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Ask HN: Can calculus be taught without differentiating or integrating by hand?

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Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?

#91

Post by chris_wot »

ordu wrote: post #84
kingkongjaffa wrote: post #79

Earlier quoted context omitted.

I feel like you are trying to border on philosophical 'knowing' rather than the very established body of evidence around cognition and understanding, and the ability to share and pass information to another. > no one knows what is knowledge, how it works, and how it can be acquired Chimps literally teach each other to use rocks as tools. Clearly knowledge or 'how to accomplish a task' is something that can be communi…

> Chimps literally teach each other to use rocks as tools. Clearly knowledge or 'how to accomplish a task' is something that can be communicated to another. Yes, and no. Lets take calculus as an example and the skill to take derivatives. If a teacher show you how to take derivatives, you'll kinda get the idea but if you try it, you would face a lot of problems and you probably would need teacher's help. To become pro…

> To become proficient with derivatives, you need to teach yourself by taking a lot of derivatives. But... wait... you are taking derivatives, you're not doing something special that can be called "learning".

Actually, that is the process of learning. Doing it yourself and checking your results is part of a feedback loop essential in learning anything. That's why the answers in the back of most math textbooks are so important, if you can't tell you've got the problem wrong, then you have "taught" yourself something wrong.

Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?

#92

Post by sargstuff »

dual numbers[0] in lisp/lambda calculus context aka let a=lisp car and b-epsilon be lisp cdr. Then type of number/calculation can also be symbolic. Although, per Lewis Caroll's concept of imaginary numbers with no concept of 'time', this doesn't quite work per functions/lists needing 'time to compute' a result. Guess why Lewis Carrol's 'turing' rabbit was always late.

[0] : https://www.youtube.com/watch?v=ceaNqdHdqtg

Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?

#93

Post by globular-toast »

seanmcdirmid wrote: post #87
globular-toast wrote: post #61

Earlier quoted context omitted.

This is also a good example of why we teach things that might seem "useless". You just don't know what someone will find useful in their life. The teachers probably didn't even think of video games having physics and needing to understand it for that. In my uni class I think I was the only one to use forces and masses (and therefore acceleration) instead of just speed. I initially used speed but then my school learni…

I'd rather start the other way around: teach things steeped in the context that they are useful in. Everything is useful, so teach them usefully. Not all of us are great at storing away knowledge and accessing it much later when we find it actually useful.

But then you won't be able to teach as much. I think this whole thread boils down to pedagogy is difficult.

Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?

#94

Post by Tainnor »

chris_wot wrote: post #33
Tainnor wrote: post #23

Earlier quoted context omitted.

I'm sorry, but if you don't know how to solve x+1=7, you don't actually understand maths.

I agree, but how far do you take this? Do you say if you don't know the Peano axioms you don't understand maths?

No.

Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?

#95

Post by Tainnor »

WheelsAtLarge wrote: post #45
Tainnor wrote: post #23

Earlier quoted context omitted.

I'm sorry, but if you don't know how to solve x+1=7, you don't actually understand maths.

I made it an easy equation to make a point. Pick your favorite multivariable set of equations. Depending on the situation a computer can solve it in a fraction of the time and with better accuracy than a human with pencil and paper. And it can create thousands of variations in a matter of minutes.

> I made it an easy equation to make a point

It was a bad example then. Obviously there are equations that are difficult to solve by hand and you don't need to remember all tricks in order to do successful modeling, but if you're not able to solve x+1=7, it means you don't even understand what an equation is, so you won't be able to use it for modeling anything.

Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?

#96

Post by Tainnor »

kqr wrote: post #46
Tainnor wrote: post #23

Earlier quoted context omitted.

I'm sorry, but if you don't know how to solve x+1=7, you don't actually understand maths.

Maybe that's fine? I mean I'm sure a lot of people who fix stuff up in their homes know very little of the physics behind their hammers, but they can still use them just fine. Maybe a good tool does not need to be understood on a deeper level to be used. If we gave students more exposure to maths as a tool to be used, rather than arcane formulae and symbolic manipulations, they could build an intuition and appreciati…

> I'm not saying x+1=7 is a good example of that.

Exactly, it's not. I'm not arguing with what you're saying, I'm arguing with what GP wrote.

Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?

#97

Post by Tainnor »

max_ wrote: post #48

Numerical Methods already made it pointless for me to learn how to solve different kinds of integrals. I don't need to master any of the dozens symbolic integration techniques. I am not sacred of any equation looking too complex to solve. I just write less than 30 lines of python and have the solution to the problem via numerical methods. At the end of the day what I care about is that answers, not symbolic technique…

If you use numerical methods routinely, you need to know about error propagation too, otherwise you may get results that are completely wrong. That's theory too, although of a different kind than the typical symbolic calculus.

Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?

#98

Post by impendia »

I'm a math professor. In my opinion, no. This would be possible, but I don't believe it would be a good idea.

I firmly believe that, if you want to learn any math subject, whether elementary or research-level or anywhere in between, you have to do a bunch of computations. Trying to skip this would be like trying to master chess by reading books and learning theory, without ever playing against live opponents.

I think it's very plausible that we could do with much less symbolic computation by hand, and more use of computers and calculators. Doing integrals and derivatives by hand is often overemphasized, because that's the easiest thing to test for. But I believe that even in an ideal world, this should have a prominent place in the curriculum.

Re: Ask HN: Can calculus be taught without differentiating or integrating by hand?

#99

Post by Optimal_Persona »

I'm probably an atypical calculus learner. I'm middle aged and brushing up on my math(s)/physics so I can do audio DSP, understand the acoustics/physics of musical instruments, and explore patterns in nature - all for personal/artistic reasons rather than job-related (for now :-)).

I had an "a-ha" moment when I came across Ivan Savov's "No Bullshit Guide to Math and Physics" about 6 weeks ago in terms of finding a resource that meets my needs perfectly. I personally find it useful to work through a certain number of problems by hand so I understand the concepts, as well as how folks did the work before the advent of computers. I'm also the kind of person who will type out examples by hand when learning a new computer language so I can get a feel for it.

https://minireference.com/static/excerpts/noBSguide_v5_previ...

I find learning calculus & physics together is a good mix for me especially in light of my musical/DSP interests. FWIW just in these 6 weeks - I've been able to read through things like Claude Shannon's classic Theory of Information or Manfred Schroeder's books on chaos/fractals and number theory and get the gist of the math in a way I couldn't before.

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