This is the realisation I had (20 years late). Especially beyond high school, maths is hard. You have to put the effort in, and you have to do the example problems.

I think its expressiveness and conciseness makes it more dispiriting to study: one page of a maths textbook can contain so much information, and it feels like you're just stupid when it takes an hour to understand that page fully.

This is very important. Apparently some people get frustrated by this because they are used to studying lower-information-density textbooks. Maybe there would be value in teaching these meta-skills, like how to approach learning each subject. People may apply techniques, like rote learning or brute-force cramming, reading the same page for hours on end, expecting to magically grok it, where these are ineffective of even counterproductive. Not sure how well this is teachable though.

- illustration — whether drawings, gestures, animations, using objects, whatever works, but have a visual equivalent to 'translate' notation; the typical example would be fractions which are as simple as it gets with drawings and hard as hell using only numbers, for most people discovering them. The same is true for most common functions and transformations (which, when you reduce it to the maximum, are the entire body of applied math).

- real-world application (typically the best candidates are simple physics or statistics in high school). The point here is to give a point of reference, a "feeling" of the mechanism or logic or paradigm. The deeper problem is that this application must feel 'interesting', like a puzzle, and not everyone is interested by the same things. That's where, imho, narrow AI could help craft customized courses with ad hoc examples that fit people's "world view" and "approach" to problems, and show animations for every variable, equation, let people play with those and see the results.

Both probably fit in the "faster feedback" approach, i.e. have the teacher/teaching material give feedback as fast and as often as possible, to guide the learning mind on the right path. This is an extremely important discovery of UX in the last 50 years and I believe education has much to learn from these very applied insights into human cognition.

This approach breaks down when you go into higher math. You end up working with these abstract objects that have no visual representation. You may be able to find an equivalent object from geometry or graph theory, but it may be so complicated you could never draw it. Heck, even basic shapes like the Platonic solids are difficult to visualize, let alone draw, correctly.

I solved a problem recently requiring me to count the number of vertex colourings of an icosahedron, up to rotational symmetry. Trying to draw one of these things and visualize all of the possible rotations was basically impossible for me. I ended up loading Blender and playing around with one in 3D, using the coordinate planes to see all the axes of symmetry.

If I had been given a higher dimensional object like a tesseract then all bets would have been off. I would have had to find another approach entirely, likely algebraic.

It doesn't have to be literally visualization. Just some "feeling" for the objects. Some conception of them as actors, actees, things interacting, or having emotions or whatever. Again, whatever works. Maybe some people can work with concepts just based on strictly memorizing the definitions and drilling through proofs, but others need to conceptualize it all into a narrative. I don't mean dumbing it down, but making it easier to "grab" mentally. Hard to describe.

The problem with math is that it's seems much harder than it actually is. Every time I have to look into some specific math I haven't worked with before I spend a lot of time decoding what they actually mean. Usually what is happening is not that difficult at all, mathematicians just insist on writing it down in the most convoluted and incomprehensible way imaginable.

It's like they took every bad coding practice and applied it all to math. Why have descriptive variable names if you can just use random greek letters ? Why give a function or operation a name at all if you could make up some weird symbol instead ? And of course you want those symbols to have different meanings depending on context. Imagine if programmers worked like that and wrote everything as obfuscated C++ code, because fuck you.

This is a common complaint I see from programmers all the time. However, I've seen people trying to do real maths with descriptive names, and it fails very, very quickly.

This is by no means comprehensive, but as a (probably thought) experiment, try writing reasonable complex code without variable name completion. You write the same variable name over and over and over, and after a while you realise that since this variable is only use in this screenful or two of code, it's easy enough just to use a short name and know what it is, because you're only thinking about it here, in this context.

I'm not going to try to defend mathematical notation in general because there are enormous inconsistencies, many of which are historical, accidental, and indefensible.

But when you're doing the maths it becomes tolerable, then usable, then actively helpful. It's like the parentheses in Lisp - all newbies complain about them, and those versed in the art know that after a while they not only don't matter, they are a genuine positive.

But unless you take the time to do the math, that won't happen for you. That makes it sound like a deliberate barrier, but it's not, it really isn't.

So why don't math tools have variable name completion ? and/or why insist on still doing things on paper in 2019 ?

The other things about variable names is that they force you to think about what a variable actually contains. This in itself is very helpful not only for others reading your work, but also for yourself when trying to grasp a problem.

But your everyday research mathematician will just stare in disbelief.

I don't know your background, your profile is empty, but it sounds like you are someone who genuinely has no idea of how research in math works, and therefore feel that you really must have a better way of doing things. And maybe you have. But speaking as someone who has a PhD in pure math, and who has worked in safety critical software, I can only say that so far everything you're suggesting just really doesn't make sense.

The reason that for centuries mathematicians use single letter glyphs to represent the things they're dealing with is because it is, for the purpose of doing the work, the most effective thing to use.

I'm not talking about computer assisted proofs or anything like that. Just using a readable syntax and the mathematical equivalent of a word processor would be an enormous step forward. No one is writing books in cursive with a fountain pen anymore either, which is basically analogue to what mathematicians are still doing.

No, I'm not talking about proof assistants either, I'm talking about actually doing math using any kind of computer system.

> Just using a readable syntax and the mathematical equivalent of a word processor would be an enormous step forward.

Do you have any idea of how to do that? I've done research in math, and I've written software for safety critical systems. I don't know how to create a system like a word processor for math that would let me actually do the math.

Do you know how to do that? If so, please, let me know.

Because math is abstract. The variables and constants don’t mean anything in most cases and are only given different symbols to distinguish one from another. Descriptive variable names or even descriptive subscripts get OLD very quickly when you’re working through a page of calculations to solve a problem.

The beauty of math, if you’re using it to solve some real world problem, is that you can forget about what the symbols mean and focus on solving the equations, evaluating the integral or derivative, or whatever other calculation you’re doing.

Once you’ve completed all of the calculations and gotten your answer simplified as much as possible, then you can go back to the problem you were trying to solve and interpret your result in context. Since you’ve done all of your calculations purely symbolically, you can see the relationships between the quantities you’re working with. In many cases, some quantities you thought were important actually cancel out and don’t appear in your final answer at all. This tells you something fundamental about the independence of your quantities. Perhaps the most famous case of this comes from physics: the acceleration of an object in free fall is independent of its mass. You would not see this so clearly if you substituted numerical quantities as soon as possible.

Don't they really mean anything or are mathematicians just unable to grasp what they mean ? To me this seems like a symptom of an underlying bigger problem in our understanding of math rather than those variables not having a meaning.