MA196x On-Line Materials -- Knowing with Certainty

Last modified: April 20, 2009

This page contains some material related to the course MA196x, Knowing with Certainty: Proofs in Contemporary Mathematics.
(Jump now to Week 1, Week 2, Week 3, Week 4, Week 5, Week 6, Week 7.)

Office Hours

D Term, 2009. Please come by at any time. Unless there is an unavoidable conflict, I will be in my office 2:00-2:50 each Monday, Tuesday, and Thursday.

Course Overview

This is a different kind of math course. Attendance in class is mandatory. Therefore, there will be little need to post materials on this web page. But we will all have access to it just in case we need it.

Text: Write Your Own Proofs in Set Theory and Discrete Mathematics by Amy Babich and Laura Person.

This course is an introduction to mathematical thinking aimed not only at the beginning mathematics, statistics or actuarial major, but also at students seeking to further their mathematical interests and those simply curious about logic and reason.

The course has two broad aims. The students in the course will be expected to explain, justify, defend, disprove, conjecture and verify, both verbally and in writing, mathematical ideas. One expected by-product of this training is that the student will develop concrete proof-writing skills which will enhance his or her chances for success in more advanced mathematics courses.

The second goal of the course is to survey the role of proofs and related constructs in contemporary mathematics and other parts of science. Students will complete team projects on topics such as computer-generated proofs, logic programming, the role of proofs in physics, the Kepler Conjecture, and Fermat's Last Theorem. Regular classroom discussion will draw the connection between the concrete problem-solving theme of the assignments and the more philosophical context in which most projects will be carried out.

A more detailed rationale appears below.

Prerequisites: Some college-level mathematics Antirequisite: MA1031 (I.e., students who have successfully completed MA1031 should not sign up for this course.)

Notes on Material Below

On this page, you will find on-line versions of only some of the material distributed in class.

All files are in Adobe's PDF format. You will need to install Adobe Acrobat Reader in order to access these documents.
In some cases, a link may not be active until the date given in parentheses.

Knowing with Certainty -- MA196x

Section D01

Jump now to Week 1, Week 2, Week 3, Week 4, Week 5, Week 6, Week 7.

KNOWING WITH CERTAINTY, WEEK 1 -- March 16 to March 20.

March 16 handout (completed in class March 20);
March 17 worksheet on logical statements;

Course syllabus;
Rules/Guidelines for homework assignments in the course;
CLOSED: Problem Set 1 (March 20).

KNOWING WITH CERTAINTY, WEEK 2 -- March 23 to March 27.

One of our class activities is to explore current events in mathematics. Therefore one of our assignments is to attend and take notes on this year's Levi L. Conant Lecture to be given in HL116 at 4:00 pm on March 30th.

CLOSED: Problem Set 2 (March 28).

KNOWING WITH CERTAINTY, WEEK 3 -- March 30 to April 3.
April 2 Worksheet (umm, April 2?).
Problem Set 3 (April 4).
KNOWING WITH CERTAINTY, WEEK 4 -- April 6 to April 10.
KNOWING WITH CERTAINTY, WEEK 5 -- April 13 to April 17.
Problem Set 4 (April 14).
KNOWING WITH CERTAINTY, WEEK 6 -- April 20 to April 24.
Problem Set 5 (April 20).
Sample Solution with a proof by induction (April 20).
KNOWING WITH CERTAINTY, WEEK 7 -- April 27 to May 1.
Problem Set 6 (May 3).
KNOWING WITH CERTAINTY, WEEK 8 -- May 4 to May 5.

A more detailed rationale

As at most universities, mathematics courses at WPI are roughly partitioned into "service courses" and "rigorous courses". (With a talented population of students enrolled in technical majors, WPI has a large number of non-majors taking upper-level math courses. So I would not not label the second category "courses for math majors".) Many talented students find the transition from service courses to these latter course quite difficult. To some, it seems unfair that they are suddenly expected to know principles of logic and reasoning at the abstract level. Particularly in proof-oriented courses, all too many talented students get blind-sided by the sudden change of focus from problem-solving to theory-building.

Unfortunately, this leads too many good students to the conclusion that they "don't like proofs" or "can't do proofs".

Over the years, the Department of Mathematical Sciences has taken various steps to help solve this problem. We designated five 2000-level math courses as "transition courses". These courses --- with course numbers 2073, 2271, 2273, 2431 and 2631 --- have as their main goal the introduction of some interesting and useful mathematical subject, but also as a subtext, a goal to introduce the student to logical thinking, to concepts like the contrapositive, quantifiers, mathematical induction, counterexamples and so on. These courses are very popular, but some students take them too late, or don't get enough of the foundational material out of the subtext to prepare them to be successful in Advanced Calculus (MA3831, MA3832) or modern algebra (MA3823, MA3825).

Another component to the department's approach to this issue was the creation of the "analysis sequence", MA1031 through MA1034. These courses also have two aims: first to cover the same material (at least at the core level) as the ordinary calculus sequence, but also to introduce the math-oriented freshman to mathematical thinking, as well as proofs and logic. These courses are quite highly recommended by those who complete them and provide an efficient way to cover both basic calculus and the basic ideas of proofs and logic. But it is hard to advise students while they are still in high school. Sometimes the course attracts the wrong audience: students enroll who do not belong there, and students who should be taking such an introduction to mathematical thinking do not enroll in MA1031. Since the courses build on one another, it is sometimes too late to join the analysis sequence after A Term has begun.

So why not offer another "path to enlightenment"? Another freshman-level introduction to proofs? That's what we are trying. We are offering it in D Term so that students who arrive on campus in Fall 2008 will have time to learn about mathematics, time to see if they might some day want to explore math at a deeper level, and will have met their advisor on Academic Advising Day before the course starts.

The course should also be fun! It would be nice if the course could bring students to the cutting edge of research, but one must be realistic about how much one can accomplish with only minimal exposure to university-level mathematics. So the compromise is to include a sort of "current events" component where students will read about some of the latest results in mathematics (particularly in Euclidean geometry) and still have manageable proof-oriented homework assignments.

Students will find Professor Martin a tough grader, who expects students to not only work out the answer to a problem but to explain it in a simple, clear and efficient way, using proper English sentences. All who enter the course should be forewarned that this is not a cake walk!

William J. Martin / WPI / martin.deletethis@wpi.edu