[graycat]

My wife was at the top of the class in nearly every course she took in school, K-Ph.D. I was at or near the top of the class in material I liked, essentially just math, physics, and chemistry. From those two examples, here are some lessons on how to make good grades and/or do well.

I will start with what my wife did: First, care about making good grades; care a LOT. Second, actually do at least most of the reading and homework. Take good notes in class and study them. Work with other good students in the class to share notes and thoughts on the content of the course. It helps to have a terrific memory, and for that it helps to really care a LOT. It helps to cut out nearly everything except course work and to have the ability to do well with relatively little sleep.

Then, with the basic learning done, work with other good students in the class to get a good view of what the teacher likes to see on tests, term papers, etc. I.e., figure out how to please the teacher, i.e., 'read' the teacher. So, before tests, work with those other good students to write out likely test questions and together work out good answers. Then on a test with, say, four questions, may have written out good answers to two or three of the questions the night before. Tough to compete against that.

For the other good students she worked with, she had a room in the girls dorm on the 'academic unit' with other astoundingly bright girls, and they shared draft test questions, etc.

Lucky was not in a class with any of those girls because would have come in at best second. For making As in college, those girls were fantastic.

Of course they were beyond belief in the humanities courses, e.g., could actually make some sense out of the mush. But for a while my wife was in pre-med so also made As in the 'filter' courses in organic chemistry and comparative anatomy.

It also helps to have some just fantastic academic talent: My wife wanted to take a course in European history but did not want to have to work to make a good grade so just audited the course. The prof asked audits also to take the tests, so my wife did. At the end of the course, a lecture course with 300 students, the prof told my wife that she should have taken the course for credit because she made the highest score in the class. And she didn't even seriously try. Tough to compete against that!

For a term paper, she organized all the information on index cards, arranged them, then typed the paper.

She made high school Valedictorian, in college was 'Summa Cum Laude', Woodrow Wilson, PBK, and won two years of NSF graduate fellowship in one award.

Then what I did: In K-8, the girls were much better students than I was in penmanship, spelling, writing, memorizing poetry, artistic drawing, working in groups, clerical accuracy, etc., all as should be expected. Also all the teachers were women and clearly liked the girls much better than the boys. So, I gave up on trying to get good grades from the teachers and just pursued what interested me.

I was so often treated with such contempt by the teachers that for all the rest of my time in school I was unconsciously terrified of criticism from the teachers. So, if something didn't go just right in a course, then I was terrified that the situation was hopeless so gave up. So, really, I could be comfortable in a course only if the material was pure math with a level of precision about like that of Bourbaki so that I could be iron clad sure that my knowledge could not be questioned. It was also good if I had carefully read an excellent text before the course and, thus, really already knew the material. K-12 teachers: Quit hurting good students.

Things made a big change in the ninth grade in math: I liked the math and really cared. I had no study skills but began to develop some. The teacher sent me to the state math tournament; I was likely at the top of the class. He realized that mostly I was learning just from the book and told me that for the tournament I should learn the last two chapters on trigonometry he was not going to be able to cover in class, so the weekend before the tournament I did.

That pattern went on: I really liked math and learned mostly from the book. I really made no attempt to get good grades from the teacher, but I understood the material so well I got A or B from the teacher but led the class or nearly so on state standardized tests.

That pattern continued in college:

It worked out that I never took freshman calculus! The college I went to for my freshman year didn't want me to start with calculus but put me in some course beneath what I'd already covered in my high school (that had a relatively good math sequence). So, I showed up only for the tests and otherwise got a good calculus book and dug in. For my sophomore year I went to a much better school, with a quite good math department, and just started with their sophomore calculus. Did fine.

In freshman physics, I really liked the material and led the class, effortlessly.

It was nice: Often I studied the physics in the gorgeous reading room of the library. There some of the really pretty girls were wearing some short, slightly full, plaid, heavy wool 'wrap around' skirts, each held together with a huge, chrome diaper safety pin! Still I got some physics done! Got to really like physics to learn under such circumstances!

