The hardest (ie, best) math prof I ever had (I was in EE, he was in the math department) used to say that "engineers can teach calculus, but their students cannot go on to teach calculus". I'm impressed by someone who knows a subject deeply enough to take that long of a view.
Personally, I just like the engineering view of dx and dy as simply being new variables (with caveats that we immediately forget). Which is why I'll probably not teach calculus any time soon.
But, still, the best "revelation" moment I had was when he off-handedly said "An integral is the inner product of a function and a suitably-dimensioned unit". Light bulb came on, and I "got" integration for the first time. Too bad they can't lead up to it that way in high school.
(Side rant: why do they teach trig before calculus in high school? That's completely backwards. Trig is a bunch of arbitrary formulas if you don't have the calculus behind them.)
> Trig is a bunch of arbitrary formulas if you don't have the calculus behind them.
Why do you claim this? You don't have to have seen calculus to appreciate how trig functions are defined, how to manipulate them, or how to use them in applications.
As a math professor, I personally like the fact that we teach trig and exponential/logarithmic functions before calculus. They are (as you well know) exceedingly rich examples which illustrate why calculus is interesting and useful, and knowing them already enables the student to study calculus without excessive digressions.
Trig is the math of triangles and circles — it can be fully understood in geometric terms. Calculus requires far more foundation.
The linked article is interesting but the definition of differentiability looks wrong to me — maybe my brain needs more coffee but it looks like only linear functions are differentiable as defined.
I learned my calculus the pure math way — axioms and analysis. Epsilon delta arguments make more sense to me than "Ball". The fact that to clarify the examples the author resorts to epsilon delta description suggests to me that this approach is clever rather than clear.
"(Side rant: why do they teach trig before calculus in high school? That's completely backwards. Trig is a bunch of arbitrary formulas if you don't have the calculus behind them.)"
I'm fairly convinced the answer is that That Is What The Curriculum Does, and Do Not Question It.
None of the subsequent 9 hours of debate since you posted this convinces me otherwise. It's not possible that there's a better way, and we can marshal all sorts of rationalizations about how this is the best way, and the possibility that it might not be simply can not be conceived. If you think that there might be a better way, you just must not be aware of how what we already have is perfect.
Our math curriculum could be significantly improved in many ways, except that this is the general societal attitude towards it that I see, and it turns out that "Fix the math curriculum" becomes an unsolvable problem when you add the constraint "But don't make any changes to it of any kind, not even to merely reorder a few topics". Feh.
> Side rant: why do they teach trig before calculus in high school? That's completely backwards. Trig is a bunch of arbitrary formulas if you don't have the calculus behind them.
Math is hardly the only place where arbitrary facts/formulas are taught, and people taught to apply them, before learning the underlying math/reasoning behind the arbitrary facts/formulas.
And trig is useful in lots of places in the science curriculum without the backing calculus, so teaching it in the math curriculum early to support the broader curriculum makes sense from that perspective.
As much as I respect djb, the title is a huge misnomer. If you're a mathematician you want analysis and topology. You want definitions of continuity and differentiability that generalize nicely to arbitrary dimension, to manifolds, to Lie groups, to whatever. You want to marvel at the connections between certain special integrals and infinite sums, and you want to see the full construction of the real numbers for its own sake. You want a list of equivalent definitions of differentiability so you can get a better intuition and use whichever one is the nicest for what you're trying to do.
In a very strong sense, writing a document that "focuses purely on calculus" is antithetical to mathematics.
Since real analysis is arguably the core of mathematics and of applications of mathematics, it has been formalised in many ways. See [1] for an overview of the state of the art.
You would need to get to the reals, which takes already a fair amount of time and includes a lot of machinery. Depending on how you constructed the reals, you might be able to reuse some concepts. So, it depends on where you start and which path you take.
Also, arguing by reference to the choice of appropriate values is incomplete argumentation and so it won't be accepted by a formal system. You'd have to fill these holes.
I'd suggest to start by evaluating the existing formalized constructions of the reals.
If you're taking things like that back in time, make sure to translate them to French. Also, learn French. Not knowing French in the 17th century is like not knowing English today.
I remember the "concept" part of my high school calculus class. The teacher had us recite out loud the epsilon-delta definition of continuity until everybody memorized it so they could regurgitate it on the test.
I always wondered why analogies and pictures weren't used more often:
examples:
A 100m sprint is a continuous function of time (f(t) = distance from starting line) because sprinters cant teleport. In fact it is uniformly continuous because people have a maximum speed.
