Math is The Coolest Invention In the Universe


#1

This is a bit of a twist on the motorcycle and bike threads. There’s a lot of fascinating things about math. A trick I like that’s easy to explain to just about anyone is to find a cup or glass (occasionally a squat beer bottle) and to bet someone that it’s bigger around than it is tall. This is usually true for all but the tallest skinniest glasses and it hold for some beer bottles. It’s based on an old grade school buddy: C=pi*D

Multiply the diameter of a glass little over 3 and it quickly grows to be a larger length than the height of the glass. You can convince yourself of this with a piece of string or your hands.

But despite the mastery of time and space granted by mathematics (at least the most commonly studied kind) we are plagued by innumeracy and a fear of algebra. I feel we are too often forced to contend with two different options: Either math is intrinsically difficult, or math is universally poorly taught.

Neither or these explanations satisfies. If math were intrinsically difficult, then why does fear of math start with arithmetic or the barest basics of algebra? No mathematician considers these operations difficult (though plenty consider arithmetic tedious- because it is.) And can it really be true that most math instructors are terrible? This just strikes me as incredibly unlikely.

So what are the other options? I’m really asking here. My current personal hypothesis is that the symbolic language of math is a language you need to learn to read fluently, but learning to read it is not emphasized the way that learning to read English or French is. Oftentimes people need to sit down and “sound out” what’s happening on a page. This tends to snowball with the ordinary difficulty of acquiring any new skill and students end up behind, stressed out, and demotivated. By contrast, students are quite familiar with English before they’re assigned reading for history or science. Throw on some cultural baggage about how “math is hard” and you’ve built yourself a nice little layer cake. Whether my hypothesis is any correct or not, I suspect the real answer is also a layer cake.


Things That Really "Grind My Gears..."
#2

Do you mean learning to read French if you are French? Because here in the US, where neither French nor Calculus are required to eat or poop, many more people make it at least to Calculus than are fluent in French.


#3

Ouais, exactement.

But since you mentioned French:


#4

Eh, I’m pretty sure we mostly teach it wrong. I figured out how to add, subtract, multiply, and divide with positive and negative integers before they tried to reach me. It was easy and intuitive (and whatever I was doing worked).

Then they tried to teach me and it’s been a confusing chore ever since.


#5

Part of the problem is that many folks apparently consider it somehow cool to be shit at maths. Somehow, it’s not at all embarrassing to completely suck at basic arithmetic, despite the fact an eight year old can master that shit.

People are wilfully stupid; intelligence is more about attitude than ability.


#6

I think it’s also the way it’s taught. If you try to teach languages by immediately breaking them down to total abstraction, then you fail to create competent speakers, too. Kids learn their first language because there are obvious and direct benefits to doing so - it helps them to communicate feelings, needs, etc and that helps them to solve problems. The way maths is taught appears to be to first settle upon the abstract concept or process that you want to train users in, then (sometimes) to clothe that in vestigial real-world examples - “If Jenny has 8 beans and trades them for peas, and each pea is worth 1.25 beans, how many peas does she get?” kind of stuff. It isn’t compelling and it doesn’t communicate the importance of the skill in any way. I had a stats book at university where the opening gambit was worm length. The other one - intended for non-mathematicians - didn’t even get as far as standard deviation, so was useless.

The best way to teach anything is with reference to meaningful problem-solving. The way I was taught Pi, for instance, has stuck with me for 20-odd years and counting, because we were were asked to work it out from the diameter from the start, and, finding that 3 was a bit too small, we focussed more and more on working out what this little additional bit was. Clearly that was the intent of the teacher, but only once we had tried to resolve the size of the remainder to several decimal places did she reveal pi in all its majesty.

Similarly I never had any problems with economics, accounting, forest mensuration, etc, which are all pretty intensely mathematical (with discounting, depreciation, sampling, cruising methods, etc), because it was clear what the problem to be solved was, and the problem led to the mathematical solution.


#7

This hints at the emphasis of process over results, but in a way, I’m not sure that’s a bad thing. I think oftentimes people get mad that kids are marked “wrong” despite getting the right answer, but think about it: How many other subjects do people have that kind of latitude? Virtually none. A history exam question might be, “Explain the causes of the First World War.” Is it enough to state, “alliance system, militarism, ethno-nationalism, and long-standing border conflicts brought about by the disintegration of the Hapsburg Empire?” That’s not an explanation, it’s a list. It’s “the right answer” after a fashion, sure. But no one would start talking about the utter collapse of American history education if a teacher marked it wrong and their parents put it on Facebook.

The misconception is that with math, “there’s a right or wrong answer.” This is false. There are many different answers that lead to the same place. You can define pi in a seemingly limitless number of ways including in terms of arccosine or the ratio of the diameter of the circle to its circumference or through a power series. Each type of answer is suited to different purposes and ideas.