Some of the more advanced physics was badly taught or from a poor book, and then my grade fell to a B; I had no patience with poor quality material. But in math the books were much better, and I did very well on both learning the material and grades. I got Honors in math and 800 on the GRE test of math knowledge and got sent to an NSF summer program in axiomatic set theory, modern analysis, and differential geometry. The differential geometry was lectures by a Harvard graduate and student of A. Gleason. He said that I needed only the inverse and implicit function theorems, but so far I'd not seen either of those (I later got them from Fleming's book). I was too intimidated to realize that I could have hit the library for an afternoon and evening and walked out with good knowledge of both theorems. The theorems are just local non-linear versions of the standard and fairly obvious general solution of a system of linear equations. There is a nice proof using contractive mapping. Alas, due to those two theorems, I walked out of the class -- bummer, it was material I would have liked to have learned, especially for relativity theory.

In my career I continued a lot of independent learning from some of the best math texts, e.g., I took a second pass through Rudin's 'Principles', went carefully through Halmos's 'Finite Dimensional Vector Spaces', Fleming's 'Functions of Several Variables', the math parts of von Neumann's 'Quantum Mechanics', and much more, a big stack more.

When I went for my Ph.D., what I had learned before I entered was nearly enough for the course work. For the research, I brought my own problem with me to graduate school, had an intuitive solution I'd worked out on an airplane flight, got enough math in my first year to turn my intuitive solution into some solid applied math, later wrote some corresponding illustrative software, and that was my Ph.D. research. For a Master's, there was a question in a course without an answer. I thought for two weeks in the evenings and saw a first solution and asked for a 'reading course' to attack the problem. When the course was approved, I gave my first solution right away. Two weeks later I had a much nicer solution, wrote it up, and that was the end of the 'reading course' and the last I needed for a Master's. Later I published the paper. When I published I did some more library work and discovered that I'd invented a theorem comparable with the classic Whitney extension theorem, that is, H. Whitney long at Harvard. I also discovered that I'd solved a problem stated in the famous Arrow, Hurwicz, Uzawa paper in mathematical economics. So, that work I did in that 'reading course' was publishable.

Eventually I concluded that working the more challenging exercises in the best pure math texts -- Rudin, Halmos, Royden, Neveu, etc., is good training for doing original research. Eventually I discovered that if not afraid of being whacked in the neck with a bad grade by a prof in a course, then usually can get what need for a given research problem from stacks of books and papers much more quickly than the rate of coverage in a course.

Eventually discovered a 'way' to do research: Do a lot of intuitive guessing; then test the intuitive guesses with some intuitive filters or checking on special cases.

For a simple outline, "Is A true? Okay, likely if A is true, then B is true. Is it believable that B is true or would that be asking too much? Naw, likely B is false. So likely A is false. So, check A on some simple special cases. Okay, still A seems false. So, try C. Is C true? If C is true, then likely D is true. Okay, maybe D is true. Check out C on some simple special cases. C might be true! So, maybe try to prove C is true. Now to prove C is true, likely the proof has to make essential use of all the hypotheses we have for C, so in looking for a proof, be sure can make good use of all the hypotheses. Else are trying to prove something stronger than C which is likely not true. Now, if C is true, just what, intuitively, is going on?"

Can do a lot of this while writing little or nothing. When the intuitive work seems good, then try to write out some actual math with careful derivations, any new definitions and then theorems and proofs. If the intuitive guessing goes well, then have a good shot of seeing how to do the proofs right away.

Mostly don't write longer than trivial algebraic derivations without having a fairly good idea what is going on intuitively. That is, mostly don't expect to get much just from pushing symbols around.

In nearly all of K-12 and college, what my wife did worked much better than what I did. For graduate work, a Ph.D., publishable original research, and applications in a career, what I did worked much better.

