Read “A Mathematician’s Lament” by Paul Lockhart, it’s free online.

He lays out a brutal critique of the modern mathematical curriculum in the Unites States but in summary:

We teach mathematics to children as a huge set of rules to memorize and use to get good scores on standardized tests so that they can “get into good colleges.”

We don’t treat mathematics with any reverence or care, like we do with the arts. Math is taught as a bunch of arbitrary brute facts that old wise men came up with centuries ago and we spend all of elementary and high school relentlessly drilling them into students heads no matter how much pain and suffering it causes.

There is no actual exploration of mathematical beauty, or mystery. There isn’t any discussion of the underlying philosophy of mathematics, or how any of the rich and fascinating history of its development as a field. It’s like if we taught music as just a way to write notes on a page in certain time signatures and keys, but never actually let students listen to a piece of music or discuss the great composers or cultural movements of music through the ages.

Of course that seems ridiculous to people, but for some reason when we do that exact same thing with mathematics, nobody bats an eye. In fact, people think it would be strange to do it any other way.

• Teaching math is mostly done w/o context and history, IMHO a lot of math makes much more sense when the original problem is understood, before the level of abstraction is being raised.
• Math is a also a language and a notation. Unless one uses math regularly, there is simply not enough practice/repetition to read/speak this notation.
• Math is a tower of abstractions, depending on other abstractions. A lot of topics in math depends on people understanding a lot of basic parts, which means if a student just got by with a prior topic, it is near impossible to catch up/understand what is currently being taught. (Compare to other topics: For example, if a student is bad in their Greek history, they get a fresh start when the topic is industrialization in England w/o any penalty.)
• Math in the primary and secondary schools is mostly computation, ‘real math’ is only taught to people studying MINT.

tl;dr

• we need a better curriculum in the primary/secondary schools
• we need more exercises in reading/writing the mathematical notation (sorry, just understanding math is not enough, because understanding doesn’t make one fluent)
• at least in my school years, math was not repeated enough.
• reading/understanding math is really hard, at the higher levels, understanding 2-3 pages on a textbook per day is an acceptable pace. I guess all the entertainment nowadays makes it not easier to sit still in a room and get math into ones brain

For me the ‘breakthrough’ with math was, simply to accept that at the higher levels we are speaking about symbols (abstractions) that follow certain rules and everything else is derived by pure logic. Just accepting that one is manipulating symbols with rules to get to other symbols and learning the rules, made it click for me. Disclaimer: Was lucky with great math teachers in university, but even in my university there were people who simply could not accept the game of mathematics and were frustrated, because they wanted easy question/answer style formulas in the sense: When you see this, substitute PI with 3.14 and multiply r by r and write down the number that your calculator shows. They never made any effort to understand where PI comes from, where the radius comes from and why it makes sense.

What is insane, is how many people studied computer science but are totally unable to apply mathematics to the problems they try to solve. Supposedly most of them learned relational algebra and discrete mathematics during their studies (and formal languages/complexity theory)… it is like something is missing in their ability to transfer what they learned in the university to basically the same problems where the symbols have different names. That is something I would love to understand.

Mathematicians are shitty communicators who like feeling special because they can understand their obscure language.

I’m a programmer and in this field there have been tons of books published, conference talks, and heated internet arguments about how to make your code as readable as possible: formatting, function length, naming of variables and functions, keeping number of cross references low, how to document intent, etc. Mathematicians do none of that - it’s all single-character names (preferably from the Greek alphabet to complicate it further) and they rarely communicate intent before throwing formulas at you. You can easily tell when a mathematician has written code because it’s typically hot garbage in terms of readability.

Personally, my problem was always that math concepts were never presented in a way that actually made sense in the “real world.”

I was taught that complex numbers were real numbers with imaginary parts that had something to do with the square root of -1. Yeah, I get it, but… why?

Fast forward a few decades and I’m writing code that processes a digitized waveform. Now it makes sense. Math isn’t hard when you have a frame of reference. Learning math concepts solely for the sake of learning them is very hard.

Part of it has been how it is/was taught. Math was always the subject I had to work harder on than any other. The problem is that I was never taught to really conceptualize the problems. Once I started taking physics and real world applications came into play, it all sort of clicked and got much easier.

