How to Ace Calculus: The Streetwise Guide
By Colin Adams, Abigail Thompson, Joel Hass
Henry Holt and Company Copyright © 1998 W. H. Freeman and Company
All rights reserved.
If you are reading this introduction then this book is probably not for you. This book is directed at calculus students who have better things to do with their time than read wordy preambles that won't be on the exam. But just in case you haven't actually bought this book yet and are considering a purchase while flipping through the pages in a bookstore, we'll tell you what it's all about.
If you want to know the tricks of the trade that will make learning the material of first-semester calculus a piece of cake, then this is the book for you. If you want to learn lots of cool things while having a good time, then this is the book for you. If you want to carry around a book that makes people think you are surfing the wave of knowledge, then this is the book for you.
Do you remember being in a class and being hopelessly confused? Perhaps your attention wandered at some important moment, or the lecturer thoughtlessly slipped into ancient Greek when explaining the basic idea. After class, you asked your brainy friend over a cup of coffee, "What was going on in that class?" Your friend explained it all in five minutes flat and made it crystal clear. "Oh," you said, "is that all there is to it? Why didn't they just say that in the first place?" Later, you wished that friend was around to explain all the lectures to you.
This book aims to play the role of your friend. It gives informal explanations of the key topics of calculus, getting across the ideas without the technical details and fine print that would be found in a formal text. This book does not substitute for a calculus textbook, but it should make it much easier to figure out what the textbook is talking about.
If you approach it with the right point of view, learning calculus can be not only a mind-expanding experience but also fantastic fun, just about as good as something not involving whipped cream and maraschino cherries can get. This book is going to tell you how calculus is taught, how to get the best teachers, what to study, and what is likely to be on exams. This is the stuff we wish we'd known when we had to take calculus. So, enough stalling. Why don't you go up to that nice cashier, plunk down some money and buy this book, and we can talk more after?
Exactly Who and What Is Your Instructor?
2.1 Choosing an instructor
Here we give a brief introduction to mathematicians, their pecking order, and their identifying characteristics.
READ THIS BEFORE CHOOSING AN INSTRUCTOR.
Understanding mathematicians is a lot like bird watching. You need to know enough distinguishing features to say, "Ah, a yellow-bellied sapsucker" with conviction.
Choosing the best instructor is the single decision most likely to determine whether your calculus experience will be a series of intellectual delights or whether you schedule dental appointments during calculus lectures because they are less painful.
You can often figure out the specifics about your instructors by looking at their doors. Generally a small sign will be posted containing some clue to the instructor's official title. There are several possibilities:
A. Permanent faculty, tenured (sign on door says Professor or Associate Professor). Tenured means that they cannot be fired, even if they are grossly incompetent. Associate Professors are a rung below Professors. Sometimes this is because they are at an earlier stage in their career, sometimes because their career stalled after they were discovered hiding in the chimney of the dean's apartment.
B. Permanent faculty, untenured (sign on door says Assistant Professor). These people can be fired, but if they are, it will not be for reasons related to their ability to teach calculus. In Europe, Assistant Professors really are assistants, whose job is to mow the Professor's lawn, carry the Professor's briefcase, and teach the Professor's class. In the United States, the Assistant title just means that they are in the preliminary, untenured stage of their career.
C. Visitors (sign on door says Visiting Professor, or Visiting Assistant Professor). "Visiting" means that their welcome is due to expire at the end of one or two years. It does not necessarily mean that they have anywhere to go afterward.
D. Temporary faculty (sign on door says: Lecturer or Instructor or Adjunct Professor). Some colleges hire temporary faculty mainly to teach classes. This may mean that they really care about their teaching.
E. Graduate students (sign on door just gives their name, with no title, or has some pseudo-title like Adjunct Instructor).
F. No sign on door: A very bad omen. It may mean that the instructor is too disorganized to post a sign or that revengeful former students keep ripping it down. Perhaps the instructor is trying to avoid previous generations of students? Investigate further.
G. No door: Danger, danger. Could mean that the instructor is deemed unworthy of an office. This makes it hard to hold office hours. Also could mean that you're looking in the wrong building.
