
#19
Mar1711, 12:39 PM

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My previous lengthy post was just trashed by this frustratingly inefficient browser.
I was giving an example of standard lowering one can read about in today's Atlanta Journal Constitution, where the superintendent defends a recent decision to offer graduation credit for remedial courses taken by students who fail the basic courses previously required for graduation. 



#20
Mar1711, 12:49 PM

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Testing and grading are very difficult. More difficult than deciding what they should know, is deciding how to tell whether they do know what you want.
E.g. if you want people to understand "why" things are true, you may begin by asking them to give proofs. But this results in memorized proofs that are mainly computational or symbolic, and still not understanding the reason for the steps. If you ask students to give a proof of a simple special case of a given result, intending for them to specialize the argument to that case, most students instead will regurgitate the full general argument, then say with a staright face that your question is a corollary of the lengthy general result. True enough but they have igniored the fact that a simpler proof using the same ideas is better here. How to test whether someone has understood an idea is the hard part. It can only be done by testing the ability to do something they have not seen done, that uses the same idea. One has to invent ones own questions for this to work, since taking them out of a different book may just favor those students who have seen that book. One good way is to ask for examples of concepts they have learned. Once I wrote down the expression for a Riemann sum of an integral of a continuous function f from a to x, wrote limit in front of it, and asked for the derivative of this function. If they knew what they were doing they would say f(x). Only one person got this, even all were supposed to know the FTC. More shocking is to ask whether something explicit like e^(x^2) has an antiderivative. Almost all students will say in question 1, that yes every continuous function is the derivative of its indefinite Riemann integral, but then in question 2, no this continuous function does not have an antiderivative. Of course I know that confusion rests on the common inability to understand what a "function" is, and to understand that the abstract looking limit of a Riemann sum of a continuous integrand from a to x, is just as much a function as a word like "sine" or a symbol like x^2. How can anyone who does not know what an abstract function is understand the FTC? And how do we deal with that. We must try hard not to fool ourselves by devising tests that students can pass without understanding anything. There is some interesting current research on this, in the peer instruction movement. If you have not seen it I recommend the video by Eric Mazur? at MIT 



#21
Mar1711, 01:02 PM

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This all goes exactly counter to the attempt to devise tests that the students will pass. I have often sat frightened the night before a test, faced with a class that cannot do anything, wondering how to ask questions someone might get right. In truth my job is devise questions that will not be got right except by those who understand.
Still experience shows that most students will only get questions for which they have been specifically prepared, ones which look like the ones in class, and require no additional thinking. Whether to ask such questions is a matter for ones own conscience,a s the outsider cannot tell whether one has prepared ones class for the questions or not. Still even with the best intentions, we all know that students get right those tests which were written by his own teacher more often than those written by others. For this reason we used to make sure the teacher of a grad course never wrote the prelim in that subject. years later we did the opposite, the teacher who prepared the students also wrote the prelim. we got a lot more passes that way. 



#22
Mar1711, 01:09 PM

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I once had a class so weak that even when I prepared them by asking on take home homework for proofs of certain basic theorems, then hand corrected them, gave them back, and then tested on those same theorems in class, the results were still miserable. This class could not even regurgitate correctly proofs they had practiced with my criticisms. My evaluations were very low that semester.
Another phenomenon that has accompanied lower standards of teaching os the class evaluation. In the early 1960's, Harvard had no evaluations. Indeed the Dean told us proudly when we started there that, probably for the first time in our lives, our teachers were not being graded on our performance. I bragged about this to my colleagues at my state school when we obsessed over the fact that pleasing the students was rated higher than teaching them. Then I called Harvard to enquire the current practice and learned they now use evaluations. 



#23
Mar1711, 01:19 PM

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Still, I love Andy's comment: :"I respect my students." Out of respect for those students I just criticized as weak, no doubt it was not their fault they could not prove anything. Their teachers from high school on up had probably never tried to teach them reasoning, continually watering down their courses from geometry on up, and this course was their first encounter with a difficult concept.
It is hard for an old guy like me to adjust to, or accept, the enormous deterioration that has occurred since 1956, and my retirement makes way for more energetic and more flexible people to jump in and try their hand. More power to you Andy. Of course respecting students includes believing they can do more than they may think at first, and finding ways to inspire them to try. I know you include this in your mantra as well. 