[Tycho]

In high school, I kind of looked down upon the culture of 'caring a lot' about getting top grades. Partly because I saw much of it as suckers jumping through arbitrary hoops, and partly because I reckoned that my memory was easily capable of holding all the necessary information, if I just put in a meagre number of hours studying (which I hated, so usually didn't bother). I focussed on trying to impress the teachers by answering the difficult questions in class which (seemingly) stumped the 'straight A' students, and also I loved English class because that was the one where disciplined study yielded little advantage. If someone got top grades but didn't outwardly show any flare in any of these areas, I considered them 'not that intelligent.'

Not saying this was justified, but it's definitely how I felt. What I probably should have realized is that in the long run, self-discipline is an extremely important trait no matter what your natural aptitude is. But I also think schools should make more effort to make subjects appealing on an intuitive level, not just a stack of exercises to slog through - I would have done much better in say maths if that was the case. I have to admit though, even though my attitude is different these days, what your wife did in college to get top grades still seems... kind of lame.

[graycat]

Once I understood what she did to get top grades, it did surprise me, was maybe "kind of lame".

But there was a 'reason' she did what she did: There were some strong influences from her family that anything less than an A would be 'shameful'.

So, yes, might notice that in K-college she didn't "outwardly show any flare". Right. She wanted the As. They meant a LOT to her. She knew that she didn't really get any extra points for "flare".

But eventually I learned that away from a course with a prof, credits, and grades, really, out of the view of anyone powerful, influential, and potentially critical, she had plenty of 'flare', really was just brilliant. E.g., computing wasn't her field at all. But at one point she wanted to do some computing, in of all things artificial intelligence. Well, I was on a team of three that had designed and developed an AI language. So I gave her a one hour lecture covering everything from how to use the computer, file system, text editor, and scripting language to our AI language.

Two weeks later she had a good, first program running. I looked at it, explained some of the 'theme' of how that AI approach was intended to work, and let her try again. In two more weeks she had one of the nicest AI programs our group ever saw. Just brilliant. We had some really bright computer science grad students in our group; in computer science, she was brighter. Astounding.

There is much more to human performance in school, research, and a job than meets the eye: In front of powerful, influential, potentially critical people, she was just terrified to appear less than Little Miss Perfect in the sense she got from her family. In our home, in front of just herself and me, she was free to show 'flair' and be brilliant.

If you sense that the "lame" part might not be so good in some respects, you are correct. I spent a lot of time around high end academics and saw a lot of people with spectacular grade point averages, some of whom maybe never failed to dot an 'i' since before kindergarten. Commonly these people were from very bright up to brilliant, but it seemed that their grade point averages were from various reasons and not just brilliance. Some of those various reasons were, in the end, not so good.

But if want to make the As my wife made, then what she did worked, and not much else does.

There was a time in high school plane geometry where apparently I did something like you did: I was totally in love with the subject, ate the exercises in the book like popcorn, by the hand full. The teacher was the most offensive person I've ever met in a classroom, so I refused to appear to do her assigned homework and mostly slept in class. Each day she assigned three not very difficult exercises. What I did was all the non-trivial exercises then turn to the back of the book for the more difficult supplementary exercises and do all of those. I never once failed to do a non-trivial exercise during the whole course. To save time, I didn't write out all the proofs but usually did small versions just in my head, in the margin of the book, on scraps of paper, etc. For the few exercises that actually took some effort, say, two hours, I'd write out a proof carefully.

For one of the supplementary exercises, I started on Friday afternoon and just kept going and finally got it late Sunday evening. Nice exercise! On Monday in class, there was one of her assigned exercises, easy, with the same figure. So, after she discussed the solution to that exercise, for the first and last time I 'participated' in the class and mentioned the exercise in the back of the book with the same figure. She was thrilled and had the class turn to that exercise. Time passed .... After about 20 minutes she was getting frustrated and nasty, was exhorting the class to "think", etc. Since I didn't want to be accused of ruining the class, I raised my hand and said, "Why don't we ..." at which time she angrily interrupted me nearly shouting "You knew how to do it all the time". Yes, I was 'guilty' as charged! Of course I knew how to do it. I knew how to do every non-trivial exercise in the book. If I didn't know how to do it, no way would I ask her.

