<p>Ive noticed that quite a few times it's been suggstd that in order to make the APs harder, proofs should become part of the test. This way, the exceptionally bright students would be able to shine by demonstrating that their knowledge of the subject goes well beyond blindly following rules and that they understand the concepts behind the math.
This is a good idea in theory, but I feel it's a flawed idea when applied. I mean, let's say you do put proofs on a test. What you're going to wind up getting is a bunch of students memorizing the steps to do these proofs. They're going to be learning tricks and tips, mnemonics, whatever it takes so that they don't actually have to understand the proof to do it. The solution to the previous problem becomes that same problem incarnate.
No matter how difficult you make a test, or how much you try to make it so that those who understand concepts will do better, people <em>will</em> find a shortcut.</p>
<p>To ramaswani:</p>
<p>One of the points I have emphasized is that the multifactorial system is great, so great, in fact, that it overshadows the less-than-satisfying testing curriculum. By that, I mean that college reps succeed despite the decrepit AP/SAT system, not because of it. If those tests were improved, the academic ability of a person will be much less mysterious.</p>
<p>For example, I consider that once you are past 750 on any SAT, it does not really give any more information (the difference between 750 and 800 is usually 3-4 mistakes). So for top students, their is very little objective distinction between scores, and thus it makes it harder to discern academic abilities among top students for college reps. And as you know, most applicants to Harvard will match that "top student" label, given the current system of AP and SAT.</p>
<p>If those tests were improved so that you actually had to be talented, not just a grade-grubbing speed guy, then the admission office will have a easier time. People who find it hard currently will have a lower grade, but grade-deflation is not bad in itself, because college rep will regard more highly of a 3 on an AP, knowing the test is actually college level.</p>
<p>*** I meant "easier time" in the last post :D </p>
<p>And I'm sure mathboy will have a more cogent response to Hippo, but I think proofs are much less subject to brute-forcing memorization than plug-and-chug problems. In fact, the IMO competition, testing the brightest math students around, is basically a bunch of proofs problems. Doing math is doing proofs.</p>
<p>Adding proofs will improve the test. that doesn't mean the test will become perfect, but it will be much more improved.</p>
<p>"What you're going to wind up getting is a bunch of students memorizing the steps to do these proofs. They're going to be learning tricks and tips"</p>
<p>You raise a sort of strange concern, because it seems like you're saying one can memorize how to do the questions in college math, which of course includes the aspect of reading and writing mathematics. I'd be interested to know what leads you to this idea, but let me try to set your worries at rest.</p>
<p>To my experience, college math success involves less of natural talent [once a threshold is crossed, when the ability to actively think about the ideas is reached] and also less of any brute memorization of technique. One solves the problems if and only if one <em>really</em> has thought about the nuances of the theory presented. Problems aren't designed for you to show off how smart you are, but rather to help you wrestle with the ideas presented. They're actually more comfortng than terrifying in this sense.</p>
<p>Now, if the proofs are, say direct questions like "Show this limit of so and so messy, atrocious function exists," one is more likely to succeed by memorizing shortcuts than anything else. But why do that? A mix of more routine checks like this and a few more conceptual questions [which are <em>Proofs</em> in the sense they need be answered with a certain clarity of logic] will really check for understanding.</p>
<p>Conceptual questions, not examples, constitute the majority of college math work. They ask one to ascertain the validity of a general statement. Not something you can package ahead of time, but not something which everyone will just freeze on if they've thought about the <em>kinds of things</em> certain results in whatever subject imply. </p>
<p>To some degree, practice of problems will definitely help people see the patterns in how to solve the problems better, but when you're doing conceptual problems, that represents
more that you're gaining a good grip of the ideas. </p>
<p>Now, calculus theory rests on a form of mathematics I am less fond of than other forms, which seem <em>even more</em> based on "pure thought" -- there are some important little inequalities you need to know when dealing with calculus, and they certainly save you when you want to show things. If you don't like learning these inequalities, you're taking issue with calculus, not with the principle of rigor! Because I can tell you that most of the math I deal with is just ALL pure theory and conceptual stuff. </p>
<p>In any case, a healthy balance of concreteness and theory is good for a first level course like calculus, but I'm saying not to exclude the latter, for fear of foundations being shaky.</p>
<p>If you want to distinguish the conceptual and concrete proofs: think about how different the problems of showing a general fact like "differentiability implies continuity" as opposed to showing for some <em>specific</em> f(x) that its limit is so and so. The solving of conceptual problems can't be packaged beyond a degree, is the gist of my point.</p>
<p>May I respectfully also correct: "to make the APs harder" - and say, to make it more <em>thorough</em> is the goal. I could probably imagine making up AP Calculus questions with the current level which are INSANELY hard. Just make really messy integrals that're hard to evaluate. That'd be harder. Not the goal.</p>
<p>Thorougher, and more reflective of the subject, is the goal.</p>
<p>You raise good counterarguments. I didn't mean that students will literally memorize proofs step-by-step, but high schoolers are sneaky and most of them don't like to have to think, so they'll do whatever they can to minimize how much critical thinking is necessary to get by. It's sad but true. </p>
<p>I realize that this becomes more difficult to do with proof-styled questions, but don't forget that the APs are standardized tests, so the questions change very little on a year-to-year basis. So how are you going to be able to consistantly every year come out with new conceptual proofs? There's only so many topics that can be fairly covered (remember only freshman year math is allowed) so sooner or later the questions will start to get repetitive, and this is where the "memorizing" of sorts will kick in.