Process matters. At the university level, the professors are typically okay with students using whatever process they like, not because they’re more concerned with correct answers, but because that’s what university is about: Finding appropriate ways to answer novel problems. When it comes to elementary and middle school students, it might not be doing the student any favors to let them develop their own way to get to an answer. For example, (because I’m apparently stuck on pi today, which isn’t that interesting to me, ironically) if I want to quickly mentally calculate something to do with circles (which comes up in my hobby of carpentry a lot) I use 22/7 as an approximation of pi. It works 9 times out of ten and is a handy way to get rid of factors that are multiples of seven mentally because seven is annoying. But 22/7 does not equal pi. I understand that, and I can make educated decisions knowing that. Kids can come up with kludges and ideas that only work some of the time, only work for small numbers, or only work with certain numbers.

So I’m sympathetic to teachers who say that they really insist on specific techniques and ideas that are well-developed, understood, and rigorous instead of just grading on answer alone. Especially because the archetypes and algorithms themselves are part of mathematics. I do think there is a right and a wrong way of handling students who seem to have an intuitive grasp of the concept, but oftentimes the right way of doing things runs up against things like class sizes and time per student. These are very real considerations that cannot be ignored as a factor in mathematical pedagogy.

Part of what I’m getting at with, “I’m not convinced all or most math teachers are bad,” is that math is important. It is economically and militarily important that the US generates mathematically able people. We’ve been investing time and money into it ever since the Space Race. This is something we as a society care about. Compared to art, music, and PE, the amount of consideration given to this topic is phenomenal. So I’m not convinced that it’s just a matter of people not caring and not doing anything about obvious problems. Generations of math teachers have been taught various methods and ideas to address these problems, so why aren’t they working?

I think you might be getting at something with this. I’ve long thought of Thales theorem as the “napkin and taco” problem. Where you consider the intersection between the corner of a square napkin with the edge of a taco. I think it does have some limits, but I think you hit on something even more important:

How many times a day do you read something, minus the number of times a day you calculate or logically solve a problem. In that delta is a world of difference in terms of fluency and “naturalness.”

I can show a typical high school graduate a passage from a book discussing technical ideas about Frisian grammar, and as long as it’s in English, they can probably make some sense of it. If I showed them a basic Maclaurin series, their eyes would instantly glaze over. Even if they took some calculus in high school. (I’m excluding exceptional cases of high school students who take calc II in high school.)


#8

I think you’ve got me ass-backwards on this. What I meant is that if you try to teach language the way maths is taught (which was for many years the way language was taught) you get people that can spit out verb tables, but aren’t capable of articulating anything in the target language.

There is a difference between maths and languages in that as you say, being able to do things the right way is more critical in maths - in most languages you can be more or less intelligible whilst making lots of mistakes. What I mean is that the way in which the process being taught is arrived at is very important to comprehension, uptake, and reproduction. That’s not to say that the student should be forced to rebuild maths from the ground up, but supplying the answer to a question the student hasn’t asked is as good a way of teaching maths as just shouting random verbs at a student is for language.

Frisian grammar is also quite similar to English, and I still think a lot of students would struggle. The description of language is a very specific skill-set in itself, and not everyone receives an education in which it is clear. English in particular, being a language that has forgotten most of its rules and structures, and mixed Germanic and Romance components, doesn’t lend itself to an abstract understanding of language.


#9

I actually think we’re just having a little trouble agreeing with each other, or at least I suspect we agree more than disagree. With the Frisian example, I didn’t mean that a high schooler could flawlessly learn it from a grammar with no previous groundwork, just that they could make sense of some of it despite coming to it somewhat blind. I’m comparing this to what I think is the equally likely situation they they wouldn’t be able to make any sense of an equation or expression that might only be one or two levels above their heads. Perhaps Frisian is a bad example, but my point is that people can, with some difficulty, read above their level in a limited way that has a great deal to do with the way we’re inured to communicate through English text. This seems to be less true for mathematics, and I suspect it has a lot to do with a steeper intelligibility curve due in large part to a lack of emphasis on “reading” math.

I definitely agree. So I work at a children’s museum right now, and a big part of my job right now has been to get children to learn to problem solve and ask the right questions. We have a bubble display that fills a flat glass container with bubbles, which promptly form straight walls against each other and tend to form hexagons. I always ask kids to look around the museum for other examples of the pattern (I’m actually thinking of our beehive display.) Sometimes I have to lead them to it, but then I ask them how many sides shapes need to have to fit together. Do you think triangles fit together well? How about squares? We have blocks they can play with. Penrose tiling is absolutely waay above their level, but I sort of try to get them to that general type of thinking. But my primary goal throughout is to ensure the kids are interested in the questions- otherwise it’s really just a waste of both our times.


#10

I think there’s the possibility that math is intrinsically difficult for most people but comes very easy to a small minority - that mathematical ability could be seen as an aspect of neurodiversity.

