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> I found teaching Calculus I was able to draw in students so much more if I worked in what problems Newton was trying to solve and why. It gave them a story to

> I found teaching Calculus I was able to draw in students so much more if I worked in what problems Newton was trying to solve and why. It gave them a story to follow, a reason to learn this stuff. Be careful. In undergrad, I worked as a math tutor at their tutoring institute. We had two calculus classes - one for engineers/science and the other for business/economics. If you're into Newtonian stuff or it's relevant to your degree, then your approach is all great. What I consistently saw is people in degrees like biology (or even industrial engineering) had a harder time learning because of those physics applications in the book. They came in with one problem: Having trouble with the calculus. And then they discovered they had two problems: To understand calculus they suddenly had to understand physics as well. You'd get students who were totally adept at differentiating and integrating, but would struggle with the problems that involved physics. Yes, it is important to be able to translate real world problems to math ones, but it's a bigger problem when you don't care about that particular field. Same problem with the business students. Since all the tutors were engineering folks, they had trouble tutoring the business students because their textbook was full of examples that required basic finance knowledge (and it really doesn't help that financial quantities are discrete and not continuous, adding to the tutors' confusion). If you can find an application the student is interested in, then by all means, use that approach. For a general purpose textbook, though, it hinders learning for many students who don't care for the particular choice of application the book decided to use.

dgacmu 2y ago
I got to see that yesterday with my daughter: her algebra book used PV=nRT as an example of joint proportion and asked some questions about it. She, having never seen the ideal gas law, was quite thrown by it. That said, after we went through it and had a brief physics lesson, it worked quite well and I'm glad they used the example instead of just making something up -- but it required having a tutor (me) on hand to help make the context make sense.
- HiroshiSan 2y ago
  Ah yes I know that exact problem. AoPS intro to algebra. That question was great for getting an intuitive sense of proportion, though I do remember one of the sub problems gave me some trouble.
  - dgacmu 2y ago
    That's the one! It was a very nice example. I suspect for some students they could ignore the physics, but daughter needed to walk through the physical interpretation of the components before getting into the math. From my perspective as a tutor, it was a good use of time (gotta learn it some day anyway, and it provides useful physical intuition throughout life), but I could see it causing frustration if someone just wanted to learn algebra or didn't have a resource to turn to. (Love those books. I went and asked all of my colleagues who had won teaching awards, what books they recommended, and all of them said aops)
    - arjvik 2y ago
      I owe a great deal of my mathematical maturity to going through nearly every AoPS book published in middle and high school :)
- acchow 2y ago
  > required having a tutor (me) on hand to help make the context make sense. I wonder how good ChatGPT would be as that tutor. You can ask it to “explain like I’m 12”
  - CamperBob2 2y ago
    It's almost unbelievably good as a tutor. Still, you need to check everything it tells you. Treat verification as part of the lesson.
    - dgacmu 2y ago
      So, I'm a professor, and I have, um, really strong opinions about this. :-) Perhaps too strong and long for the current forum. But I'll see if I can be brief. ChatGPT is really, really good at providing solid answers of varying levels of detail and complexity to hyper-common questions such as those used in problem sets. This is one part of the skill set of a tutor, and it's a valuable one. When I interview TAs for my classes, however, I actually put a lot more emphasis on a different skill: The ability to get into a student's head and understand where their conceptual difficulty or misunderstanding is. This is a very different skill, and it's one that ChatGPT isn't as good at, because we've gone from "maximum likelihood answers from questions that are in the middle of the distribution" into a wide range of possible sources of confusion, which the student may lack the