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Simplifying and Refactoring Introductory Calculus (2018)

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Re: Simplifying and Refactoring Introductory Calculus (2018)

#71

Post by rramadass »

johnnyb_61820 wrote: post #64
rramadass wrote: post #30

Some good books for introductory Calculus; 1) Calculus: Basic Concepts for High Schools by Lev Tarasov. Soviet-era book written as a dialogue between the author and reader. Absolutely fantastic (also see his other books on Probability etc.) - https://mirtitles.org/2018/09/04/calculus-basic-concepts-for... 2) Calculus: An Intuitive and Physical Approach by Morris Kline. A classic; any book by Morris Kline is a must-ha…

I hope you check out my book, "Calculus from the Ground Up": https://www.amazon.com/Calculus-Ground-Jonathan-Laine-Bartle...

Thanks for the pointer; will definitely take a look at it.

Btw - Your name seemed familiar and then i remembered, you are also the author of "Programming from the Ground Up".

Re: Simplifying and Refactoring Introductory Calculus (2018)

#72

Post by rramadass »

srean wrote: post #51
rramadass wrote: post #30

Some good books for introductory Calculus; 1) Calculus: Basic Concepts for High Schools by Lev Tarasov. Soviet-era book written as a dialogue between the author and reader. Absolutely fantastic (also see his other books on Probability etc.) - https://mirtitles.org/2018/09/04/calculus-basic-concepts-for... 2) Calculus: An Intuitive and Physical Approach by Morris Kline. A classic; any book by Morris Kline is a must-ha…

Thanks for the list. I have been on the lookout for a specific book I was suggested as a kid. This was when our high school Physics was traveling a few paces ahead of our mathematics curriculum. All I remember is that the Indian paperback edition had a blue cover. It was very helpful. I find it hard to understand the persistent calculus hate that I see on HN. For us it was a very enjoyable experience. We learned it t…

Try a google image search for something like "India Calculus book blue cover" (and similar phrases) and you might most probably find what you have been looking for.

When it comes to Calculus most people get traumatized due to the excessive rigor and abstraction many teachers/books start with in the beginning. There is the Heine discrete sequence approach that is far easier to understand (see https://news.ycombinator.com/item?id=49312120) than the standard epsilon-delta definition but for some reason it is not emphasized. Add the fact that many mindlessly recommend Spivak/Apostol etc. books which are highly rigorous and a beginning student is lost and starts to develop an aversion to the subject.

For some motivation see Everyday Calculus: Discovering the Hidden Math All around Us by Oscar Fernandez - https://press.princeton.edu/books/paperback/9780691175751/ev...

Re: Simplifying and Refactoring Introductory Calculus (2018)

#73

Post by rramadass »

2b3a51 wrote: post #58
rramadass wrote: post #30

Some good books for introductory Calculus; 1) Calculus: Basic Concepts for High Schools by Lev Tarasov. Soviet-era book written as a dialogue between the author and reader. Absolutely fantastic (also see his other books on Probability etc.) - https://mirtitles.org/2018/09/04/calculus-basic-concepts-for... 2) Calculus: An Intuitive and Physical Approach by Morris Kline. A classic; any book by Morris Kline is a must-ha…

> Calculus: The Princess of Mathematics by H.C.Verma et al. Garudalife have a store on Abebooks, so available in the UK at under £12 per volume (each circa 340 pages) but inevitably £18 postage. These look interesting and one day we'll have print on demand in each territory!

Don't give up so easily :-)

You can buy it from the publisher directly since they do international shipping. Contact them for the shipping charges.

You might also want to check Amazon India and see how much it costs to order from them.

Talk to some of your local schools/colleges/libraries and see whether they can get it and you borrow from them.

Finally; ask your "Indian Friend" (direct/indirect) to get the volumes on one of his/her visits to India ;-)

Re: Simplifying and Refactoring Introductory Calculus (2018)

#74

Post by johnnyb_61820 »

srean wrote: post #69
johnnyb_61820 wrote: post #62

Earlier quoted context omitted.

I'm curious if you've taught calculus? Do your students remember limits by the end of calculus? Most studies show that students DO NOT RETAIN limit concepts (ESPECIALLY epsilon-delta ones). It is used as a crutch and then largely discarded before anyone is actually comfortable/familiar with them. By being a little handwavy at the beginning, you can then tell Dorothy that she's had her ruby slippers with her all along…

> Most studies show that students DO NOT RETAIN limit concepts Is that so ? I would not have guessed. I am not being sarcastic. Going by experiences of my own high school cohort I would have claimed that limits had a more lasting impression. BTW I enjoyed your arxiv paper on 2nd order derivatives.

Here's an interesting paper on the topic for epsilon-delta proofs:

https://arxiv.org/pdf/1701.05187

Also see Table 2 of this older paper for limit thinking in general:

https://u.math.biu.ac.il/~katzmik/sullivan76.pdf

Re: Simplifying and Refactoring Introductory Calculus (2018)

#75

Post by BobaFloutist »

jgord wrote: post #13

I have strong opinions on how Calc should be introduced - visually. I think we don't cover basics like the distributive rule in school very well, and that it should be a much more nuts n bolts visual / measuring / counting experience. Ive attempted to outline how I think this stuff should be taught, by making a video tour of the concepts - from Counting, to Distributive rule / algebra, to Quadratics then the Derivati…

My experience was that the visuals were a cool distraction from familiarity with the rules/identities. People always want a shortcut to comfort, and they often want it to be that a thorough exploration of motivating examples, often with graphics, grants understanding and comfort before facility, but I've always found that a relatively cursory description of the motivating example is more than enough, and after that I just need well-organized reference materials, worked examples, and rote practice.

