YEAH, I LIKE MATH. BECAUSE IT'S THE SAME IN EVERY COUNTRY: Late in our weekend discussion of higher-level math,
our own Kim Cosmopolitan couldn't help but wonder:
So here's a question: I feel like I am far from the only person in the world who sailed along in math before being generally befuddled by calculus. (For background - full on mathlete, did ok in calculus, but only because of a friend who taught me the "how tos" before every test, which I promptly forgot thereafter.) I have heard similar stories many times. Is this because (a) calculus is hard, yo, or is it possible that (b) calculus is sufficiently harder than "regular" math that your average high school math teacher isn't necessarily capable of bringing the degree of teaching skill required to get the students fully onboard? Or alternatively (c), this wasn't others' experience and my own mathematical abilities were more limited than they seemed to be during grades K-11.
J. Bowman's thoughts, below the fold:
It's not you; calculus is that hard. I posted something similar above, but in terms of both abstraction and complexity, calculus is a big leap over algebra (which is, itself, a bit of a jump from arithmetic). There are so many concepts that most students won't have seen before: real numbers, limits, epsilon-delta (which tends to also be the introduction to mathematical proof), continuity, combination and composition of functions... and that's before we even get to the derivative. By the time you reach the Mean Value Theorem and the Fundamental Theorem (my favorite theorem, despite my distaste for the discipline), most students have, as spacewoman puts it, decided to be a lawyer.
(My story is the same. Solid, top-level mathelete, aced everything through pre-calculus, and then hit a wall somewhere during AP Calculus from which I never recovered. And then Economic Statistics in college was a disaster, a true mercy grade (back when I thought about double-majoring).)
Yeah, I did IB Stats in high school because I didn't want to even attempt the Calc class. My college tested everyone entering to place you for math, and they tried to put me in Calc then, too. I told them I was going to be a journalism major and I would be taking nothing more than they required, which happened to be Stats. I was a mathlete, got the highest score on the IB Stats test, generally really good at math ... but there was no way I was even going to attempt Calc.
ReplyDeleteI missed a day of math in 7th grade that threw off my learning for years - it took me a long time to really get that "X" was going to be different in each problem.
ReplyDeleteBut I did catch on to algebra eventually, and was a whiz with it when it came to using it for Chemistry. The same, actually, with Calculus and Newtonian physics, at a high school level. And then yes, I hit a wall, and suddenly even the Calculus I used to know became much more difficult.
The thing I'm noticing with my kids' math is that they're learning elements of algebra in 3rd and 4th grade much more explicitly than I did. Lot's of problems in the form of 3 X ___ = 12- I think this will serve them well, so that Algebra won't be presented to them as an entirely new concept in a few years. I'm sure I had similar problem sets, but it was never stated that what we were doing was Algebra. That was a mystery reserved for Junior High.
I had to take calc as a coreq for a computer programming course I was planning on taking. The class was on Saturday from 9 -1.
ReplyDeleteI was a decent math student in high school; certainly not a mathlete but I did well. I wasn't looking forward to taking precalc or calc in my mid-30's, but the coursework demanded it.
Turns out I loved calculus. I thought it was, well, fun. I even took Linear Algebra and while I found that to be pretty intimidating, it wasn't that bad.
I hit my mathematical wall with Discrete Math. Ugh.
I was an excellent math student all through school, and then BAM, Calc I Honors in 11th grade nearly killed me. (My school didn't have AP classes at the time, but it was the equivalent of AB.) For the first time in my life, I failed a class. Thankfully, I had an incredible teacher who was so patient with me, and worked with me all the time to help get at least the basic gist of what I could (and save my HS transcript).
ReplyDeleteFast forward to first year of college, and I essentially retook the class. It was like an epiphany the 2nd time around! It made sense! It was easy! All the things that were stumbling blocks in high school suddenly came together. I really think Calculus is one of those things that most people don't "get" the first time around, but with repeated exposure, can eventually understand.
Fast forward a few more years -- now I TEACH AP Calculus (which I'm sure my HS teacher never would have imagined!). I always open the year by telling my students that I failed it in high school. I tell them that it's hard, that often they won't understand many things, that they'll hate me at times. BUT, I also tell them that when they get to college, and have to take the class again, it will be SO much better to have seen it before and they will have an advantage over their classmates who didn't get to take it in high school.
I am a high school math teacher and I did find that some of my students had difficulty with the level of abstraction that was required once they got to Calculus. It can be especially challenging because each topic really does build on the last, so if you haven't mastered a concept, you're going to have trouble going forward. Additionally, the mastery of complicated algebraic, geometric, and trigonometric concepts is absolutely essential. Calculus really combines everything that has been taught to that point.
ReplyDeleteDuring my undergraduate years, the other math majors and I joked about every person having a "wall"when it comes to their ability to master mathematical concepts. Luckily for me, I didn't hit that wall as an undergrad, but I imagine if I had done my masters or PhD in math, I would have hit it then.
Yep - did fine in Algebra, Geometry, Trig. Got Bs in AP Calculus, but only made a 2 on the AP exam itself, so I had to take it in college. And take it I did - three times without success. It's about 20% of the reason I switched to a design-your-own major program that meant NO MATH requirement.
