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Becoming a mathematician part4) Starting College
Becoming a mathematician, part 4) College training.
I suspect it does not matter greatly which college you go to, as they all have their strengths and weaknesses. Places like Harvard or Stanford or Berkeley offer famous lecturers on a high level, incredibly advanced courses, and brilliant highly competitive students. For many of us, this can be more intimidating than inspiring. And often the famous professors are simply unavailable for conversation outside of class. In the early 60’s at Harvard, I found the lectures were wonderful, if I got the best professors, and then they walked out and I never saw them again until next time. Office hours were minimal and if I tried to see some of them, they were frequently busy or uninterested. Even intelligent questions in class seemed as likely to be met with sarcasm as a helpful answer. I suspect things have changed now with people like Joe Harris and Curt McMullen there, who are great teachers as well as researchers, and who enjoy students. Of course there were outstanding teachers like Tate and Bott there in the old days too, but not everyone was like them. As a result, I had to go away and get back my enthusiasm for math at a more supportive place.
It is helpful to go somewhere where you will enjoy your time, enjoy the courses and the other students, and get help from professors who think students matter. Today this is more common everywhere, even at famous universities, than it was long ago, but ask around among the student body. And be prepared to work very hard. Some if not most of my own undergraduate frustrations could have been lessened, possibly solved, by better study habits.
As to what courses to take, this is tricky and complicated by the almost worthless AP preparation most kids get today in high school. In general an AP class is a class taught by someone with nowhere near the training or understanding of a college professor, although they may be a fine teacher. But to expect a calculus course taught by an average high schol math teacher to substitute for a honors introduction to calculus taught by Curt McMullen or Wilfried Schmid or Paul Sally, is ridiculous. Nonetheless, so many students have bought this ridiculous idea that Harvard and Stanford do not even offer an honors introduction to calculus anymore for future math majors. There simply are none out there who have not had AP calculus in high school. Thus the student entering from high school is faced with beginning in one of many choices of several variable calculus courses. The most advanced one, the one taught a la Loomis and Sternberg, realistically requires preparation in a very strong one variable course a la Apostol, but which Harvard does not itself offer. So the only students prepared to take it are those elite ones coming in from Andover or Exeter or the Bronx high school of science, but not the rest of us coming in with our inadequate AP courses from normal high schools.
Thus the jump from high school to college has been made harder by the existence of AP courses. So in my opinion, even with AP calculus preparation, it is often helpful for a prospective mathematician to try to begin college in an introductory, but very challenging, one variable calculus course, modeled on the books of Spivak or Apostol, if you can find them. These do exist a few places, such as University of Georgia, and University of Chicago, which still offer beginning Spivak style calculus honors courses. To quote the placement notice from Chicago: “The strong recommendation from the department is that students who have AP credit for one or two quarters of calculus enroll in honors calculus (math 16100) when they enter as first year students. This builds on the strong computational background provided in AP courses and best prepares entering students for further study in mathematics.”
(I am not positive, but I assume that 16100 is the spivak course. But do your own homeworkl to be sure.)
The point is that AP preparation provides no theoretical understanding, so plunging students into advanced and theoretical calculus courses of several variables, as they do at Harvard and Stanford, by beginning in Apostol vol 2, or Loomis and Sternberg, without background from Apostol volume 1 or Spivak, is academic suicide even for most very bright and motivated students.
If you go to a school where there is no Spivak or Apostol vol. 1 type course, where the calculus preparation is from Stewart, or some such book, you are perhaps getting another AP course, only in college. Then you have to choose more carefully. Many such college courses will indeed be no more challenging than a high school AP course, and should not be repeated. Just ask the professor. They know the difference, and will help you choose the right level course. Either get in an honors section, or an advanced course suitable to your background. And join the math club. Try to find out who the best professors are, and do not be scared off if weak students say a certain professor is tough. You may not think so if you are a strong student. Once you get there, try to sit in on courses before taking them, to see which professors suit you. Student evaluations are notoriously hard to interpret correctly. The professor with the worst reputation among students, Maurice Auslander, was in my grad school days at Brandeis my absolute favorite professor. He cared the most, offered the most, and taught us the most. He also worked us the hardest.
Once you get a semester or two under your belt, it will get easier to find the right class, as hopefully the colleges own courses prepare you for their continuations, although this is not guaranteed! There is no way to force one professor to included everything the enxt one expects, nor to exclude material he/she loves that is outside the curriculum. Do your own investigating. Ask the professor what is needed for his/her course and try to get it on your own if necessary. After leaving the honors program temporarily as an undergraduate, I got back in by studying on my own over the summer from an advanced calculus book (David Widder), to make up my theoretical deficiencies and survive the next course.
Everyone should study calculus, linear algebra, abstract algebra, ode, and some basic topology. If you have no background in proofs from high school, you will need to remedy that as soon as possible. It is best to do this before entering, even if they offer a “proofs and logic” course. Such courses are often offered to junior math majors, whereas they are needed to understand even beginning courses well. For this reason it is extremely helpful to read good math books on your own that contain proofs. Today especially it is important to know some physics even if if you only plan to do math. Much of the inutition and application of math comes from physics. Even if you only want to do number theory, sometimes viewed as the purest and most esoteric branch of math, many of the deepest ideas in number theory come from geometry and analysis and even statistics, so nothing should be skipped. Work hard, read good books, seek good teachers, and try to have fun. College is potentially the most exciting and fun time of your life, and the one where, believe it or not, you have the most freedom and free time.
