This is exactly the calculus that I was taught in school - right down to the pizza slices analogy.
Calculus they won't teach you [video]
31–40 of 124 posts
Re: Calculus they won't teach you [video]
#32I.e. mathematical history. I think one power of calculus (and everything else) is that you don't need the history. The same way I don't need the history of the hammer to use a nail. What drives my interest is the examples. At the end, the book author talks about students not paying attention until the teacher starts talking about examples. To me, that suggests you should first contextualize and create a need for the…
As others have argued, one can learn mathematics without its history, but it is very helpful to motivate many of its abstractions. A good example of this are some of the textbooks of David Bressoud. In a "Radical approach to Analysis", concepts are introduced in historical order. The book starts with the heat equation and its solution by Fourier series which was rejected at the time, in constrast to most books that start with the real numbers and limits.
Re: Calculus they won't teach you [video]
#33Earlier quoted context omitted.
I disagree. The biggest thing that can help you learn something complex is by introducing it as it was discovered, and making someone feel like they could have come up with it on their own.
I don’t see why the history of discovery of something should be a particularly good way to rediscover it. As a concrete example, if someone asks “how do we know the Earth is round” or “how do we know the planets orbit the sun”, one could go through ways to figure this out with math and minimal technology. But one could also point out that we have spaceships and probes and have actual pictures! You can see the Earth f…
“How do we know the world is round” is an interesting example to use. Eratosthenes got an estimate for the radius of the Earth that was pretty close using, I dunno, sticks and his eyeball or whatever ancient Greeks had.
The history is useful there, I think. First off, it is motivating — that’s a pretty big question, really shows what math can accomplish without much hardware. Second, it provides some sort of useful constraints if you are trying to work out the computation yourself. In a textbook the problems are always a bit artificial, some info will be given, some hidden. “Anything an Ancient Greek would have” is a bit more real-world, you can see where the constraints come from.
Re: Calculus they won't teach you [video]
#34Earlier quoted context omitted.
I don’t see why the history of discovery of something should be a particularly good way to rediscover it. As a concrete example, if someone asks “how do we know the Earth is round” or “how do we know the planets orbit the sun”, one could go through ways to figure this out with math and minimal technology. But one could also point out that we have spaceships and probes and have actual pictures! You can see the Earth f…
Another concrete example of the conflict is classical vs quantum physics which is nearly exclusively taught as the historical story of the quantum revolution or whatever you call it around 1900. It would be trivial to skip the historical story and just "yo here's a FET transistor, a solar cell, and a vacuum tube, and the math just works, so get used to it" but nobody is ever taught that way. Would be interesting for…
> Would be interesting for someone to attempt to write a non-historical physics textbook.
I recall some prominent physicist saying they've never read Einstein's original papers on relativity; and they don't have any real desire to, since we understand it much better now, and have better examples, notations, etc.
Re: Calculus they won't teach you [video]
#35Earlier quoted context omitted.
I don’t see why the history of discovery of something should be a particularly good way to rediscover it. As a concrete example, if someone asks “how do we know the Earth is round” or “how do we know the planets orbit the sun”, one could go through ways to figure this out with math and minimal technology. But one could also point out that we have spaceships and probes and have actual pictures! You can see the Earth f…
Another concrete example of the conflict is classical vs quantum physics which is nearly exclusively taught as the historical story of the quantum revolution or whatever you call it around 1900. It would be trivial to skip the historical story and just "yo here's a FET transistor, a solar cell, and a vacuum tube, and the math just works, so get used to it" but nobody is ever taught that way. Would be interesting for…
A downside to the modern approach is that “plug the equations into a computer and let it churn for a bit” plays a big role I think. This is seems a bit annoying from a teaching point of view
* “here’s what the computer came up with” doesn’t really build as much intuition as doing the equations out by hand
* there’s enough material in “how to plug it into a computer and make it run quickly” for a whole degree
Re: Calculus they won't teach you [video]
#36Earlier quoted context omitted.