ReplyDeleteIt was functions what brought me low.
I hit a similar wall with chemistry - I loved chemistry in high school and thought that meant I should take Honors Chemistry as a freshman in college. Did great until second semester when we started getting into organic chemistry. Somehow I just couldn't get the hang of what the hell the carbon was doing.
My own personal theory is that with both calculus and chemistry, the things I did in the beginning I could understand WHY they worked (maybe not as fully and elegantly as a full-blown mathematician, but enough) so I was happy to then put it to practice. But with calculus, I wasn't willing to NOT understand it and just memorize the "how tos" that KimCosmopolitan referenced earlier, but not smart enough to actually understand it.
I have a feeling that if I'd gone further than high school physics I'd have hit the same wall there.
<span>One of my math professors in college told me that everyone (even the Fields Medal winning mathemeticians) has a "math wall" beyond which their brain simply cannot go. For some people, it can be at simple arithmetic (like my best friend from high school, the one who got a 250 on the math section of her SATs, then was excited that she got 150 points all by herself). For the lucky fancy types, that wall may be beyond our current mathematical understanding and they won't hit it in their lifetimes.
ReplyDeleteFor me, it was Calc III. I had a similar experience to Emily - Calc was hard for me in high school, though I did ok, and then it clicked in college (once my physics professor uncle was able to make infinite series make sense to me). But Calc III nearly killed me. True, I didn't have a particularly good prof, but I'm convinced that the best teacher in the world couldn't have helped me with triple integrals. I could mechanically do the problems when presented to me numerically, but ask me to set one up from a graph and it's game over. Physics was a similar problem - I can't take the "real world" and express it as a math problem. The only reason I passed Calc III (which I needed for my major) was that the course was graded on a curve and most everyone else understood even less than I did.
So I majored in chemistry, where this doesn't come up that much. ;-)
The funny thing is that I really, really love number theory. We did a smattering in high school - Euler phi functions and abundant and deficient numbers and all that. I loved it - still do. But I couldn't take those courses in college without taking linear algebra, which I was pretty sure I wouldn't survive.</span>
Ann-Marie, see below - you and I were clearly writing about the "wall" at the same time!
ReplyDeleteThe thing is, I don't think the "wall" is what we would think of as linear, either, at least not at certain points. Like, my brain refuses to understand calculus (it seems like it's all predicated on fake stuff, in the same way that I feel like geometry is--ain't no such thing as one dimension for anything practical, and I include a bunch of quantum operators in "practical"). But I do fine at linear algebra and discrete math. But I mostly think of those as logic problems, anyway.
ReplyDeleteI majored in history and biology, needed math through calculus for the biology part and got lucky that the semester I took it we had a terrible substitute professor who cried almost every day in class and gave anyone who didn't earn an "A" a "B".
My experience, I guess, is similar - can't do Calc III but can do number theory, which generally seems to be considered "higher math" than calculus. So yeah, it's not linear, but there's definitely a big place in the mathiness that my brain will not go.
ReplyDeleteThere is, by the way, a similar theory of chemistry - if you break down chemistry into it's four "pure" areas (organic, inorganic, analytical, physical), the theory goes that no one is good at all four, or at least no one is equally good. As you might expect from my math-physics woes, I excelled in organic and inorganic, did much less well in physical (which has too much in common with, y'know, actual physics) and was pitiful at the analytical stuff, which is basically chemical math.
I do not honk when I see the bumper sticker, "Honk if you passed P-Chem." I passed, but not by enough to feel I've earned the honk.
1. I had no problems with basic calculus (limits, basic derivatives, etc.), but when you started getting into the following level (integrals), hit the wall pretty damn hard.
ReplyDelete2. I had one acquaintance in college who the sole reason she didn't get into Phi Beta Kappa was because she was scared to take a math class--PBK requires at least one semster of college math (or placing out of the equivalent). Everyone who was in the PBK ballpark got a reminder letter at the end of junior year that "hey, if you don't have a math credit already, you need to get one to be eligible."
I'm really relieved that it wasn't just me. I never understood why math was so easy for me except for Geometry, which took a lot more effort for me, and then BC Calculus, which just about killed me and was the first class I ever dropped (went back to AB Calculus and managed to get through okay).
ReplyDeleteMatt, I somehow slipped under that wire? PBK let me in despite my thrice-dismal performance in Calculus and never taking another math class after that.
ReplyDeleteHi Emily, what a fantastic comment. My background started out similar with straight A's in math until 11th grade AP Calc I. First quarter B-, second quarter I was heading for a D, and then one day I was talking to the teacher about the definition of the derivitive, staring at the blackboard, and it just clicked. I'd never had anything happen like that before, One second I was a D student, and the next second I was back to an A. I took the BC exam in in 12th grade, got a 5 and thought I had Calculus wired.
ReplyDeleteThen I made my big mistake. With my AP score I opted to skip past of Calc 1&2, and worse, took Linear Algebra my fall semester of college. When the spring semester rolled around I was thrown right into DiffEq, after almost a year without thinking about Calculus, and I never caught up again. I was able to get Bs in it and Advanced Calculus, but I never really grokked the higher levels of math, and as a Physics major that turned out to be a serious limitation. That decision to skip Freshman Calculus is on a very short list of life/career decisions that I actually regret.
