<p>cgm:</p>
<p>could you be a little more specific? When you say two years of calc, is that HS level or college? Is it Calc AB in year 1 and Calc BC in Year 2?</p>
<p>JHS:</p>
<p>Being the parent of a math-phobic and a math-lover, I can talk on both sides of my mouth! :) The math-phobic took discrete math instead of AP-Calc in high school; it was a set of discrete math topics that were deemed to be useful. The math-lover flew through calc.</p>
<p>I, too, think that Calculus is over-rated for those not going into math sciences, and that statistics would be far more useful for the ordinary citizen. Unfortunately, many schools do not consider statistics as being "real math" or being the most challenging level of math (it requires Algebra II), so it is not promoted as much as it should be. Our high school did not offer AP-Stats.</p>
<p>To go off on a tangent and bring the discussion back to the issue of English skills, if colleges required stats instead of calc, we'd have the same situation as regards profs and TAs. Those departments are full of foreign-born profs and grad students.</p>
<ol>
<li><p>Yes, in light of the other thread (where this discussion began), it's worth noting that deciding which type of math you're going to require does nothing to solve the problem of not having enough people who speak English well to teach it.</p></li>
<li><p>I am very confused about Statistics (never having taken that course, although I wish I had). Lots of people seem to be saying that calculus is necessary for understanding statistics (and that makes sense to me). Essentially, that is a centerpiece of the argument for a calculus requirement for pre-meds and social science people (or for people in general). But the assumption seems to be that AP Statistics and general college Statistics courses neither require nor teach any calculus. What's the deal?</p></li>
<li><p>I want to make clear once again that I, at least, do not intend to be making a "pity the poor math-phobe" argument. I am inclined to believe that every educated person should have some measure of math-literacy (and literacy-literacy, too). I am just confused by the pervasive assumption that it's important to learn calculus, measured against my experience that it's not important to remember it at all, and by the assumption that the majority of calculus-learners are unmotivated because they don't expect to use it.</p></li>
</ol>
<p>I think if there's going to be a requirement, there ought to be some kind of consensus about why that requirement exists, and what one hopes to accomplish with it. There seems to be a fair amount of evidence that the calculus requirement does not meet that standard.</p>
<p>Don't you think bringing the whole discussion of which types of math we decide to value in our society down to a judgement of which types "foreigners" are bad at is at best, incredibly shortsighted, and at worst xenophobic?</p>
<p>Which type of math is important to us, and which type of math can we avoid non-native english speakers in should not be related questions.</p>
<p>What do you think is the standard for this requirement JHS? To me, calculus is not the strongest candidate content-wise in all of math, but is certainly a strong methodological subject area. In higher level math, the proofs and abstractions, while they can be a very useful area of thought, are less likely to be used in a straightforward manner in every day life. I feel like the skill set and questioning that is involved in calculus is a more methodologically useful tool for someone who's not going to be regularly engaging in math, though someone can feel free to disagree with me. Since the idea is to present methods of thought, the content of introductory courses is not as important as ensuring that the process of the course is a worthwhile endeavor.</p>
<p>That being said, I don't think calculus should be a high school requirement (it's certainly not at my HS), and I go to a university that doesn't have requirements because I believe in that educational philosophy (for some, not all). Universities that do have a core, however, do have the obligation of explaining why they feel it's necessary, as you say. It shouldn't be this entire discussion on CC, to be honest, the argument should be made and found directly in a universities published materials on educational philosophy (and no, "quantitative proficiency" is not adequate). That being said, there are many arguments a university can make for it (some of which arepresented here, even if you don't feel they're very strong), but that's the end of hte discussion really. By choosing a university that feels it's essential for you to have some subset of knowledge which they've identified, you're choosing to agree with that educational philosophy and to take part in it.</p>
<p>As for number 2, it's because the style of stat in those statistics classes is a less power, less descriptive, and completely limited way of looking at statistics designed because they want to be able to offer it to people who don't have calc. All of that material gets retaught using calculus later on (which most agree is ridiculous and redundant). Even in my PChem class, where we have to use a lot of stats and calculus, the calculus is taught in class, the calc is on the HW set, but there is typically one question that basically says, "Assume this is NOT continuous, but has discrete..." blah blah blah, and it's up to us to figure out how the math goes from the continuous to discrete case (since both exist in quantum mechanics), and believe me, it's very easy going in this direction once you've learned the calculus.</p>
<p>Discrete math could very well be a better topic to discuss content-wise, but I would point out that the point of these courses in college (I'm not sure why we're confusing college versus high school in this thread... maybe I missed something) is not content, it's methods, and it doesn't really matter what they're teaching, rather it matters how they're making you think.</p>
<p>modestmelody:</p>
<p>I have to say, you do yourself and Brown proud in this thread. I am impressed every time with the thoughtfulness, passion, content, and style of your posts.</p>
<p>I don't think I am confusing college and high school, but I'm not certain what the difference should be. I don't think any high school actually requires calculus. What happens is that it becomes an informal requirement for students who want to apply to the most selective colleges, because they think those colleges want to see it, or it becomes a function of a four-year math requirement for students strong enough to complete pre-calculus in three years or less. And then many colleges require calculus for a variety of majors whose students reportedly don't see much utility in the calculus requirement, and a few colleges effectively require it for everyone.</p>
<p>I am agnostic. I want to understand what's going on, that's all. I pretty much agree with everything you've said. But I note that most of the defenses of calculus here are based on content, not method, and a pervasive theme seems to be that an emphasis on content, not method, is what most non-math majors prefer. And its strange, to say the least, if calculus is being taught to improve the consciousness of students, and they don't generally understand how they are being improved by it.</p>
<p>
[quote]
Universities that do have a core, however, do have the obligation of explaining why they feel it's necessary, as you say.