Yea, it has gotten worse in the US thanks to the common core bullshit. My daughter’s math education has been eroded with shitty ed tech apps like the ones from MHM[0]. I’ve been able to support her but there’s many kids that are not so fortunate. [0] https://www.hmhco.com/
What's wrong with common core?
Most of the hate seems to stem from the changes in mathematics teaching. Instead of focusing on rote memorization, the curriculum wants students to understand the concepts, thus it teaches more ways to tackle a smaller pool of topics. I believe this idea comes from the way that mathematics is taught in countries whose students outperform Americans. A friend of mine who was schooled in eastern Asia claims his daughter is learning the same methods he did growing up.
FWIW, I learned some cool techniques that I didn't know about when my friend showed me his daughter's homework. But I can see how parents of fifth graders who aren't able to help their kids with their homework because they don't understand it / never learned that way would be angry.
States all had their own standards, which were largely the same, but just kind of different enough. Prior to Common Core, private companies sprang up to fill the need of mapping all of the different state standards to a common standard.
Re: Calculus they won't teach you [video]
#37I.e. mathematical history. I think one power of calculus (and everything else) is that you don't need the history. The same way I don't need the history of the hammer to use a nail. What drives my interest is the examples. At the end, the book author talks about students not paying attention until the teacher starts talking about examples. To me, that suggests you should first contextualize and create a need for the…
I think something quite different than you, but that isn't important. The important thing is that different people are learn differently and are motivated in different ways. Curriculums should take this into account and accommodate different learning styles. Unfortunately, they usually don't.
Re: Calculus they won't teach you [video]
#38Earlier quoted context omitted.
What's wrong with common core?
Most of the hate seems to stem from the changes in mathematics teaching. Instead of focusing on rote memorization, the curriculum wants students to understand the concepts, thus it teaches more ways to tackle a smaller pool of topics. I believe this idea comes from the way that mathematics is taught in countries whose students outperform Americans. A friend of mine who was schooled in eastern Asia claims his daughter…
Modern "teaching" practices are just pretending to teach concepts; they're mostly based on the expectation that the students will somehow "rediscover" everything on their own. You can't truly understand concepts without doing quite a bit of solid practice. Many teachers will fail to provide the needed guidance, if only because they often never properly learned the subject themselves; and "Common Core" ends up being used as an excuse for substandard teaching.
Re: Calculus they won't teach you [video]
#39Earlier quoted context omitted.
I don’t see why the history of discovery of something should be a particularly good way to rediscover it. As a concrete example, if someone asks “how do we know the Earth is round” or “how do we know the planets orbit the sun”, one could go through ways to figure this out with math and minimal technology. But one could also point out that we have spaceships and probes and have actual pictures! You can see the Earth f…
“How do we know the world is round” is an interesting example to use. Eratosthenes got an estimate for the radius of the Earth that was pretty close using, I dunno, sticks and his eyeball or whatever ancient Greeks had. The history is useful there, I think. First off, it is motivating — that’s a pretty big question, really shows what math can accomplish without much hardware. Second, it provides some sort of useful c…
Erastosthenes did it by comparing the angle of the shadow cast by a stick on the summer solstice at noon in two cities, Alexandria and Syene (now Aswan), which are about 800 km apart. What was special about Syene was that it is so close to the Tropic of Cancer line that the sun would hit the bottoms of wells on the solstice, meaning that he only needed one stick, in Alexandria, and the distance between the cities, to do his estimate. His error was about 2%.
He's also famous for the sieve of Erastosthenes for finding prime numbers and figuring out the earth's axial tilt.
Re: Calculus they won't teach you [video]
#40Earlier quoted context omitted.
I disagree. The biggest thing that can help you learn something complex is by introducing it as it was discovered, and making someone feel like they could have come up with it on their own.
I don’t see why the history of discovery of something should be a particularly good way to rediscover it. As a concrete example, if someone asks “how do we know the Earth is round” or “how do we know the planets orbit the sun”, one could go through ways to figure this out with math and minimal technology. But one could also point out that we have spaceships and probes and have actual pictures! You can see the Earth f…
I think it can depend on the particular discovery whether we should teach "here's how the first person figured it out" or skip right to "here's how a later person found a much better, easier to understand approach".