It's frustrating, since it feels like an intelligent enough approach should be able to skip the rote practice, but it turns out that conceptual mastery requires mastery of execution, which requires familiarity, which requires practice.

And I say this as someone with very strong intuition, who was always asking why I needed to practice if I already understood, who often grasped concepts immediately - turns out, I still needed practice to understand thoroughly.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#76

Post by srean »

rramadass wrote: post #72
srean wrote: post #51

Earlier quoted context omitted.

Thanks for the list. I have been on the lookout for a specific book I was suggested as a kid. This was when our high school Physics was traveling a few paces ahead of our mathematics curriculum. All I remember is that the Indian paperback edition had a blue cover. It was very helpful. I find it hard to understand the persistent calculus hate that I see on HN. For us it was a very enjoyable experience. We learned it t…

Try a google image search for something like "India Calculus book blue cover" (and similar phrases) and you might most probably find what you have been looking for. When it comes to Calculus most people get traumatized due to the excessive rigor and abstraction many teachers/books start with in the beginning. There is the Heine discrete sequence approach that is far easier to understand (see https://news.ycombinator.…

Thanks, image search is a good idea. Why did I not think of that. It was probably Silvanus' classic. I can't seem to find a picture of the cover from that age.

I think for us our first exposure was through the notion of instantaneous velocity in Physics. Our Physics classes were usually a step or two ahead and there the calculus was a little handwavy. The mathematics classes in the coming weeks would soon follow making the ideas a little more grounded.

It worked for us is all I can say. Were us is my group of high school friends.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#77

Post by srean »

johnnyb_61820 wrote: post #74
srean wrote: post #69

Earlier quoted context omitted.

> Most studies show that students DO NOT RETAIN limit concepts Is that so ? I would not have guessed. I am not being sarcastic. Going by experiences of my own high school cohort I would have claimed that limits had a more lasting impression. BTW I enjoyed your arxiv paper on 2nd order derivatives.

Here's an interesting paper on the topic for epsilon-delta proofs: https://arxiv.org/pdf/1701.05187 Also see Table 2 of this older paper for limit thinking in general: https://u.math.biu.ac.il/~katzmik/sullivan76.pdf

Thanks.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#78

Post by [deleted] »

johnnyb_61820 wrote: post #61
johnnyb_61820 wrote: post #60

Thanks for the shout-out! I teach calculus to homeschool co-op students regularly, and I got tired of the existing books on the market (I used to teach from Saxon Calculus) so I wrote my own, "Calculus from the Ground Up". This paper represents the principles I used for writing the book. Interestingly, my reformulation of the Leibniz notation for the second derivative was actually the result of writing the book. I wa…

On the second derivative side, a fuller treatment (including applying the approach to partial differentials) is given in the paper "Total and Partial Differentials as Algebraically Manipulable Entities". https://arxiv.org/abs/2210.07958

[deleted]

Re: Simplifying and Refactoring Introductory Calculus (2018)

#79

Post by anthk »

rramadass wrote: post #30

Some good books for introductory Calculus; 1) Calculus: Basic Concepts for High Schools by Lev Tarasov. Soviet-era book written as a dialogue between the author and reader. Absolutely fantastic (also see his other books on Probability etc.) - https://mirtitles.org/2018/09/04/calculus-basic-concepts-for... 2) Calculus: An Intuitive and Physical Approach by Morris Kline. A classic; any book by Morris Kline is a must-ha…

Calculus Made Easy, but renewed by Martin Gardner. Yes, that one.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#80

Post by cyberax »

rramadass wrote: post #70
cyberax wrote: post #40

Earlier quoted context omitted.

I don't disagree. Sequences are important, and the bridge between sequences and functions (Bolzano–Weierstrass theorem, mean value theorem, etc.) is crucial. But they are not immediately needed to understand the limits. Try to see how far you can get just with the epsilon-delta formulation of limits of functions.

My point is that the ideas of Sequences/Convergence/Infinity (Heine's definition) provides a better intuition than the epsilon/delta limits of function definition. The former is discrete so you could literally take any interval and demonstrate how an infinite sequence of real numbers within that interval can converge to a "limit". The student can now understand the idea of a "difference" i.e. a finite change that get…

I'm in the opposite camp. Most of the calculus foundation actually depends on continuity.

And one thing that has changed recently: mobile phones and zooming. Every child now has an almost instinctive understanding of being able to find the correct zoom level on a map so that some feature can fit completely on the screen.

And now the key insight: you can zoom-in on a continuous function indefinitely.

So the epsilon-delta formulation for the limit of functions becomes almost trivially easy to explain. And you can build from there.

I understand that the notion of continuity in itself requires limits to define it properly. But you can do that _later_. I'm speaking from experience of helping a friend's child understand calculus.

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