I've always wondered whether my complete pity grade senior year of college in statistics was due to my not having taken any math in almost three years, or simply hitting a conceptual wall. From the experiences relayed here, sounds like I was never going to get it.
ReplyDeleteWhen I was getting ready to head off to college, I asked my calc teacher (at that point I had had her for 2 years in HS) which of the math offerings at my school would be "next" for me. She read the descriptions, and indicated which one would follow. I decided, however, to just take my school's placement test and let them decide where I should be. They recommended that I take the class before the one my teacher suggested, and I decided to go on their suggestion, figuring that
ReplyDelete(ah, stupid enter button)
ReplyDeletefiguring that what I could remember for the placement test was a better indication of where I should be. Good decision on my part!
When I learned I would be teaching AP Calc back in 2007, I spent the WHOLE summer reviewing and essentially relearning the course, as I hadn't taken the college class in nearly 10 years. Now that I've been teaching it for a few years, I definitely have a better sense of how to break it down for my students -- but I always encourage them to retake it in college if they aren't confident!
Calculus? I barely made it through algebra. Apparently I hit that math "wall" really early. My math woes can be traced back to the fourth grade when Sister Clarence fell and broke her kneecap a few weeks into the school year. She was replaced by a series of temporary substitutes who had questionable qualifications (this was a Catholic school in the 80s). Upon her return in the spring, she promptly fell, broke her hip, and retired - thus bringing back the subs. I got off track with math that year and developed severe math-related anxiety that continues to this day. My lack of math skills hurt my SAT scores and is a big part of why I've never gone to grad school. In other words, math is not my thing.
ReplyDeleteHowever, I voluntarily took physics in high school because I find it fascinating and I really wanted to understand the hows and whys of things. It's the only science class I've ever loved even though I barely understood it. And I do mean barely. The fact that I even managed to pass the class is due to having a great lab partner and a remarkably patient teacher (and some would say generous grader). In college I took a statistics class in which I worked harder for a B than I did in all my other classes combined.
In other words, even if you hit your math wall with calculus, I'm still super-impressed by your mathiness.
If what y'all say is true, then I hit the wall in the middle of Calculus in college. It was so damn furstrating because I had LOVED math up to that point. I attended a college that was in the middle of some experimental math teaching program and I decided to blame my breakup with math on that. But it makes sense.
ReplyDeleteMy issues with physics are related to a teacher who was a religion teacher...my poor engineer father was so angry the whole year....
My wall was Calculus III-IV. Did fine in high school, AP Calculus was OK, scored a 4, opted not to skip to Calc II in college and did well enough in Calc I, good in Calc II (best teacher I had in college, and the only college professer I remember by name--Beals at Rutgers, just a great, great professor), stumbled with a C in Calc III and then failed Calc IV (think I got a C or D the second time). The three-dimensional stuff (x,y,z) killed me - my brain doesn't work that way. This was all when I thought I was going to be an engineer so it seemed important. Ended up graduating with a Theater major/Chem minor, so . . . not so much.
ReplyDeleteI did the same thing the summer before I took precalc. It had been about 15 years since I stepped foot in a classroom, so I picked up a couple of Algebra review books and spent the summer reviewing and finding "x".
ReplyDeleteI hit the wall in Calculus II in my first semester of undergrad. I was fine with AP calculus and physics in high school, but hit the wall in Calc II. I suspect that my understanding was compounded by the class meeting early in the morning in a far away building with a professor who may have been a briliant mathematician, but neither a great teacher nor speaker of English. And then proceeded to forget most of the Calculus I had ever learned until taking econ classes in junior year that required some basic calculus. If I could get a re-do on undergrad, I'd take some more science, engineering and comp.sci classes. Of course, if I had, I'd be here wishing that I'd taken more foreign policy, literature and poli sci classes. I guess I just like me some schoolin'
ReplyDeleteI relate. I did well in math in HS all the way through calculus, with only a minor blip of confusion in geometry until it clicked. At college I took "calculus for people who have already had calculus" (18.011, does that still exist?) and did really well until we hit vectors, which I completely tanked. Then we finished vectors and matrices and I was fine again. I can't do dimensionality, which also explains my struggles with organic chemistry, even though I love it.
ReplyDeleteI'm wondering, everyone who struggled with statistics: weren't you allowed a "cheat sheet" of the formulas to reference, or did you have to memorize them? I aced stats but only because I didn't have to memorize the formulas -- I just had to know which ones to use, when, and we were able to make our own reference sheets. But then I took stats at the b school, not in the math department... but the b school students were always whining about it.
This has been very enlightening. My daughter had the same problem. Did very well through high school in math. Scored high enough on her ACT for strong engineering schools to recruit her. (NO interest in an engineering major!) Got to her Calculus class (I believe her first Calc class) in college and hit the wall. Could not wrap her brain around it and it was a pre-req. Ended up dropping before it hit her transcript. Try #2 - was going better but not well when she was diagnosed with bacterial meningitis and had to withdraw from the class. 3rd time was the charm. Finally passed - I think with a B. She is done with math forever.
