johnnyb_61820

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

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

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 :)

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.

I hope you check out my book, "Calculus from the Ground Up":

https://www.amazon.com/Calculus-Ground-Jonathan-Laine-Bartle...

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").

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

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.

DNA is a lot more like an FPGA than a traditional programming language. Basically, everything is wrapped in a while(true) that has conditionals as to whether or not it engages. Promoter sequences are the condition statements, and transcription factors are very similar to variable states, where the concentration of the transcription factor is essentially its value.

The documentary covered some important ground but leaned too heavily on the idea that Tesla is responsible for customer abuse of their platform.

For those looking for more general reasons why "No Code" works for some tasks but doesn't scale (i.e., to present to a boss), I did an article series on No Code:

https://mindmatters.ai/t/no-code-software/

Additionally, as an anecdote, I've seen organizations get burned on the security side because they had people who knew nothing about web security trying to build apps.

Makes you wonder who you can trust out there.

Social media companies only work when they attract a mass of people, who could have been spending their time elsewhere or on another site. Does this imply a responsibility on the part of the social media company to their early users who were promised one thing but received another?

Big publishing houses like to throw the term "predatory journals" around a lot, but a look at their own publishing practices raises questions.

I thought you all would be interested in my latest book, "Learn to Program with Assembly". This is a 64-bit rewrite of my previous assembly language book, "Programming from the Ground Up."

Why the "universal algorithm" is the modern equivalent of the perpetual motion machine, and researchers don't seem willing to give up on it despite its theoretical problems.

While the concept of "code debt" is a useful metaphor, many people take its implications too seriously. This article looks at the biggest misapplications of the "code debt" concept, and how sometimes code debt is like getting money for free.

Looks like they are aiming for level 4 and still using safety drivers, though.

Here, it is shown that blockchains aren't intrinsically required for NFTs, and, in fact, create additional problems for them.

The new NXP NTAG 424 DNA chips are pretty amazing. Additionally, I've been working on an open-source library for writing to the chips with iOS here:

https://github.com/johnnyb/nfc-dna-kit

Many of the recent "developments" in technology are actually delivering the opposite of what technology is supposed to offer.

This is an article on some of the reasons why computer programming should be taught to everyone at a high school level.

Walks through the problems inherent with attempting to make something "no code".

A short history of number systems that got added to mathematics (and math education), as well as the proposal for having hyperreal numbers be the next addition to the curriculum.

Article discusses various tweaks to the math curriculum which would help students learn it easier. Some of the problem of math education is that it has accrued "code debt" over the years and needs some "refactoring."