To my mind, the premature formalization of the math is the principal contributor to gas lighting and alienation of people from maths. The reduction of concepts to symbols and manipulation thereof, is an afterthought. It's misguided for them to be introduced to people right at the outset.

People need to speak in plain English [0]. To some mathematicians' assertion that English is not precise enough, I say, take a hike. One need to walk before they can run.

Motivating examples need to precede mathematical methods; formulae and proofs ought to be reserved for the appendix, not page 1.

[0] I mean natural language

I'm an adult who's been programming computers professionally for 20 years, and went to school for it, and I've lost most mathematical skills past what I'd learned by 6th grade or so, from lack of use.

People who aren't even working in a field that's STEM-adjacent have even less use for stuff past simple algebra and geometry (the latter mainly useful just for crafting hobbies and home-improvement projects) and a handful of finance-related concepts and formulas.

I expect to go to my grave never having found a reason to integrate something, at this rate.

The result is that any time I try to get back into math (because I feel like I should, I guess?) it's not really motivated by an actual need. The only things that don't bore me to tears for sheer lack of application ends up being recreational math problems, and even that... I mean, I'd rather just read a book or do almost anything else.

I feel the opposite.

Before high school, math is just a grind of memorization and unmotivated manipulation of numbers.

Many students (ok, me, but I expect the same was true of others), get turned on to math for the first time when they encounter proofs in high school geometry and also actual applications in high school physics.

It's a revelation to students that math can be a way to go from one truth to another and thus find new truths. It's a way of thinking and that can be very exciting.

Tragically, many students disengage before this can happen because of sheer boredom and the tedium of endless math drills. Once they develop a gap in their knowledge it becomes difficult to progress unless those gaps are addressed. For lots of students, it all ends with fractions. You'd be surprised how many adults don't really understand fractions. For others it ends with algebra, and for the college bound it ends with calculus.

Only math majors and a minority of engineering/science/CS folks get past the "standard sequence" of math courses in college and gain an appreciation for the really interesting stuff that comes AFTER all that.

Mathematics is the conversion of a large number of object languages in to a single meta language that lets us talk about all of them.

The sin of modern mathematics is that it's meta language is so ill define that you need towers of software to manipulate it without contradiction. Rewriting all of it into s-expressions with a term rewriting system for proofs under a sequent calculus is an excellent first step to making it accessible.

We do not need to go back to the 16th century when men were men, an numbers positive. If people want to look at what math talks about instead of how it talks about let them pick up stamp collecting.

What does premature formalization mean and when does it occur? Do you mean formal in the sense of using formulaic, rote manipulations or formal in the sense of proofs and rigor?

As someone who went on to study mathematics at the graduate level, I was bored out of my mind in high school math and most subjects. What's missing from a lot of primary and secondary school education is context, and that's what makes it boring. Math wasn't easy because I was particularly good at it. It was easy because it was just blindly following formulas and basic logic.

Something is very wrong with our educational system because almost all math at the primary and secondary levels is basic logic. So when people with this maximum level of mathematics education say they're "bad at math" or "don't get math", it means that they lack extremely basic logic and reasoning skills.

In my mind, we need to teach mathematics in a contextual way (note that I don't necessarily mean applications) in a way to enrich the reasoning and exploring of it. This should include applications, yes, but not be fully concentrated on applications. Sometimes one needs to just learn and think without being tied to some arbitrary standard of it being applied.

This sentiment comes up all the time here. Mathematics uses formalism because it's easier.

It's easier to read "a(b+c+d) = ab+ac+ad" than

> If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

It's well known that good notation is exactly the one that elevates good intuition. For example, the Legendre symbol has the property that (a/p)*(b/p)=(a*b/p), an important visual cue that you wouldn't get from writing down (in way too many words) what the Legendre symbol actually means.

Also, most actually good mathematical textbooks aren't just dumps of formulae and proofs and they do contain motivation, examples, pictures, etc. You're attacking a strawman. But you can't just relegate the formalism and proofs to the appendices, that's crazy.

The ironic thing is, I swear that this must have been how math (at least more advanced math) was taught a century ago. Or at least, nowadays I've taken to relying on textbooks from the early-to-mid 20th century to learn new math. Maybe it's survivor bias and the only textbooks from back then that anyone remembers are the good ones.

I hate new textbooks because they're so built around instant gratification. They just come out and tell you how to solve the problem without building the solution up in any way. Maybe afterward they take a swipe at telling you how it works, but that's just completely the wrong way around IMO. It robs me of the chance to mull things over, try to anticipate how this will all come together in the end, and generally have my own "aha" moments along the way.

But, getting back to what you say, I think that it also engenders this tendency to reduce math to symbol manipulation. Because if they give you the formula in the first paragraph, then all subsequent explanation is going to end up being anchored to that formula. And IMO that's just completely wrong. Mathematical notation is at its best when it's a formalizing tool and mnemonic device for cementing concepts you already mostly understand. It's at its worst when it's being used as the primary communication channel.

(It's also an essential tool for actually performing any kind of symbolic reasoning such as algebraic manipulation, of course, but I'm mainly thinking about pedagogical uses here.)

I believe you're right, even though I don't have any evidence except for my own experience.

This issue becomes very clear when you see how many ways there are to express a simple concept like linear regression. I've had the chance to see that for myself in university when I pursued a bunch of classes from different domains.

The fact that introductory statistics (y = a + bx), econometrics (Y = beta_0 + beta_1 * X) and machine learning (theta = epsilon * x, incl. matrix notation) talk about the same formula with quite different notation can definitely be confusing. All of them have their historical or logical reasons for formulating it that way, but I believe it's an unnecessary source of friction.

If we go back to basic maths, I believe it's the same issue. Early in my elementary school, the pedagogical approach was this: 0. only work with numbers until some level 1. introduce the first few letters of the alphabet as variables (a, b, c) - despite no one ever explaining why "variable" and "constant" are nouns all of a sudden 2. abruptly switch to the last letters of the alphabet (x, y, z), two of which don't exist in my native language 3. reintroduce (a,b,c) as sometimes free variables, and sometimes very specific things (e.g., discriminant of a quadratic equation) 4. and so on for greek letters, etc.

It's not something that's too difficult to grasp after some time, but I think it's a waste to introduce this friction to kids when they're also dealing with completely unrelated courses, social problems, biological differences, etc. If you're confused by "why" variables are useful, why does the notation change all the time, and why it sometimes doesn't - and who gets to decide - this never gets resolved.

Not to mention how arbitrarily things are presented, no explanation of how things came to be or why we learn them, and every other problem that schools haven't tackled since my grandparents were kids.

I have never disagreed more with a comment. You can fully decide that you're not interested in mathematics, after having taken all of the math classes that you could possibly be offered before university, without ever encountering a proof, or even a mathematical definition.

I do agree that explaining why mathematical concepts are useful is something that's often lacking in mathematical curricula, but not that the problem is premature abstraction. Like another commenter said, the opposite is true. The way children are first introduced to (and therefore soured to) something that adults call "math" is by performing pointless computation that has as much to do with actual math as lensmaking has to do with astronomy.