Please, print your name on all assignments and tests: your professor is not a decoding device.

3          If you have questions you MUST ask, you have the right to interrupt lecture or discussion at any time  (see also II.1).

4          I am always open for all questions and discussions during the class and office hours. You can always arrange meeting with me at any other time suitable for both sides.

5.                   No multi-choice tests, all tests will consist of problems you have to solve from the beginning to the end. Partial credit will be given for any parts of problems solved. The policy is open books and notes, no talking, no cheating.

6.                   No open books and notes during quizzes.

7.                   Papers must be turned on time.

8.                   There is no social promotion in my classes. Grades are given only for knowledge acquired (see also II.9.).

 

II.        Unofficial part.

II.1.      There are no "stupid" questions, there are only bad teachers.

II.2.      All you do, you do it for yourself, not for the professor.

II.3.      Do not be concern about grades, be concern of knowledge, because grades are the steepest increasing function of knowledge (here is an example of math language).

II.4.      Do not be afraid of problems, let them be afraid of you.

II.5.      Only doing nothing may be without mistakes. If you don’t make errors, you don’t learn anything. 

II.6.      Do not be nervous - it may be only worse.

II.7.      Common sense is the base of all decisions, together with knowledge they can do almost everything (even pass this course!).

II.8.      Keep your particles together.

            II.9.      The only valid excuse for not knowing the subject is a sudden death.