Byers W., Hillel J. (1998) A Course on Mathematical Thinking Working document, Department of Mathematics and Statistics, Concordia University
© Byers & Hillel

Introduction The experience of our department at Concordia is probably not very different than that of most Canadian mathematics departments. Our entering students are a heterogeneous group and while some of them are very well prepared for university mathematics, the large majority is not. Even those incoming students who have done well in CEGEP (Quebec's compulsory two-year colleges where students do their introductory calculus and linear algebra) seem to have trouble with the transition to rigor and in adapting to the style of teaching and style of discourse which characterizes our subject. Some of the students have encountered proofs in previous mathematics courses yet it is painfully obvious to those of us who teach first year university courses that many students have a very vague notion of proofs and proof techniques. In fact, many are even unclear about the need for proofs at all.    We felt that there was a need for a bridging course that would help students make the transition; a course that would put emphasis on the necessary "tools of the trade" for coping with university mathematics rather than cover a lot of new mathematical terrain. In 1993 we co-taught such a course which we called "Introduction to Mathematical Thinking". Among the things we discussed in the course were: 1. Different types of argumentation in mathematics, when and why they can be used: - examples, diagrams, analogies - plausible or heuristic arguments, informal proofs - formal/rigorous proofs, counter-examples (In particular, following Polya, we distinguished between "convincing oneself", "convincing a friend", and "convincing an enemy".) 2. Proof techniques: Direct, Contradiction, Contrapositive, Induction 3. Types of Proofs: Necessary, Sufficient, iff, Existence, Uniqueness, Constructive, etc. 4. Quantifiers and Logical symbols 5. Definitions, Notations, and Terminology Another seemingly big gap in the mathematical background of many students is their knowledge of the structure of the real number system and their knowledge of functions (i.e. functions which are not given by a single analytic expression) . How many of your students think that ˆ2, e, and ¼ are the only irrationals? How many think that the rationals and the irrationals alternate on the real line like even and odd integers? So for the underlying "mathematics content" we drew examples of proofs mostly from the following mathematical topics: - Real numbers with emphasis on properties of rationals and irrationals such as their decimal representations, denseness of the rationals,... - Set Theory: cardinality, finite, countable and uncountable sets, proofs of the countability of Q and uncountability of R. - Elementary number theory: divisors, division algorithm, prime numbers, g.c.d., etc. - Functions and sequences. Teaching Style The course was not taught in the usual lecture format. Rather, a typical class might have started with the students working in small groups (or individually, if they preferred) on a prepared worksheet which was then followed by a general class discussion. A typical worksheet might ask students to take a "convince a friend" argument and to make it into a formal argument, to complete an incomplete proof, to analyze proofs for implicit assumptions and for correctness, or to rewrite proofs by using the appropriate quantifiers.    Here is an example of one such worksheet: Each of the following proofs is flawed. Debug the problem. (a) Statement : If x and y are irrationals so is x+y. Proof: By contradiction. Suppose x and y are irrational and x+y is rational so x+y = m/n for some integers m and n, n ‚ 0. Hence, y = m/n - x = (m - nx)/n so y is written as a quotient which implies that y is rational. This gives the desired contradiction. (b) Statement: A sufficient condition for k2 to be odd is that k is odd. Proof: By contrapositive. Suppose k is even then k = 2m for some integer m. Hence k2 = (2m)2 = 4m2 which is an even number, contradicting the premise that k2 is odd. (c) Statement: For each positive integer n, 6 divides n3 - n. Proof: By induction. If n = 1 then n3 - n = 0 which is divisible by 6. Assume 6 divides k3 -k. Since k is an arbitrary integer, k can be replaced by k+1 so 6 divides (k+1)3 - (k+1). Hence, by induction , 6 divides n3 - n for all positive integers n.    