I want to take an unscientific poll on the number of students in the technical fields now vs. the recent past. This is engineering and the sciences (please indicate which field of engineering or science). If you all would be so kind, could you indicate the relative number and quality of students majoring in your department in the following years: 2006 2003 1999 1989 Thank you.
You can find statistics on enrollments and degrees for physics at http://www.aip.org/statistics/ Their sample base is surely a bit wider than you'd get from responses here.
the big difference is from the 60's when the students, at least in math, were much stronger. the use of calculators, and emphasis on AP tests, and rampant grade inflation to keep them out of the army in vietnam, watered down quality of work. also in those days kennedy inspired good students to want to "go to the moon", and there were merit scholarships for bright students with full ride scholarships to harvard. even walt disney had werner von braun on his tv show, explaining rocket science to children. now we have different national leadership, to say the least. instead of training our own talent we import it. so grad stduents are still good, and profs, but they are mostly immigrants. some money is being put back into training US students, but it is in the form of bribes to admit lesser qualified grad students who are us citizens, instead of real money for improving the preparation of US students to actually compete. this is the mentality of bureaucrats, instead of upgrading quality, make it profitable to universities to accept lower quality.
hmm I would agree with the emphasis on calculators and technology degrading students abilities thing. but I wouldn't agree with the AP tests degrading quality, if anything i'd say they are tougher than most college calculus courses, and require a greater knowledge of the techniques and theory than most students get from a standard calc course. or at least the techniques, there isn't that much difference in theory.
That has not been my observation at all. For the AP test, memorizing a few formulas will have you set. Many teachers would rather just give the formulas without explaining them in order for the students to pass the AP exam. The calculus teacher at my high school taught that in order to get the area of a curve rotated around a horizontal axis, you did pi times the integral from the first point to the last, of the distance from the curve to the axis of revolution, squared.... He not once explained where that came from or why it works. Students just accepted it. Also, he never provided a derivation or proof of anything even as simple as the product rule. Students accept it because they then can do well on the AP exam. What good does that do for them? Did they really learn anything? The Calculus II course I took at the University of Arizona was far more difficult and useful in the sense that most things were derived and/or proven in front of us all. There were problems on the final exam which were extremely difficult and required some actual thinking.
there are many different levels of difficuklty of calc classes in college, but if yours was easier than an AP course, you got a very weak college course. mine are much harder than AP courses and most successful AP students drop out. that said, there are many more weak college courses now than there were, precisely because AP courses have down graded preoparation so much that we have to water down our college cousres to acomodate AP students. so before AP coiurses college calc was much harder than it is now. but a decent college calc course is still much harder than an AP course. again it depends here you go. the calc course i taught in a private high school was much harder than the subsequent non homniors calc course at univ of florida, according to a student who dropped my cousre and cruised in the one at fla. but successful students from my course went to harvard and duke and brown. the one at harvard said the course there was hard but he survived because i taught differential forms and exterior derivatives, and stokes theorem, in my high school course. did your AP course cover that? and winding numbers? and vector geometry? I taught my high school course from the sophomore calc book used then at berkeley, by marsden and tromba. take a look at it and compare to your AP or college courses and report back.
well all of those things are part of a calc 3 course, with some of those things being reserved for complex analysis and differential geometry. ^and if you said yourself that you used a sophmore college textbook from a top university, this is not indicative of what any freshman calculus courses teach normally. also if you taught those things in a calculus 1&2 course at a pace where people were able to understand, than you could quite possibly be one of the best math teachers to ever exist. but keep in mind that it would be rare for a college calculus course to be taught at that level anywhere. And to a limited extent harmful, its not about the speed at which people learn things but the depth of their understanding, by using a sophmore text you required them to learn twice as much as they would normally learn if they wanted to learn it properly. Also am I correct in assuming this book was proof based? and that it focused far more on creating rigourous mathematical proofs instead of giving a conceptual understanding first (it was a sophmore book so I would guess the author assumed prior understanding of derivatives, even though he may have gone over the material again in a new light), I've had teachers who went straight into rigour, its a poor teaching method for lower division stuff, if your good you can keep up with it and do everything that they show you, but you won't learn how to really understand the material. I personally am a big fan of the idea that you take an AP calculus or physics course and then you take it again in college except at a more advanced and indepth level. but I never took a calc 1 or 2 course, I learned the material on my own and took the ap test and did well enough to go on. and right now I see other students who have been in college calculus for a year struggling with problems that I have an easy time on. So i'd have to say that i'm not that impressed with the college systems ability to produce better students, partly because of the larger class sizes making it impossible for a professor to check any work or really do any problems with the classes with students. This changes in the upper level classes in my experience, so maybe its better for students to learn calculus in high school as it gives them more teacher interaction.
