111–120 of 124 posts
I am gonna bookmark this and rewatch this video again 8 years from now when my son reaches that age.
If the web still exists. That's how you become a data hoarder.
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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.
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.
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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.
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.
Learning should be like peeling onions. You peel off one entire layer, from all sides, before moving on to a deeper layer of complexity.
Different strokes for different folks and all that but the concept of infinity doesn't really bother me that much because it doesn't really exist to me as a number more than it is a fact of the unbounded-ness of some set with a total ordering. It is more a statement of potential rather than actual existence, that given an infinite set, you can always find a "larger" element, which to me is just a result of the way it…
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Darn, at some point I started imagining he had a protocol for synchronizing clocks over a distance. assuming the difference was in longitude. (E.g. Symmetric NTP over carrier pidgeon.)
Longitude was a much tougher problem to solve.
I can't say I was always good at math as my marks varied all over the place but I never really had problems with calculus. Sure, some of its mechanics were a pain—awkward integrals for instance—but the concepts usually made sense to me.
As I've mentioned on HN previously, I'm a strong believer in teaching a simplified calculus to kids at a young age (in primary school). Early familiarization and appreciation of concepts such as limits and rates of change before they enter high school I reckon would greatly ease their burden of learning the subject.
This needn't be complicated, introducing calculus to kids with, say, versions of Zeno's dichotomy paradox can be easy and lots of fun for them—especially so when they're asked to solve the seemingly intractable problem of the frog in the center of a pool who never makes it to the bank as each successive jump he makes is only half the distance of the previous one.
Kids can clearly see the poor frog never makes it to the bank given those rules but their eyes light up when you tell them there's a 'nice' branch of math that can save him called calculus which says his feet are too big for him to drown.
Thus we've introduced calculus as a friendly helpful subject at a young age and not scary frightening subject it's so often portrayed to be. In this example we've introduced them to the concept of infinite series (and infinity), limits and how calculus can solve problems that cannot be solved by simple arithmetic alone.
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For the formal underpinnings of the infinitesimal approach, the phrase to look for is "Nonstandard Analysis": https://en.wikipedia.org/wiki/Nonstandard_analysis Fun stuff.
I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis? I loved real analysis in college, for example I found cantors set kind of crazy. Does that "exist" in nonstandard analysis? Or are there other mind-bending implications?
This is a bit of a philosophical question which hinges on what the word "equivalent" means. I disagree with the other comment. It is not totally equivalent, or even at all equivalent.
I think about math in terms of set theory. You create sets, like the real numbers, the integers, or otherwise. You can add other elements to those sets. For example:
* You can add a variety of infinities (countable, uncountable, positive, negative, etc.). You have a self-consistent system.
* You can add imaginary numbers, think of the whole thing as a ball, with a zero on one end and infinity on the other. That gives you complex analysis, which is self-consistent as well.
All of those are helpful, useful, and powerful, and generally lead to the same place where they overlap, they're not very compatible with each other, either formally or intuitively.
I view the two formulations of calculus as similar. Even if they lead to the same results, that's not the same as saying they're equivalent. They're different formalisms.
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It could be worse but one thing that constantly bothers me is the stupid new vocabulary. One of the worst examples: "number sentence". It means what you, an adult, might guess it does (well, it means one of the things you might guess it means) but why use it at all ? They introduce it before the kids have a decent idea of what a normal sentence (what, a "word sentence", I suppose?) is. Many of the symbols in them are…
"Number sentence" does not appear in the common core standards. https://www.nctm.org/uploadedFiles/Standards_and_Positions/C...
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It could be worse but one thing that constantly bothers me is the stupid new vocabulary. One of the worst examples: "number sentence". It means what you, an adult, might guess it does (well, it means one of the things you might guess it means) but why use it at all ? They introduce it before the kids have a decent idea of what a normal sentence (what, a "word sentence", I suppose?) is. Many of the symbols in them are…
> One of the worst examples: "number sentence". It means what you, an adult, might guess it does (well, it means one of the things you might guess it means Specifically, it means “equation or inequality”. > They don't really act like sentences or serve the same purpose. They act exactly like sentences, and serve exactly the same purpose, because they are declarative sentences about the relationships between numbers.…
"Exactly" is simply incorrect in both occurrences, here.
> It also, as a sibling comment notes, has nothing to do with Common Core, its an orthogonal development in pedagogical approach to the Common Core standards.
All this hit alongside Common Core. Ask teachers and they won't quibble with lumping this practice in with Common Core, since that's how it's expressed in practice, though some might be aware that it's not in the standards. Source: I know a lot of teachers.
> What you offer there is an analogy, not an example.
It's both, so you're technically wrong, which I gather you think is the worst kind of wrong. Analogy's probably the better word here, though, sure, if I was only going to use one.
> Analogies don't really act like examples and don’t serve the same purpose.
They act exactly like them. Using your sense of "exactly" from above.
Let me know, not if anyone here can do Calculus (that is ordinary; who can't? ), but rather whether you actually personally use Calculus, and please describe generally an example of needing Calculus to figure something out in the last 5 years. I think my point is that the history of mathematics is exponentially more interesting than mathematics. We like mathematics, but we like stories far more than we like whatever…
To your second point, I would disagree that we learn maths that we may likely never directly use not because of historical fascination, but because progressively expanding student's mathematical knowledge serves the purpose of meta-learning, learning to learn. As to your question I've touched on a little for manipulation of some of the classic equations in ecology and population biology, growth curves, population gro…
That was not my point. No one learns math because of historical fascination. The point was simply the stories about the math are more interesting than the math itself, which isn't any stretch of the imagination.