Edit: Also, math is really all about relationships and conceptualizing interesting problems or ideas. If it had been presented to me that way, I think I would have been more adept at it earlier.

Because it was taught wrong to most adults when they were children. Pedagogy has changed, though, and gen alpha are actually becoming numerate instead of being told to just memorize things like my generation was. Maybe the zoomers got lucky with that, too.

But seriously, as a mathematician and a teacher, you’re not bad at math because of something inherent to you. You’re bad at math because you weren’t taught numeracy.

I can only speak for myself, but honestly I’ve never been able to figure out that root of why it’s so complex to me and difficult to keep track of / understand. The only thing that seems to have a “rational” explanation to me is… Selective memory. It has been a burning question to myself for so long.

For a while I just said “It’s too arbitrary and not logical” except math is built upon logic - 1 + 1 is clearly 2 because if I hold one finger on one hand then bring another finger from my other hand I have two fingers held.

(Imaginary numbers though can fuck off)

I got into programming long ago because it is logical - there’s (almost) always a reason why a computer does $THING even if I can’t tell you, someone surely can. Though generally the answer is “someone told it to do the wrong thing”. If I dig deep enough, I can usually find the answer. My life is full of so many questions that I’ll probably never have the answer to, and I found refuge in the fact that I can get the answers here.

However… computers follow a set of rules, just like mathematicians do. So for me to call it arbitrary would just be wrong. I mean sure, a lot of the rules and formulas certainly seem arbitrary to me, there’s a reason why they are the way they are and it can be tracked down just like you can track down why a computer does $THING.

When it comes to numbers though, my brain just doesn’t seem to hold on to it properly. I can randomly recall weird functions and quirks in libraries that I use - even remember plenty of arbitrary “things” like Vim motions… Yet ask me what nine times seven is and I can’t tell you what the answer is without doing the weird finger trick.

So the only explanation that I can come up for that is just selective memory. I like computers and as such my brain is willing to actually memorize these things. Whereas I’ve never liked math and so my brain doesn’t see a reason to “memorize math”.

It really frustrates me because math and computer science intersect in a lot of ways, and I’ll always be held back by this. Games for example, they run really well on your GPU because GPUs happen to be excellent at math, specifically in parallel. Encryption? Fancy math equations! Almost everything at a low level comes down to math.

Similarly, for as much as I love logical things, I could never hold the concepts of logic gates in my head. I mean, logic is literally in the name! Even when I was heavily into Minecraft I couldn’t pick it up through Redstone.

As such, I think for me, the “logic” argument doesn’t hold up as much as I like to think it does. The analyst in me says that I want it to be something as logical as “math is illogical” because that’s easier to admit and sounds better than “I just don’t like math”. Even worse, perhaps that subconsciously stops me liking it, thus blocking myself from ever being able to excel at it… And yet, here we are (or rather, “here I am”).

I went through engineering school, and 20 years of work (not as an engineer), before finding a calculus text that explained why the derivative of x^2 is 2x. Along with many practical applications of calculus.

That book was Calculus Made Simple, published in 1914. Thanks, Project Gutenberg!

Edit: derivative of x^2 is 2x. Got my differentiation and integration confused!

In my case because I was in gifted classes so I got this idea that I was just brilliant and never needed to study for anything. Then as soon as a subject got hard enough for me to not ace it without effort I just quit instead of knuckling down and doing the work. Math was the only subject where I truly ran into a wall cause some of that stuff is just not at all intuitive, it’s loaded down with obscure rules and memorization, etc.

It felt less like instruction on how to use a vital tool to make my life easier and more like someone was intentionally making my life harder by making me learn math. It’s like someone came up to me and said ‘Oh, you’re walking 10 miles uphill? Here, since you’re going this way, carry this 40lb rock with you. it’ll be real useful at the end, trust me bro.’ And I was like ‘This is already a hard enough walk, the fuck am I carrying this rock for?’ so I set it down.

I have since picked some of it back up, and I now recognize the utility of learning it and wish I’d learned it when I was younger cause it’s even harder now.

For me specifically it is because I have dyscalculia. Which made all mathematics is school quite difficult. I could understand reasonably complex ideas and through my own methods could solve equations but I couldn’t show my work because it was all done it my head.

