Uncovering the Secrets of Calculus 2 and Physics: Strategies for Success

[Brad Barker]

I have always studied 1 year head than what i am learning right now. (i studied 10 hours a day)
And try to relate different subject together to think deeply, always think about the sense behind the text.
Life is often easier if you are able to look from higher aspect.

yeah, i do that, too.

but not for 10 hours a day. yeesh.

 

[mathwonk]

Here is some advice I once gave a class of the best students in one of the best private high schools in my state, after completing a Spivak style precalculus course with me. One of them has phD in physics now and another is full professor of math at an Ivy league school.

Comments:

I've enjoyed this class a great deal, and I've learned a lot about the attitude of high school kids (i.e. young adults) towards math. and towards school, and a bit about the people who have been in my class. One of the things that came as a bit of a surprise to me, oddly enough, was that math apparently does not mean nearly as much to the students as it does to me. In a way some of the students seem to have expected me to be a salesman of mathematics and to be prepared to sell them on it. I had rather expected to find people as prepared to "buy" as if I were selling water to people lost in the desert. I remember the shock I felt the day one student responded indignantly to my suggestion that one way to scavenge time to do math was to skip lunch. Other students apparently thought nothing of missing the state math meet for various other commitments. It became clear that students in high school have so many activities to participate in that they cannot easily find time for a really intensive involvement in perhaps any of them.

In spite of the original intent, it was not feasible to give the class in the way it would be given at a state University. University students, even ones with much less ability than those in this class, are motivated to work much harder than did this class and thus to cover much more ground. I understand that it is difficult for a middle aged, settled, professional with only one academic focus in his life to recall the point of view of a young person, with all the world of options open before him, and with greater personal concerns confronting his mind than mulling some mathematical problem. Still I would have expected somehow in all a school like this to find a few who were as much in love with mathematics as I seem to recall having been even as a young student.

I remember regularly staying after school to practice for the math team, spending study halls doing math, and going to university libraries to read math books that contained what seemed like rare and beautiful information not available in our textbooks. (I also found time to play on the basketball team and the track team, sing in the madrigals and chorus and play in the band and the orchestra). I think this is a wonderful school, but in a way there may have occurred here some loss of innocence as to learning and discovery and its intrinsic joy and value, from the scramble to prepare ones vita for college admission, and possibly even to live up to someone else's values. There are also wonderful things going on, provoked by teachers who know better than I how to reach and hold the imagination of teens.

Well anyway, I appreciate the kids for sticking with me, especially in my most aridly humorless moments. The class finally evolved to where I was not really trying to cover any particular quantity of material, but simply to regularly expose the class to some beautiful and deep ideas. Also I did not have the time to grade a lot of homework, which probably means they did not do a lot of it. Consequently I do not really think this class should substitute for a similar class at the university level, and I would recommend that the same material be studied again in an atmosphere in which the students are really expected to master it. Of course one can conceive of schools or courses where time spent restudying this material would be wasted, so I am recommending a high level, well taught, honors course at a good school.

Ideally one should follow this course by a Spivak calculus class modeled on the old math 11 at Harvard, which is no longer offered there because Harvard faculty believe there is no longer a population of bright potential math majors there who are also so unsophisticated as to need a first year calculus course. A second option would be to take a more advanced course like differential geometry or differential topology or even beginning analysis. The advantage, or disappointment depending on your view, of taking an especially sophisticated early course is that later courses usually don't really expect you to know that stuff. This means you will have the prerecquisites for more courses than you think you do, but that a lot of courses will seem tiresomely condescending.

A student from this course who finds himself at Harvard might do well to take math 21, if they still have that. The much more difficult math 55 still remains from the old math11-55 sequence but 55 would be quite a challenge in my opinion. Of course if a student feels the way the students at Harvard did in the old days it would probably be impossible to discourage him from taking the most difficult course available. Students in the 60's seemed to fight for space in those courses, even if not always wisely so. We thought it was a badge of honor to be admitted to the most ridiculously hard courses, and took the most well meaning contrary advice as a sad underevaluation of our abilities, sometimes rightly, sometimes with painful results.

My advice is to never underestimate yourself, but neither should you underestimate how hard you are going to have work to realize your true potential. Getting into the right course in college is important and deserves some study. Always interview the professor, look at the book and interview students who have taken the course. But if you are an ambitious hard working math lover, don't listen to the negative comments of some lazy friend or math hater about how hard so and so is. The advice probably won't apply to you. Also attend the first meeting of a course you are interested in taking to see if you like it, or audit it the year before if it is particularly difficult and important. I sat in on and studied an entire semester analysis course in addition to my regular load just because what I learned helped me in my own courses. Afterwards I didn't bother to take the audited course, but I learned more from it than many I did take.

A rule I always followed, which has exceptions, was to take courses from full professors and avoid courses from graduate students. Grad students may be wonderful teachers but they usually are not, and they don't know nearly as much as professors. People who brag on how good some grad student is as a teacher of some lower level course may really mean that he teaches an easy course. Of course there are schools where people get to be full professors just by being the oldest person around, and where the younger professors are the best. In general the better you think you are, or want to be, the more highly qualified teachers you should seek out.

