Minimalist Mathematics Lectures
The idea is that these lectures act like crystals of silver iodide do in Cloud Seeding, as nucleation points. I haven't given these nearly enough study to do them justice.
Now the viewer, without feeling under any pressure, can turn to YouTube, Google and Wikipedia to find out more about the ideas Toby has presented. They can then make them the basis of a blog post with links to relevant videos like this ...
... and other random thoughts that come up, such as the observation that this process Nancy describes is almost the reverse of the process of finding the derivative at a point from the limit of secant lines:
Then you might note that the final problem here (11 minutes 34 seconds) has an intriguing self-similarity. So you might suggest that people first watch this video on the chain rule to see what you mean:
And then you see integration by substitution often has this same form, when it is the inverse of the chain rule, as in the first example Nancy gives here:
And then you watch this video and suddenly you understand implicit differentiation and all that weird Leibnizian dy and dx stuff which was always totally mysterious before. It's just the chain rule applied to a (sometimes trivial) parameterization of the curve.
This whole process of learning new ways to explain how to do stuff, is, I think, what some people call teaching. I have no idea whether it actually works or not, as far as the student is concerned... You tell me.
And you might note that this process is relevant to dividing up a section of a curve into N pieces where N is an ever-increasing integer quantity, ...
And so you start to imagine that one day you might even be able to connect the ideas of the Fundamental Theorem of Calculus together, so that you will understand why it is that the algebraic process of taking the derivative and the inverse process of finding the indefinite integral relate the slope of a graph to the area underneath the graph. Because, despite having a first-class honours degree in Mathematical Sciences, you have never really been able to see intuitively how the inverse operation to finding the slope of a curve relates to the area underneath it. See Stokes' theorem for an idea of what a miracle this sometimes seems to be.
Now you note that maybe there is a less scary way of introducing students to the idea of "the right way to integrate" than they do at Oxford. See Oxford Lecturers on How to Make Maths Seem Boring.
Now ask yourself whether this style of teaching, even with triangle-critics like "SMH" looking over your shoulder, would leave you feeling like this in the third year:
Or like this in the first year:
But she went back. See And then I had a Million Views.
'But you're not going to get an LLB from YouTube!' you might say. This is true, but you're not going to get a Bar Association worth joining from Cambridge! See More Ludicrous Nonsense from Wall Street, Hannah Ahrendt on Pathological Political Associations and On Progressive and Degenerative Research Programs.
Now the viewer, without feeling under any pressure, can turn to YouTube, Google and Wikipedia to find out more about the ideas Toby has presented. They can then make them the basis of a blog post with links to relevant videos like this ...
... and other random thoughts that come up, such as the observation that this process Nancy describes is almost the reverse of the process of finding the derivative at a point from the limit of secant lines:
Then you might note that the final problem here (11 minutes 34 seconds) has an intriguing self-similarity. So you might suggest that people first watch this video on the chain rule to see what you mean:
And then you see integration by substitution often has this same form, when it is the inverse of the chain rule, as in the first example Nancy gives here:
And then you watch this video and suddenly you understand implicit differentiation and all that weird Leibnizian dy and dx stuff which was always totally mysterious before. It's just the chain rule applied to a (sometimes trivial) parameterization of the curve.
This whole process of learning new ways to explain how to do stuff, is, I think, what some people call teaching. I have no idea whether it actually works or not, as far as the student is concerned... You tell me.
And you might note that this process is relevant to dividing up a section of a curve into N pieces where N is an ever-increasing integer quantity, ...
And so you start to imagine that one day you might even be able to connect the ideas of the Fundamental Theorem of Calculus together, so that you will understand why it is that the algebraic process of taking the derivative and the inverse process of finding the indefinite integral relate the slope of a graph to the area underneath the graph. Because, despite having a first-class honours degree in Mathematical Sciences, you have never really been able to see intuitively how the inverse operation to finding the slope of a curve relates to the area underneath it. See Stokes' theorem for an idea of what a miracle this sometimes seems to be.
Now you note that maybe there is a less scary way of introducing students to the idea of "the right way to integrate" than they do at Oxford. See Oxford Lecturers on How to Make Maths Seem Boring.
Now ask yourself whether this style of teaching, even with triangle-critics like "SMH" looking over your shoulder, would leave you feeling like this in the third year:
Or like this in the first year:
But she went back. See And then I had a Million Views.
'But you're not going to get an LLB from YouTube!' you might say. This is true, but you're not going to get a Bar Association worth joining from Cambridge! See More Ludicrous Nonsense from Wall Street, Hannah Ahrendt on Pathological Political Associations and On Progressive and Degenerative Research Programs.
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