[/quote]
</p>
<p>I haven't read the rest of this thread, but being one of a few spokespeople for core (I'm a current U of C student), I'll try to approach this one.</p>
<p>(Full disclosure: I love math and mathematical thinking. I love delta-epsilon proofs, logic proofs, etc. and was in a class in high school that allowed me to get intimate with math. I hated calculus with a fiery passion, and on my calculus placement test in college, I skipped all of the integration questions and went straight for the proofs. This approach gave me all the math credit I needed for core. Even though I'm done with math requirements, I plan to take both stat and econ as electives to attempt to match up my quantitative mind with my analytical mind).</p>
<p>Phew.</p>
<p>Okay.</p>
<p>For Chicago, one does not need to take calculus to fulfill a math requirement. One can take the more useful stats, the more straightforward comp sci, or the more theoretical Studies in Math. Studies in Math is not rocks for jocks-- it's math professors Paul Sally's and Diane Herrmann's lovechild, and it's a course on really, really, really fascinating concepts, but designed to be taught to a non-calculus crowd. Here is the method over content approach at work-- nobody is going to ask you what a triangle is in your career, but asking yourself what a triangle is will get your brain doing acrobatics in ways it might not be doing otherwise had you not been asked.</p>
<p>Most students (maybe 80-90%) take calculus or pass out of it, and those that didn't do it in high school often want to do it in college. (There are three tracks of it: one with few proofs and lots of tutorials, one that's designed to pick up where AP left off, and one that's highly theoretical).</p>
<p>And even though I got lucky myself and don't have to take another calc class ever again, I'm currently revisiting another high school enemy: biology. Though the class is kicking me in a way I didn't expect (it's comparable to an AP bio class in challenge level), I'm finding it really fascinating, if only because it has me thinking about the world differently.</p>
<p>A POV from someone simpler and closer to the trenches of whom we're trying to address: Here's how a I, as grade school educator (with 2 masters degrees so don't picture me stupid, just math-challenged from bad education long ago), look at Post #79 from the PhD mathematician, describing "Discrete Math":</p>
<p>
[quote]
Teach them methods of proof, number theory, some graph theory, discrete probability, automata, logic, etc...
[/quote]
</p>
<p>Substantial and varied curriculum, worthy of a year's study. </p>
<p>
[quote]
more interesting, IMHO, and are definitely more mind-expanding than mere calculus...
[/quote]
</p>
<p>The motivation is there. There is no learning without motivation. And it motivates for its own sake, so is worthy of the higher education learning environment</p>
<p>
[quote]
gives one a much better picture of what mathematics actually is,
[/quote]
</p>
<p>the very definition of "literacy"</p>
<p>
[quote]
whereas calculus is usually watered down into a problem-solving course.