Almost everyone in categories A, B, C, and D has a Ph.D.
Permanent faculty members, with their ranks, should be listed in the course catalog, where you can look them up. At large universities, permanent faculty are usually either the people who do research in really high-level mathematics or the people who have been around forever and control the math department political machine. Sometimes the best teachers are found in this group. The worst teachers are often found here. These faculty will be teaching all the advanced mathematics courses, as well as calculus. Some find teaching calculus a chore, an obligation that must be tolerated in order for them to be able to do their research. Others truly enjoy it.
At many smaller institutions, the faculty aren't expected to do much research. Instead, the emphasis is on teaching, and you will find many faculty members who put a tremendous amount of time and thought into their teaching. This sometimes translates into better teachers. It is also true that professors at such schools often have to teach two to three times as many classes as their counterparts at the research universities. So, although they are not distracted by their attempts to prove that the convex core of a hyperbolic 3- manifold is compact, they are often distracted by the demands of hundreds of students they are teaching in a given semester in four different classes.
Distinguishing between categories C (visitors) and D (temporary faculty) is tricky but can be important. Some visiting faculty members are on sabbatical or on leave from their real home somewhere else. They may be visiting a colleague at your university, or perhaps your campus is located near a good surfing spot. They will return home at the end of the year, blissfully unconcerned with the eviscerating essay you turned in on your student evaluation. Pleasing the students ranks about 1.5 on a scale of 100 for these faculty. On the other hand, you can sometimes stumble across a real gem here, someone who comes from a campus where teaching is job one.
Other visiting faculty may be new faculty who have just finished their Ph.D.s and are currently teaching their first class. They may have some title like "Visiting Assistant Professor" or something with the word "postdoc" in it. Their main concerns are their research and getting another job the following year. They can't totally blow off teaching if they want to get another job, and though they usually try to do a good job, their teaching abilities vary wildly.
Lecturers and temporary instructors are usually hired on a year-to-year basis, primarily to teach. Whether their job is renewed at the end of their contract probably depends heavily on their teaching skills. Their future meals may depend on their student evaluations. Pleasing the students is extremely important to them. If your instructor has been at the university for more than a year and does not have a title containing the word "professor," the instructor is probably temporary. These instructors usually teach only calculus and precalculus classes, and if they have been around for awhile, it's very likely they do a superb job. If not, they may be related to the department chair.
Graduate students can be great instructors, but they usually don't have much experience. Their teaching ability is all over the map, from very high to very low. One problem with graduate students is that it's hard to get anything past them. They remember taking calculus all too well. Some graduate students do have years and years of experience — their minds were addled in the 1960s, and they've been in graduate school ever since. You can usually spot these people by their snarly gray hair, slurred speech, and the ratty ponchos they wear when it rains.
In most cases grad students have little in the way of a teaching track record for you to research. Your best strategy with them is to attend the first class or two, and if it looks disastrous, hightail it out of there.
Often, professional status correlates with amount of office space — if all else fails, you can deduce the status of an instructor by checking out the office. Estimate its square footage and divide by the number of names on the door. Add the floor number and multiply by -1 if there are no windows. The bigger the resulting number, the more important your instructor is in the departmental hierarchy.
There are, of course, many exceptions to these wild generalizations. We've known visiting professors who are dedicated and brilliant teachers, and tenured faculty members who devote their lives to their students. Fortunately, there's a more accurate and quite easy way to find out who the brilliant lecturers are.
Trade SecretTo find out who the best lecturers are: Ask!
Everyone around the department knows the good teachers. Try dropping in on a couple of faculty members and ask them. Try the younger ones; they don't know enough to mislead you. The secretaries will know who generates waiting lists and who generates a lot of dropped classes and incompletes. The graduate students will know. If there is a math club, its members will be dying to talk to you about this. (Get an excuse ready so you can leave after 10 minutes. They will be prepared to discuss it all afternoon.)
Some colleges publish student reviews of instructor performance. You (or your parents or the taxpayers) are paying a fortune for four years of college. Shell out a few bucks if necessary and take the time to help make the difference between enlightenment and boredom.
Trade SecretTo find the best instructor: Go to several classes and stay with the best!
