To the instructor
At many universities and colleges in the United States of America, a course which provides a transition from lower-level mathematics courses to those in the major has been adopted. Some may find it hard to believe that a course like Calculus II is considered “lower-level” so let’s drop the pejoratives and say what’s really going on. Courses for math majors, and especially those one takes in the junior and senior years, focus on proofs — students are expected to learn why a given statement is true, and be able to come up with their own convincing arguments concerning such “why”s. Mathematics courses that precede these typically focus on “how.” How does one find the minimum value a continuous function takes on an interval? How does one determine the arclength along some curve, etc. The essential raison d’etre of this text and others like it is to ease this transition from “how” courses to “why” courses. In other words, our purpose is to help students develop a certain facility with mathematical proof.
It should be noted that helping people to become good proof writers — the primary focus of this text — is, very nearly, an impossible task. Indeed, it can be argued that the best way to learn to write proofs is by writing a lot of proofs. Devising many different proofs, and doing so in various settings, definitely develops the facility we hope to engender in a so-called “transitions” course. Perhaps the pedagogical pendulum will swing back to the previous tradition of essentially throwing students to the wolves. That is, students might be expected to learn the art of proof writing while actually writing proofs in courses like algebra and analysis2. Judging from the feedback I receive from students who have completed our transitions course at Southern Connecticut State University, I think such a return to the methods of the past is unlikely. The benefits of these transitions courses are enormous, and even though the curriculum for undergraduate mathematics majors is an extremely full one, the place of a transition course is, I think, assured.
What precisely are the benefits of these transitions courses? One of my pet theories is that the process one goes through in learning to write and understand proofs represents a fundamental reorganization of the brain. The only evidence for this stance, albeit rather indirect, are the almost universal reports of “weird dreams” from students in these courses. Our minds evolved in a setting where inductive reasoning is not only acceptable, but advisable in coping with the world. Imagine some Cro Magnon child touching a burning branch and being burned by it. S/He quite reasonably draws the conclusion that s/he should not touch any burning branches, or indeed anything that is on fire. A mathematician has to train himself or herself to think strictly by the rules of deductive reasoning — the above experience would only provide the lesson that at that particular instant of time, that particular burning branch caused a sensation of pain. Ideally, no further conclusions would be drawn — obviously this is an untenable method of reasoning for an animal driven by the desire to survive to adulthood, but it is the only way to think in the artificial world of mathematics.
While a gentle introduction to the art of reading and writing proofs is the primary focus of this text, there are other subsidiary goals for a transitions course that we hope to address. Principal among these is the need for an introduction to the “culture” of mathematics. There is a shared mythos and language common to all mathematicians — although there are certainly some distinct dialects! Another goal that is of extraordinary importance is impressing the budding young mathematics student with the importance of play. My thesis adviser3 used to be famous for saying “Well, I don’t know! Why don’t you monkey around with it a little…” In the course of monkeying around — doing small examples by hand, trying bigger examples with the aid of a computer, changing some element of the problem to see how it affected the answer, and various other activities that can best be described as “play,” eventually patterns emerged, conjectures made themselves apparent, and possible proof techniques suggested themselves. In this text there are a great many open-ended problems, some with associated hints as to how to proceed (which the wise student will avoid until hair-thinning becomes evident), whose point is to introduce students to this process of mathematical discovery.
To recap, the goals of this text are: an introduction to reading and writing mathematical proofs, an introduction to mathematical culture, and an introduction to the process of discovery in mathematics. Two pedagogical principles have been of foremost importance in determining how this material is organized and presented. One is the so-called “rule of three” which is probably familiar to most educators. Propounded by (among others) Gleason, Hughes-Hallett, et al. in their reform calculus it states that, when possible, information should be delivered via three distinct mechanisms — symbolically, graphically and numerically. The other is also a “rule of three” of sorts, it is captured by the old speechwriter’s maxim — “Tell ’em what you’re gonna tell ’em. Tell ’em. Then tell ’em what you told ’em.” Important and/or difficult topics are revisited at least three times in this book. In marked contrast to the norm in mathematics, the first treatment of a topic is not rigorous, precise definitions are often withheld. The intent is to provide a bit of intuition regarding the subject material. Another reason for providing a crude introduction to a topic before giving rigorous detail revolves around the way human memory works. Unlike computer memory, which (excluding the effects of the occasional cosmic ray) is essentially perfect, animal memory is usually imperfect and mechanisms have evolved to ensure that data that are important to the individual are not lost. Repetition and rote learning are often derided these days, but the importance of multiple exposures to a concept in “anchoring” it in the mind should not be underestimated.
A theme that has recurred over and over in my own thinking about the transitions course is that the “transition” is that from inductive to deductive mental processes. Yet, often, we the instructors of these courses are ourselves so thoroughly ingrained with the deductive approach that the mode of instruction presupposes the very transition we hope to facilitate! In this book, I have to a certain extent taken the approach of teaching deductive methods using inductive ones. The first time a concept is encountered should only be viewed as providing evidence that lends credence to some mathematical truth. Most concepts that are introduced in this intuitive fashion are eventually exposited in a rigorous manner — there are exceptions though, ideas whose scope is beyond that of the present work which are nonetheless presented here with very little concern for precision. It should not be forgotten that a good transition ought to blend seamlessly into whatever follows. The courses that follow this material should be proof-intensive courses in geometry, number theory, analysis and/or algebra. The introduction of some material from these courses without the usual rigor is intentional.
Please resist the temptation to fill in the missing “proper” definitions and terminology when some concept is introduced and is missing those, uhmm, missing things. Give your students the chance to ruminate, to “chew”4 on these new concepts for a while on their own! Later we’ll make sure they get the same standard definitions that we all know and cherish. As a practical matter, if you spend more than three weeks in Chapter 1, you are probably filling in too much of that missing detail — so stop it. It really won’t hurt them to think in an imprecise way (at first) about something so long as we get them to be rigorous by the end of the day.
Finally, it will probably be necessary to point out to your students that they should actually read the text. I don’t mean to be as snide as that probably sounds… Their experiences with math texts up to this point have probably impressed them with the futility of reading — just see what kind of problems are assigned and skim ’til you find an example that shows you “how to do one like that.” Clearly, such an approach is far less fruitful in advanced study than it is in courses which emphasize learning calculational techniques. I find that giving expressed reading assignments and quizzing them on the material that they are supposed to have read helps. There are “exercises” given within most sections (as opposed to the “Exercises” that appear at the end of the sections) these make good fodder for quizzes and/or probing questions from the professor. The book is written in an expansive, friendly style with whimsical touches here and there. Some students have reported that they actually enjoyed reading it!5
At the University of Maryland, Baltimore County, where I did my undergraduate work, these courses were actually known as the “proofs” courses.↩
Dr. Vera Pless, to whom I am indebted in more ways than I can express.↩
Why is it that most of the metaphorical ways to refer to “thinking” actually seem to refer to “eating”?↩
Although it should be added that they were making that report to someone from whom they wanted a good grade.↩