ReplyDeleteFunny related story - I got a call toward the end of the semester the third time she took it and she announced that it had officially been one semester since she had meningitis. I asker how she knew and she said that they had just started brand new material that she had never seen before. :-)
Yoicks. I hit my math wall in third grade -- single-digit multiplication, aka the times tables. I'm not kidding -- I had anxiety attacks every single Thursday (test day). Cheated, got caught and shamed almost to death by my truly beloved teacher. Made better cheat sheet. Gutted it out, never cheated again. Stayed in the top three in my class for the next 9 years, graduated as Salutatorian, but suffered DREAD/LOATHING/ANXIETY nearly every math class during that entire time. My SAT math and verbal scores were 250 points apart, and this at the absolute apex of my math skills and preparedness ... then I got into Columbia, and o my Shatner, the dance of joy and relief I did when I realized that this Ivy (maybe the only one? unless maybe Brown?) had no math requirement for graduation -- would that you had seen it.
ReplyDeleteThe thing is, I could never trust that the numbers would do what the teachers and non-math-idiots said and believed that they would. I had to triple-check everything, always -- I could do the exact same problem four times and come up with four different goddamned answers, which they said is "impossible." There was no room for be-boppin' and scattin' in math the way I did in every other subject, including science, and that made it absolutely unreachable for me. I worked so hard at it, and still, everything I ever wrote down and turned in was purely on a wing and a prayer.
I have since developed a theory that all humans are made with an operating system that is fundamentally Words or Numbers, and that this is how we understand the world/approach all problems. Guess which one I am. Heh.
I'm not entirely sure if I hit the wall or not. I was going to major in math. (Sidetrack - my vague career plan at that point was to teach HS math. It is obvious to me now that was a bad plan, because I am fundamentally ill-equipped to try and control a classroom of teenagers. Just glad I didn't find that out the hard way.)
ReplyDeleteI got through Calc II and III with no problems (as I recall), and then fall of sophmore year I took Linear Algebra. And what I came to realize very quickly was that while I was able to solve the problems, when they asked me to try and prove something, I just had no idea where to start. I could look up the answer and it would all make sense, but there was simply something missing in my ability to figure things out on my own. I got through the course with a decent grade, but I knew this wasn't the way for me to go. So I switched majors to history and things worked out better.
I did take a couple more math classes relating to statistics (if the NYU catalog hasn't changed much in 20 years, I'm guessing Theory of Probability and Mathematical Statistics), the first was fine, the second rough but I got through. Not that I even remember much of what it was about now. I'm pretty sure that was the class I finished the take-home final on the morning my mom was coming to move me out of the dorm, which meant when she showed up I had nothing packed. She was NOT happy. Didn't help that I was in a single that year and the place was a real mess.
I had an interesting time with math. Pretty much enjoyed it the whole way through (except for geometry, which I partially attribute to a sadistic, merciless and pitiless teacher), though I struggled with advanced algebra. And of course, BAM...AP calc hit, and I was floored. (Cue another sadistic, merciless and pitiless teacher.) But here's the thing: not only did I find the class tough, but I also experienced firsthand the idea that what I had considered "cheating" was considered by my teacher to be merely "finding the answer by any means possible." She considered it legit that some of my classmates got together, found outside sources to find the answer to some of our problems, then worked backward. My world was rocked something fierce. And to this day, partially because of that day when our AP calc teacher gave her blessing to "cheating," I now believe that what is considered cheating up through high school is considered collaboration at the college level and networking as an adult.
ReplyDeleteOh, yeah...took a calc class in college, and was among the top students in it. I've been wondering lately how I would fare were I to take the class again. (Might I be good enough to actually tutor kids in this realm at some point?)
Marsha: I've heard such horror stories about P-chem that, at least from an outsider's point of view, if you interacted with it to the point where you got a D, you've fully earned the right to honk as loudly as you want! I know people who have taken it THREE TIMES and only barely passed the third time. My hat's off to you.
ReplyDeleteFor a time, I considered majoring in math. But I had already hit one wall in high school, and that wall was mathematical proofs. I could work out most proofs without a problem, but certain types of problems still make me nervous just thinking about them. While I was able to go around that particular wall and succeed in high school and college calculus (which I loved), my inability to understand those proofs and the memory of that awful feeling of never being able to wrap my head around them was enough for me to know I didn't want to spend the rest of my education feeling that way.
ReplyDeleteI came up against the shoals in Calculus BC. I ended up retaking Calculus AB my senior year, and did fine on it (4 on the Calc AB AP test) (after all but failing junior year -- although there were extraneous reasons for that too), then ended up taking first year calculus in college, doing great the first half and then floundering a bit as we got beyond what I had done for a third time. I think I could have gutted it out into the math beyond and done fine, but until that point it had all come so very easy for me. And at that point I realized I could potentially be great at history (although that itself detoured into law school) and probably never really be great at physics, which is what I thought I was going to do.
ReplyDeleteI had that same problem with organic chem - started as a chemical engineering major, but the right-hand and left-hand stuff and visualizing the 3D-ness of the molecules killed me.
