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I agree with this. But I don't see the students rejecting this, but the education degreed peoples who choose texts and the publishers want to make all learning

I agree with this. But I don't see the students rejecting this, but the education degreed peoples who choose texts and the publishers want to make all learning for all people. This is foolish. Most people don't need to know calculus. And if you do learn it, do so with rigor so you actually learn it and not just the appearance of it, which is much much worse.

DrSAR 11mo ago
Not sure I agree with 'appearance [...] is much worse'. Given the choice between a class room of first years who believe (in themselves and) an appearance of calculus knowledge or a room of scared undergrads that recoil from any calculus-inspired argument they 'have never learnt it properly', I'll take the former. I can work with that much more easily. Sure, some things might break - but what's the worst that can happen? We'll sort out the rigour later while we patch the bruises of overextending some analogies.
godelski 11mo ago
> Most people don't need to know calculus I don't like this line of argument. It applies to many things, many of which we'd laugh at for suggesting. Most people don't need to know how to read. Most people don't need to know how to add. Most people don't need to know how to use a computer. The foolishness of these statements are all subjective and based on what one believes one "needs". Yet, I have no doubt all of these things can improve peoples lives. I'd argue the same with calculus. While I don't compute derivatives and integrals every day[0], I certainly use calculus every day. That likely sounds weird, but it is only because one thinks that math and computation is the same. When I drive I use calculus as I'm thinking about my rates of change, not only my velocity. Understanding different easing functions[1] I am able to create a smoother ride, be safer, drive faster, and save fuel. All at the same time! The magic of the rigor is often lost, but the magic is abstraction. That's what we've done here with the car example. I don't need to compute numbers to "do math", I only need to have an abstract formulation. To understand that multiple variables are involved and there are relationships between them, and understanding that there are concepts like a rate of change, the rate of change of the rate of change, and even the rate of change of the rate of change of the rate of change! (the jerk!)[2]. That's still math. It may not be as rigorous, but a rigorous foundation gives you a greater ability to be less rigorous at times and take advantage of the lessons. So yes, most people "don't need calculus" but learning it can give them a lot of power in how to think. This is true for much of mathematics. You may argue that this is not how it is taught, but with that I'll agree. The inefficiency of how it is taught is orthogonal to the utility of its lessons. [0] Is a physicist not doing math just when they do symbol manipulation? I can tell you with great confidence, and experience, that much of their job is doing math without the use of numbers. It is about deriving formulations. Relationship! [1] https://easings.net/ https://easings.net/ [2] https://en.wikipedia.org/wiki/Jerk_%28physics%29 https://en.wikipedia.org/wiki/Jerk_%28physics%29
- altmanaltman 11mo ago
  I get the argument you're making but that's a bit like saying cavemen used to do calculus as they hunt, which is a valid way of looking at this maybe but they didn't really "use calculus" just intuition. Simillarly, when learning calculus, most people do not do so at a driving course, they do it in the classroom. If you're willing to stretch the definition of what "using" maths is then it can apply to everything and that devalues the concept as a whole. I'm not on the toilet, I'm doing calculus!
  - godelski 11mo ago
    I understand that interpretation but it's different from what I meant. The difference may be in two different cavemen. One throws his spear on intuition alone. The other is thinking about the speed he throws, how the animal moves, the wind, and so on. There is a formulation, though not as robust as you'd see in a physics book. > the definition of what "using" maths is then it can apply to everything In a sense yes. Math is a language, or more accurately a class of languages. If you're formulating your toilet activities, then it might be math. But as you might gather, there's nuance here. I quoted Poincaré in another comment but I'll repeat here as I think it may help reduce confusion (though may add more) Math is not the study of numbers, but the relationships between them. Or as the category people say "the study of dots and arrows". Anything can be a dot, but you need the arrows
    - altmanaltman 11mo ago
      Yeah, I do understand your point of view. I'm just doubting if it applies universally, like you may superimpose that assumption on the thinking caveman, but is the thinking caveman really doing the same? Yes, technique is one thing, but being really good at throwing spears doesn't make you really good at math, is my argument. And most people will encounter maths in a formal setting while lacking the broader perspective that everything is technically "math". Yet, we need to see the argument from the common person's view, if we're talking about calculus and learning in the traditional sense. The view you stated is quite esoteric and doesn't generalize well in this setting imo. It's like a musician saying they see music in every action, but to most non-musicians (even if the stated thing is kind of true) that doesn't make a lot of sense etc.
rramadass 11mo ago
> Most people don't need to know calculus. People should have at the minimum a conceptual idea of Calculus. A good motivation is Everyday Calculus: Discovering the Hidden Math All around Us by Oscar E. Fernandez - https://press.princeton.edu/books/paperback/9780691175751/everyday-calculus https://press.princeton.edu/books/paperback/9780691175751/ev... > And if you do learn it, do so with rigor so you actually learn it This is not strictly necessary for everybody. The conceptual ideas are what are important; else you are merely doing "plug-and-chug" Maths without any understanding. You need to focus on rigor only based on your needs and at your own pace. Concepts come first Formalism comes second. A good example; In the Principia Newton actually uses the phrase Quantity of Motion for what we define today as Momentum. The phrase is evocative and beautifully captures the main concept instead of the bland p = m x v definition which though correct and needed for calculations conveys no mental imagery. In Mathematics one should always approach a concept/idea from multiple perspectives including (but not limited to) Imagination, Conceptual, Graphical, Symbolic, Relationship, Applications, Definition/Theorem/Proof.
- wing-_-nuts 11mo ago
  Thank you for the book link! One major problem I always had with calculus was that we didn't focus nearly enough on applications of it, outside of some optimization problems, which I loved. I originally went for engineering, and took cal I and II, then switched to cs and poof all of the math heavy stuff vanished (except maybe some LA for graphics)

