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Starting out with integration makes so much more sense, and it's way more graspable from a geometric standpoint (what's the area under this line?) as opposed to

by dvt 4y ago

Starting out with integration makes so much more sense, and it's way more graspable from a geometric standpoint (what's the area under this line?) as opposed to the weirdness (and relative abstraction) of the derivative.

77pt77 4y ago
I completely disagree. Also, there's basically one way to differentiate (not really, but kind of) wheres integration is so tricky there are many, many ways to do it.
- dvt 4y ago
  I totally understand your position, and I completely agree that integration is way more complex, often doesn't work, and very quickly gets you in the Gamma function danger zone, but I still think it should be taught first (maybe just looking at elementary functions) because it's way more intuitive in my opinion. There's a reason it was developed first, historically speaking.
- abdullahkhalids 4y ago
  Freshmen calculus I does not have to immediately jump into mechanically difficult integrals. You can leave those for Calculus II and concentrate on the fundamentals of integration and differentiation in Calc I.
  - 77pt77 4y ago
    You completely missed my point. I mean the actual mathematical analysis concepts involved in integration (measure theory, Rieman integration, principal value) are far more intricate than the concepts involved in differentiation.
    - abdullahkhalids 4y ago
      Oh I see. That is indeed correct, but even a difficult book like Apostle largely concentrates on Riemann's theory for the introduction. Learning should be like peeling onions. You peel off one entire layer, from all sides, before moving on to a deeper layer of complexity.
8note 4y ago
Derivatives have nice physical problems to solve like "find the maximum" and "find the relationship between these two variables" which you can find by measuring a linear slope a lot of times. Learning derivatives starts with linear regression, which you do when learning about lines on a Cartesian plane. By the time you know about drawing lines on planes to find the integral area underneath, you've already figured out the derivative so you know what direction to move your pen in next while drawing the line
- dvt 4y ago
  > problems to solve like "find the maximum" and "find the relationship between these two variables" I think these are actually pretty abstract, so it's hard exactly to grasp what a derivative is until maybe Calc II or III -- imo, the main problem is that it just seems like mathematical black magic: you need to define a limit, you need to use that definition to subtract two slopes, the complexity of which magically kind of vanishes, and you end up with a "method" for spitting out derivatives. Whereas finding area under a curve seems like a common sense thing to do: you take smaller and smaller slices of a thing and add it all up. I do agree that if you really learn anti-derivatives (which, by all accounts, is harder), you basically get derivatives for free.
watwut 4y ago
Derivatives are massively easier. Starting with easy concepts makes more sense

by dvt 4y ago

77pt77 4y ago

I completely disagree. Also, there's basically one way to differentiate (not really, but kind of) wheres integration is so tricky there are many, many ways to do it.

dvt 4y ago
I totally understand your position, and I completely agree that integration is way more complex, often doesn't work, and very quickly gets you in the Gamma function danger zone, but I still think it should be taught first (maybe just looking at elementary functions) because it's way more intuitive in my opinion. There's a reason it was developed first, historically speaking.
abdullahkhalids 4y ago
Freshmen calculus I does not have to immediately jump into mechanically difficult integrals. You can leave those for Calculus II and concentrate on the fundamentals of integration and differentiation in Calc I.
- 77pt77 4y ago
  You completely missed my point. I mean the actual mathematical analysis concepts involved in integration (measure theory, Rieman integration, principal value) are far more intricate than the concepts involved in differentiation.
  - abdullahkhalids 4y ago
    Oh I see. That is indeed correct, but even a difficult book like Apostle largely concentrates on Riemann's theory for the introduction. Learning should be like peeling onions. You peel off one entire layer, from all sides, before moving on to a deeper layer of complexity.

8note 4y ago

Derivatives have nice physical problems to solve like "find the maximum" and "find the relationship between these two variables" which you can find by measuring a linear slope a lot of times. Learning derivatives starts with linear regression, which you do when learning about lines on a Cartesian plane. By the time you know about drawing lines on planes to find the integral area underneath, you've already figured out the derivative so you know what direction to move your pen in next while drawing the line

dvt 4y ago
> problems to solve like "find the maximum" and "find the relationship between these two variables" I think these are actually pretty abstract, so it's hard exactly to grasp what a derivative is until maybe Calc II or III -- imo, the main problem is that it just seems like mathematical black magic: you need to define a limit, you need to use that definition to subtract two slopes, the complexity of which magically kind of vanishes, and you end up with a "method" for spitting out derivatives. Whereas finding area under a curve seems like a common sense thing to do: you take smaller and smaller slices of a thing and add it all up. I do agree that if you really learn anti-derivatives (which, by all accounts, is harder), you basically get derivatives for free.

watwut 4y ago

Derivatives are massively easier. Starting with easy concepts makes more sense