At UGA we teach calculus to freshmen in classes of 35 or less. I have 16 survivors right now after after 18 AP students droped out, of the non honors class. So thigns are different everywhere, but I obviously have more experience after 40 years of teaching at the college level than many people. I agree with you that the appropriate course for a good AP student (surely you realize that most AP students did not elarn anything, and provide the bulk of the college AP enrollment), is to retake the calculus at an honors level. Nonetheless many of us from the 1960's entered college with no caclulus and took rigorous real analysis level calculus courses, such as Spivaks book was written for, and Apostol. These books ere written in the 60' for beginning calc courses for students with no calc background. My first freshman hw set asked me to prove e is irrational using the taylor series, and to calculate it by hand out to 9 or more decimal places, i.e. featured both high level theory and detailed calculation. By the second semester we were doing abstract vector algebra and several variable calc and differential equations, including hilbert spaces. The sophomore course was on manifolds (Loomis - Sternberg), much higher level than the berkeley book I taught out of to hs students. I will not evaluate my own hs teaching, and some parents of unsuccessful students called it "simply bad teaching", but some of the students were certainly excellent. Of the 6, one of them is now a well known full profesor of math at an ivy league school, and another was phi beta kappa at harvard and took a phd in physics before changing fields. he said my course was essential to his survival in harvards sophomore math course from raoul bott, and he did not take the loomis sternberg version. all that is necesary for students to learn advanced material is to want to, and for the teacher himself to understand te stuff. this is what is missing formhigh school AP courses usually. Having someone teach AP who took calc in college is very diferent from having an international level researcher teach it. I have also taught eulers characteristic for polygons to third graders, using models, and one of those students, became an astrophysics major and aeronautical engineer. It is not harmful but good for young students to be taught something deep, if it is fun and intellectually stimulating. Other hs students i have taught did projects in galois theory before going to such schools as mit. as a teacher i had some serious failings, which i mention for your use as a teacher yourself. I did not praise my students enough, and I did too much explaining, instead of letting them do more for themselves. Those failings are still visible here. But my enthusiasm went a long way, and the beauty of the higher level material had its own fascination for them.
mathwonk: what should students do who would like to learn the material at a higher level than is taught in classes? i can learn a lot from reading books on my own, but sometimes i need to talk to someone who really understands what is going in order to get a good feel for what the concepts are. do you think it's possible to learn strictly from books?
try to get it from a book or eprson that has more to offer than you are getting in class. it may be that the book itself has more to offer than is being demanded by the elcturer. if there is an appendix with proofs that are being skippwed yuo could read that. it may be that the lecturer has more to offer than he is saying, because he believes it is not wanted. so go talk to him/her and find out. it may be that you need to consulkt a better book such the ones recommeded earlier on this site. if you have an advanced person available talk to them. if not, and even if you do, it is wise to read good boks, books written by matehmaticians, not just college teachers. it is possible to get quite a lot from good books. then after trying it helps to discuss what thinks one has learned with someone who can help clarify and deepen it. i always liked discussing the material with a good friend in the sme class.
I wish I went to the HS that you taught. I learnt that back when I was a freshman. I can't believe they have become one of my concerntrating topics because of its usefulness. I think the book by marsden and tramba is weak when you introduce the subject with exterior calculus. I really wish to write a vector calculus with n-dimension generalisation instead of 3, and of course with the use of linear algebra and differential forms. It is because that is how I was taught in vector calculus and I found how useful it has become in my further education.
Very interesting. But new generations of students acquire some new skills,driven by the technology development. Althought they can't take cubic roots from numbers by hand,I think they are smarter overall.
when I first broke into multivariable calculus on my own I found a set of online course notes that broke into an n dimensional formulation, I wholeheartedly agree thatits th best formulation, from the mathmatical perspective. because the three dimensional formulation is used in the sciences, its usually necessary to teach it along with the n dimensional formulation. Mathwonk you do sound like a ood teacher and I wish I had you, out of curiosity did you have more class hours for math in the 60's? it seems to me that part of the problem with modern technical education is the emphasis on communication and "artsy" education, maybe its just me bu there seems to be an impression out therre that technical educations are cold and practical. This makes it difficult and unwise for a student to take more than 1 years worth of math their freshman year. Or maybe its always been that way and its just me.
new skill like what? putting cheat sheet and formulas into a programmable calculator? I dont agree with this at all. In my HS, my pre-cal teacher used calculator in class extensively. Once all students moved to calculus, which calculator usage was limited, encounter many troubles because they dont know how to do the algebra. Many student don't even know how to draw a graph without a calculator. Moreover, in higher mathematics even computer is powerless. It can give you a better picture of what the problem is, but it doesn't solve a problem itself.
I see three driving factors that have resulted in a decrease in the quantity and quality of students today. The first factor is a reduced emphasis on the sciences in the US today compared to the 60s and 70s (my youth). For example, The National Science Foundation actively supported many summer programs for gifted high school students during the 60s and 70s. One summer I studied non-Euclidean geometry, the next, digital electronics and nuclear physics. They crammed calculus down our throats the first two weeks of the latter program. Fast forward to the 21st century: I didn't see anything of the sort for my kids, all of whom are in college now. The second factor is that there is far too much reliance on standardized testing today. Communities, businesses, and politicians place extraordinary pressure on the schools to ensure students perform well on standardized test. Teachers teach the tests and little else. Rote memorization is the key to high performance on a test with a large number multiple guess questions. Students are not taught to learn, to think. Standardized tests are evil. The third factor is money. Science and engineering aren't particularly lucrative careers anymore. Intellectual property lawyers rake in the money. Freshly minted IP lawyers are paid more than a tenured science professor 25 years out of college. Freshly minted MBAs do quite nicely also. A lot of smart young people are attracted to those big bucks.
Standardised test is okie in one sense, if the test was designed to force student to learn materials. The difficulty of the standardised test is way too low. I can't believe I could pass the english part of the TAKS without knowing any english..........