Comments: We see that the examples above deal with several types of commonly observed student difficulties with proofs. Example (a) incorporates an incorrect statement into an otherwise correctly-set proof by contradiction. Example (b), also looking like a familiar correct proof, proves the converse of the statement. We have spent a fair deal of time in the course dealing with the different forms of mathematical statements ("necessary", "sufficient", "if ...then", "P whenever Q", etc.). Example (c) tackles what is probably the most difficult problem for students; dealing with the, often implicit, quantifiers attached to a statement. When asked to associate the quantifiers "there exists" or "for every" to statements, students' responses often revealed serious gaps in their understanding. On an elementary level, knowing, say, that the odd numbers make up the set { 2k+1 / k an integer}, some students would write "m is odd if m = 2k+1 for every k" or try to prove the product of two odd numbers is odd by putting "let m = 2k+1 and n=2k+1 be any two odd numbers" . In other words, there was a lack of awareness about the "change in status" of the symbol k. When asked to rewrite the statement that two sets have the same cardinality with quantifiers, students' response indicated that they believed that every map between the sets must be a 1-1 correspondence. The problem of implicit quantifiers was particular acute with regards to proofs by induction. Even though some students have already seen such proofs in previous courses they were not convinced that the induction hypothesis does not simply assume that the result is true, nor were they immune to false arguments such as example (c) above, though we spent nearly two weeks on induction. However, from the moment we started to address the question of quantifiers (about half-way through the course) and had several worksheets devoted to it, students started to improve, particularly with induction proofs. What lingered, however, was the the fuzziness about the distinction between a description of a set and a description of an arbitrary element in the set. In general we found that the hard work for us as instructors involved designing appropriate worksheets which would lead to productive discussion. (In truth the hardest task for the instructor was to refrain from intervening prematurely in the students' discussions.) Evaluation procedure A large weight (50%) of the final grade was based on the "portfolio" handed in by each student which consisted of all the work done on each of the assigned problems. Students initial work on a problem was commented on without assigning a grade and students were expected to revise their work and resubmit it (this process sometimes had several iterations). The portfolios were collected twice, in the middle and end of the course. Overall Evaluation of the Experience Some of the things which we were not happy about in the course included: (1) the textbook which we picked for the course, "How to read and do proofs" by D. Solow (John Wiley & Sons) which tends to treat analysis of proofs with such a degree of detail that one can not see the forest for the trees. It also does not have a particularly good coverage of the mathematics topics that we wanted to bring in so it was not really helpful even as a source of problems. It has the merit of being inexpensive so it is not bad as an accompanying text. (2) The standard layout of many classrooms has unmovable desks which is simply not conducive to breaking the class up into groups. As well, some students were often reticent to participate actively in these groups. (3) The evaluation of the portfolio was extremely time-consuming even though we only had 14 students in the class. In a normal class of 25-30 and one instructor it would be virtually impossible to manage. This doesn't mean that we feel that the idea of the portfolio should be abandoned but only that it should be restricted to a set of representative problems. It is obviously crucial that the class size remains (relatively) small since individual contact with and feedback from the instructor is an important part of the students' experience. However we were, in general, pleased with the course. We felt that it essentially met the objectives which we had set at the beginning and we were convinced that it provided our students with an important mathematical experience. Our feeling about the usefulness of the course were collaborated by students' responses to interviews which were conducted after the course was finished. Here is a quote from one student in response to the question "Was this course useful in other courses?" : "Yes, absolutely! There is a whole new world that has opened up to me. The asides that often dot lectures in mathematics are more understandable to me. It's like I've seen how s/he [the professor] is setting up arguments, how they [the teachers] are thinking. I suppose I feel more comfortable with the culture of mathematics." (Not all of our students were either that positive or that articulate.) Our recommendation to the department was to make such a course part of the core programme of all mathematics majors. We felt that it should be taken by students in their first year, preferably, in their first semester of study. This recommendation was in fact incorporated into the large scale curriculum changes which the department is currently proposing.   We would be pleased to hear from CMS members about similar experiences in other Canadian universities.