[/quote]
</p>
<p>improves the situation, therefore it's worth the effort and time to consider making a change in approach</p>
<p>THANK you, quicksilver40133. I found your post illuminating.</p>
<p>Here's a short list of things I would teach students in a discrete mathematics course, at either the advanced HS or undergraduate level:</p>
<p>Methods of proof: proof by contradiction, proof by induction, proof by cases (at least). These three methods of proving something apply not only to mathematics but to everything: science, philosophy, etc. It's a new way of looking at problems, a way of thinking, that can be applied infinitely to almost any situation.</p>
<p>Logic: boolean algebra, propositional logic, predicate calculus (nothing to do with regular calculus), etc. These are also a way of thinking about everything: does an author actually provide enough proof to support his claim? Does A follow from B? It's applicable to everything.</p>
<p>Graph theory: linked structures and graph colorings. The game of dots is an application of graph theory - if you like dots, you'd get a kick out of it. It's also immensely useful in scheduling problems: making schedules subject to many constraints. Applications abound.</p>
<p>Algorithms: Solving problems by breaking them down into their component parts, expressing real world concepts via their mathematical relationships, and understanding different constructs and abstractions which can be used to model problems. Maybe some programming! Wouldn't it be refreshing to write a bit of code in a math class? Knowing how to program a bit could never hurt, and knowing how to think to solve problems in the way that one would do so with algorithms... well, the usefulness of knowing how to solve problems should be self evident.</p>
<p>Compare this to calculus: the study of functions. Ok, so... where do we see functions? Calculus lovers will say "everywhere!". Calculus haters will say "nowere!". Both are actually correct. Yes, calculus has applications in almost all fields. But you must be working in that field to ever use calculus for it. And, often, calculus is just an approximation of discrete mathematics... as is the case in economics, where one uses calculus to describe what is actually discrete data. But that's rambling.</p>
<p>I started out as a math major, but deltas and epsilons drove me over the edge. Echoing an earlier post, I found one of my old calculus blue books, circa 1969, and couldn't believe that I had actually written that! I don't remember any calculus, but I have used algebra a lot. I understand the argument that calculus gives you a new way to think about the world, but I think qucksilver's proposal in #89 has a lot of merit. [I am an accountant by trade who does a lot of number/logic puzzles for fun]</p>
<p>We cross-posted, Quicksilver! So.. Where do I sign up for your course as outlined in Post #89? :) </p>
<p>JMO: If you have personal interest in becoming a professor, please keep these thoughts in your mind when you interview for teaching positions. Academicians might advise otherwise, but I'm speaking from "out here in the real world" of education consumers. I don't know if such a line of reasoning would threaten an interview or search committee, so please don't take this as career advice, just a compliment from someone who'd resemble your humanities students.</p>
<p>quicksilver's description of discrete math is what should be taught at the college level to those who do not intend on going into a math or math-heavy science major.
S took discrete math (it was called Advanced Math) in his high school; it was not taught at the level described by quicksilver. He was admitted to several top LACs without having taken Calculus. In college, he fulfilled the Gen Ed requirement by taking a Stats course.</p>
<p>Counter-questions to JHS:</p>
<p>Do you think that it is beneficial for all students to have the experience of encountering problems that they cannot solve or ideas that they cannot understand? What about problems that they cannot solve at first, but do solve hours, days, or weeks later, because the ideas suddenly become clear?
What about problems for which the students believe that their answers are correct and iron-clad--when the students can then be shown counter-examples that lie outside the conceptual frameworks they've developed to that point?</p>
<p>I think all of these experiences have intrinsic intellectual value. I don't think that calculus has to be the course "vehicle" of choice, though. A good proof-based geometry class would do it, or number theory, or the discrete math being described on this thread. Much of the benefit could be derived outside of math and science entirely--even the third category above, although I think that it's easiest in mathematics to provide a student with a clear-cut and convincing disproof of a belief that the student had held and defended strongly (and that can be really mind-opening).</p>
<p>Students who participate in math competitions generally derive these benefits by the end of high school; others, in proof-based mathematics later. Calculus might have traditionally been the course of choice to convey these benefits, because the material can put students on unfamiliar ground. Calculus can be taught in many ways, though, and these comments would not usually apply to Calc AB/BC or the equivalent course now.</p>
<p>Wow- so much out of my Calculus versus Great Books comment. I was being specific- a pet peeve of mine is the thought that certain specific books are required to be well educated whereas knowing how to think as is taught in math and science courses is not. Some added comments- remember this is a forum, not a college paper to be graded (or none of us would ever have time to post...).</p>
<p>CGM- you do not need to read any of the books mentioned to be able to do well and discuss well- there are plenty of other books one can read that have a richness of thought and command of language. People who have not learned science using calculus are doomed to less understanding of the world we all live in- learning the calculus relationship between speed and acceleration makes them both so much more understandable instead of learning them as separate equations to be memorized. Every time you drive a car you are making use of these two- ever wonder why it is harder to stop when accelerating?... (please, no one nitpick the details here, think concept). </p>
