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I wholeheartedly agree, I used to tutor calculus in college and getting students to focus on the fact that the derivative was the slope of the curve or the rate

I wholeheartedly agree, I used to tutor calculus in college and getting students to focus on the fact that the derivative was the slope of the curve or the rate of change and the integral was the area under the curve and the significance of that made calculus so much easy. Drawing the unit circle to explain the cos, sin, and tan was also so enlightening to students. Most students just see a list of characters when the see math problem, but "math people" understand what is going on underneath and become excited to solve a problem and see multiple paths of potentially solving it. If you can unleash that excitement the problem solves itself. Once you understand the why, the how becomes so much easier.

LeifCarrotson 5y ago
> Once you understand the why, the how becomes so much easier. The interesting thing is how each student has a different path to understand the "how". It sounds like you have particularly vivid mental imagery, and gravitate towards visualizations for solving problems. This isn't true for everyone; it's not necessarily aphantasia but a unit circle or slope/area curve just isn't as useful for some as it is for you, others might prefer to think in terms of limits or series or patterns or a myriad of other techniques. You can repeat "The derivative is the slope of the curve" until you're blue in the face but some people just won't 'get it', finding the means by which they can get it is the fun part.
- monkeywork 5y ago
  https://www.youtube.com/watch?v=rhgwIhB58PA https://www.youtube.com/watch?v=rhgwIhB58PA
  - jiriro 5y ago
    What’s the TL;DR?
    - AStrangeMorrow 5y ago
      I watched it a while ago, but the TLDR is more or less that the learning styles (ex: visual learners, or auditory, reading/writing, by doing etc...) are a myth. People do not learn better when we use their preferred learning style. But overall, people do learn better with multi-modal (mixed support / style).
      - AKluge 5y ago
        Yep - multiple representations, and interactivity, for the win. It is incumbent on the instructional designers and teachers to construct a process where the learner actually learns the concepts rather than learning processes and using computational resources to cover a lack of understanding.
- frankbreetz 5y ago
  >>You can repeat "The derivative is the slope of the curve" until you're blue in the face but some people just won't 'get it'. Then you have to explain why the slope of a line is significant. For example if you have the position of something in the form of an equation you can take the derivative and get the velocity. Anybody who understands why the slope of the line is interesting will fair far better.beyond that the area under the curve of a velocity graph will give you the distance traveled.

LeifCarrotson 5y ago

> Once you understand the why, the how becomes so much easier. The interesting thing is how each student has a different path to understand the "how". It sounds like you have particularly vivid mental imagery, and gravitate towards visualizations for solving problems. This isn't true for everyone; it's not necessarily aphantasia but a unit circle or slope/area curve just isn't as useful for some as it is for you, others might prefer to think in terms of limits or series or patterns or a myriad of other techniques. You can repeat "The derivative is the slope of the curve" until you're blue in the face but some people just won't 'get it', finding the means by which they can get it is the fun part.

monkeywork 5y ago
https://www.youtube.com/watch?v=rhgwIhB58PA https://www.youtube.com/watch?v=rhgwIhB58PA
- jiriro 5y ago
  What’s the TL;DR?
  - AStrangeMorrow 5y ago
    I watched it a while ago, but the TLDR is more or less that the learning styles (ex: visual learners, or auditory, reading/writing, by doing etc...) are a myth. People do not learn better when we use their preferred learning style. But overall, people do learn better with multi-modal (mixed support / style).
    - AKluge 5y ago
      Yep - multiple representations, and interactivity, for the win. It is incumbent on the instructional designers and teachers to construct a process where the learner actually learns the concepts rather than learning processes and using computational resources to cover a lack of understanding.
frankbreetz 5y ago
>>You can repeat "The derivative is the slope of the curve" until you're blue in the face but some people just won't 'get it'. Then you have to explain why the slope of a line is significant. For example if you have the position of something in the form of an equation you can take the derivative and get the velocity. Anybody who understands why the slope of the line is interesting will fair far better.beyond that the area under the curve of a velocity graph will give you the distance traveled.