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Definitely agree - I struggled beyond basic integration in High School, and my "why" and "what is it" type questions about calculus were met with annoyed "read the book" replies and young me lost interest in higher maths at a time when my mind was more plastic.

As an adult having these personalities enthusiastically breaking down the fundamentals and using analogies to help cement the concepts intuitively has been a taste of what I imagine having a brilliant and engaging math professor would be like.

For me it clicked once it understood that integration is the area "beyond" the "curve".

because so many people were exposed to the topic through a resource (typically another person) who had little experience (or interest) in effectively teaching the subject.

I used to homeschool and one of my sons had terrible baggage about math by the time I pulled him out of public school when he was 11. And somewhere along the way, I learned that a lot of elementary school teachers are women who weren't doing well in math and got encouraged to go become an elementary school teacher because, basically, sexism is alive and well and the world goes "Oh, I know what a woman who can't cut it in college is qualified for: Dealing with small kids all day!"

So a lot of elementary school teachers are women who suck at math and were also never told 'Get over it and get a degree you want.' Instead, they were told 'Oh, go teach kiddies. That's what you are qualified for.'

And I know this is a gender difference in terms of the world treating men and women differently -- a social difference, not a biological one -- because I'm a woman who is good at math (or was at one time) and was married to a man who was terrible at it. No one ever told him (a career soldier) "Oh, go become an elementary school teacher!"

Instead, he got his wife to tutor him in college math even though he found it so difficult he threw his pencil across the room one day when I showed him the right way to do it and then I told him "You can be nice or go find another tutor."

It's interesting you mention the solider aspect because the US Marine Corps went through a similar evaluation.

For a long time, being a drill instructor (the guy yelling at you in boot camp) was considered a low status position and therefore a career dead end. This in turn led to the DI spots being filled primarily by people who were in trouble, didn't care or were "bad apples".

At some point, the questions were raised: "Wait, why are we putting some of our worst Marines to train new Marines? Wouldn't it it make more sense to have our BEST Marines train the next generation?". The follow up was then: so how do we actually make that happen?

The answer was to make being a DI one of the stepping stones to higher level ranks. This dramatically increased the quality of DIs and in turn, raised the quality of the training and finally the Marines.

There is a similar story in some countries with socialized medicine. E.g. to get a license to practice medicine privately, you first have to spend N years working in a public hospital. This ensures that you can get top quality people working even in a public healthcare based system.

> a lot of elementary school teachers are women who weren’t doing well at math and got encouraged to go become an elementary school teacher because, basically sexism is alive and well

I hear your frustration, and I’m glad that there has been progress in this regard. However I think that what the GP was saying and what you’re saying are two different issues:

The GP is referring to particular teaching styles which might be overly formal (in the mathematical sense) or declarative (this is convolution, that’s how it is, remember it) as opposed to intuitive (which might not convoy the whole truth but helps some if not most people understand and remember the concept better). This issue can and often takes place in higher education where the teacher or lecturer is expected to be highly qualified.

I remember being frustrated with the way my linear algebra and multivariable calculus classes were taught. Both generalize to n dimensions but many operations and theorems can often be understood intuitively in two or three dimensions. The courses I took were extremely formal with not a minute spent on the graphical interpretation which would have helped so much. When I raised the issue with two separate professors, both were of the opinion that the graphical approach was frivolous because it wasn’t precise. But that argument misses the point of the lecture theatre space ...

To my mind, a system that shunts people into teaching elementary school in a way that actively creates early educators with a math phobia has got to have serious long term consequences, even for a lot of people who are good at it. We all get told "math is so hard," not just the bad students.

People who think a thing is hard and must be dealt with in a serious manner will use formal methods to ward off their fear of the topic or their fears of social perception. It takes a great deal of confidence to say "We are just going to have fun today and play with this wonderful topic!"

They exist, but they aren't the norm.

I agree in general its an issue when basics are being thought by any person who does not really understand what they are teaching. They end up tecaching the rote method and not the intuition. Happens all the time in maths.

I think the larger issue here is that a lot of elementary school teachers are people who have a math phobia because their difficulties with math derailed their career dreams. So they not only are bad at teaching it, the primary thing they teach their students is that "Math is scary stuff and something to be feared."

And they teach it to really young kids who are very impressionable and don't have some means to avoid being brainwashed into believing it.

My one and only goal with homeschooling my son was to get him past the math phobia he had developed and teach him "Math is your friend." He's genuinely bad with numbers, but he no longer has a math phobia and has a better grasp of the concepts than a lot of people, even though he can't calculate anything in his head and routinely asks me or his brother to calculate it for him.

A math phobia is crippling and many people never get over it.

I wholeheartedly agree on the point about decentralizing and modernizing education, and thank you for reminding me of this. We still have a massive bias towards academic institutions to work against, but the fact that knowledge is so easily accessible, and of such a high quality, is pretty amazing.

Why do you think academic institutions are something to work against to? I'm in academia and here everybody is enthralled and really happy about the explosion of high-quality, freely available learning resources.

The only kind of complain that I have ever heard (and very rarely so) is because some of this material is supported by videos in a platform with shitty terms of service.

it's the bias that has to be worked against, not the institution

But who has that bias? Is it you? People in academia certainly not. It would be absurd. As if medical doctors and nurses would be against the education of the population for lifesaving techniques and for healthy eating.

> medical doctors > healthy eating

Wrong example, lol

I'm split on this. On one hand I appreciate these kinds of posts because they bring math with intuition to broader masses.

On the other hand, it makes people believe that they have understood what's there to understand and move on. In a sense, they contribute to the Dunning-Kruger effect.

The article illustrates this quite well. The intuition is presented well, but is no substitute for actually learning math. The first two questions in the end:

> Why is one function reversed?

> Why is convolution commutative?

With having studied math "properly", it's immediately obvious that they are answers to each other. Defining it with the "reversal" on one function is what makes the operation commutative by introducing a somewhat hidden symmetry and that's a quite common trick in mathematics.

The other two questions are also quite clear with some actual math background. Not M.Sc. level math, just some proper calculus. You just need to sit down and do the math, which can take half a page of pen-on-paper. That's what math is, you need to work it.

Again, don't get me wrong, I appreciate the effort. But those types of posts don't make you more fluid in math than reading a popular science article about Alpha Centauri makes you an astronomer or an expose about Covid-19 makes you an epidemioloist. It's popular science and can't replace the actual study of the subject.

It's a good goal from just a humanistic standpoint and because learning should be recreational but from an innovation standpoint it's honestly likely overrated.

Innovation is the result of very particular institutions and resources being coordinated and people networking and clustering tightly, which is why, despite incredibly dispersion of knowledge over the internet, innovation and VC money actually still is so geographically concentrated.

I think these online learning resources are good but the expectations should be correct, they're not tools to kickstart innovation, which is much more tacit and reliant on close interaction between people rather than knowledge.

Internet has some amazing learning resources these days. Not just articles like these and videos, but also interactive explorables. Even my 7yo was able to understand basic aspects of game theory and complex systems with the help of explorables made by NickyCase, MeltingAsphalt etc. Most recently, I taught myself to not just solve 4x4 Rubik's cube but understand its connections to ideas in abstract algebra such as commutators. I feel extreme gratitude for all the people who create such enlightening material and offer it freely, without asking for a paid subscription to their Udemy course.

It inspired me enough that I started creating a DMOZ-style catalog of learning resources as a FOSS project: https://github.com/learn-awesome/learn

> and the entire body of high quality MOOCs are just available __for free__ over the Internet

nobody alerted me to these