Why YOU Should Take Math 251 !

A Course to help students learn the language of mathematics...

If you have already signed up for math 251, welcome to what I know will be an outstanding experience in your education!! This course is going to be lots of fun for both myself and my students – I have no doubt. This course is truly different!! I will try to elucidate what I mean by this curious comment below. Some of you have already made it through several semesters of calculus and others of you are taking this course concurrently with calculus. Regardless of which of these categories describes you, this course has much to offer you. Despite the course number, this course is not “more advanced” than calculus – in fact in a very fundamental way, its more basic than calculus! If you have not already taken calculus, don’t worry – there is no calculus in this course!

For now, let me make mention of an often overlooked bit of information regarding calculus students visa vi math 251: Students often have more difficulty than necessary in calculus because some parts of calculus belong to a conceptual part of math which is not taught or emphasized in courses prior to calculus. This includes understanding proofs of theorems and abstract thinking-yucko!! Because one is confronted head on with some of this type of thinking in the context of learning calculus, Calculus feels to many students like trying to function within a “foreign language” that one has not specifically learned to speak very well yet. If you already have taken some calculus and feel that much of what the professor did on the board in lecture went right over your head, you know exactly what I am talking about (That professor could have even been me…yikes!!!.).

Really a major focus of math 251 is exactly this transition to higher math. Math 251 itself is not high level math, but rather a slow and hopefully pleasant introduction to “learning to speak the language of math” in the context of presenting the preliminaries to learning results in discrete math. So whether or not you are required to ever take math 251 as part of your official program of study, math 251 is a course that is designed to greatly help you learn how to learn many of the aspects of math that typically confound students of calculus and higher level math courses of all sorts. My own feeling is that taking math 251 concurrently with calculus will greatly enhance your experience in learning math and make it all the more pleasant and enjoyable.

What will I study in Math 251 and why is this course so different?

Ok – how can this math course really be all that different than previous math courses? One way is the idea that although math 251 covers specific topics, you will be asked to engage in some different kinds of thinking than in previous math courses. While courses like algebra, trigonometry, pre-calculus and statistics are highly focused on learning some very specific methodology, you were pretty much told which results/theorems were true and then told to practice certain techniques that were to be used for certain kinds of problem types in rather routine ways – not necessarily always easy, but nevertheless somewhat routine. You were probably not asked too often questions of the sort “make a conjecture” or “give a plausibility argument” or “give a counterexample” or “generalize a certain result”.

These types of questions and others are all part of proving theorems and gaining a better conceptual understanding of math. Scary as this may sound, when broken down bit by bit and taken at a slow pace, this can be a fun and intriguing aspect of math –it’s really the heart of mathematical thinking! Mathematicians and other types of creative problem solvers know that this way of thinking about math can be lots of fun compared to memorizing formulas, and is really the way new math is discovered before it is refined and printed in the textbooks. You see when new math is being discovered, it’s often done by curious and very enthusiastic people with half-baked ideas sitting around drinking coffee* and shouting out ideas –some good and some utter nonsense – somewhat akin to a party game where everyone is excited and having a good time. Often ideas are quickly jotted down on napkins and further explored later….but by the time it gets all sorted out and carefully developed in all its careful detail and printed up in textbooks, it changes from the fun and excitement it once was to something very formal and precise and foreboding looking. However, once it’s printed in books, it becomes very hard for students to imagine it as the exciting endeavor it really was by the people who first did it. One of my goals in math 251 is to provide both the freedom and coaching to help you explore the excitement of discovering mathematics for yourself. Of course you will be “discovering” things other people already know –but that should not take away the fun and the thrill for you! Who knows – someday you could be discovering something for the first time that nobody else knows…and if you discover just the right things that people really care about in the mathematical field (such as proving the so called Riemann Zeta Conjecture, for example), you might just be the one to claim a million dollar prize!!

What is expected and how will I be Graded in math 251?

I expect a good playful attitude about the idea of being asked to think about things that are not always easy. You can not get enough out of this course unless you are physically present and mentally participate as best you can – so I expect regular attendance and a willingness to participate. This does not always mean knowing everything (as a student, I almost never knew anything until spending a good long while thinking hard about things first)– but it becomes obvious to me who is making an effort and who is just coasting and waiting for other people to do the thinking…I expect everyone to make a strong effort and not come up with reasons why you have not had the time to make a strong effort for weeks or why you had to miss a large number of classes. It is not possible to function reasonably in the mathematics learning process by cramming for tests – that misses all those things that really are cannot be tested very well, but are a very important part of the learning process.

