8 ms·

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 cann

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.)

k__ 11y ago
Maybe to teach the people how to use formulas they didn't make up themself. This is the part of Math that most of us do after school.
impendia 11y ago
> 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.
- stephencanon 11y ago
  sin and cos are the natural basis of solutions of y'' + y = 0. How do you define them?
  - tarentel 11y ago
    Ratios in a right triangle perhaps?
    - threatofrain 11y ago
      How did you get that information? How did you, in a non-magical way, go from information about an angle to information about a ratio? If students don't know this information, then perhaps they are studying applications. So, what applications are students taught in typical trigonometric texts? Periodic behavior perhaps? Like sound? Only perhaps a brief blurb in the text that application is even possible. Perhaps they look at something about an incline plane. It is unlikely that they will touch projectiles. It appears that trigonometry is there to give students some sense of mild comfort for future work in physics or engineering. This makes me think, "Why not statistics instead?"
      - AnimalMuppet 11y ago
        > How did you, in a non-magical way, go from information about an angle to information about a ratio? By having a right triangle? The rest of your post seems to show that you want trig to be about periodic behavior, when it really is about triangles. That's what trigonometry means - measuring triangles. Yes, trig has applications to periodic behavior, projectiles, differential equations, inclined planes, and all kinds of other stuff. But the point of a trig class is not to teach the applications. The point is to teach the tools, and maybe touch on the applications.
        btilly 11y ago
        The problem is this. Using compass and ruler constructions there is a set of angles you can construct, and you can calculate sin and cos for those angles. You can even write the values for those out explicitly. However no part of this construction sheds light on how to find sin and cos for angles that you don't know how to construct. Or even gives good intuition that no matter how you do it, you can define it in a way that makes sense for all angles. In fact we draw a picture, people look at it, and their intuition tells them that things will work out. Very few students will notice the logical gaps. But to close the logical gaps, you need to start with Calculus first, and then derive trig formulas from that. (Yes, I'm aware of the history here. Euclid presented trig reasonably rigorously a very long time before Calculus. Newton invented Calculus in the 1600s, and then used it as a heuristic to figure out answers that he then rederived using trig in The Principia. Leibniz reinvented Calculus in part based on inspiration from Newton's work. None of this was made formally correct until the late 1800s.) (I have no opinion on pedagogical arguments about which is best to present first. I believe that we present trig first as a holdover from a curriculum where The Elements was the standard textbook until very recently.)
- tokenrove 11y ago
  I don't know, I agree with the parent. I hated trig in my first encounters with it; the thing that made it practical was writing videogames and graphics demos, and the thing that made it interesting was calculus. For many people, it's the last math they're taught, they don't get the applications, and it's no wonder they don't hunger for more.
Tloewald 11y ago
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.
- adamtj 11y ago
  You can find your epsilons and deltas in the definition of an open ball. They are just two different ways of writing the same idea. For example, in Definition 2.1, the 'h' in the open ball F is your epsilon and the 'h' in B is the delta. In an epsilon-delta proof, you show that for all |x - c| < delta, |f(x) - f(c)| < epsilon. The open ball B is just the set of all x satisfying |x - c| < delta and the open ball F is the set of all f(x) satisfying |f(x) - f(c)| < epsilon. Any epsilon or delta that you choose implies a set of numbers satisfying those conditions. Those sets are open balls. By using them, you don't have to say things like "all 'x' such that ...". Which method you prefer probably depends on what you're more familiar with and how you tend to think. Open balls can be easier to visualize, if that's how you think. Note that not just any ball will do. Closed balls are open balls that also include their boundary. That is, they use a less-than-or-equal-to instead of less-than. Which you use can make a big difference. An open ball on the real number line is just an open interval, an interval excluding its endpoints. It's easy to generalize: an open ball in a Cartesian plane is a circle excluding its border. In three dimensions, it's a sphere excluding its surface . . . and that's why it's called a ball.
  - Tloewald 11y ago
    I understand what an open ball is (and surely if you read my original post carefully this would be clear), I just don't think it actually makes the discussion clearer. In particular, when explaining something to people, starting with creating unfamiliar concepts is a Bad Idea if those concepts don't have a significant payoff. In particular, because this is not a discussion about arbitrary spaces, the use of the word "Ball" is counter-intuitive. (But I admit I am probably biased by my own experience.)
rndn 11y ago
> An integral is the inner product of a function and a suitably-dimensioned unit. 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?
- stephencanon 11y ago
  Integral[f(x)g(x)dx] defines an inner product on a space of functions (glossing over exactly what functions) on an interval. So Integral[f(x) dx] is the inner product (f, 1), where "1" is the constant function g(x) = 1 on the interval.
  - rndn 11y ago
    So ∫f(x)g(x)dx = (f.g)·dx?
    - stephencanon 11y ago
      No, f·g = ∫f(x)g(x)dx.
      - rndn 11y ago
        But don't you need to account for the width dx of the slices you sum up? I.e. wouldn't f.g = f(x_0)g(x_0) + f(x_1)g(x_1) + … over a real interval blow up?
        JadeNB 11y ago
        The point is that f.g (or, as a mathematician or physicist might write it, ⟨f, g⟩ or ⟨f|g⟩) has no independent meaning, but must be defined; and one way to define it (for `L^2` functions, the only one compatible with the `L^2` norm) is as stephencanon did at https://news.ycombinator.com/item?id=9620263 https://news.ycombinator.com/item?id=9620263 .
        bandrami 11y ago
jerf 11y ago
"(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.
dragonwriter 11y ago
> 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.