ReplyDeleteThat contrast between understanding a problem and simply memorizing and implementing a method to solve the problem was always key for me in math. My best and worst math moments from school (really from first grade all the way through Pre-Calc, as far as I went, which was enough to test me out of any college math) always hinged upon whether I could connect the method to what was actually happening in the problem and, if not, whether it was acceptable for me to use a different method.
ReplyDeleteWow - you're making me feel great. I did better than a D, but again, helpful curve. I actually took Advanced P-Chem as well - had to if I wanted a BS instead of a BA. Advanced P-Chem was somehow "easier" than regular P-Chem. Or maybe it just connected better with my strange brain. Who knows.
ReplyDeleteI'm the girl who not only aced Orgo but loved it and spent three years working in an Orgo lab and taking every Orgo class i could, so I cannot be trusted.
I used to tutor high school math and helping kids with proofs was always my favorite part. Because if a kid is having enough trouble in math to need a tutor, it's pretty much a sure thing that they do NOT understand the whole proof thing. (No idea why, but the two seem to go together.) And proofs CAN be taught, and they CAN be learned if both teacher and student are patient enough. The light that goes on in a kid's eyes when they finally "get" the whole concept of proofs is just awesome to behold.
ReplyDeleteSing it, sister. I hit my wall somewhere around fractions in elementary school, and it was a constant struggle to keep up after that. I did ok in classes because I didn't have a choice, and I survived algebra, but geometry was just a nightmare, though I think we can blame my teacher as much as my faulty brain. He was TERRIBLE. I never understood a thing he said. My parents finally got me a tutor, and it helped, but honestly, I couldn't even get the information to stick in my brain. I'd understand it when it was taught to me, but when I'd try to go back and do, say, a proof for a test? It was all gone. (I am also terrible at remembering the rules for poker--I can't keep the different types of combinations in my head. They just trickle back out.) My method of avoiding Pre-Calc was to take a remedial math class my junior year, so that I could take Algebra II my senior year, and be done with math forever. Sadly, the remedial class was also terrible, and I usually got my homework done while still sitting in class. Didn't learn a thing for one entire year. (Florida public schools are, in what, the mid 40s in academic achievement rank?)
ReplyDeleteSo, yeah. I just survived math. I got good grades because my mom insisted, but I often had no idea what I was doing. I'd just go through the motions. I did a little better in Physics because the teacher there was much better. When I got to college, I took a placement test at orientation that placed me out of College Algebra, which was a relief. I then had just one more semester to take, and since my major didn't require any math, I learned I was allowed to substitute a computer class in, so I managed to avoid all math in college.
My father the engineer was super-annoyed that I wasn't better at math, and was completely confused by my efforts to avoid Pre-calc in high school, but I just kept telling him that I was NOT going to take a class just so I could fail it my senior year. He kept saying he would help me, but unfortunately, math just made too much sense to him. He couldn't break it down enough for me. All y'all that can wrap your brains around math amaze me. You should watch me try to calculate the tip on a restaurant bill. It takes full MINUTES. And yes, I know that moving the decimal over and doubling it will work. That still takes me TIME.
One weird quirk of my math high school career: I never officially took Trigonometry. I took Geometry at one school where Trig was done with Pre-Calc, but then switched to a different school which had Geometry & Trig in one course. I guess I picked up what I needed here & there.
ReplyDeleteBrown had no math requirement for graduation. Thank goodness. (Though I did take Statistics and Econ.)
ReplyDeleteI LOVE proofs.
ReplyDelete<span>"Calculus? I barely made it through algebra."</span>
ReplyDelete<span></span>
<span>Oh, I hear that. I got through elementary school math just fine, did okay in jr. high with some work, and just lost it completely in high school. I STILL struggle with percentages - I keep a post-in over my desk with the formula for calculating % growth and I still have trouble with it. And I know I learned that in 4th or 5th grade. </span>
My dad, the mechanical engineer, would only give me this as help for my math homework: "Algebra is like a seesaw, Amy." And then he'd pat my head and walk away. Thanks, Fa, thanks a lot.
ReplyDeleteOh, gosh yes - they even brought in these things that looked like tinker toys to help us build the molecules and I could sort of get it then, but then when I was supposed to recognize it from a 2-D drawing, or worse, DRAW IT MYSELF, I was totally lost again.
ReplyDeleteI very much enjoyed the proofs in geometry - I even liked that the more you pared it down and whittled to the most concise proof the more points you got.
ReplyDeleteNot at all convinced twenty years later that I would have the faintest idea how to do it now.
I enjoyed math, including high school calculus. My Honors Calculus with Theory prof wanted me to major in math. I did not want me to major in math, so we parted ways after one quarter. (Oddly enough, I got a B in that class, which begs the question why he wanted me to major in math. Although he did intimate that he gave me the B to motivate me. I suppose I was motivated... to stop taking math.)
ReplyDeleteThe thing is, I think that I am a person who likes numbers, but I don't have whatever "it" is that causes one to become (and enjoy being) a mathematician. I knew a guy in college who just loved math, ultimately getting a PhD in math. That just wasn't me. Ultimately, I think that things all turned out as they should, with me wandering off to major in philosophy and poli sci.