Beating usain bolt's record is a discontinuous function of completion time because f(world_record + epsilon) = 0 while f(world_record - epsilon) = 1.
Fundamental theorem of calculus: If you want to know how fast a guy is running at time t, look at how much ground he covered in 1 second. To get more and more accurate, look at how far he traveled in 0.5 seconds and so on...
I was lucky enough that this is how I was taught calculus in high-school. It definitely wasn't easy at the time, but I feel for all the students for whom calculus is taught as a mindless set of algebraic rules.
This focus on calculating derivatives as opposed to actually understanding the concept and why the calculations work that way is, I think, why so many struggle with it.
Definition 5.1/5.2 is interesting. It defines derivative at point c, not the derivative function of f. Note that f1(x) is not equal to f'(x) for all x, but f1(c) = f'(c).
Yeah. The paper says: The derivative of f at c is written f'(c). The derivative of f, written f', is the function c -> f'(c).
So the derivative (f') is the result of substituting c for x in f1. For example, if f1 = (x -> x + c) then we would have f' = (c -> c + c) = (c -> 2c).
One thing that bugs me about this, is that a lot of theorems seem to be formulated backwards. To give an example:
Theorem 9.1. Let f be a continuous real-valued function. Let y be a real number. Let b ≤ c be real numbers with f(b) ≤ y ≤ f(c). Then f(x) = y for some x in [b, c].
Why is this well-formed? Once you say "Let y be a real number", I'm free to pick any real number, which means that there might not be a b and c such that f(b) ≤ y ≤ f(c). Now, I obviously understand what is said, but shouldn't this be formulated more like:
Let f be a continuous real-valued function. Let b ≤ c be real numbers from the domain of f. Let y be a real number in the closed interval bounded by f(b) and f(c) ([f(b), f(c)] or [f(c), f(b)], depending on whether f(b) ≤ f(c) or not). Then there exists an x in [b, c] such, that f(x) = y.
The way this and other theorems, definitions, etc. are formulated in the article bugs me, because I must go back and re-qualify variables based on information deduced from things introduced, after the variable in question was introduced.
[+] [-] bandrami|10 years ago|reply
Personally, I just like the engineering view of dx and dy as simply being new variables (with caveats that we immediately forget). Which is why I'll probably not teach calculus any time soon.
But, still, the best "revelation" moment I had was when he off-handedly said "An integral is the inner product of a function and a suitably-dimensioned unit". Light bulb came on, and I "got" integration for the first time. Too bad they can't lead up to it that way in high school.
(Side rant: why do they teach trig before calculus in high school? That's completely backwards. Trig is a bunch of arbitrary formulas if you don't have the calculus behind them.)
[+] [-] impendia|10 years ago|reply
Why do you claim this? You don't have to have seen calculus to appreciate how trig functions are defined, how to manipulate them, or how to use them in applications.
As a math professor, I personally like the fact that we teach trig and exponential/logarithmic functions before calculus. They are (as you well know) exceedingly rich examples which illustrate why calculus is interesting and useful, and knowing them already enables the student to study calculus without excessive digressions.
[+] [-] Tloewald|10 years ago|reply
The linked article is interesting but the definition of differentiability looks wrong to me — maybe my brain needs more coffee but it looks like only linear functions are differentiable as defined.
I learned my calculus the pure math way — axioms and analysis. Epsilon delta arguments make more sense to me than "Ball". The fact that to clarify the examples the author resorts to epsilon delta description suggests to me that this approach is clever rather than clear.
[+] [-] jerf|10 years ago|reply
I'm fairly convinced the answer is that That Is What The Curriculum Does, and Do Not Question It.
None of the subsequent 9 hours of debate since you posted this convinces me otherwise. It's not possible that there's a better way, and we can marshal all sorts of rationalizations about how this is the best way, and the possibility that it might not be simply can not be conceived. If you think that there might be a better way, you just must not be aware of how what we already have is perfect.
Our math curriculum could be significantly improved in many ways, except that this is the general societal attitude towards it that I see, and it turns out that "Fix the math curriculum" becomes an unsolvable problem when you add the constraint "But don't make any changes to it of any kind, not even to merely reorder a few topics". Feh.
[+] [-] dragonwriter|10 years ago|reply
Math is hardly the only place where arbitrary facts/formulas are taught, and people taught to apply them, before learning the underlying math/reasoning behind the arbitrary facts/formulas.
