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Simplifying and Refactoring Introductory Calculus (2018)

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Re: Simplifying and Refactoring Introductory Calculus (2018)

#61

Post by johnnyb_61820 »

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

Re: Simplifying and Refactoring Introductory Calculus (2018)

#62

Post by johnnyb_61820 »

xiphias2 wrote: post #15

,, additionally, moving limits to the end of a first-year course allows students to develop intuitions around the derivative first before seeing the formal proof of their validity’’ Waiting a year to get from intuition to theorems is a perfect way to ruin math. Math is not supposed to be easy/simple, but it supposed to be a great way to understand systems based thinking. At the same time there could be more examples…

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, and by this time they recognize the power and importance of the concept.

In some ways, it is good to be able to explain, from the ground up, why each piece is in place. But, sometimes, understanding how to build a student requires knowing when we need to temporarily handwave something away so that it is actually meaningful when they get it. And then, doing that so they don't maintain a false conception is also important, which is why handwaviness is often helpful ("this is kind of like dividing by zero, but kind of not, and we will get to the distinctions later so just trust us for the moment").

Re: Simplifying and Refactoring Introductory Calculus (2018)

#63

Post by johnnyb_61820 »

bee_rider wrote: post #5

Looks like this came out nearly 8 years ago, so… how’d it work out? Given the way job titles work these days I guess we could have some Senior Engineers here who learned calculus from this paper…

Not really. I teach high school calculus to homeschool co-ops. My earliest students are just a few years out of college. I usually only teach 2-10 students per class, and I don't think I managed to get any school to adopt my "Calculus from the Ground Up" book.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#64

Post by johnnyb_61820 »

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...

Re: Simplifying and Refactoring Introductory Calculus (2018)

#65

Post by johnnyb_61820 »

light_hue_1 wrote: post #9

> Again, by using differentials instead of derivatives, we have transformed a number of processes that students find unintuitive into a single process where the intuition is supplied by the student’s knowledge of algebra. Ah yes. Algebra. The subject all students love dearly. If only we could get students to love and appreciate calculus as much they love algebra! I have a hard time even imagining an article that is m…

So you are saying that disconnecting calculus from algebra improves the average student's calculus ability? What planet are you from?

We spend a lot of time and effort teaching kids how to do algebra, and how to manipulate equations algebraically. By the time they get anywhere near calculus, they know how to do this. Thus, rather than trying to build a whole mathematical world from scratch, the idea is to *build on what they are already practicing* rather than try to drop them in a wild, uninhabited country and say "good luck".

What's funny is the number of adult parents of students who tell me they took four semesters of calculus in college and *never understood what it was about*. This is the real crisis I'm trying to solve. We are teaching. People are learning just enough to pass tests, but aren't internalizing any of it.

I work with engineers on a daily basis. Many never fully grasped what calculus was trying to teach. But they are extremely fluent in algebra. The reason for this disconnect is that no one bothered connecting them strongly.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#66

Post by johnnyb_61820 »

jjmarr wrote: post #22
cool_dude85 wrote: post #7

Got to the place where he says "As you can see, this is identical to the d/dx() operation except that the result is not divided by dx." What does it mean with his d() operator to "divide by dx"? All of a sudden it seems like he has changed dy/dx from unfortunate notation that sort of looks like a division into something that actually is dividing two meaningful things, dy and dx? And so what the hell are dy and dx?

> And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? -- George Berkeley, namesake of UC Berkeley, in 1734, critiquing infitesimal approaches to calculus. Math uses limits because "dx" as a concept is hard to defin…

I wrote an article about Berkeley's critique here:

https://mindmatters.ai/2021/03/the-needless-complexity-of-mo...

I actually quite enjoy Berkeley. I wish he had framed his critique slightly differently, but the past is the past :)

Re: Simplifying and Refactoring Introductory Calculus (2018)

#67

Post by dj_rock »

richard_chase wrote: post #16

I think Stewart's Calculus is excellent and it is rightfully the standard textbook. No modifications needed in my opinion.

James Stewart made a lot of money off his textbook.

Here is an article about the interesting house he designed:

https://torontolife.com/real-estate/look-inside-integral-hou...

Re: Simplifying and Refactoring Introductory Calculus (2018)

#68

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)

#69

Post by srean »

johnnyb_61820 wrote: post #62
xiphias2 wrote: post #15

,, additionally, moving limits to the end of a first-year course allows students to develop intuitions around the derivative first before seeing the formal proof of their validity’’ Waiting a year to get from intuition to theorems is a perfect way to ruin math. Math is not supposed to be easy/simple, but it supposed to be a great way to understand systems based thinking. At the same time there could be more examples…

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.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#70

Post by rramadass »

cyberax wrote: post #40
rramadass wrote: post #35

Earlier quoted context omitted.

> the removal of sequences and all the associated theorems from the introductory calculus. Instead, start with limits of functions and the notion of continuity. Strongly disagree. Sequences(discrete) and Convergence are vital to understanding Calculus. Only then the idea of converging to a limit from left or right makes intuitive sense. Pair it with a graphical view of secants converging to a tangent(continuous) and…

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 gets smaller and smaller in concrete terms.

You do the above for x (an independent variable over the above sequence yielding a sequence of delta_x's) and y (a dependent function yielding a sequence of delta_y's).

Now the limit of the sequence of the ratios of the above two sets of differences (i.e. sequence of delta_y/delta_x) can be calculated and defined as the "derivative" i.e. rate of change of one w.r.t. another.

Everything is direct and there is no confusion. They can then easily map the idea of the discrete "difference" to a "differential" in a continuous domain/range and see that the exact same techniques/ideas hold.

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