She never let me finish the solution!

I didn't know that the exercise would be difficult for the teacher. I wasn't even sure I was one of the best students in the class. I guess she wasn't working all the exercises!

Uh, apparently on the state test in plane geometry I did well! Another guy and I were 1-2 in the class. We were also 1-2 in the school on the Math SAT. The teacher who read me my SAT scores had known me since the sixth grade, looked at my Verbal SAT, 538 (654 the second time I took it) and said "Good". It wasn't good, and I knew that. Then she looked at my Math SAT, stopped, looked afraid, and said, "There must be some mistake". Yes, sweetheart, there was, and had been for 12, long, painful years, you ditzy bimbo. I was a good candidate for the best math student in the school, and the school never knew it (actually apparently the principal did know; apparently he looked at some of my standardized test results -- but the teachers did too much gossiping among themselves).

So, yes, for that plane geometry teacher, there was a strong sense of 'competition' even with the students. So, she didn't want me to show any 'flair'. But I demonstrated that my knowledge of the subject was from my efforts in learning and not from her efforts in teaching. Of course this was in part my reaction to all those K-8 teachers who treated me with contempt.

Contempt? Of course, my Ph.D. and research were in mathematics: Well my eighth grade arithmetic teacher strongly advised me never to take anymore math in school! Why? I didn't do well on her tests. Why? I understood the material quickly but for the exercises, say, multiplying two four digit numbers, had poor accuracy. Why? Not from lack of understanding. But at the time, common for boys, my 'clerical accuracy' was not good; my handwriting was awful which meant that commonly I misread my intermediate results; and no one explained to me that I needed to be sure to work carefully to write clearly, keep the vertical columns lined up, and get correct results. Finally a college physics prof told me bluntly that I had to work carefully enough to get correct numerical results since a mistake could ruin a physics career. From then on my accuracy was from okay up to good enough.

Even now I do not trust my clerical accuracy. So for anything important, I do it, let the results 'age' hopefully for a few days, do it again independently, and compare the results. Also I no longer want to do anything like 'manual' arithmetic or use a calculator where I have to copy between the calculator and, say, paper. Instead I essentially just program all arithmetic. Otherwise I'm too prone to simple mistakes.

But the contempt from those K-8 teachers did damage. One graduate course started off with some axiomatic set theory. The prof gave a pop quiz. One of his questions was tricky, and I saw how to do it only at the end of the time. So I wrote quickly and used a symbol without defining it but used it with the meaning I'd seen in an earlier NSF course in set theory. So, later the prof called me for a 'conference'. He was convinced that I was a poor student. I explained the symbol and said that I thought that its meaning was standard in axiomatic set theory, and he then saw that my solution was one step shorter than his and good. Then he relented. But that contempt from him was too close to what I'd gotten in K-8; I concluded that there would be no way to please him; gave up; and never went back to that class. He wasn't teaching the course very well anyway, and later I got a just brilliant presentation of that material, best course of any kind I ever took in school.

These examples can show students some of the challenges in school: Not all the challenges are obvious on the surface. Your remark about 'discipline' is correct: Even if a teacher dumps on a student unfairly sometimes, the student needs to continue on and not just walk away. The course is not all just about competition for 'flair' but also has some material should get through, flair or not.

But, for a Ph.D., one way for a grad student to 'polish their halo' is to do some publishable, original research, independently or nearly so, early on, i.e., show some 'flair' for research. Why? The main difficulty for a Ph.D. is just the research, and showing that can do research, especially independently, can do wonders at getting the faculty on the side of the student.

We've now solved all the problems in higher education!

[captn3m0]

Thanks for writing this.

[graycat]

Welcome.