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Calculus they won't teach you [video]

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Re: Calculus they won't teach you [video]

#41

Post by bmitc »

alexb_ wrote: post #13
tommiegannert wrote: post #5

I.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 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.

The way calculus was discovered was via highly geometric methods that are beyond the comprehension of people today. So the historical development of calculus needs to be modernized into modern notation and concepts, which at that point basically becomes examples.

While calculus is actually easy to develop and motivate, the way we "discover" it today is nowhere close to how it was originally discovered.

In my opinion, calculus is most easily motivated via physical examples. Here is a curve, it represents some information, and what are some things we'd like to know about this curve and thus the information it represents? What are some things we can do with that information? Such motivation can be more abstract and not rely on physical examples, but I think the physical examples help tie it to the so-called real world and somewhat mirrors, in spirit, how and why calculus was originally developed.

Re: Calculus they won't teach you [video]

#42

Post by jwsteigerwalt »

Integrals -> Derivatives -> Series -> Limits?

That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).

Re: Calculus they won't teach you [video]

#43

Post by comte7092 »

zozbot234 wrote: post #38
mywittyname wrote: post #36

Earlier quoted context omitted.

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 u…

In the past, teachers would just hand out endless worksheets to drill students.

I don’t see how that is any less of an excuse for substandard teaching.

Re: Calculus they won't teach you [video]

#44

Post by hyperjeff »

chriswarbo wrote: post #34
VLM wrote: post #30

Earlier quoted context omitted.

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.

Most physicists have never read them, fwiw, as interesting as they may be for their historical role. Most classic papers go unread, and it’s not a bad thing pedagogically. How things get sorted out is usually a huge mess and following the real history can be an enormous distraction (and can confuse and misdirect students).

Re: Calculus they won't teach you [video]

#45

Post by zozbot234 »

comte7092 wrote: post #43
zozbot234 wrote: post #38

Earlier quoted context omitted.

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 u…

In the past, teachers would just hand out endless worksheets to drill students. I don’t see how that is any less of an excuse for substandard teaching.

Well, if we're going to have substandard teaching either way, let's pick the kind that actually works. Drill baby drill.

Re: Calculus they won't teach you [video]

#47

Post by blagie »

jwsteigerwalt wrote: post #42

Integrals -> Derivatives -> Series -> Limits? That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).

That's the historical progression, not the teaching progression. That's the point. Integrals were conceived over 2000 years ago. Derivatives came Newton-ish. Formalism with limits came last.

Re: Calculus they won't teach you [video]

#48

Post by JackFr »

earlgray wrote: post #12
tommiegannert wrote: post #5

I.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…

Learning names for the sake of learning names is never helpful. However the history of mathematics can help to contextualise the material and its motivation. For example, group theory has myriad modern applications. It can be (and usually is) presented in the standard mathematically-polished form, beginning with the axioms and perhaps using matrix groups or simple geometric symmetries as examples. For many learners t…

While I agree with your sentiment, studying abstract algebra at the undergraduate level we never got more than a broad sketch of a proof of the general insoluability of the quintic. It more seemed an opportunity for the professor to tell the life story of Galois -- mathematical genius and political firebrand killed in a duel over a woman at age 20.

That being said, it is difficult to find a motivating example for much of elementary abstract algebra -- which is doubly hard, because for many students abstract algebra is a new level of abstraction, math without numbers and it takes something to make that leap.

Re: Calculus they won't teach you [video]

#49

Post by SllX »

buildsjets wrote: post #46
connectsnk wrote: post #29

I am gonna bookmark this and rewatch this video again 8 years from now when my son reaches that age.

I am gonna recommend that you youtube-dl it. You really should not expect it to be accessible 8 years from now.

I second this.

While it’s still amazing which videos I can still find from 2005 or 2006, there’s not a week that goes by where I’m not kicking myself for not being quicker to the trigger to download something.

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