We had our final exam in Analytic Geometry. It was hard. I hardly answered the last problem. One of my classmates is a student assistant of our instructor. She had assisted in the checking of our papers this morning. It was hilarious because, according to my classmate, even our professor doesn't know how to answer the last problem. He had to inquire to the math department about it (he's an engineer teaching math in the College of Engineering.) Funny and somehow sad that he gave us a question that even he, the instructor himself, couldn't answer. Oh the quality of education!
That doesn't necessarily mean the teaching quality is bad. "Must the maker of firecrackers pop?" -BF Skinner In other words, whether he's a good teacher or not (and he may be a terrible teacher for all I know) it's independent of whether he can answer a particular question in the field he's teaching.
It doesn't mean his teaching ability is bad. However, assuming the test is actually testing objectives for the course, the instructor should be competent in the objectives he's teaching. Or, more likely, the test question isn't actually testing one of the course objectives, except perhaps indirectly. In this case, the teacher still lacked thoroughness in preparing for the course, since he failed to notice the test included someone's pet "challenge question" instead of questions that actually tested course objectives. The attitude of using test questions as a teaching opportunity isn't as unusual as it should be; nor is the attitude that the test is some sort of competition that should differentiate between the exceptional students and the merely good. It's hard for some teachers to maintain enough discipline to limit their tests to what they're designed to do - measure whether or not the students met course objectives. In this case, the test question in question was some other teacher's pet question and the teacher in question just inherited it without really looking at it ahead of time.
That is strange. Did he himself write the questions? On a sidenote, I've had excellent teachers that otherwise messed up. For instance, a Physics lecturer I had insisted that a vanishing curl is the only requisite for a conservative vector field, forgetting that the domain of the vector field could be multiply connected. Another professor I have is a Ph.D in Engineering, but knows Differential Equations like the back of his hand - after all, he does research in that field. His field is more applied mathematics than "pure", but still, he knows what he's talking about. What I mean to say is, the academic degree your instructor may have isn't particularly relevant, he may as well have become an expert in the field through research, but as you seem to put it, maybe this guy is in way over his head!
I think it's good he expects his students to be better than he is. That is the mark of a true teacher. How many of us here have eclipsed the abilities of our high school math and science teachers? I don't think any the less of them, and am thankful for their high quality teaching. I remember one good university Prof that would take his own exam during the same time the students took it. He never got a perfect score and always there was a student or two that did better than he did.
I've also had exams like that. We were asked to prove something, and eventually I found a counterexample. When I showed the professor, he was very embarassed. I actually regretted showing him the counterexample Can you post the question that he asked? I'm intrigued...
It immensely surprised (disgusted) me to learn, in grade 12, that a teacher needs only a teaching certificate, which generally implies a degree in Education. Knowledge of the subject that they are teaching is irrelevant. One of our choices in high-school was to take either chemistry or biology in grade 11. You would have to take the other in 12. Since I preferred chemistry, I took biology in 11 to get a higher level in chem. Along came grade 12, and... pssssthhhhthht. A quiz question half-way through the chemistry course asked for 3 uses of hydrogen. One of my responses was "rocket fuel". It was marked as wrong because it wasn't in the textbook. The "proper" answer was "peanut butter". My only salvation from that was that I was in the midst of a research project for a novel that I was writing, and so had all non-classified details of the NRDS facility and the work done there on the KIWI and NERVA series nuclear rocket engines, including a map of Jackass Flats and the connecting rail lines, in one of my notebooks. I essentially waved that in the teacher's face and forced him to increase my mark. He couldn't understand it, and so couldn't argue against it. I could just as easily have mentioned Apollo, but I had no paperwork to back that one up. Even more disgusting is that the other chemistry teacher, whose class I didn't get, was an actual chemical engineer from India. He led our school science team to a provincial (maybe national?) championship. His name was (is?) Paul Sethi. I'm not going to mention the name of the slipgarbage that I got stuck with. Alternatively, I teach pool better than I play it. I'm good, but not close to being a Master. Some of my previous students (young teammates at the time) have become Masters and play for big money. I'm prouder of that than I would be if I achieved that status myself.