It took me until grade 10 or so to come up with a method to know the multiplication table but it still takes more than a moment to work a simple calculation out.

Oddly I can work out and calculate the weight of a trailer of freight within a few hundred pounds with very little information in order to balance it for shipping but 9 x 8 involves 10 x 8 = 80 - 8 = 72

I failed math multiple times until I was put into math B for the slow people. I did better than the teachers assistant on tests. Went back to regular math the next year thinking I could do it if I was so good at remedial class and failed within a few weeks and stopped going.

Somewhere in middle school they give up on the “if you have two apples and buy two more apples you have four apples” or “if you have 3 pizzas cut into 8 slices each and your family eats 13 slices how many pizzas do you have left?” and they start trying to teach 11 year olds by making them memorize proofs and phrases like the transitive property of equality.

To a lot of high schools and colleges, the aesthetic of academia is much more important than students actually learning anything useful, so they teach math class with a chalkboard full of squiggles rather than any kind of practical approach.

From Algebra class on up, it’s taught as a rules following exercise. “Okay, now we do this, and then who knows what we do next?” And it is amazing how many of them are set up as trick questions, how often out of the infinite span of numbers there’s often one right answer and one wrong answer. “How many of you got five? Well you’re wrong, it’s negative 3.”

Meanwhile, I was learning how to fly. In flight school, you learn how to navigate by dead reckoning. I want to fly this course on the map, which is x distance and x degrees from true north as measured from the chart. Given a weather brief and the performance characteristics of the plane from the pilot’s operating handbook, calculate: true airspeed given indicated airspeed, altitude and temperature wind correction angle, given true course, true airspeed, wind direction and wind speed ground speed, given true course, true airspeed, wind direction and wind speed true heading given true course and wind correction angle magnetic heading given true heading and local magnetic variation time aloft given distance to travel and ground speed fuel consumed given time aloft and fuel consumption

The tool you’re taught to use to calculate all of this looks like this:

It’s basically a circular slide rule, that has a vector plotter on the back. The trigonometry is done by accurately drawing and measuring the triangle, the ratio problems (anything “per hour”) is done by rubbing a couple of logarithms together, and you’re on your own for the addition and subtraction. Ever used a slide rule? They don’t keep track of the decimal point for you. So you have to do these built-in sanity checks, like “Wait, no, the plane doesn’t even hold 70 gallons of gas, there’s no way I’ll burn that much in ten minutes.”

I learned how to do that before I took physics class, and surprised my physics teacher that I knew how to do “boat crossing a river” problems with a weird piece of cardboard. Later on, when I was teaching flight school, I taught that procedure to “It’s been 30 years since math class” boomers and “Trigonometry is next semester” teenagers. They all picked up on it without much problem, because “the wind is blowing you to the right” is a real thing they’ve felt in their own asses by now.

IMO it’s cultural. In my country (Canada), it’s considered unpopular and indeed socially expected for one to find it uninteresting or even useless. In Russia, mathletes are popular like football players.

A lot of elementary school teachers don’t go into teaching to teach math.

Disclaimer: I have always been good at math, and have asked myself the same question. This is what I came up with by asking others:

It heavily depends on how your brain works plus your teacher’s abilities.

1. Most stuff can be done by taking educated guesses, that’s just not possible in math/physics etc. There is only right or wrong, no in between, thus if you are good at creative thinking, it doesn’t help at all. Also, since one rule is based on other rules, you really need to get the first rule to get the next. This requires a ton of abstract thinking to get the depenencies. A lot of people are good at memorizing stuff, which can be done without understanding it.

2. For you to get and stay interested in a topic, it needs proper presentation. Lots of logical thinkers are not good at presenting stuff to people who’s brains work differently. They just try to eyplain things the way THEY understood it, which is why their presentations are simply boring to others.

Studying mathematics is a difficult but also rewarding activity. This requires having a positive relationship with the effort. By analogy we could compare this to sport. To give up practicing mathematics because it is difficult is equivalent to giving up sport because it tires.

For those interested in the education of mathematics, I would recommend this book by mathematician David Bessis.

Mathematica: A Secret World of Intuition and… by David Bessis

https://www.youtube.com/watch?v=KJafasuk4NQ