Math 260 Syllabus
Course: Math 260, Introduction to Abstract Math, 4 credits
Instructor: Dr. Larry Krajewski, MC 525
Phone: 796-3658 [office], 782-1648 [home] (call before 10 p.m. please)
Office Hours: 12MW, 9 TTh & by appointment
Prerequisites: Grade of C or better in Math 180
Text: A Transition to Advanced Mathematics, 6th edition, by Douglas Smith, Maurice Eggen, and Richard St. Andre, Brooks/Cole, 2006
Description: Sentential and quantifier logic, axiomatic systems, and set theory. Emphasis is on the development of mathematical proofs.
- Learn to read and construct mathematical proofs
- Learn to communicate mathematical concepts and proofs, using appropriate mathematical notation and language
- Become knowledgeable about logic, set theory, relations and functions, mathematical induction, and the real number system.
- Students shall demonstrate a basic understanding of axiomatic-deductive systems.
- Students shall understand proofs and be able to judge the correctness of an argument.
- Students shall demonstrate the ability to reason inductively and deductively.
- Students shall demonstrate the ability to apply appropriate mathematical tools and methods to novel or non-routine problems.
- Students shall demonstrate the ability to use various approaches to problem solving and to see connections between these varied mathematical areas.
- Students shall use the language of mathematics accurately and appropriately.
- Students shall present mathematical content and argument orally.
- Students shall present mathematical content and argument in written form.
Methodology: Lecture, class discussion, small group work, student presentations.
One cannot benefit from or contribute to a class discussion or activity unless one is physically present (this a necessary condition, not a sufficient one). Attendance is required. Call me (796-3658) if you will not be in class. A valid excuse is necessary to miss class. Unexcused absences may lower your grade for the course.
Assigned readings of the texts and handouts need to be done if meaningful discussion can occur.
Your active participation makes the course go. Math is not a spectator sport. Assigned problems and textbook exercises are ways for you to develop problem solving skills and reflect on your learning. Do the problems when they are assigned.
I. Logic and Proofs
A. Propositions and Connectives
B. Conditional and Biconditionals
D. Basic Proof Methods I
E. Basic Proof Methods II
F. Proofs Involving Qunatifiers
II. Set Theory
A. Basic Concepts
B. Set Operations
C. Extended Operations and Indexing
A. Cartesian Products
B. Equivalence Relations
A. Functions as Relations
C. One-to-One, Onto Functions
V. Cardinality (if time permits)
A. Equivalent Sets
B. Infinite Sets
C. Countable Sets
I will use a 90 – 80 – 70 – 60 framework for grading. I will give you written assignments on Thursday and these will be due to the following Thursday. You may consult each other but the write-up is your responsibility. I suggest you go to separate rooms to write up your answers.
There will be three exams in class. I need to see what you can do all by yourself. The dates will be determined during the first week of class.
A Note to You
Mathematics can be an intellectual adventure, a powerful tool, and a creative experience Some of you may have had mathematics courses that were based on the transmission, or absorption, view of teaching and learning. In this view, students passively "absorb" mathematical structures invented by others and recorded in texts or known by authoritative adults. Teaching consists of transmitting sets of established facts, skills, and concepts to students. I do not accept this view. I am a constructivist. Constructivists believe that knowledge is actively created or invented by the person, not passively received from the environment. No one true reality exists, only individual interpretations of the world. These interpretations are shaped by experience and social interactions. Thus, learning mathematics should be thought of as a process, of adapting to and organizing one's quantitative world, not discovering preexisting ideas imposed by others. It is one way to make sense of the world.
Consequently, I have three goals when I teach. The first is to help you develop mathematical structures that are more complex, abstract, and powerful than the ones you currently possess so that you will be capable of solving a wide variety of meaningful problems. The second is to help you become autonomous and self- motivated in your mathematical activities. You will not "get" mathematics from me but from your own explorations, thinking, reflecting, and participation in discussions. As independent students you will see your responsibility is to make sense of, and communicate about, mathematics. Charles Schultz, creator of "Peanuts", compared people to multispeed bikes and noted that "most of us have gears we do not use." Hopefully you will see mathematics as an open-ended, creative activity and not a rigid collection of recipes. And the last is to help you become a skeptical student who looks for evidence, example, counterexample and proof, not simply because school exercises demand it, but because of an internalized compulsion to know and to understand.
As a teacher I have come to realize that when I teach mathematics I teach not only the underlying mathematical structures but I am also teaching my students how to develop their cognition, how to see the world through a set of quantitative lenses which I believe provide a powerful way of making sense of the world, how to reflect on those lenses to create more and more powerful lenses and how to appreciate the role these lenses play in the development of their understanding.
So I ask your help in establishing a mathematical community where one uses logic and mathematical evidence as verification rather than the teacher, where mathematical reasoning replaces the memorization of procedures, and where conjecturing, inventing, and problem solving are encouraged and supported.
You may find this experience frustrating at times. Persevere! Eventually I hope you will own personally the mathematical ideas you once knew unthinkingly or only peripherally (and sometimes anxiously). I want you to become competent and confident using mathematical ideas and techniques.
In training a child to activity of thought, above all things we must beware of what I will call "inert ideas" - that is to say, ideas that are merely received into the mind without being utilized, or tested, or thrown into fresh combinations . . . Education with inert ideas is not only useless: it is, above all things, harmful. Except at rare intervals of intellectual ferment, education in the past has been radically infected with inert ideas . . . Let us now ask how in our system of education we are to guard against this mental dryrot. We enunciate two educational commandments, "Do not teach too many subjects," and again, "What you teach, teach thoroughly." . . . Let the main ideas which are introduced into a child's education be few and important, and let them be thrown into every combination possible. The child should make them his own, and should understand their application here and now in the circumstances of his actual life. From the very beginning of his education, the child should experience the joy of discovery. (Alfred North Whitehead, The Aims of Education)
Americans With Disability Act. If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me or Wayne Wojciechowski (MC 320, 796-3085) within ten days to discuss your accommodation needs.