I'm assuming you mean AP- one of the central tenets of the IB math curriculum is that topics such as Calc, Stats, and Vectors shouldn't be taught separately. The only math classes they offer are Math Studies (General), Math Standard Level, and Math Higher Level.
ReplyDeleteNever had a problem- 5 on BC Calc, Majored in Math undergrad & grad, took the Putnam (and did not score anything spectacular there, but neither does anyone else).
ReplyDeleteI now teach IB Higher Level Math, which includes Calc AB (and a good portion of BC). Iwould agree with Ann-Marie's comments; Calc doesn't just require an understanding of everything that came before, it requires absolute mastery.
I tell my kids that instead of running into a wall, there is a point where everyone swims through Jell-o. It will get obnoxiously difficult for eveyone eventually, but it is possible to keep moving. Sometimes, if you back up, you'll find that the part you've been over is much easier the second time through, and may give you momentum to get further along.
I've been enjoying the past two days of Math, but if I could beg for a third day, I'd love a discussion of the actual use of any of it. This is coming from a guy who loves the topic and uses it everyday, whose job it is to make it appealing to teenagers (and does that job well, frankly.) What bothers me is that most every student will never need even the most simple parts of it, and the ones who do find themselves using it in completely foreign ways.
The real problem is that there are so many beautiful and interesting aspects to the topic, but we bury those in years of 'finding x.' Like Marcus duSautoy says, we spend all our time in the grammar, and never show them the Shakespeare. What, if any, math do you actually use? How could your math education have been better?
If you're interested, a good place to start would be A Mathematician's Lament by Paul Lockhart, which is available in many places online. Still, I'd be curious to hear other, non-math persons' ideas.
<span>Closer to the OP, let me add in that part of Calculus' problem is that we teach it in such an awful way. We waste so much time with Continuity and Delta-Epsilon type crap that by the time we get to the blessed point of any of it, any interest in the subject has been drained away.
ReplyDeleteMy students who struggle are incredibly relieved to hear that Newton & Leibnitz, who invented (discovered?) Calculus, never thought about any of that stuff- some guy named Cauchy came back and plugged up their holes 150 years later. Point is, you don't NEED to know all that nonsense in order to do some pretty amazing things.</span>
Trig proofs were my fave. Getting it down to 1=1? So satisfying. The part of my brain that hogged all the logic problems when we'd get a Variety Puzzles book, I guess.
ReplyDeleteI was never really a math guy, but I got an A+ in Calculus I first year of college. Maybe it was because I had taken essentially the same course in high school the year before. Maybe it was because my study partner was an absurdly attractive member of the Michigan dance squad. Either way, I got a C- in Calculus II, and the prof was generous. That was the end of my math career.
ReplyDeleteThis was me as well. Integrals is where it all fell apart. That also happened to be the second semester of my senior year when my class rank was already determined and I had already gotten into the college I wanted to get into. Combine all that with a first-period honors Calculus class and, well, when Mr. Mandelbaum asked me if he'd ever see me in class again, I was honest when I said, "Probably not."
ReplyDeleteWhen I got to college, I did okay in the first half of Calc 115 but integrals got me again. Only class I got a C in--both in high school and college. Does not compute.
So much good stuff here. I considered just replying to an early comment, so people wouldn't get bored before getting to my continued ramblings, but there's too much.
ReplyDeleteGetting it the second time: My father has described that as his exact experience with college mathematics; every semester, the previous course suddenly made sense, which allowed him just enough insight to pass the current course, which would make sense to him the next semester. He ended up getting a business degree. My experiences with calc and differential equations were similar; the second time I saw it, I wondered what my problem had been the first time around.
On walls: Linear algebra, on the other hand, has always been a problem for me. Too much to keep track of, too many things that don't quite work the way you want them to. When I took it as an undergrad, I went to see my TA for help, and he asked me what I had against mathematics. I had to tell him I had just left the registrar's office, where I had dropped off the forms declaring it it my major. Awk-ward.
Another note about this: Regardless of the subject, the idea of the wall is actually a very difficult concept for graduate students to grasp, because this has (usually) all been so easy for them. It's especially hard to apply to their peers.
When I started the PhD program, I heard horror stories about studying for qualifying exams. I have friends who put in 100+ hours studying for a single exam (and you take three). My exams were Tuesday, Wednesday, and Thursday of the same week. My roommate and I started studying Saturday afternoon, and I took one of mine in pen. We figured they were just doing it wrong. It wasn't until my last class, when I actually ran into things I didn't immediately and intuitively understand (and put in more time studying for the final than for all of my qualifiers), that it occurred to me that I didn't have some magic formula for studying.
I was discussing this thread with a former math teacher colleague (now with his math Ph.D and going back to teaching HS in the fall) and he shared the following two pieces with me that I thought many people on here might enjoy.
ReplyDeleteNational Council of Teachers of Mathematics/Mathematical Association of America joint statement on high school calculus: http://launchings.blogspot.com/2012/04/maanctm-joint-position-on-calculus.html
Bressoud's thoughts on the rise in AP Calculus enrollment: http://www.maa.org/columns/launchings/launchings_03_10.html
I have to agree with the assessment that success in calculus does require true mastery of algebra and trig, among others. I think that's a big part of why I struggled in high school -- my 9th and 10th grade experiences were lackluster, at best.