#24
Mar1811, 09:46 AM

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#25
Mar1911, 09:26 PM

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I don't care if people get through college easily, I do care when instructors do a garbagety jobs which screwover dedicated students.
As an example, I recently finished a complex variables course where the instructor blatently admited he didn't know anything about the subject, of course he is a tenured proffessor. The class was not difficult, but it didn't develop sufficent understanding of the subject. Most annoyingly the other students didn't care, some even supported the instructor. 



#26
Mar2311, 12:01 AM

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its not the ones who admit how little they know that you have to look out for. from them at least sometimes you can learn how to learn.
sometimes the ones who are learning it along with you do a better job because they see the difficulties a beginner has and point them out. old hands have often forgotten where those problems are. usually it is ignorant or insecure people who brag about how much they know. it is also a teaching tactic to admit one knows very little, designed to give the students courage to ask questions, but sometimes as here, it backfires. 



#27
Mar2311, 12:41 AM

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I honestly don't really think college is sold to students as a means of improving themselves. There really is this kinda vagueness about college. How do you become successful? College. How does college do that? uhhh, it just does. I have begun telling my students that they really need to take advantage of the astounding resources the university provides you in order to make yourself better! I have a feeling students don't really understand this because we've all been trained to believe that all ability is innate and that people CAN'T become better at anything. Except cooking.
I'm also getting the feeling that professors and lecturers typically don't care. One professor at my university, based on what a few students told me, has a course where he gives no homework and no tests. Of course, they say that no one learns anything (it's not a physics course obviously). I feel most professors just feel their job is to read a textbook to people and the students are on their own to learn. This is unfortunately in contrast to the student's perception that all they're suppose to do is go to lecture and the professor is suppose to teach them and if at the end of the semester they don't know the material, it's the professors fault. How in the world did this idea about college come to be? 



#28
Mar2311, 07:29 AM

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http://usgovinfo.about.com/od/moneym...ndearnings.htm That said, it's true that a college education is rarely discussed in terms of nonmaterial benefits: learning how to think critically, use of logical arguments, etc. 



#29
Mar2311, 01:59 PM

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For example, the same questions (we all have our favorites) are often asked again and again here on PF. This, in spite of there being a FAQ, access to previous threads, and the whole Google thing. The way we each deal with this is often illuminating. I have to (attempt to, anyway) treat each student learning the material the same just because I have heard the same question over and over again doesn't mean I should tell the student to 'go read the book' even though it's *really* tempting to do so. One way to deal with this is to not teach the same class over and over and over again that's fine when there's lots of faculty qualified to teach a particular class and everyone can rotate around. More typically, faculty are more qualified to teach some classes than others, and nobody really *wants* to teach classes perceived to be 'bad/boring' (e.g. Physics I for premeds) because it's a drag. There's a lot of noise in AAPT about strategies to improve those giant intro classes nobody wants to teach I'm not convinced there is a single right or wrong way, but knowing that I'm not the only one interested in trying new approaches is good, and having some freedom to try new things is essential to combat boredom both mine and the students. 



#30
Mar2311, 03:03 PM

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I always tried to rethink the material every time I had to teach the same class, to keep it as fresh as possible. Or teach it in a different way. there are lots of approaches to a subject, although it does seem sometimes the most popular teachers are those who do it over and over and always the same way.
It is very hard to do but I always tried to think of at least one new insight I had never understood before and share it with the class. Or teach from a different book. One of my favorites was teaching from Euclid in my last year of teaching geometry for teachers. It turned out to be the best year I ever did it and the best book I ever used. Ironic that it took me 50 years of study of mathematics to find out Euclid has the best geometry book, when the whole world knew this for centuries, before it was forgotten in the modern era (1960 and later). Another favorite was trying to figure out what the fundamental theorem of calculus should really say for riemann integrals of functions that may not be continuous. It turns out to be difficult to state well. In one direction already Riemann knew that a function has a Riemann integral on [a,b] iff it is bounded and continuous off a set of measure zero. But a function IS an indefinite riemann integral iff it is Lipschitz continuous and its derivative equals a riemann integrable function almost everywhere. That last condition is not so natural, but at least if you start with a riemann integrable function f, then you can recognize the indefinite integral G because G is Lipschitz continuous and its derivative equals f a.e. I.e. there is a MVT type theorem that a Lipschitz continuous function whose derivative is zero a.e., is constant. I thought this was really cool, and helped me appreciate the radon nikodym theorem for the first time. 



