johnnyb_61820

Indeed! Glad I have fans :) I also have an electronics book "Electronics for Beginners" plus a few others.

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.

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.

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").

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 was wanting to write a piece on why the second derivative looks the way it does. I had about 10 different calculus books I was looking through trying to find a solid answer and there was none. So, I eventually decided to try and derive the formula myself. I was quite surprised when I was able to derive a formula, but it was different. I tried to figure out for a while how to get from my formula to the standard one, until I eventually realized that the standard formulation was itself problematic.

It's in the "Calculus from the Ground Up" book as "Appendix B", but I don't use it in the main text so as not to confuse students who take further calculus courses. I found a middle ground for the book which neither forces the new notation nor commits the mistakes of the previous one. The book is not heavy in higher-order derivatives anyway, so the usage is minimal.

What I did in my book to keep everything algebraic but not introduce weird notation is just set the derivative equal to a variable. So, say m = dy/dx. Then, the second derivative is just dm/dx.

The advantage to the revised notation is that you can describe things that are difficult or impossible to describe in the other notation. For example, you can legitimately look at d^2y/d^2x (note the placement of the 2 on the denominator to see how this is different). This is a valid ratio under my system but invalid under the standard system (though I actually consider my system to be the standard system just with prior mistakes corrected).

Indeed! I'm glad you enjoyed the book! I hope you wrote it a nice Amazon review :)

If you are going to use infinitesimals, though, it requires some additional doing for notation. The standard notation for higher-order derivatives (and partial derivatives) needs to be modified in order for them to work (but they do work great once you do this).

Instead of the second derivative being "d^2y/dx^2" it is "(d^2y/dx^2) - (dy/dx)(d^2x/dx^2)" and the differentials can be manipulated just like any other entity. Additionally, you can infer this notation by simply applying the quotient rule to the first derivative (which is a quotient of infinitesimals).

See more:

"Extending the Algebraic Manipulability of Differentials" ( 10.48550/arXiv.1801.09553 )

"Total and Partial Differentials as Algebraically Manipulable Entities" ( 10.48550/arXiv.2210.07958 )

Much of the calls for AI legislation have been based on fear of the technology. Instead, we should establish rational policy goals for AI regulation that align with the technology's usage, not our fears about the future.