Integrals are tough, because the formulas to calculate them aren't nearly as straightforward as for derivatives, and most of them feel like tricks (trigonometric substitution, I'm looking in your direction).
ReplyDeleteThe Fundamental Theorem of Calculus tells us why: integrals are the inverse of derivatives, and inverting things is hard. It's harder to subtract than add (think about the six-year-old trying to figure out why 3 - 5 isn't 2), harder to divide than multiply, harder to take a square root than to square something. Definitely wall-worthy concepts.
Another light bulb moment. I just remembered that my daughter never had Trig/EA as one of her math classes. Maybe they include that in something else now? Back in my day it was a separate class. But her high school classes were all Algebra and Geometry-type classes until AP Stat her senior year.
ReplyDelete@Rick - I had a very odd situation in my schooling. Being a military brat, I started 6th grade in the US, moved to Australia (out in the boonies of the outback) and they skipped me up to high school which was 8th grade. Their approach to math was not to have separate classes for Algebra, Geometry, etc. Instead you had a little of each every year so that by the time you matriculated, you had the full gamut of classes. I was only there 2 years and then decided to skip back to my original grade when I returned so I then went through the usual US system for math classes. It was kind of fun because each year I always had some great insight into the first 2-3 months of the subject. I've wondered how my math experiences might have differed if I had completed school under that system.
On a related note, in Australia they taught French as a conversational class rather than learning the elements before you start speaking the language. I started my third year of French on my return to the US and did not know the French alphabet but could speak better than anyone in the class.
Very interesting! I teach out of an Australian book (Haese & Harris) for exactly that reason. Most of my kids enjoy it, though there are admittedly some for whom the year feels too disjointed.
ReplyDeleteWe had a student transfer in from Ghana, and my entire department drooled over his book. It was similar to the Australian book, but with a very no-nonsense 1950's style presentation. At least 3 of us have offered to buy it from him, but he's holding onto it- possibly because he realizes how much better it is than the US Pre-Calc book we use with the picture of the St. Louis Arch. (Ooh, it's a curve- Math must be important! And Fun!)
My story is the opposite. I still have trouble with linear algebra. In fact, when I had to teach a matrix algebra unit in the math classes at ITT Tech, it helped my students out a lot to see me stumble through it, too. It helped them see that everyone has to work at, not just them.
ReplyDeleteDiscrete math, on the other hand, I just get. Combinatorics? Love it. Graph Theory? It's just connect-the-dots for grownups. Coding theory? Well... coding theory's okay. It would be better with fewer matrices.
I actually got an A in both halves of pchem. I much preferred thermodynamics to the quantum mech section, though.
ReplyDeleteMy approach to calculus extensively utilizes wonderful technological tools called "TI-89" and "Wolfram Alpha."
ReplyDeleteBut in all seriousness, I experienced the same phenomenon. I was something of a math whiz in middle school. Finished Algebra 1, 2, and Geometry by the time I was 12, won the regional MathCounts competition at 13 and placed in the top 1/3rd at state. Fall of 9th grade, took college trigonometry, aced it. Fall of 10th grade, took Calculus 1 at community college, barely got an A, also got a fear of mathematics so strong that I didn't take Calculus 2 until fall semester of 12th grade. Calculus just requires an entirely different mode of thinking. I'm actually okay with a lot of the theory of it, but I just hit a wall when I'm required to memorize what are [to me] seemingly arbitrary tables of integrals and trig substitutions and the like.
I derive absurd amounts of joy from the very reasonable fact that graphing calculators are still used by students today. For some reason, I figured technology had advanced far beyond that point since my high school days. To what, I have no idea. Personal robots?
ReplyDeleteI'm just gonna keep liking all these, because I'm so glad it's not just me who has trouble in three dimensions. (So, how are y'all spatially challenged folks at parallel parking? or navigation? I'm pretty bad at both, as well as just plain lacking in kinesthestic intelligence, i.e. frequently bruised from whacking into things I thought I had room to go around, or putting things down half off the edges of tables.)
ReplyDeleteMy dad taught computers and math, so I always had a tutor at home. I took honors math throughout high school and I did fine in math mostly because I had a very patient father. However, I drove my teachers crazy because my homework would be perfect and frequently I sat in the back of the class and explained concepts to other students, but I did abysmally on tests. I have horrible test anxiety. What I would finish would be correct, but I always ran out of time.
ReplyDeleteMy senior year they didn't offer Advanced Math Topics--which I think I would have loved--only Calculus. I decided that I didn't want to take it (I had early acceptance to the only school I wanted to go to, so it didn't matter). In college I took the placement test should have taken Calculus, but I opted to take whatever class was just below it. To graduate I needed either two math courses and one science course, one math and two science courses, or one math, one science, and one computer course. I chose the last option. My computer teacher loved me because I ruined the curve in that class.
I still enjoy math, but from what you're saying I'm glad I avoided Calculus. Now I spend a lot of time transcribing old handwriting (from the 1500s) and I love it because I see it as a puzzle.