#31
Mar2311, 08:20 PM

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From my experience, people who have never gone to college feel that college educated people are smarter and all around more capable and skilled than noncollege graduates. However, in my experience as a college graduate, quite the opposite is true! I know the truth  college is just a place to sit through a bunch of classes and try to memorize things. They don't really learn how to be.... well, better! For example, my group communications course was taught like we were all 4 year olds (which most of the students were maturitywise). We learned very little about group communication. We had to do 2 group presentations which in general were awful and nothing from the course was actually forced upon us. They may have mentioned "this and this and this helps get a crowd reaction" or "never do this in a speech as it offends the audience" or things such as that, but it was never applied. We went up, gave a talk of the quality you'd see in a high school and that was that. A+ for all. Incidentally, one of my favorite courses was my undergraduate condensed matter lab course which had an attached lab where we actually fabricated things in the department's machine shop! We were told how to do something, why it was done that way, told what tolerances had to be met, and we went and applied that knowledge. Use the wrong drilltype? Do it over. Are your holes threaded incorrectly? Do it again. Are you outside of the tolerances required? Do it over. We did it until we became good. All while at the same time knowing half the machines you worked on could rip your hand off if used incorrectly. That was a great class for me! Not so much because I'm some machinist deep down (or am I?), but because I was taught how to do something, told to do it, and not allowed to stop doing it until I got it right. At that point I felt "hey, I can do something a vast majority of people can't!" and I knew if I for whatever reason felt like pursuing fabrication even further, I would be able to. In my eyes, all I see at my university is "here is some stuff, memorize it and if you cant, oh well, you still pass" or "try to do this, if you can't, oh well". No effort into getting students to really achieve anything. Also, I hate when people feel being good at something is simply genetic or innate. "I'm just not good at math" or "I'm not a good writer" or "I can't speak in public". BLAH. I failed my first 2 calculus courses and after 4 years of just toughing it out, I ended up being better at math than almost every physics major and some of the math majors. I was also terrified of speaking publicly or in front of a class. I'm not great at it now (I still have issues losing track of what I'm saying) but I'm a hell of a lot better after teaching labs for 2 years. 



#32
Mar2311, 09:22 PM

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I struggle with making the material relevant to nonmajors; I want to use everyday examples that illustrate the underlying concepts, but even the most basic device references topics well beyond an introductory class. Even the simple camera in cell phones goes beyond what I can talk about I can barely discuss the hologram on credit/debit cards. I've thought about introducing more demos during lecture I know those can be *very* effective, but I'm concerned that I would have to reduce course content. 



#33
Mar2311, 09:33 PM

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Maybe there should be a distinction between "education" (selfimprovement) and "training" (learning to perform specific tasks). Both are needed in order to earn a degree, but sometimes there's too much emphasis on one or the other too much "education" results in navelgazing 4year olds, while too much "training" results in lab zombies. 



#34
Mar2311, 10:20 PM

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#35
Mar2511, 12:39 PM

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#36
Mar2511, 01:22 PM

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there is always a tension between doing a good job and reducing course content. most people agree it is impossible to teach the standard curriculum in any course and do it thoroughly enough for most students to learn it, but it takes a lot of moxie, or foolhardiness to omit any sanctioned topics. thats why its more fun to teach number theory than calculus, you get to pick the topics and its ok to choose fewer of them. of course if the professional number theorists hear me say this they rise up and demand that i include more quadratic reciprocity or whatever their favorite topic is.



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