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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?

#51

Post by alok-g »

I would love experimentation and innovation in this direction.

I see students solving complex differentiation and integration problems without understanding what's going on, or being able to apply calculus to the real world problems.

I also wish for Lagrangian Mechanics to be the default instead of Newtonian in middle and high school. In my understanding, the core thing in the way is that Lagrangian uses calculus much more prominantly than Newtonian. Being able to teach calculus in a simpler way could be useful for that too.

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

#52

Post by ForOldHack »

ordu wrote: post #41

I believe that the only way to find the answer is to get a group of volunteers and try to teach them calculus without teaching them to differentiate and integrate by hand. It is futile to ask people who was taught the other way, they absolutely predictable will say that it is important to learn calculus the way they were taught. People do not know how education works. People just do to younger generations what was do…

This already has been done. It was denonstraitable. Project seed and a brilliant educator, Dr Steven Giavante.

The person who explained how it worked,without ever seeing it, was Dr Fernando Flores.

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

#53

Post by sambapa »

santiagobasulto wrote: post #27
pentaphobe wrote: post #25

Earlier quoted context omitted.

> music is not like math That's going to age beautifully

Yeah, my bad, I was mostly referring to "music industry". Music is 100% like math. Music is math, and math is music. Edit: inb4, I did read Hofstadter's Gödel, Escher, Bach.

No, music is not like math in that sense. Music is like math in the sense that carpentry is like music or math, or sports are like music or math. You can't be a spectator, you have to grind thousands upon thousands of exercises to be good in it.

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

#54

Post by GistNoesis »

Depends if you want to teach a mathematician or an engineer or a physicist.

If you only need an engineer, numerical approximation is good enough. You teach about slopes, and area under the curve. Automatic differentiation and numerical quadrature.

Alternatively you can teach about function representation, by introducing Taylor and Fourier series, and then you have an easy closed formula to integrate and differentiate a function.

If you want to teach a mathematician, integration by part is a great first introduction to the freedom of choice that you encounter when playing the find the proof game. There may be multiple paths to the solution.

If you want to teach a physicist, you have to teach change of variables, and how to strike the infinitesimals to simplify them. Also teach them variation of constants and separation of variables.

If you want a statistician or financier, just teach Monte-Carlo integration and Ito's formula.

Education is pick and choose, depending on what you need it for.

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

#55

Post by xigoi »

Perenti wrote: post #16
ggm wrote: post #2

Pedagogy has at least two aspects. The rote component, which people usually deprecate, and the inductive/reasoned component, where you learn "how it is" not "what to do" The thing is, the rote component is (in my personal experience) much more important than people give credit to. You need to learn some axiom applied rules to be able to use "mental muscle memory" to perform things, and then once you can operate as a…

I'm one of those people who find maths really easy. I've also had extensive experience working in mathematics education, from primary to tertiary levels. I've even written policy for our state education department. You are spot on the money. I've seen "progressive" educators who believe that technology, including CAS is the answer to teaching calculus to high school and tertiary students. I think the tech is mostly u…

> only fractions with denominators with only 2 and 5 as factors can be expressed exactly in binary without using custom datatypes and arithmetic logic.

You mean decimal, not binary. For binary, it’s only 2.

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

#56

Post by iancmceachern »

GistNoesis wrote: post #54

Depends if you want to teach a mathematician or an engineer or a physicist. If you only need an engineer, numerical approximation is good enough. You teach about slopes, and area under the curve. Automatic differentiation and numerical quadrature. Alternatively you can teach about function representation, by introducing Taylor and Fourier series, and then you have an easy closed formula to integrate and differentiate…

The challenge is you often have all 3, and more, all om the same class.

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

#57

Post by okaleniuk »

I tried that in Geometry for Programmers https://www.manning.com/books/geometry-for-programmers

I needed a short chapter on calculus so I go on and explain Bezier curves and splines, but I couldn't afford a full introduction. So, I only explained how differentiation works and gave a few examples. The rest was delegated to SymPy.

Funny thing. There are errors in differentiation formulas in the final edition of the book. Some final edits went wrong or something. Anyway, so far, we had exactly 1 (one) complaint about that.

Apparently, programmers don't care about formulas if the code is already there.

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

#58

Post by globular-toast »

skhunted wrote: post #39
WheelsAtLarge wrote: post #6

Yes, it can. The biggest failure in math education is that we spend an inordinate amount of time solving equations without understanding how to apply them to solve real-world problems. Thanks to the calculator we can now spend less time figuring out how to solve x+1=7 and put that time on how to understand when the equation is needed in the real world or how to create an equation that will help solve a real-world pro…

Students don’t like “real world” problems. They say they want more of that but when you actually do those types of problems they complain or do poorly on them. Word problems are even more confusing to students than non-word problems. The vast majority of so called real world problems aren’t things people who use math in their jobs actually do.

I guess you are referring to the "Steve has 17.5 credits, how many pizzas can he buy?" type problems?

For me, when learning calculus it was that it seemed pointless. They were teaching me to do this mechanical task, but why? Why not another task like increasing every other number in the equation by 6? It wasn't until I learnt about velocity and acceleration that it all started to make sense. The task of differentiating/integrating seemed far less important than the understanding that functions have derivatives and anti-derivatives and what that means.

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

#59

Post by tarsinge »

My experience is the opposite can be true: I learned how to differentiate and integrate in high school but did not get a deep intuitive understanding until 10 years after when I got back at Math for video games and machine learning. If I were to teach someone I would first work through practical problems with acceleration, speed and position, going from one to the other with computer calculations (I remember the concept really clicking with the use of accelerometers inputs), and only going into the symbolic notation and hand calculation after already being able to use them. But that’s just my way of learning.

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

#60

Post by globular-toast »

exabrial wrote: post #34

Imho, the better question is should calculus be taught without physics? None of it clicked for me, I stumbled to make any meaning out of any of the hand wavy explanations my poor teachers gave me… until I took my first calculus based physics class and a hundred light bulbs went on as to why any of this stuff needed to exist, but more importantly why.

I was exactly the same. I was good at integrating but had no idea why we were doing it. Then we did a physics-based module (called mechanics) and it was like you: a hundred light bulbs went on. Nowadays I "see" numbers in a totally different way. If I see a time axis I can't help but see the rates of change etc. I wonder if it could be taught in a more top down way: start thinking about rates of change in some discontinuous functions, like finance. How growth of assets over time relates to income. Then move to continuous with distance, speed, acceleration. Then do the thing where you differentiate symbolically.
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