DrSAR 11mo ago

Not sure I agree with 'appearance [...] is much worse'. Given the choice between a class room of first years who believe (in themselves and) an appearance of calculus knowledge or a room of scared undergrads that recoil from any calculus-inspired argument they 'have never learnt it properly', I'll take the former. I can work with that much more easily. Sure, some things might break - but what's the worst that can happen? We'll sort out the rigour later while we patch the bruises of overextending some analogies.

godelski 11mo ago

> Most people don't need to know calculus I don't like this line of argument. It applies to many things, many of which we'd laugh at for suggesting. Most people don't need to know how to read. Most people don't need to know how to add. Most people don't need to know how to use a computer. The foolishness of these statements are all subjective and based on what one believes one "needs". Yet, I have no doubt all of these things can improve peoples lives. I'd argue the same with calculus. While I don't compute derivatives and integrals every day[0], I certainly use calculus every day. That likely sounds weird, but it is only because one thinks that math and computation is the same. When I drive I use calculus as I'm thinking about my rates of change, not only my velocity. Understanding different easing functions[1] I am able to create a smoother ride, be safer, drive faster, and save fuel. All at the same time! The magic of the rigor is often lost, but the magic is abstraction. That's what we've done here with the car example. I don't need to compute numbers to "do math", I only need to have an abstract formulation. To understand that multiple variables are involved and there are relationships between them, and understanding that there are concepts like a rate of change, the rate of change of the rate of change, and even the rate of change of the rate of change of the rate of change! (the jerk!)[2]. That's still math. It may not be as rigorous, but a rigorous foundation gives you a greater ability to be less rigorous at times and take advantage of the lessons. So yes, most people "don't need calculus" but learning it can give them a lot of power in how to think. This is true for much of mathematics. You may argue that this is not how it is taught, but with that I'll agree. The inefficiency of how it is taught is orthogonal to the utility of its lessons. [0] Is a physicist not doing math just when they do symbol manipulation? I can tell you with great confidence, and experience, that much of their job is doing math without the use of numbers. It is about deriving formulations. Relationship! [1] https://easings.net/ https://easings.net/ [2] https://en.wikipedia.org/wiki/Jerk_%28physics%29 https://en.wikipedia.org/wiki/Jerk_%28physics%29

altmanaltman 11mo ago
I get the argument you're making but that's a bit like saying cavemen used to do calculus as they hunt, which is a valid way of looking at this maybe but they didn't really "use calculus" just intuition. Simillarly, when learning calculus, most people do not do so at a driving course, they do it in the classroom. If you're willing to stretch the definition of what "using" maths is then it can apply to everything and that devalues the concept as a whole. I'm not on the toilet, I'm doing calculus!
- godelski 11mo ago
  I understand that interpretation but it's different from what I meant. The difference may be in two different cavemen. One throws his spear on intuition alone. The other is thinking about the speed he throws, how the animal moves, the wind, and so on. There is a formulation, though not as robust as you'd see in a physics book. > the definition of what "using" maths is then it can apply to everything In a sense yes. Math is a language, or more accurately a class of languages. If you're formulating your toilet activities, then it might be math. But as you might gather, there's nuance here. I quoted Poincaré in another comment but I'll repeat here as I think it may help reduce confusion (though may add more) Math is not the study of numbers, but the relationships between them. Or as the category people say "the study of dots and arrows". Anything can be a dot, but you need the arrows
  - altmanaltman 11mo ago
    Yeah, I do understand your point of view. I'm just doubting if it applies universally, like you may superimpose that assumption on the thinking caveman, but is the thinking caveman really doing the same? Yes, technique is one thing, but being really good at throwing spears doesn't make you really good at math, is my argument. And most people will encounter maths in a formal setting while lacking the broader perspective that everything is technically "math". Yet, we need to see the argument from the common person's view, if we're talking about calculus and learning in the traditional sense. The view you stated is quite esoteric and doesn't generalize well in this setting imo. It's like a musician saying they see music in every action, but to most non-musicians (even if the stated thing is kind of true) that doesn't make a lot of sense etc.

rramadass 11mo ago

> Most people don't need to know calculus. People should have at the minimum a conceptual idea of Calculus. A good motivation is Everyday Calculus: Discovering the Hidden Math All around Us by Oscar E. Fernandez - https://press.princeton.edu/books/paperback/9780691175751/everyday-calculus https://press.princeton.edu/books/paperback/9780691175751/ev... > And if you do learn it, do so with rigor so you actually learn it This is not strictly necessary for everybody. The conceptual ideas are what are important; else you are merely doing "plug-and-chug" Maths without any understanding. You need to focus on rigor only based on your needs and at your own pace. Concepts come first Formalism comes second. A good example; In the Principia Newton actually uses the phrase Quantity of Motion for what we define today as Momentum. The phrase is evocative and beautifully captures the main concept instead of the bland p = m x v definition which though correct and needed for calculations conveys no mental imagery. In Mathematics one should always approach a concept/idea from multiple perspectives including (but not limited to) Imagination, Conceptual, Graphical, Symbolic, Relationship, Applications, Definition/Theorem/Proof.

wing-_-nuts 11mo ago
Thank you for the book link! One major problem I always had with calculus was that we didn't focus nearly enough on applications of it, outside of some optimization problems, which I loved. I originally went for engineering, and took cal I and II, then switched to cs and poof all of the math heavy stuff vanished (except maybe some LA for graphics)