6 ms·

This is a good book, but the first time I read it, I was puzzled. There's all this stuff about "infinitely small pieces" like every other calculus resource. It

by woah 6y ago

This is a good book, but the first time I read it, I was puzzled. There's all this stuff about "infinitely small pieces" like every other calculus resource. It wasn't until I had practice using the derivative rules and the chain rule that calculus clicked for me. I think calculus should be taught like "here's a set of tools to help you rewrite your equations to make them output the slope of their graph". Then, afterwards, you can get into what makes it all tick. Sort of like teaching a student to write "hello world" before diving into the low level mechanics of a compiler.

ducttapecrown 6y ago
The low level "what makes it all tick" of calculus is known as analysis, set theory and the topology of the real line, and is the subject of upper division college math classes. The stuff about "infinitely small pieces" is a mnemonic made to help you remember and intuit the tools of calculus.
- hodgedual 6y ago
  You know, I never understood this. Limits (how a function behaves as dx approaches zero) are much easier (for me) to understand than infinitesimals (an idea which obviously can't even exist on the real line), and it's much harder to end up in contradiction.
  - todd8 6y ago
    I agree, I was a math major and remember how high school calculus using infinitesimals ended up confusing my friends in college.
  - Koshkin 6y ago
    Yet, the idea of infinitesimals is very popular in applications. For example, in undergrad physics courses, when deriving formulas (essentially, doing math!) they keep talking about a "small area" or a "small volume" (where a given function is assumed to stay constant).
  - madhadron 6y ago
    When you're actually computing things, such as numerically approximating an integral, you do so by breaking it into small pieces and adding them together. Infinitesimals are the limit of that process as the smallness goes to zero. I think the key problem here is that students use the symbolic rules but never actually do numerical computations of this kind.
  - HPsquared 6y ago
    That's the beauty of mathematics - two people are able to approach a question from totally different perspectives, and produce the same result.
ookdatnog 6y ago
> I think calculus should be taught like "here's a set of tools to help you rewrite your equations to make them output the slope of their graph". Then, afterwards, you can get into what makes it all tick. I feel the exact opposite way. The method you're describing is how I was taught calculus in high school, that is, with a strong focus on practicing mechanical manipulation of formulas rather than building deep understanding. And I just disconnected from that, I got no mental reward from procedurally finding the derivative or the integral of a function for the 100th time. Lack of motivation meant lack of focus, which in turn made my performance drop, and at the time I assumed I probably just wasn't as good at mathematics as I thought I was. That lasted until I got mathematics classes in college, where many of my mathematics courses used a more first-principles-based approach, and I performed well again. > Sort of like teaching a student to write "hello world" before diving into the low level mechanics of a compiler. I found Structure and Interpretation of Computer Programs (https://mitpress.mit.edu/sites/default/files/sicp/index.html https://mitpress.mit.edu/sites/default/files/sicp/index.html) the most enlightening intro to programming book I ever read. Its approach is also strongly first-principles-based. I'm not writing this comment to say that you're wrong; the method you describe may very well work well for you, and many others with you. Rather, I'm using the opportunity to spread an opinion that I feel quite strongly about: the best way to learn a topic is completely student-dependent and there is no such thing as the one true way in which a topic should be taught. The most powerful motivator is always intrinsic motivation, and different minds are motivated by different challenges. Unfortunately it is hard to bring into practice in a batch-processing based education system.
- wegs 6y ago
  An intermediate approach I've been taking with my own son is to use calculus and initially not bother with either the manipulation OR the formalism. You have something moving, have the position and want the speed. Or vice-versa. You use it in a bunch of places to build out vocabulary and fluency, and build up both the mechanics and the formalism later. And we derive a few formulas along the way. integrals of x and derivatives of x2 (x^2, depending on your programming language) pop up all over the place.
  - 6y ago
HPsquared 6y ago
I was first exposed to calculus by a couple of visual analogies for the meaning of differentiation and integration. Keeping those firmly in mind, the more formal parts (algebraic manipulations, limits, etc.) come more easily.

by woah 6y ago

ducttapecrown 6y ago

The low level "what makes it all tick" of calculus is known as analysis, set theory and the topology of the real line, and is the subject of upper division college math classes. The stuff about "infinitely small pieces" is a mnemonic made to help you remember and intuit the tools of calculus.

hodgedual 6y ago
You know, I never understood this. Limits (how a function behaves as dx approaches zero) are much easier (for me) to understand than infinitesimals (an idea which obviously can't even exist on the real line), and it's much harder to end up in contradiction.
- todd8 6y ago
  I agree, I was a math major and remember how high school calculus using infinitesimals ended up confusing my friends in college.
- Koshkin 6y ago
  Yet, the idea of infinitesimals is very popular in applications. For example, in undergrad physics courses, when deriving formulas (essentially, doing math!) they keep talking about a "small area" or a "small volume" (where a given function is assumed to stay constant).
- madhadron 6y ago
  When you're actually computing things, such as numerically approximating an integral, you do so by breaking it into small pieces and adding them together. Infinitesimals are the limit of that process as the smallness goes to zero. I think the key problem here is that students use the symbolic rules but never actually do numerical computations of this kind.
- HPsquared 6y ago
  That's the beauty of mathematics - two people are able to approach a question from totally different perspectives, and produce the same result.

ookdatnog 6y ago

> I think calculus should be taught like "here's a set of tools to help you rewrite your equations to make them output the slope of their graph". Then, afterwards, you can get into what makes it all tick. I feel the exact opposite way. The method you're describing is how I was taught calculus in high school, that is, with a strong focus on practicing mechanical manipulation of formulas rather than building deep understanding. And I just disconnected from that, I got no mental reward from procedurally finding the derivative or the integral of a function for the 100th time. Lack of motivation meant lack of focus, which in turn made my performance drop, and at the time I assumed I probably just wasn't as good at mathematics as I thought I was. That lasted until I got mathematics classes in college, where many of my mathematics courses used a more first-principles-based approach, and I performed well again. > Sort of like teaching a student to write "hello world" before diving into the low level mechanics of a compiler. I found Structure and Interpretation of Computer Programs (https://mitpress.mit.edu/sites/default/files/sicp/index.html https://mitpress.mit.edu/sites/default/files/sicp/index.html) the most enlightening intro to programming book I ever read. Its approach is also strongly first-principles-based. I'm not writing this comment to say that you're wrong; the method you describe may very well work well for you, and many others with you. Rather, I'm using the opportunity to spread an opinion that I feel quite strongly about: the best way to learn a topic is completely student-dependent and there is no such thing as the one true way in which a topic should be taught. The most powerful motivator is always intrinsic motivation, and different minds are motivated by different challenges. Unfortunately it is hard to bring into practice in a batch-processing based education system.

wegs 6y ago
An intermediate approach I've been taking with my own son is to use calculus and initially not bother with either the manipulation OR the formalism. You have something moving, have the position and want the speed. Or vice-versa. You use it in a bunch of places to build out vocabulary and fluency, and build up both the mechanics and the formalism later. And we derive a few formulas along the way. integrals of x and derivatives of x2 (x^2, depending on your programming language) pop up all over the place.
- 6y ago

HPsquared 6y ago

I was first exposed to calculus by a couple of visual analogies for the meaning of differentiation and integration. Keeping those firmly in mind, the more formal parts (algebraic manipulations, limits, etc.) come more easily.