I think this is a side-effect of the way our educational system is structured. We rely on big, standard tests to evaluate schools and their students. It would be too expensive, and somewhat subjective, to evaluate students' reasoning abilities, their conceptual knowledge. Can you imagine if these tests were evaluated by rooms full of people reading proofs, rather than by bubble-sheet scanners? So we test what we can test, which is procedural knowledge, and we teach that at the expense of deeper understanding. And because the best way of measuring procedural knowledge is to measure how many procedures you know, we race students forward to as much calculus as they can memorize.
The Calculus Trap (2005)
51–60 of 60 posts
Re: The Calculus Trap (2005)
#52Earlier quoted context omitted.
Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It was pretty standard at my public high school. I'm sure I didn't understand it very well at the time, but it was good to have seen it when I took real analysis.
Had to look it up to be sure, since I didn't recognize the name, but it looks like this refers to using limits to "invent" differentiation? If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same teacher both years, can't remember when exactly Calc was) I remember it involving a lot of drawings of graphs so we could understand what each term referred to, and can't really imagine an easier way to lea…
Yeah, exactly. I honestly don't know how else you'd teach the derivative either, but I guess it's done.
Re: The Calculus Trap (2005)
#53Earlier quoted context omitted.
Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It was pretty standard at my public high school. I'm sure I didn't understand it very well at the time, but it was good to have seen it when I took real analysis.
Had to look it up to be sure, since I didn't recognize the name, but it looks like this refers to using limits to "invent" differentiation? If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same teacher both years, can't remember when exactly Calc was) I remember it involving a lot of drawings of graphs so we could understand what each term referred to, and can't really imagine an easier way to lea…
The limit definition of differentiation is definitely intuitive, but epsilon-delta refers to the most common definition of what it means to take a limit. It's also simple, because when you say it in words it makes perfect sense, but it's not at all obvious why someone would define limits in such a way, or what the alternatives are.
Re: The Calculus Trap (2005)
#54Earlier quoted context omitted.
No way high school or college level classes are teaching intro calc with delta-epsilon proofs, it just won't make any sense. It's generally introduced in Real Analysis, which is generally a sophomore level class. Develop the intuition, then crystallize and formalize the idea with a proof. Otherwise, it's just not going to make any sense.
Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It was pretty standard at my public high school. I'm sure I didn't understand it very well at the time, but it was good to have seen it when I took real analysis.
Same here. That's how I learned it as well. That said, I didn't really "get it" until I was well into the class. I still remember that "aha moment" very vividly, even though it was in 1992 or so. I was sitting at a table in the school library working on a problem about a pool being emptied while it was raining and my brain said "Holy crap! This is just about how fast things are changing!". With all of the proofs, graphs, etc. things were never put into such a simple statement.
The end result of months of flailing followed by the "aha" was a grade of C for the class AND being the only one to get a 5 on the AP test.
EDIT: As a fun corollary, I really hated Calculus class in high school. Ironically, I ended up getting a Masters in EE, so I ended up spending the next several years doing Calculus in > 50% of my courses.
EDIT EDIT: Also fun to note was that Calculus was only offered to Seniors at the time and required 4 years of math beforehand: Algebra, Geometry, Advance Math, and Trigonometry. That meant you had to double-up on math in either Sophomore or Junior years, which basically limited class size to a small handful of people.
Re: The Calculus Trap (2005)
#55Earlier quoted context omitted.
I was taught the Epsilon-Delta definition of the limit in Calc 1, right off the bat in first year (University of Waterloo). Real Analysis here is a third year course run by the Pure Math department and from what I've heard it's one of the hardest undergrad courses the university offers, period. I'm nearing the end of my second year as a math student, finishing up calc 3 and linear algebra 2. When I take a look at a f…
Real analysis is not actually that hard, you do however have to learn a bunch of notation, concepts and processes before you can grasp things. The people who struggled with it at my course (different country, similar content I suspect) were, entirely unsurprisingly, the people who didn't attend all the lectures.
I don't think the difficulty is due to anything inherent to real analysis. I think it's due to the fact that most of the professors in the pure math department participate in the development of the math contests for CEMC [1]. Thus they tend to love creating extremely complicated and challenging homework problems.
If you have time, click around on that page and check out some past Euclid (grade 12) math contests. Even if you've come through real analysis I bet some of the later problems in those Euclids will give you trouble.
Re: The Calculus Trap (2005)
#56Earlier quoted context omitted.
Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It was pretty standard at my public high school. I'm sure I didn't understand it very well at the time, but it was good to have seen it when I took real analysis.
Had to look it up to be sure, since I didn't recognize the name, but it looks like this refers to using limits to "invent" differentiation? If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same teacher both years, can't remember when exactly Calc was) I remember it involving a lot of drawings of graphs so we could understand what each term referred to, and can't really imagine an easier way to lea…
Epsilon-delta is the formalization introduced by Cauchy, Bolzano and Weirstrass during the 19th Century.
It is not the way calculus was invented by Newton and Leibniz. They thought in terms of infinitesimals. However, this approach is relatively hard to formalize. It was only done by Robinson in the 1960s.
To see why Cauchy et al had to work on epsilon-delta, take a look at [1], an excellent book.
Re: The Calculus Trap (2005)
#57Earlier quoted context omitted.
As someone who tried teaching themselves math with Khan Academy, I feel like Khan Academy suffers from the same problem most sites do, in that they tell you the information before you can discover it for yourself. This is different than how AoPS teaches math, where they ask you questions which then guide you towards self discovery, and as a result, you start asking your own questions which create paths for you to go…
You, and AoPS’ target audience of current and potential math nerds are very far from the norm. Inquiry based learning is actively harmful to learning for many students, and much less efficient for more or less everyone. > Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching > Evidence for the superiority…
Your link is to a very controversial polemic which IMO sets up a straw man, and then makes its own argument in a (both theoretically and empirically) questionable way, based on the authors’ pet “cognitive load” theory (which I personally think is bunk, but YMMV). It should be read in the context of its critics, and taken with a heap of salt.
To understand the problem with US-style mathematics pedagogy, I would recommend reading http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd...
For some advice and materials based on an alternative theoretical framework, let me recommend https://www.map.mathshell.org/trumath.php
* * *
I think Khan Academy should be thought of as a consistently average-quality US-high-school-style lecture, combined with US-style trivial exercises. It is a slow, unimaginative, pedantic curriculum.
But it has the advantages of being free, always available, and self-paced (in the sense that students can keep going through as much of it as they want without needing to wait, and can return to previous sections any time). I’m glad it exists, because it sets a quality floor; live teachers have variable quality, and while many are better than KA lectures, some are certainly worse.
Re: The Calculus Trap (2005)
#58Earlier quoted context omitted.
I expect it was more of a teacher issue then. A solid math teacher is worth their weight in platinum.
I had an excellent Calculus teacher in high school, and I did very well on the AP exam. It was still vastly more challenging than the trigonometry class that preceded it. Even today, I use trigonometry all the time almost without thinking, but I can't for the life of me remember how to long-divide polynomials (or picture a scenario where I'd need to). There's just so friggin much deeply abstract symbol manipulation i…
Long division of polynomials is literally exactly the same algorithm as long division of base-10 integers, except with no carrying.
Re: The Calculus Trap (2005)
#59Earlier quoted context omitted.
You, and AoPS’ target audience of current and potential math nerds are very far from the norm. Inquiry based learning is actively harmful to learning for many students, and much less efficient for more or less everyone. > Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching > Evidence for the superiority…
Your link is to a very controversial polemic which IMO sets up a straw man, and then makes its own argument in a (both theoretically and empirically) questionable way, based on the authors’ pet “cognitive load” theory (which I personally think is bunk, but YMMV). It should be read in the context of its critics, and taken with a heap of salt. To understand the problem with US-style mathematics pedagogy, I would recomm…
If you can refer me to three relevant articles that critique or falsify Kirschner et al. I’ll be happy to read them. I’ve read Problem-Based Learning is Compatible with Human Cognitive Architecture: Commentary on Kirschner, Sweller, and Clark (2006) already.
Re: The Calculus Trap (2005)
#60If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom. It is unfortunate that the prevailing educational trends are to get rid of tracks, lanes, and advanced classes, and dumping all the kids into mixed ability classes.
I believe it's been roughly found that tracking helps smart kids moderately and hurts everyone else with the brain drain. It's a question of to what degree we are individuals, and to what degree we are a community. I think it's a question that cuts right down the middle of many people's value systems.
The data isn't very strong to support the idea that leaving advanced kids in the room with those that need more attention is of any help, and there is now growing evidence of achievement gaps widening, possible because it can be demoralizing for the slower kids, possible because the more advanced kids are now more likely to get outside supplementation which boosts them even further ahead.