In large universities you may have a choice of five or ten different classes in the same subject. In smaller places, hope that teaching is taken seriously. By asking around you should be able to find two or three likely candidates who do a great job. Try them out.
Given a choice, why not pick the best instructor? You will usually be able to tell the first day who is going to be good and who will put you to sleep. Set your standards high. Don't make the mistake that many students do of assuming that it is your own fault if the class is incomprehensible.
Trade SecretIf a class you are prepared for is completely incomprehensible, it is probably the fault of the instructor.
If more than 10 percent of the class is asleep on the first day, that is a BAD sign. If a clear handout is distributed explaining the class, and if the instructor holds lots of office hours, that is a GOOD sign. If you are unsure whether you are in a French 101 class or a math class, that is a BAD sign. If this book is required in the course, that is a GOOD sign. A VERY GOOD sign.
Colleges do not like students checking out classes like this because it makes more paperwork for them and makes it harder to predict class size. Some college administrators would like you to pay your tuition and stay out of sight. Don't worry about them — get your money's worth.
If you happen to be taking calculus in high school, then a lot of this advice is probably irrelevant. You have no choice. You will be assigned a particular teacher, usually the only one teaching calculus that year, and it doesn't matter how much you kick and scream, that's who you get. You'll just have to cross your fingers and hope for the best.
On the other hand, look at the bright side. You won't be learning calculus in an auditorium that seats 1000, with a professor who looks to be about an inch high from your seat in the upper balcony.
2.2 What to expect from your instructor
Now that you have an instructor, and you know what he or she is, let's move on to the more refined topic of who he or she is. Here are a few examples of what you might expect.
Famous Mathematician Story John Von Neumann was a Hungarian mathematician who came to the United States in the 1930s and in his spare time invented the concept of computer programming. He was also a little unusual.
One time a student went up to him after a calculus lecture. "Professor Von Neumann," the student said, "I don't understand how you got the answer to that last problem on the board." Von Neumann looked at the problem for a minute and said, "ex." The puzzled student thought he had been unclear. "I know that's the answer, Professor Von Neumann. I just don't see how to get there." Von Neumann looked at the student for a minute, stared into space, and repeated, "ex". The student started to get frustrated. "But how did you get that answer?" Von Neumann turned to the student and said, "Look kid, what do you want? I just did it for you two different ways."
Moral Sometimes professors have a hard time remembering what life was like before they knew calculus inside out. Having taught the same material over and over again, year after year, they just don't understand why the students haven't mastered it yet.
Famous Mathematician Story Norbert Wiener was perhaps the greatest U.S. mathematician in the first half of the twentieth century, revered among his colleagues for his brilliance. He was also famous for his absentmindedness.
After a few years at MIT, Wiener moved to a larger house. His wife, knowing his nature, figured that he would forget his new address and be unable to find his way home after work. So she wrote the address of the new home on a piece of paper that she made him put in his shirt pocket. At lunchtime that day, the professor had an inspiring idea. He pulled the paper out of his pocket and used it to scribble down some calculations. Finding a flaw, he threw the paper away in disgust. At the end of the day he realized he had thrown away his address. He now had no idea where he lived.
Putting his mind to work, he came up with a plan. He would go to his old house and await rescue. His wife would surely realize that he was lost and go to his old house to pick him up. Unfortunately, when he arrived at his old house there was no sign of his wife, only a small girl standing in front of the house. "Excuse me, little girl," he said, "but do you happen to know where the people who used to live here moved to?" "It's okay, Daddy," said the little girl. "Mommy sent me to get you."
P.S.: Norbert Wiener's daughter was recently tracked down by a mathematics newsletter. She denies he forgot who she was but admits he didn't know his way to the house.
Moral 1 Don't be surprised if the professor doesn't know your name by the end of the semester.
Moral 2 Be glad your parents aren't mathematicians. If your parents are mathematicians, introduce yourself and get them to help you through the course. (Continues...)
Excerpted from How to Ace Calculus: The Streetwise Guide by Colin Adams, Abigail Thompson, Joel Hass. Copyright © 1998 W. H. Freeman and Company. Excerpted by permission of Henry Holt and Company.
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