k__ 11y ago

Maybe to teach the people how to use formulas they didn't make up themself. This is the part of Math that most of us do after school.

impendia 11y ago

> 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.

stephencanon 11y ago
sin and cos are the natural basis of solutions of y'' + y = 0. How do you define them?
- tarentel 11y ago
  Ratios in a right triangle perhaps?
  - threatofrain 11y ago
    How did you get that information? How did you, in a non-magical way, go from information about an angle to information about a ratio? If students don't know this information, then perhaps they are studying applications. So, what applications are students taught in typical trigonometric texts? Periodic behavior perhaps? Like sound? Only perhaps a brief blurb in the text that application is even possible. Perhaps they look at something about an incline plane. It is unlikely that they will touch projectiles. It appears that trigonometry is there to give students some sense of mild comfort for future work in physics or engineering. This makes me think, "Why not statistics instead?"
    - AnimalMuppet 11y ago
      > How did you, in a non-magical way, go from information about an angle to information about a ratio? By having a right triangle? The rest of your post seems to show that you want trig to be about periodic behavior, when it really is about triangles. That's what trigonometry means - measuring triangles. Yes, trig has applications to periodic behavior, projectiles, differential equations, inclined planes, and all kinds of other stuff. But the point of a trig class is not to teach the applications. The point is to teach the tools, and maybe touch on the applications.
      - btilly 11y ago
        The problem is this. Using compass and ruler constructions there is a set of angles you can construct, and you can calculate sin and cos for those angles. You can even write the values for those out explicitly. However no part of this construction sheds light on how to find sin and cos for angles that you don't know how to construct. Or even gives good intuition that no matter how you do it, you can define it in a way that makes sense for all angles. In fact we draw a picture, people look at it, and their intuition tells them that things will work out. Very few students will notice the logical gaps. But to close the logical gaps, you need to start with Calculus first, and then derive trig formulas from that. (Yes, I'm aware of the history here. Euclid presented trig reasonably rigorously a very long time before Calculus. Newton invented Calculus in the 1600s, and then used it as a heuristic to figure out answers that he then rederived using trig in The Principia. Leibniz reinvented Calculus in part based on inspiration from Newton's work. None of this was made formally correct until the late 1800s.) (I have no opinion on pedagogical arguments about which is best to present first. I believe that we present trig first as a holdover from a curriculum where The Elements was the standard textbook until very recently.)
tokenrove 11y ago
I don't know, I agree with the parent. I hated trig in my first encounters with it; the thing that made it practical was writing videogames and graphics demos, and the thing that made it interesting was calculus. For many people, it's the last math they're taught, they don't get the applications, and it's no wonder they don't hunger for more.

Tloewald 11y ago

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.

adamtj 11y ago
You can find your epsilons and deltas in the definition of an open ball. They are just two different ways of writing the same idea. For example, in Definition 2.1, the 'h' in the open ball F is your epsilon and the 'h' in B is the delta. In an epsilon-delta proof, you show that for all |x - c| < delta, |f(x) - f(c)| < epsilon. The open ball B is just the set of all x satisfying |x - c| < delta and the open ball F is the set of all f(x) satisfying |f(x) - f(c)| < epsilon. Any epsilon or delta that you choose implies a set of numbers satisfying those conditions. Those sets are open balls. By using them, you don't have to say things like "all 'x' such that ...". Which method you prefer probably depends on what you're more familiar with and how you tend to think. Open balls can be easier to visualize, if that's how you think. Note that not just any ball will do. Closed balls are open balls that also include their boundary. That is, they use a less-than-or-equal-to instead of less-than. Which you use can make a big difference. An open ball on the real number line is just an open interval, an interval excluding its endpoints. It's easy to generalize: an open ball in a Cartesian plane is a circle excluding its border. In three dimensions, it's a sphere excluding its surface . . . and that's why it's called a ball.
- Tloewald 11y ago
  I understand what an open ball is (and surely if you read my original post carefully this would be clear), I just don't think it actually makes the discussion clearer. In particular, when explaining something to people, starting with creating unfamiliar concepts is a Bad Idea if those concepts don't have a significant payoff. In particular, because this is not a discussion about arbitrary spaces, the use of the word "Ball" is counter-intuitive. (But I admit I am probably biased by my own experience.)

rndn 11y ago

> An integral is the inner product of a function and a suitably-dimensioned unit. 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?

stephencanon 11y ago
Integral[f(x)g(x)dx] defines an inner product on a space of functions (glossing over exactly what functions) on an interval. So Integral[f(x) dx] is the inner product (f, 1), where "1" is the constant function g(x) = 1 on the interval.
- rndn 11y ago
  So ∫f(x)g(x)dx = (f.g)·dx?
  - stephencanon 11y ago
    No, f·g = ∫f(x)g(x)dx.
    - rndn 11y ago
      But don't you need to account for the width dx of the slices you sum up? I.e. wouldn't f.g = f(x_0)g(x_0) + f(x_1)g(x_1) + … over a real interval blow up?
      - JadeNB 11y ago
        The point is that f.g (or, as a mathematician or physicist might write it, ⟨f, g⟩ or ⟨f|g⟩) has no independent meaning, but must be defined; and one way to define it (for `L^2` functions, the only one compatible with the `L^2` norm) is as stephencanon did at https://news.ycombinator.com/item?id=9620263 https://news.ycombinator.com/item?id=9620263 .
        bandrami 11y ago

jerf 11y ago

"(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.

dragonwriter 11y ago

> 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.