Very interesting, though I'm the reverse of most folks here. I can add and subtract, but only by sheer force of will, and I still use my fingers. I can multiply thanks to a cobbled together variety of methods (including rote memorization and some random jury rigged methods that seem to work), but oh god division. Ugh.
ReplyDeleteGeometry I made through only because I'm pretty visual and spatial ( I can unwrap/unfold a 3-d object in my mind pretty easily), but trig nearly killed me.
Calc was not a problem at all. I breezed through it, despite thinking I had no math skills. When I went to take the ap test, I didn't bother studying at all, walked in and somehow got a 4/5. Who knew? Seems that higher math that had not much to do with actual "math" totally clicked.
My folks and I suspect that I have a form of dyscalculia, a math dyslexia. I routinely transpose numbers, struggle with simple addition, have a horrible sense of direction (uses related thinking), and despite 12 years of music lessons, have the damnedest time reading music or playing it (music and math are very much related). all of these are symptoms, and symptoms, interesting my dad shares. Compare this to my brother, who never gets lost, plays 5 instruments and taught himself how to play 3 of them, never gets lost, and breezed through all is math classes with never a second thought.
I had an argument with my department that frankly got out out of hand about this exact subject. I refuse to force my students- of whom, again, very few are likely to do any math past Freshman year- to spend $120+ on a calculator when a $4 app on their phone will do the same thing at better resolution on a touch screen in color. SAT/ACT be damned.
ReplyDeleteTo be fair, the latest model- the TI-Nspire CX- is a huge leap in quality and abilities, but it's very much too little, too late (not to mention completely user unfriendly).
My mom has a form of that. She frequently transposes numbers, and has a terrible time with even the most simple of calculations. It wasn't diagnosed until I was in jr high, when our assistant Girl Scout leader happened to work in the county special ed program. My mom had been trying to complete a BS for YEARS, but couldn't make it through the algebra requirement. The assistant leader recommended Mom get tested, and she was subsequently diagnosed. The community college made special allowances for her, and she did finally make it through a pre-algebra course, but was too daunted to continue.
ReplyDeleteInterestingly, her other symptoms have nothing to do with numbers. She has a terrible time hanging on to proper nouns, especially names, and when her mind gets stuck, it will often move on to the next thing. Meanwhile, she will have stopped talking literally in the middle of a sentence, and will have no idea what she was talking about. It is NEVER IRRITATING.
J. Bowman, I SWEAR I took these classes and learned this stuff, but it's been a long time, and now when you post all I see is "blah blah blah Ginger blah blah blah Ginger." This makes me sad. I've lost a lot of knowledge that was really hard to acquire in the first place.
ReplyDeleteI'm wondering how many of us hit the mathematical wall because of poor instructors. My tenured Discrete Math prof had been teaching the course for 20 years and truly lacked the patience for those of us who didn't quite get the course material.
ReplyDeleteI've often wondered that too. My class ended up having the same terrible teacher for pre-calculus that we'd had for Algebra I. It turns out that you can learn algebra on your own with a textbook, but not so much with the pre-calc. By the time we got to calculus, no one in the class knew anything we needed to know, and none of us ever recovered, much to the chagrin of our excellent calculus teacher. By the end of the year he had given up on us and spent the class time teaching us card tricks. Not one person in my class scored above a 3.
ReplyDeleteOn the other hand, from everyone else's stories I'm guessing I would have biffed calculus anyway, so no harm done, I guess.
Marsha, I'm not entirely sure about that.
ReplyDeleteIt seems I've either had great math teachers or awful ones. For me, there didn't seem to be anyone in the middle.
My calc teachers in college were phenomenal, but my junior and senior high school algebra teachers were just terrible. As you said, it's entirely possible to learn algebra on your own and that's what I had to do prior to my precalc course in order to be prepared.
I forget who commented on a professor (I apologize for not remembering who it was) that encouraged students working together to solve problem sets. My precalc and Calc 1 prof loved that we created study groups and emailed each other (this was in 1999) with solutions and actually helped each other out.
For personal reasons I'd love to finish my bachelor's degree (I'm considering becoming a math teacher, of all things) but I'm extremely leery of online courses and some of these "colleges-for-hire". I really enjoy a full-on classroom experience.
I feel the same way about Spanish. :)
ReplyDeleteRemembering the names of formulas and the steps to take a derivative isn't terribly important; if you actually needed to find a derivative, ever, you'd still remember how to do it. But you picked up, or at least honed, some problem-solving skills in those classes, like how to apply things you already know to figure out this new thing. Those, I'll wager, stuck with you.
My Junior year in the US was really hard for that very reason. When I left Australia I was a Sophmore equivalent but I got dropped into Algebra at a Junior level and was completely lost. With some extra tutoring and a bit of nous I go through it finally.
ReplyDeleteA little bit of everything every year makes sense when you're in Middle and lower higher school. It should be noted that in Australia when you reach Senior (Year 12) maths classes are streamed for completion of your high school certificate. At that level you can do Calculus, Algebra and Trigonometry and Geometry at an AP level or do a Discrete mathematics stream which is for your less mathematically minded students. When I returned from the US I dropped straight into Discrete mathematics and was very happy to no longer specialise.