In return for making a concerted and continued effort to learn the language of math as the semester progresses, I will be patient with you and provide a variety of good and fair opportunities for you to earn a good grade. Some of these opportunities may be take-home quizzes/tests or mini-projects, some of these opportunities may be occasional short in-class quizzes that are very similar to things that we have been recently stressing in class.

In-class quizzing and testing will be on the more routine aspects and take–home work will usually involve aspects that are more demanding both time wise and in other ways.

The intention is to make this course a good experience in learning more about the problem solving process and language of math –

To give you better tools to take with you to higher level math courses -not to intentionally make it a “weed-out” course (as some math courses tend to be-such as calculus perhaps) – but this does not imply that you can get a good grade by slacking off and having performance under par. Like any college course, there are expectations that must be met. I would surmise that most serious minded students who have made it this far in math (at least through pre-calculus) are more than capable of earning a good grade –so the rest is up to you!! As in other situations both in school and in the workplace, I stress neatness, following directions and taking pride in your work, attention to detail. Organization and presentation also are important aspects.

In additional to continually stressing learning “the language of math”, I will also attempt to impart to you some sense of the “culture of math”. Getting somewhat of a sense of the way in which the creators of math think (the “psychology of mathematicians”) can also be somewhat important. I may, on occasion, tell you about some intriguing anecdote or little piece of mathematics history or even a math joke in order to try to give you a little bit of a cultural sense of math. For example it may surprise you to know that mathematicians care a lot about aesthetic beauty and not only formal technicality, or that what is accepted by some mathematicians as a proof may not be accepted by all mathematicians – so in some weird sense, mathematicians have a lot in common with lawyers (who need to learn how to carefully use language and logical thinking to argue a case) and artists as well as having some things in common with engineers and scientists. Like most subjects, math does truly have its own unique culture. On occasion, I may decide to show you a video for the same reasons. Above all, I want to not only teach you something about mathematical thinking, but also I hope you will learn to enjoy and appreciate math.

* The Hungarian definition of a mathematician is a person who turns coffee into mathematical theorems – at least according to the late great number theorist Paul Erdös.

What Kinds of Mathematics Topics are Covered in Math 251?

While it may not be that meaningful to just list topics before you have studied them, I don’t blame students for wanting a rough list…so here it is:

Aspects of mathematical logic such as conditionals and connectives, tautologies, quantifiers, truth tables, negation of statements, direct proof, proof by contradiction, counterexamples, sets and set operations such as union and intersection, product set, induction and the well ordering principle, equivalence relations and partitions of sets, functions, compositions and inverses, infinite sets and cardinality, elements of number theory, combinations and permutations, elements of probability theory, pigeonhole principle.

Although there is all kinds of interesting overlap between different branches of math, generally speaking, mathematicians make a distinction between discrete math and continuous math. An analogy that is somewhat helpful in understanding the distinction is to think about an “old fashioned” clock (with minute and second hand) vs. a clock that has only a digital display. In theory, one can think of time as “continuous” and therefore infinitely divisible (so we could talk not just about 1 second or 2 seconds, but about 1.5 seconds or 1.9 seconds or 1.99 seconds – even if its not possible to actually read it on a given clock). On the other hand, the digital display jumps discontinuously from one second to the next with no suggestion that there is some time between one and two seconds…well, don’t take it too literally –its just an analogy, but it sort of gives the idea…or another way of thinking about it might be to think about the number line – it has no “gaps” – that is numbers on it are packed together “infinitely tightly” – no matter how close you take two numbers together, there are still infinitely many other real numbers between them. On the other hand, just the set of natural numbers does not share this property, so we think about the natural numbers as forming a discrete set whereas the entire real number line can be thought about as an example of a “continuous” type of set. These types of explanations are just abstract examples, but you will get a much better feel for the distinction between discrete and continuous math as you further study and explore various math courses.

Again, I can not stress enough, that its not the particular topics that are nearly as important as the problem solving skill you will develop by spending a semester thinking about math from the point of view that is stressed in the course – this will be much more helpful to you in future harder math courses than any memorized methods or formulas I could ever give you! What I would have given as a student to have had the opportunity to take a “user friendly introduction to higher math” course of this type – how much easier other math courses would have been!