Becoming a mathematician, part 4) College training.
I suspect it does not matter greatly which college you go to, as they all have their strengths and weaknesses. Places like Harvard or Stanford or Berkeley offer famous lecturers on a high level, incredibly advanced courses, and brilliant highly competitive students. For many of us, this can be more intimidating than inspiring. And often the famous professors are simply unavailable for conversation outside of class. In the early 60’s at Harvard, I found the lectures were wonderful, if I got the best professors, and then they walked out and I never saw them again until next time. Office hours were minimal and if I tried to see some of them, they were frequently busy or uninterested. Even intelligent questions in class seemed as likely to be met with sarcasm as a helpful answer. I suspect things have changed now with people like Joe Harris and Curt McMullen there, who are great teachers as well as researchers, and who enjoy students. Of course there were outstanding teachers like Tate and Bott there in the old days too, but not everyone was like them. As a result, I had to go away and get back my enthusiasm for math at a more supportive place.
It is helpful to go somewhere where you will enjoy your time, enjoy the courses and the other students, and get help from professors who think students matter. Today this is more common everywhere, even at famous universities, than it was long ago, but ask around among the student body. And be prepared to work very hard. Some if not most of my own undergraduate frustrations could have been lessened, possibly solved, by better study habits.
As to what courses to take, this is tricky and complicated by the almost worthless AP preparation most kids get today in high school. In general an AP class is a class taught by someone with nowhere near the training or understanding of a college professor, although they may be a fine teacher. But to expect a calculus course taught by an average high schol math teacher to substitute for a honors introduction to calculus taught by Curt McMullen or Wilfried Schmid or Paul Sally, is ridiculous. Nonetheless, so many students have bought this ridiculous idea that Harvard and Stanford do not even offer an honors introduction to calculus anymore for future math majors. There simply are none out there who have not had AP calculus in high school. Thus the student entering from high school is faced with beginning in one of many choices of several variable calculus courses. The most advanced one, the one taught a la Loomis and Sternberg, realistically requires preparation in a very strong one variable course a la Apostol, but which Harvard does not itself offer. So the only students prepared to take it are those elite ones coming in from Andover or Exeter or the Bronx high school of science, but not the rest of us coming in with our inadequate AP courses from normal high schools.
Thus the jump from high school to college has been made harder by the existence of AP courses. So in my opinion, even with AP calculus preparation, it is often helpful for a prospective mathematician to try to begin college in an introductory, but very challenging, one variable calculus course, modeled on the books of Spivak or Apostol, if you can find them. These do exist a few places, such as University of Georgia, and University of Chicago, which still offer beginning Spivak style calculus honors courses. To quote the placement notice from Chicago: “The strong recommendation from the department is that students who have AP credit for one or two quarters of calculus enroll in honors calculus (math 16100) when they enter as first year students. This builds on the strong computational background provided in AP courses and best prepares entering students for further study in mathematics.”
(I am not positive, but I assume that 16100 is the spivak course. But do your own homeworkl to be sure.)
The point is that AP preparation provides no theoretical understanding, so plunging students into advanced and theoretical calculus courses of several variables, as they do at Harvard and Stanford, by beginning in Apostol vol 2, or Loomis and Sternberg, without background from Apostol volume 1 or Spivak, is academic suicide even for most very bright and motivated students.
If you go to a school where there is no Spivak or Apostol vol. 1 type course, where the calculus preparation is from Stewart, or some such book, you are perhaps getting another AP course, only in college. Then you have to choose more carefully. Many such college courses will indeed be no more challenging than a high school AP course, and should not be repeated. Just ask the professor. They know the difference, and will help you choose the right level course. Either get in an honors section, or an advanced course suitable to your background. And join the math club. Try to find out who the best professors are, and do not be scared off if weak students say a certain professor is tough. You may not think so if you are a strong student. Once you get there, try to sit in on courses before taking them, to see which professors suit you. Student evaluations are notoriously hard to interpret correctly. The professor with the worst reputation among students, Maurice Auslander, was in my grad school days at Brandeis my absolute favorite professor. He cared the most, offered the most, and taught us the most. He also worked us the hardest.
Once you get a semester or two under your belt, it will get easier to find the right class, as hopefully the colleges own courses prepare you for their continuations, although this is not guaranteed! There is no way to force one professor to included everything the enxt one expects, nor to exclude material he/she loves that is outside the curriculum. Do your own investigating. Ask the professor what is needed for his/her course and try to get it on your own if necessary. After leaving the honors program temporarily as an undergraduate, I got back in by studying on my own over the summer from an advanced calculus book (David Widder), to make up my theoretical deficiencies and survive the next course.
Everyone should study calculus, linear algebra, abstract algebra, ode, and some basic topology. If you have no background in proofs from high school, you will need to remedy that as soon as possible. It is best to do this before entering, even if they offer a “proofs and logic” course. Such courses are often offered to junior math majors, whereas they are needed to understand even beginning courses well. For this reason it is extremely helpful to read good math books on your own that contain proofs. Today especially it is important to know some physics even if if you only plan to do math. Much of the inutition and application of math comes from physics. Even if you only want to do number theory, sometimes viewed as the purest and most esoteric branch of math, many of the deepest ideas in number theory come from geometry and analysis and even statistics, so nothing should be skipped. Work hard, read good books, seek good teachers, and try to have fun. College is potentially the most exciting and fun time of your life, and the one where, believe it or not, you have the most freedom and free time.
