<p>I don’t think having exams as just another HW assignment is the norm, though. I’ve heard of take home exams here and there, but definitely not the norm (and they’re always harder than a regular HW assignment, anyway). I didn’t mean to say they count enough to replace homework basically… but if you want an A and get like a 70-75% homework average or something cause you slack off, you need more or less 100% on the exams to get an A (typically HW is 25-30% here).</p>
<p>from memory, i’d say normal problems at my school are stuff like:
show an isometry is a homeomorphism (topology), show p[n]q[n-1] - q[n]p[n-1] = (-1)^(n-1), where p[n]/q[n] are the convergents of some continued fraction (cryptography)
a couple others i remember being a little tricky offhand are showing that the topologist’s sine curve is not path connected, showing that 1/n (with n being an integer) either terminates or repeats with period of < n-1, proving that if p is a prime and divides a*b, then p|a or p|b (this was actually a lemma for some bigger proof in an intro to higher math class, but we didn’t really go over number theory in the class).</p>
<p>typically the couple times we do only get 4-5 problems per week, 1-2 of them are more difficult than normal, but usually we have closer to 10-15.</p>
<p>I am not familiar with continued fractions, but the isometry => homeomorphism problem is indeed easy enough to do 15 of them a week. But that’s the kind of mechanical verification problem that most of my math professors have been trying to avoid. They prefer to assign problems where developing a solution will force us to gain new insights on the material. The topologist’s sine curve example is more interesting and that was one of my topology assignments as well :)</p>
<p>I guess we can agree that different departments have different approaches to teaching and that the time-consumption factor of math classes varies as well.</p>
<p>By the way, shame on your Intro to Higher Math professor for selling irreducible numbers as prime numbers. A lot of my abstract algebra classmates were confused about irreducible vs prime for that very same reason.</p>
<p>I was an architecture major my first year which puts me a year behind and now I’m double majoring in Arabic and Government with a minor in Russian and Eastern European Studies and I’m finishing in three years with max 16 credits each semester - 13 credits in both semesters of my senior year (max is 17 at my school) and one and a half summers of classes (4 this summer - one online, two next summer).</p>
<p>eh, I don’t think such problems should necessarily be avoided… doing a problem like that helps you think about continuity more than just reading about it being a distance-preserving homeomorphism even if it’s not hard. there are usually problems of varying difficulty, some easy, some moderate, some pretty hard.</p>
<p>and on that other problem, actually the professor is pretty good, number theory/algebra is his field, but the class wasn’t about number theory, I think the HW problem was actually just something with modular arithmetic, and we were supposed to assume the lemma that I proved. but I’ve lost some points here and there on homework assignments for not showing some obvious things, so i’ve gotten into the habit of being completely anal about proving every little detail. regardless, the lemma I proved was beyond the scope of the particular class anyway. getting into the algebra would have been absurd for the class, the problem was out of a basic section in the book about congruence as an equivalence relation…</p>
<p>Shouldn’t students be comfortable with the concept of continuity by the time they get to topology? Regardless, I am glad that my math professors don’t think they need to hold my hand every step along the way. I am grown up now, I can do easy exercises on my own when I feel that I haven’t fully grasped a concept yet.</p>
<p>And there is no need for abstract algebra to state the correct definition of a prime number. A number p is prime by definition whenever p|ab implies p|a or p|b. It might be irrelevant for the purposes of his class, but why should he further reinforce a wrong definition? It will only confuse students more down the road once they are told that they have been working with the wrong concept of primality all their lives (and even in other college math classes!!!)</p>
<p>actually, now that I think about it, most classes here probably assign around 8-10 problems per week, intro to higher math assigned considerably more because it was a class to get people proving things. but 4-5 still seems low.
and no, you don’t need to take analysis before taking topology. i’d hardly call it hand holding. you might as well say assigning any exercises at all is hand holding, if you can’t figure everything out from going to class and reading the book to solve any problems on the exam, regardless of how hard they are, then it’s hand holding.</p>
<p>reinforce, by the way, is hardly the correct word. more like mention. to be a good teacher, you don’t start confusing people with stuff that’s beyond the course in an introductory class. when I TAed calc2, I just told people that 0! = 1. I didn’t explain why. that sort of thing would have done more harm than good. (or a better example: when you first learn about continuity when doing limits in calc1, do you think about it as drawing the graph without lifting up your pencil or whatever, or do you memorize the actual definition of continuity?) do you really think getting into algebra is a good pedagogical idea when people are just learning about congruences and getting used to the notion of an equivalence relation? and presenting a more accurate definition or treatment of something in some later course is hardly a confusion. you learn the new definition and move on. in middle school (or is it high school), everyone learns about functions as the rule. I was never confused when I got to college when learning about domains (in the true sense, not the middle/high school sense), target spaces, images, and so on.</p>
<p>When I learn new material, I need someone (whether it be a professor or a textbook) to show me the interesting examples. I can find the trivial ones on my own. By the way, I would argue that it is an important skill for a math major to be able to construct examples for a new definition. </p>
<p>
I learned the epsilon-delta definition long before anyone mentioned the pencil-lifting criterion. </p>
<p>
When I learned about functions, they were rules that assigned a value of a target space to each value of a domain. My 8th grade math teacher made sure everyone knew that a function needs a domain and a codomain. How else do you discuss properties such as injectivity and surjectivity, inverse functions, etc? (And yes, we used that exact terminology in middle school.)</p>
<p>
Getting into algebra, no. Working with the correct definition of prime, yes. I learned the actual definition of a prime number before I understood equivalence relations or saw congruences treated rigorously, and I dealt with it just fine. Maybe I am spoiled in that respect but I hate it when teachers hide half of a concept in order to make it more accessible. That reminds me of a geology professor who had us compute the impact velocity of a meteorite with constant acceleration due to the Earth’s gravity. He didn’t even bother to point out that the answer we got was 20x higher than actual impact velocities.</p>
<p>if you actually talked about epsilon-delta definitions, injectivity, surjectivity, codomains, and domains (that is, domains beyond just treating it as the largest subset of the real numbers that makes sense in the context), then you most likely went to a very good middle/high school. I would be very surprised to see that kind of stuff in middle/high school curricula, except at the few places that offer classes in real/complex analysis, etc. normally, the function-related stuff comes up in an intro to proofs class or a discrete math class, and delta-epsilon continuity definitions are typically mentioned in calculus textbooks, but no one ever really tests on that in first year calc. sometimes it’ll come up once or twice in calc3, but even then, not always. it typically comes up for real in analysis, topology, etc. The point is that good teaching practice means teach to your audience. If you’re teaching a high school algebra I class, I would be hesitant to even mention codomains. If you’re teaching a freshman calculus class, you should emphasize the geometric picture and how to do the computations, not the epsilon-delta continuity definitions or the proof of the ratio test. If you’re teaching an intro to proofs class, you don’t want to do more than touch on number theory topics. If you’re teaching a class full of math majors in analysis, topology, algebra, etc., then you can be much more precise (and as compared to certain other classes, rigorous).</p>
<p>and typically to deal with inverse functions when you don’t know about a target space (in high school, anyway), you generally assume one exists and solve for it, and it’s inherently assumed that you’re actually finding an inverse for a bijective restriction function.</p>