This may be true, and not to put down any high school teachers, but there's a difference between being a high school teacher (a position that usually only requires a bachelor's degree) and a being a university professor (a position that most certainly demands a doctorate, plus years of publishing advanced articles in decent, peer-reviewed journals). Seriously, a person that has earned a Ph.D in a particular field, has done research in that field for X years, and has been teaching it for another Y years should have the ability to solve pretty much any undergraduate question related to the subject, or atleast have the ability to know what references to consult, and not go to another department to have them figure it out for you.
That has not happened to me, at least not too often, but what has happened is that a question I wrote for last semester's exam looks impossible to me a few months later. I feel guilty when that happens since it seemed so easy when I was immersed in the material. More pertinent to this situation, it can occur that something I taught in the first half of the course looks hard to me by the end of the course. But when that happens I sometimes try not to put it on the exam, assuming it will look even harder to the class. Still someone who reviews well should still get it, and i do not want to penalize a hard working student who does that expecting to be tested on it. I am talking here about hard stuff like Riemann surfaces, but maybe also some advanced calculus or complex analysis, or anything with a trick to it that one can easily forget. It almost never happens in highly conceptual material where once you understand it you never can miss it again, like abstract algebra. E.g. no instructor could possibly forget how to prove Lagrange's theorem on groups, or maybe even Sylow's theorem at least if he uses the conceptual approach of Helmut Wielandt exploiting group actions on subsets of order p^k. But anyone can slip up on a tricky antidifferentiation problem requiring some sneaky substitution. There are even published answer books in existence where the author slips up on a residue calculation.
Often that is true, but profs don't always teach undergrad subjects in their field of expertise. Also you are ignoring my example of a prof that was teaching in his area of expertise and even wrote the text on the subject (senior level class). He did not get 100% on his own tests and an occasional student would outperform him. Personally I beat his 92% on one exam. A good teacher is going to bring you to the brink of his knowledge and then throw you off that cliff and watch you fly away.
So true. If only all professors were like this one that you mention. This one in particular that you use as an example also seems humble - I've seen way too many stuck-up, arrogant douchebags that fly off the handle if you ask them something or even suggest that something written on the blackboard is wrong.
here is an example of a question I did not see how to do on an algebra exam I was grading but did not write, that to me tests trickery (or memorizing the book's proof) rather than concepts. Prove that a surjective endomorphism T of noetherian modules is also injective. Every student knows that in a noetherian module an increasing sequence of submodules stabilizes at some finite step. The trick is to see how to use this to prove that kerT = {0}. If you think of it, looking at powers of T is the answer. I.e. If T is surjective and kerT ≠ 0, the ker T^k gives a strictly increasing sequence of submodules, a contradiction. I do not believe this tests only whether a student knows what a noetherian module is. Now it might do a better job if one gives as a hint to look at {kerT^k}. Remark: the theorem stated here is actually true for non noetherian modules too, but the proof is much trickier and uses the cayley hamilton theorem.
I think its very reasonable for an instructor to give a homework problem they can't do. I don't think its really appropriate on an exam.
:uhh: Is there any possibility of that post being repeated in English? That's the coolest description that I've ever seen. A good teacher doesn't even necessarily teach the subject matter; s/he teaches you how to learn. That, incidentally, is one of the reasons that PF is such an awesome learning environment.
Such questions can be a fair way to identify and reward those very rare and exceptional students that may not stand out from the crowd with easier questions. The important thing is to not harshly penalize those that don't get it. In essence treat it like an extra credit question, whether it is identified as such or not.
I assume you mean every student of mathematics. :tongue: I am no algebraist (or whatever branch of mathematics this belongs to) and have never had a proper Abstract Algebra course. Heh. Sometimes it might just pay off for a bright student. Yeah, unfortunately, I've seen professors sadistic enough to push you off that cliff and watch you plummet to your academic death, so to speak. Fortunately, these are in the minority.
sorry about that. actually i even stated an obviously false result, since you seem to need finitely generated for that. the result i stated generalizes the fact that a linear map of finite dimensional vector spaces of the same dimension is surjective if and only if it is injective, i.e. the row rank of Matrix equals the column rank.
The requirements for a "teaching certificate" (in the United States) varies from state to state. But in most, if not all, a teaching certificate, to allow you to teach in a secondary school, requires that you have a degree in a particular subject with a minor in education.