words to explain in a precise way. In the case of my kid, the PV=nRT question manifested as "I don't get it!" (with more exclamation points). Asking ChatGPT (well, copilot, since I have institutional access to that, but it uses ChatGPT) to help understand the problem: It digressed and introduced Boyle's Law, threw in a new symbol "I" (ok, the 12 year old) had never seen for "proportional to", and ... in some sense just added to the cognitive overload. The human approach was to ask a question: Have you ever seen this equation before? (No) Oh! Well, let's talk a little about gases.. Now, responding to ChatGPT and asking "No, that didn't help. Please ELI5 instead?" actually produced a much better answer: An analogy using a balloon. Which, amusingly, is exactly how I explained the behavior of gases to her. But even here, there's a bit of a difference: In explaining it to her, I did so socratically: "Ok, so imagine a balloon. If you heat the air inside the balloon, what happens?" "Um, it gets bigger, right?" "Yup, ..." (and now, knowing that she got that part, we could go on...) That's something you can absolutely imagine trying to program around an LLM, but it's not a native way of interacting with it. So ... I'd instead be a little more cautious here and say that ChatGPT potentially provides a really useful piece of what a human tutor offers, but it loses on the interactive exchange that helps much more rapidly zoom in on the source of confusion and correct it. Assuming that it's right. I think that for a particularly sophisticated consumer, it can be more valuable, but it requires knowing what you don't know, in some sense: The ability to already isolate what you're confused by. Once you know the question to ask, ChatGPT can often provide it -- again, assuming that some quirk of its training or your phrasing doesn't cause it to generate an answer that's wrong in some way.
- BeetleB 2y ago
  Now imagine you were reading a mathematics textbook, and they use an example from something you really have no interest in (which for me would be finance). It includes terminology you have to learn - terminology that has no bearing on the math. As discouraging it is to learn math without context, it's even more discouraging to learn it in a context you hate.
  - Koshkin 2y ago
    I have no interest in finance but I liked this book: https://nononsense.gumroad.com/l/physicsfromfinance https://nononsense.gumroad.com/l/physicsfromfinance
freestyle24147 2y ago
If someone is having trouble understanding the absolute most basic part of classical mechanics (masses undergoing acceleration) something tells me they're not going to understand Calculus no matter what lens you view it through. It's not even about the equations, it's simply using it as a backstory for the motivation of the mathematics. > If you can find an application the student is interested in, then by all means, use that approach. For a general purpose textbook, though, it hinders learning for many students who don't care for the particular choice of application the book decided to use. This seems absurd. Just because some people will find a particular application less interesting, I don't think the answer is to throw out ALL applications and turn it into a generic boring slog through which no one will be able to see when and how it's useful.
- VyseofArcadia 2y ago
  > It's not even about the equations, it's simply using it as a backstory for the motivation of the mathematics. Bingo. I rarely (if ever) used motivating examples from physics as test or homework problems. This was to ground calculus in reality somewhere. This was after years of wondering how to deal with the common student complaint of, "but why, where does this even come from?" So I started telling them where it comes from.
  - calf 2y ago
    The alternative is to ground it in philosophy and theoretical mathematics which would be even more abstruse...
- acchow 2y ago
  Most people don’t need or care to learn calculus. They just want to learn how to use calculus
  - gowld 2y ago
    Similar to how most people don’t need or care to learn statistics. They just want to learn how to say statistical words to silence critics of their pseudoscience.
mihaitodor 2y ago
That's me with statistics. I have 0 interest in gambling and sports, so most of the examples put me off instantly. I'll need to study this topic in the context of biology (bioinformatics), so I'll want to find a good stats learning guide which avoids those topics.
mp05 2y ago
Your mileage may vary, but my industrial engineering program had two semesters of proper physics and two semesters of engineering mechanics.