<p>The Great Books are so limiting- they eliminate choices because of language (they therefore can't be using any version of the Christian bible, you can translate a German mathematical text much more accurately than any religious treatise...). I would much rather have the science/math student's limited humanities credits include nonwestern lines of thought. Just as I would expect the well educated humanities person to learn how to think outside their conventional ways by taking some math/science courses beyond the elementary ones.</p>
<p>A feeling I get from humanities majors is that their field is superior because it requires thinking and creativity. How many college courses in literature are rehashing of someone else's ideas? Are people in other, nonwestern, cultures doomed to be uneducated because they use other sources to express ideas? Unless you get past the memorization of facts stage in science you won't understand the creativity involved in new ideas.</p>
<p>Most of us are terribly undereducated when compared to an Asian student who gets a US bachelor's degree- they learned Eastern and Western ideas in the course of their education (the same can be said about anyone staying within their culture for all of their education).</p>
<p>We are living in a global society, and a highly technological one. What does it mean today to be "well educated"? Does it mean knowing superficial bits about a vast array of subjects, or does it mean knowing how to think and access material in most of those fields, or something else? Part of being well educated includes understanding how our world operates and the thought processing involved- whether it be technology or the worlds of words.</p>
<p>Thanks to everyone for so many intellectual thoughts. Remember to think outside the/your box...I'll try.</p>
<p>QM: You and I are completely on the same page about what's valuable. I wouldn't have known it was supposed to apply to a calculus course, though. I got much the same thing from Derrida, and from Ronald Coase. (Not quite the same, I'm sure.)</p>
<p>I, too, want to thank all participants for their passionate and informative comments here. I have learned a lot over the last couple of days. (Including that I want to wring my daughter's departed advisor's neck for making her think she had to take calculus again. That was the least of all I have learned but . . . she should have taken Stats.)</p>
<p>
[quote]
Do you think that it is beneficial for all students to have the experience of encountering problems that they cannot solve or ideas that they cannot understand?
[/quote]
That's the crux of it. Nice job, Quant.
[quote]
I don't think that calculus has to be the course "vehicle" of choice, though. A good proof-based geometry class would do it, or number theory, or the discrete math being described on this thread.
[/quote]
Calc didn't do it for me; proofs did. </p>
<p>So yeah. I think every kid deserves the chance to butt his/her head up against something really difficult and unmerciful. Really, really difficult and unmerciful--and not because the object of the lesson is humility, either. I never found any of the humanities, arts, or social sciences as unmerciful in exactly the same way.</p>
<p>Edit: Oops. Walked away for a while and didn't see your post, JHS, before I hit submit. Agree about the universality of stats. My stats course in grad school (social sciences) used calc but it wasn't necessary. That's why God made SPSS. :)</p>
<p>^^ JHS</p>
<p>I've just read Ronald Coase's brief autobiographical sketch, written at the time he won the 1991 Sveriges Riksbank Prize in Economic Sciences in honor of Alfred Nobel. Very interesting. I wasn't familiar with Coase's work before, but now plan to read The Problem of Social Cost (at least). I'd think Coase's work fits exactly the desideratum of persuading a student--or, in his case, group of University of Chicago econ professors--that a firmly held belief is provably incorrect.</p>
<p>I'll admit--no doubt to gasps of horror from posters on this thread--that I haven't read any of Derrida, at least yet. Have made Derrida jokes, though. Partial credit?</p>
<p>Re calculus, purely a guess: In the past, a proof-based version of calculus probably served the functions I've mentioned, for students who were not accustomed to abstractions. Those who were had to wait for some later course. Non-proof-calc may not serve these functions for anyone. (If the course material is either obvious or just irritating, it's not working.)</p>
<p>. . . and thanks, Mudder's_Mudder!</p>
<p>I had to go out and buy "Calculus for Dummies" after reading this thread; I've forgotten so much. (At which point my daughter reminded me that I had just paid for her AP calculus book!)
At my daughter's school, Honors geometry used extensive proofs (the way I remember geometry from the 60's), while regular geometry didn't; I definitely think there is a valuable mental discipline to the proofs.</p>
<p>This is an interesting thread.</p>
<p>In the province where I grew up, calculus has always functioned as a "gatekeeper". The strongest students all pretty well take it in their senior year of high school. All the academically-focused elite programs pretty well have it as a pre-requisite for admission. In fact, an old prof of mine openly called our senior calculus course the "invisible sieve".</p>
<p>To me, trying to understand science without calculus is like trying to understand East Asian history without classical Chinese. You can do it, but only superficially. Am I the only one that thinks our level of scientific literacy appalling? How can we call ourselves educated when living in the beginning of the 21st century and not having some basic understand of science and math?</p>
<p>Then again, I was an Asian Studies major that took his electives in math, physics, and symbolic logic. I guess I was more than odd, at least in the eyes of my classmates anyway.</p>