 

[Eratosthenes]

My advice is to never underestimate yourself, but neither should you underestimate how hard you are going to have work to realize your true potential.

I read your whole post but that is just excellent, I could not agree more.

 

[mathwonk]

and by the way, unless his thesis is already approved, matt is obviously a shining counterexample to my old advice to avoid grad student teachers vis a vis professors.

But technically I believe he is both a professor and a grad student at the moment, or as one of my late coworkers at the meat market put it: "a card sharp and a professor too!"

 

[Manchot]

Mathwonk, just out of curiosity, why is your avatar a Pikachu? I've always wondered that.

 

[Poop-Loops]

for phys II, the prof made us read the chapter the next day's lecture was going to be about. VERY effective. you get the material first in the reading, then you get it reinforced by the lecture, plus you have an idea of what questions you'll need to be asking and what problems you'll have to solve.

This is something I don't agree with. I tried that with calc. The book would explain it one way, the teacher another, and there I was: in the middle, crying, because I didn't understand anything anymore. What's worked for me is having the teacher explain it, then try and piece together anything I didn't understand from the book. Sometimes it took me a while, but it worked.

PL

 

[bomba923]

This is something I don't agree with. I tried that with calc. The book would explain it one way, the teacher another, and there I was: in the middle, crying, because I didn't understand anything anymore. What's worked for me is having the teacher explain it, then try and piece together anything I didn't understand from the book. Sometimes it took me a while, but it worked.

PL

Hmm...I find books more informative and challenging than teachers (although I'm a HS student, so I have not the experience of college/university teaching methodology ...except for Organic Chemistry and the CalcIII I'll take this fall)
As a HS student, books (HS or college level) may explain concepts/material "minimalistically"-->give you just enough facts/information to advance further into the concept--(or the next few pages if you wish ); also, they explore the concept/material in greater depth...though that depends on the book(!). That's why I rely more on books, for example, than on notebooks/teachers; I study by the book (literally !), and any question I might have I either consult a secondary textbook...or the instructor. What I like about books is not only "how" they cover the concepts/material, but also the "depth" with which they do so.
When I took Organic Chemistry I (first semester), I studied from two textbooks--Volthardt and L.G Wade, rarely relying on notes. When I was learning carbon stereochemistry...I read L.G Wade and understood little. Then I read Volthardt...and it all made sense! Though it may take longer, I try to understand each concept as intuitively and fully as possible, reviewing each possible "case" if necessary (Organic chem practice was worth the while!)
*Basically, my advice is to understand each concept as fully as possible...that way, you will solve problems much faster with a procedure that makes sense-->e.g.,
"Practice can help you solve problems faster, understanding will allow you to tackle more difficult problems."
(practice=greater speed, understanding=greater depth , I suppose you could say?)

 

[Stephan Hoyer]

This is something I don't agree with. I tried that with calc. The book would explain it one way, the teacher another, and there I was: in the middle, crying, because I didn't understand anything anymore. What's worked for me is having the teacher explain it, then try and piece together anything I didn't understand from the book. Sometimes it took me a while, but it worked.

PL

I think this very much depends upon the teacher and the class. Last semester my professor for Vector Calculus was very good and considering the level of the course (I couldn't take the honors version) I found lecturers to be almost completely sufficient. I would review the book briefly and use it as a reference, but generally I found that with good notes I was able to pick up all or most of the concepts without it.

In contrast, the honors general chem class I took the semester before that was taught by a young professor just recently out of grad school who had some sort of fellowship. His lectures were only marginally valuable compared to studying from the book which explained concepts in more careful detail (but generally it wasn't a hard class).

In contrast, my physics seminar required learning from various texts and PDFs, as students were expected to make a presentation for the class or demonstrate a homework problem or two each week. The class served as a time to cement knowledge and review what we had indepedently studied. Of course, my physics professor was one of those types who's office hours were anytime he was in his office so he was easily accessible.

 

[devious_]

Here is some advice I once gave a class of the best students in one of the best private high schools in my state, after completing a Spivak style precalculus course with me. One of them has phD in physics now and another is full professor of math at an Ivy league school.

[...]

Great post! Thank you.

 

[Learning Curve]

Dear Noslen,
As most of the posts are informative, they're not going to help you study better. You are you. Not any of these guys. You have to find what works for you. Some like cramming. Others like to review before and after teachings. Everyone has their own way of studying and learning. You are yourself and no one else. That's why your study habits will be different than anyone else's.

Good luck.

 

[mathwonk]

To Manchot, it just seemed less scary than "the punisher".

 

[noslen]

More feedback great!

Class starts next monday and I am really excited

thanx agian!

 

[mathwonk]

with reference to the idea that learning by cramming is an option for some people, I want to repeat: cramming is never helpful except for passing an imminent test by someone with a very strong short term memory. It is very poor for actually learning, understanding, and remembering material.

Indeed the amount of time material is remembered, is roughly proportional to the amount of time spent learning it. There is no shortcut. Only young inexperienced students pressed for time to play and also pass tests ever use this method. They must also be enrolled in fairly easy courses compared to their ability to succeed in this way.

Even Fields medalists work very hard at mastering mathematics. I have known several of them and I can say this with confidence. (And they are not hapless twits like the principle character's foil in "Good Will Hunting".)