dgacmu 2y ago

I got to see that yesterday with my daughter: her algebra book used PV=nRT as an example of joint proportion and asked some questions about it. She, having never seen the ideal gas law, was quite thrown by it. That said, after we went through it and had a brief physics lesson, it worked quite well and I'm glad they used the example instead of just making something up -- but it required having a tutor (me) on hand to help make the context make sense.

HiroshiSan 2y ago
Ah yes I know that exact problem. AoPS intro to algebra. That question was great for getting an intuitive sense of proportion, though I do remember one of the sub problems gave me some trouble.
- dgacmu 2y ago
  That's the one! It was a very nice example. I suspect for some students they could ignore the physics, but daughter needed to walk through the physical interpretation of the components before getting into the math. From my perspective as a tutor, it was a good use of time (gotta learn it some day anyway, and it provides useful physical intuition throughout life), but I could see it causing frustration if someone just wanted to learn algebra or didn't have a resource to turn to. (Love those books. I went and asked all of my colleagues who had won teaching awards, what books they recommended, and all of them said aops)
  - arjvik 2y ago
    I owe a great deal of my mathematical maturity to going through nearly every AoPS book published in middle and high school :)
acchow 2y ago
> required having a tutor (me) on hand to help make the context make sense. I wonder how good ChatGPT would be as that tutor. You can ask it to “explain like I’m 12”
- CamperBob2 2y ago
  It's almost unbelievably good as a tutor. Still, you need to check everything it tells you. Treat verification as part of the lesson.
  - dgacmu 2y ago
    So, I'm a professor, and I have, um, really strong opinions about this. :-) Perhaps too strong and long for the current forum. But I'll see if I can be brief. ChatGPT is really, really good at providing solid answers of varying levels of detail and complexity to hyper-common questions such as those used in problem sets. This is one part of the skill set of a tutor, and it's a valuable one. When I interview TAs for my classes, however, I actually put a lot more emphasis on a different skill: The ability to get into a student's head and understand where their conceptual difficulty or misunderstanding is. This is a very different skill, and it's one that ChatGPT isn't as good at, because we've gone from "maximum likelihood answers from questions that are in the middle of the distribution" into a wide range of possible sources of confusion, which the student may lack the words to explain in a precise way. In the case of my kid, the PV=nRT question manifested as "I don't get it!" (with more exclamation points). Asking ChatGPT (well, copilot, since I have institutional access to that, but it uses ChatGPT) to help understand the problem: It digressed and introduced Boyle's Law, threw in a new symbol "I" (ok, the 12 year old) had never seen for "proportional to", and ... in some sense just added to the cognitive overload. The human approach was to ask a question: Have you ever seen this equation before? (No) Oh! Well, let's talk a little about gases.. Now, responding to ChatGPT and asking "No, that didn't help. Please ELI5 instead?" actually produced a much better answer: An analogy using a balloon. Which, amusingly, is exactly how I explained the behavior of gases to her. But even here, there's a bit of a difference: In explaining it to her, I did so socratically: "Ok, so imagine a balloon. If you heat the air inside the balloon, what happens?" "Um, it gets bigger, right?" "Yup, ..." (and now, knowing that she got that part, we could go on...) That's something you can absolutely imagine trying to program around an LLM, but it's not a native way of interacting with it. So ... I'd instead be a little more cautious here and say that ChatGPT potentially provides a really useful piece of what a human tutor offers, but it loses on the interactive exchange that helps much more rapidly zoom in on the source of confusion and correct it. Assuming that it's right. I think that for a particularly sophisticated consumer, it can be more valuable, but it requires knowing what you don't know, in some sense: The ability to already isolate what you're confused by. Once you know the question to ask, ChatGPT can often provide it -- again, assuming that some quirk of its training or your phrasing doesn't cause it to generate an answer that's wrong in some way.
BeetleB 2y ago
Now imagine you were reading a mathematics textbook, and they use an example from something you really have no interest in (which for me would be finance). It includes terminology you have to learn - terminology that has no bearing on the math. As discouraging it is to learn math without context, it's even more discouraging to learn it in a context you hate.
- Koshkin 2y ago
  I have no interest in finance but I liked this book: https://nononsense.gumroad.com/l/physicsfromfinance https://nononsense.gumroad.com/l/physicsfromfinance

freestyle24147 2y ago

If someone is having trouble understanding the absolute most basic part of classical mechanics (masses undergoing acceleration) something tells me they're not going to understand Calculus no matter what lens you view it through. It's not even about the equations, it's simply using it as a backstory for the motivation of the mathematics. > If you can find an application the student is interested in, then by all means, use that approach. For a general purpose textbook, though, it hinders learning for many students who don't care for the particular choice of application the book decided to use. This seems absurd. Just because some people will find a particular application less interesting, I don't think the answer is to throw out ALL applications and turn it into a generic boring slog through which no one will be able to see when and how it's useful.

VyseofArcadia 2y ago
> It's not even about the equations, it's simply using it as a backstory for the motivation of the mathematics. Bingo. I rarely (if ever) used motivating examples from physics as test or homework problems. This was to ground calculus in reality somewhere. This was after years of wondering how to deal with the common student complaint of, "but why, where does this even come from?" So I started telling them where it comes from.
- calf 2y ago
  The alternative is to ground it in philosophy and theoretical mathematics which would be even more abstruse...
acchow 2y ago
Most people don’t need or care to learn calculus. They just want to learn how to use calculus
- gowld 2y ago
  Similar to how most people don’t need or care to learn statistics. They just want to learn how to say statistical words to silence critics of their pseudoscience.

mihaitodor 2y ago

That's me with statistics. I have 0 interest in gambling and sports, so most of the examples put me off instantly. I'll need to study this topic in the context of biology (bioinformatics), so I'll want to find a good stats learning guide which avoids those topics.

mp05 2y ago

Your mileage may vary, but my industrial engineering program had two semesters of proper physics and two semesters of engineering mechanics.