71–80 of 102 posts
This is pretty good for the 80% who don't go on to study maths, and still a useful intro to the ones who redo it properly at Uni.
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…
> If you only need an engineer, numerical approximation is good enough. Fluid dynamics enters the chat. And whaddya mean "only" need an engineer?
https://en.wikipedia.org/wiki/Lattice_Boltzmann_methods
You have well defined operations and conservation laws well behaved by construction.
Don't let Navier & Stokes enter the chat, and you will be fine.
The whole problem of integration is that mathematicians historically introduced new functions, and try to relate them together, in interesting way.
Polynomials are good and well behaved. Algebras too.
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…
> no one knows how education works, now one knows what is important, why it is important, and is it possible to replace it with something else. Those are some pretty baseless statements, considering pedagogy is an entire field https://en.wikipedia.org/wiki/Pedagogy and it's an active area of research. I think it's reductive to end with 'no one knows'.
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…
> no one knows how education works, now one knows what is important, why it is important, and is it possible to replace it with something else. Those are some pretty baseless statements, considering pedagogy is an entire field https://en.wikipedia.org/wiki/Pedagogy and it's an active area of research. I think it's reductive to end with 'no one knows'.
Moreover, no one knows what is knowledge, how it works, and how it can be acquired. If it was known, then AI developers would already have build a general artificial intelligence. There are some known rituals, if you do enough of them with uneducated youth then some of them become knowledgeable. But no one knows how it works and why it doesn't work every time.
Earlier quoted context omitted.
> If you only need an engineer, numerical approximation is good enough. Fluid dynamics enters the chat. And whaddya mean "only" need an engineer?
Even in fluid dynamics the game is only as hard as you make it for yourself : https://en.wikipedia.org/wiki/Lattice_Boltzmann_methods You have well defined operations and conservation laws well behaved by construction. Don't let Navier & Stokes enter the chat, and you will be fine. The whole problem of integration is that mathematicians historically introduced new functions, and try to relate them together, in intere…
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…
So you propose to teach them different methods of solving. I think, while your mapping from solution methods to professions makes sense, the act of solving should be taught as an afterthought. The goal should be to build a robust mental model. Personally i find playing around with computer visualisations helps a lot with that. The same is even more important for differential equations, imo. I wish i would have spent…
It starts very early with things like mental calculus, where you try to get kids to compute things like 99 times 101, 98 times 102, finding tricks and rules to make the computation "easier". We could instead teach them to follow mentally some multiplication algorithm, but instead we try to promote some exploration and looking for pattern.
In chess, similarly you can train for puzzle to develop some vision without having to calculate, like multiplication table.
The education system must also take care not to prevent it from adapting to further changes by being too rigid. If the cat can't get out of the box, the box won't ever grow bigger.
But you can't have too many cats getting out, or you'll soon have an empty box and hangry cats.
I don't know what the goals should be, far from me the idea of proposing to teach, I only show paths that can be taken.
Earlier quoted context omitted.
> no one knows how education works, now one knows what is important, why it is important, and is it possible to replace it with something else. Those are some pretty baseless statements, considering pedagogy is an entire field https://en.wikipedia.org/wiki/Pedagogy and it's an active area of research. I think it's reductive to end with 'no one knows'.
Pedagogy cannot know how knowledge is generated in the human mind. Pedagogy can advise on how to construct the process of education, like classrooms or grades, but it cannot know what knowledge of calculus is. Moreover, no one knows what is knowledge, how it works, and how it can be acquired. If it was known, then AI developers would already have build a general artificial intelligence. There are some known rituals,…
> 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 communicated to another. When it is communicated the nueral activity and memory in the other persons brain are clearly doing 'something' to store and be able to recall concepts and steps needed.
> But no one knows how it works and why it doesn't work every time.
Those are massively different though, clearly transmitting the same knowledge via different approaches is possible, i.e. OP's question about teaching calculus in non-traditional ways.
Earlier quoted context omitted.
> no one knows how education works, now one knows what is important, why it is important, and is it possible to replace it with something else. Those are some pretty baseless statements, considering pedagogy is an entire field https://en.wikipedia.org/wiki/Pedagogy and it's an active area of research. I think it's reductive to end with 'no one knows'.
A charitable interpretation of what he said is that we need to do real world experiment instead of of relying on the opinion of supposed experts. Real data based on well made experiments trumps the opinion of any particular person.
Research on education, cognition, memory, and learning is definitely much more
> Real data based on well made experiments
and not just
> relying on the opinion of supposed experts
I don't think it's acceptable to throw your hands up and say 'no one knows' when clearly people have spent entire careers thinking about this stuff and testing it in experiments, and plane-scale educational techniques. We're far closer to 'knowing' than at any point in history, and true knowing is a philosophical topic anyway without a satisfying 'answer'.