And trig is useful in lots of places in the science curriculum without the backing calculus, so teaching it in the math curriculum early to support the broader curriculum makes sense from that perspective.
[+] [-] rndn|10 years ago|reply
Does that mean that ∫f(x)dx is f.(dx, dx, …) = f(x_0)·dx + f(x_1)·dx + f(x_2)·dx + … for all x in the domain?
[+] [-] k__|10 years ago|reply
[+] [-] j2kun|10 years ago|reply
In a very strong sense, writing a document that "focuses purely on calculus" is antithetical to mathematics.
[+] [-] jordigh|10 years ago|reply
[+] [-] cottonseed|10 years ago|reply
http://ocw.mit.edu/courses/mathematics/18-014-calculus-with-...
[+] [-] eccstartup|10 years ago|reply
[+] [-] bumbledraven|10 years ago|reply
Common practice in calculus books is to define continuity using limits. I define limits using continuity; continuity is a simpler concept.
[+] [-] amelius|10 years ago|reply
[+] [-] mafribe|10 years ago|reply
[1] S. Boldo, C. Lelay, G. Melquiond, Formalization of Real Analysis: A Survey of Proof Assistants and Libraries., https://hal.inria.fr/hal-00806920v1/document
[+] [-] jojo3000|10 years ago|reply
http://isabelle.in.tum.de/dist/library/HOL/HOL/Real.html
This is the classical way (and similar to HOL Light or HOL4). AFAIK in Coq the standard way is to introduce the reals axiomatically.
The bottom theories in http://isabelle.in.tum.de/dist/library/HOL/HOL/ are all about classical real analysis, based on topology and real-normed vector spaces.
[+] [-] cottonseed|10 years ago|reply
https://coq.inria.fr/library/
[+] [-] madez|10 years ago|reply
Also, arguing by reference to the choice of appropriate values is incomplete argumentation and so it won't be accepted by a formal system. You'd have to fill these holes.
I'd suggest to start by evaluating the existing formalized constructions of the reals.
I think it is not too hard.
[+] [-] hudibras|10 years ago|reply
[+] [-] anon4|10 years ago|reply
[+] [-] blahblah3|10 years ago|reply
I always wondered why analogies and pictures weren't used more often:
examples:
A 100m sprint is a continuous function of time (f(t) = distance from starting line) because sprinters cant teleport. In fact it is uniformly continuous because people have a maximum speed.
Beating usain bolt's record is a discontinuous function of completion time because f(world_record + epsilon) = 0 while f(world_record - epsilon) = 1.
Fundamental theorem of calculus: If you want to know how fast a guy is running at time t, look at how much ground he covered in 1 second. To get more and more accurate, look at how far he traveled in 0.5 seconds and so on...
[+] [-] WallWextra|10 years ago|reply
[+] [-] murbard2|10 years ago|reply
This focus on calculating derivatives as opposed to actually understanding the concept and why the calculations work that way is, I think, why so many struggle with it.
[+] [-] ww2|10 years ago|reply
[+] [-] bumbledraven|10 years ago|reply
So the derivative (f') is the result of substituting c for x in f1. For example, if f1 = (x -> x + c) then we would have f' = (c -> c + c) = (c -> 2c).
[+] [-] ww2|10 years ago|reply
[+] [-] bumbledraven|10 years ago|reply
[+] [-] anon4|10 years ago|reply
Theorem 9.1. Let f be a continuous real-valued function. Let y be a real number. Let b ≤ c be real numbers with f(b) ≤ y ≤ f(c). Then f(x) = y for some x in [b, c].
Why is this well-formed? Once you say "Let y be a real number", I'm free to pick any real number, which means that there might not be a b and c such that f(b) ≤ y ≤ f(c). Now, I obviously understand what is said, but shouldn't this be formulated more like:
Let f be a continuous real-valued function. Let b ≤ c be real numbers from the domain of f. Let y be a real number in the closed interval bounded by f(b) and f(c) ([f(b), f(c)] or [f(c), f(b)], depending on whether f(b) ≤ f(c) or not). Then there exists an x in [b, c] such, that f(x) = y.
The way this and other theorems, definitions, etc. are formulated in the article bugs me, because I must go back and re-qualify variables based on information deduced from things introduced, after the variable in question was introduced.
[+] [-] murbard2|10 years ago|reply
[+] [-] unknown|10 years ago|reply
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