y
I just came to a new realization. College tests, finals, midterms, etc are all created by professors and so can be as quirky, intellectual and fun as the professor wants them to be. APs are standardized and more or less have to follow a formula and therefore is pretty much very similar from year to year. It's very hard to avoid tricks when students know what to expect, can do countless practice problems and can sort of memorize how the collegeboard designs its questions. This is why I feel proofs don't work in a standardized setting.</p>
<p>"I didn't mean that students will literally memorize proofs step-by-step, but high schoolers are sneaky and most of them don't like to have to think, so they'll do whatever they can to minimize how much critical thinking is necessary to get by. It's sad but true."</p>
<p>I know what you mean! Trust me, I didn't want to do more work than I had to either...</p>
<p>Now, I think your point is interesting, and one I've sort of thought of -- and actually, I'd not even say it's merely a concern about AP exams, it's actually about how college exams work in general! Contrary to what you're saying, most professors will not give extremely quirky problems for homeworks and exams in mathematics. They expose students to classic type problems. Even in engineering courses (which I have by the way taken at Berkeley...heh, before I decided I'm too much of a sucker for pure math), there is some degree of truth to the fact that if you study years and years of practice exams, you'll most certainly be better off and able to answer more questions than someone who just knows the material well. In the end, we're rewarding students who've taken more practice tests a little over students with pure understanding. Is this perfectly just?</p>
<p>Well, part of the issue lies with the limitations of testing as a way of measuring understanding. You're given so many minutes to assess a student's understanding of calculus. Or any subject. NOT ideal! </p>
<p>You will find though, that beyond a point, one cannot reasonably memorize all the different ways one can ask conceptual questions...like I said, you can get awfully good at predicting what kinds of theorems you'd need to answer so and so type of question by looking at sample questions. But in the end, it would take a student <em>LESS</em> effort to just gain such conceptual understanding than to verbatim memorize every proof problem on calculus ever assigned! </p>
<p>If you want my personal view, the best courses are graded based on problem sets, with good problems written carefully by the instructor. Not based on exams. This allows instructors to give interesting problems, and be less repetitive -- on an exam, there's ALWAYS only so much freedom one has. </p>
<p>But the most ideal improvement would be to improve the AP curricula in high schools IN RESPONSE to the AP exams. </p>
<p>I think even by looking at tons of sample proofs, a student is actually learning something rather than merely memorizing techniques. Also note -- it's pretty easy to ask little variations of proof questions, which would require some extra thought. </p>
<p>Come on...I could probably take an advanced book and work every problem every written in it, then easily ace the AP. But guess what? I actually had to learn something to accomplish my end! I think the purpose is accomplished right there. And if you think you'll "run out" of questions to ask really early, well you underestimate the given subject slightly at least!</p>
<p>And if you think that it's so easy to just paste proofs together without understanding the theory...=] you have to actually try it a bit longer. What you're saying sounds like it could be a problem in principle, but it's considerably less so when you get down to seeing how hard it is to BS a conceptual proof. If you've not really practiced using the theorems actively and thinking about it, however much it seems like you could paste together stuff randomly based on cheap tricks, you'll find little success!! </p>
<p>The point is, the AP need not be a worse test than the college ones. College ones aren't even that quirky to my experience, and a perfectly standard set of questions can be a great check to understanding.</p>
<p>Lol, I am getting so sick of seeing this thread.</p>
<p>Sorry dpattzlover =] though, if I remember, from very long ago, you asked if I didn't have better things to talk about on CC. Though some of it was repetitive, that's kind of to be expected with these forums, where people consistently misread points, lots of clarification has to happen, etc. In the end, I think some good things were raised. You have to have the patience to go through it though!</p>
<p>mathboy and faraday, I cannot comment on the content, the nature and the teaching of calculus because of my total ignorance of the subject.</p>
<p>To respond to Faraday, adcoms of course transcend each and every test whilst valuing each and every test. They are but single data points, and the adcoms are trained to discern patterns, and promise within the applicant's socioeconomic context and the context of the schools.</p>
<p>That is why a promising student from the inner city, with barely precalculus in high school may get admitted over someone with Diff Eq from Philips Exeter. They are trying hard to identify those whose futures can be enhanced by their admission to Harvard, fully noting the fact that high schools shortchange many students.</p>
<p>For the benefit of all who want a concise statement of my point on the math: I was distinguishing abstract math from rigorous math (well, this point was already brought up by faraday). Abstract math being when we proceed in excessive generality, making things less tractable to your standard student, and rigorous math just when we make vague notions precise. </p>
<p>Also concisely, my point to Hippo was that there is a distinction between conceptual and example-based proofs. Conceptual proofs come in many shapes and sizes, based on the vastness of the given subject. And while it may seem like we could memorize with some vigor almost all the conceptual type questions to be asked, I stated this is slight exaggeration (cannot support that, one would have to actually TRY memorizing the vastness of conceptual problems to get what I mean), and that likely such memorization would take more effort than achieving the conceptual understanding the problems would assess. </p>
<p>Nothing new in this post, just concise summaries.</p>