I recall a similar conversation in which someone pointed out that people mostly learn through analogy. X is like Y except Z. You can learn how pass you final math exam this way, but you can’t learn math this way. The idea of working directly with axioms and not projecting any ideas about the world or yourself onto those axioms is something that I think a lot of people just aren’t very good at.

It’s something I find extraordinarily easy, and it makes me very good at solving certain kinds of problems. Having spent lots of time around some of best mathematical minds of my generation (no exaggeration) you can observe this kind of stripped down axiomatic thought process being applied to all kinds of things. When applied to social interaction the result is often disastrous (


).

So when kids sit down to learn 3 + 4 = 7, they learn that if you have 3 apples and you also have 4 apples then you can count them all and find out you have 7 apples. That gets them through the coming addition test. But math isn’t about apples, it’s about a completely abstract system that evolves to do a good job of modelling reality. If I wanted to explain what 3 + 4 = 7 means I could use the Peano axioms. Now one of the reasons we use the Peano axioms to create a system and not some other arbitrary set of axioms is because one of the things we do with those numbers is count real objects and the Peano axioms give us a set that lines up nicely with the way that real objects do. But math is about the axioms, not the apples.

I think I’ve told this story before, but in highschool calculus I was taught that limits were “where the function was going.” In first year calculus I was taught that the limit of f(x) as x→a is L iff ∀ ε > 0 ∃ δ s.t. |x-a| < δ ⇒ |f(x) - L| < ε. I never quite got limits in highschool, but when I saw that definition of them I thought, “Oh, of course”. Years later when helping my sister with math I couldn’t remember the product rule for derivatives but I could remember the limit definition of a derivative and I could derive the product rule myself from it. It’s the way my brain works - I remember the minimum amount of raw information I need to derive things I need to know instead of remembering things I need to know. I’m not sure people can be taught to think that way. I doubt I could be taught to think the “normal” way.

We could teach math better for sure (there’s a big problem with oscillating between discovery and rote based learning when you need some of each). I find it insane that we teach kids to add apples but they get to grade 9 and they can’t figure out how to add x’s, we’re clearly doing something wrong. But intuitively thinking of applying math to real world situations (i.e. I forgot to take my medicine, I’ll just look up the halflife online and calculate whether/when I should take a full or part dose to most quickly get back to the level in my system that I am supposed to have at this time of day) seems like a different way to approaching the world.


#11

I think we broadly do agree - I didn’t mean any disparagement by ‘ass-backwards’.


#12

I think, but didn’t want to say, that mathematicians seem to learn to describe maths in a way which is super useful for communicating with other mathematicians who have a shared set of background knowledge and not very useful at all to anyone who doesn’t already have that. It’s almost inimicable to teaching, fairly few seem to recognise what’s going on, and in many schools, maths lessons end up just being given books which are ‘at your level’ and working through them.

Your anecdote demonstrates this almost perfectly because I understood essentially nothing from it. Looking back, whenever I have been successful in learning maths, it has been when taught by non-mathematicians.


#13

This is absolutely my strategy, even when things that have nothing to do with math. But alas, it doesn’t seem to be a common strategy or one teachers pay much thought.

My drivers’ ed. instructor was actually insulted when I asked why you turn the front wheels one way facing uphill, another facing downhill, another if there’s no curb. He said (more or less) “You turn them the way I tell you to and that’s that.” It hadn’t even occurred to him that it might help students to know the reasoning behind the rules.


#14

No worries, I didn’t take it that way. :relaxed:


#15

Yup. I get that at work all the time with tech.

“Oh, I’m not a computer person.”
“Really? That seems almost irresponsible, since nearly everything you do day to day has a computer involved one way or another.”


#16

Somewhat similar here.

Up until my mid-teens, mathematics were effortless; I’d routinely ace the exams despite paying very little attention in class (because we’d already spent three hours repeatedly hammering at this incredibly simple thing that I already understood).

Then we got to quadratics, and I fell in a heap. It made no sense to me, and the teacher’s explanations were useless (“you’ve just got to feel around until you find the right answer”).

So I gave up on mathematics and went off to do non-school stuff for a decade. But then I decided to go to university for my neuroscience PhD.

For every subject during my undergrad degree, I scored in the 90s, often at the top of the class. Neuroscience, history, philosophy, geology; stomped all of 'em. Except for the mathematics; that, I should have failed…

I did okay in the statistics classes taught by the Psych faculty. I usually forgot it all ten seconds after the exam, but I managed to briefly retain enough to get marks in the 70s to 80s.

But because it was a B.Sci, I also had to take basic calculus, taught by the Mathematics department. Who then amply fulfilled all of the stereotypes about the teaching incompetence of mathematicians.

A notable highlight was the lecturer who would turn off his hearing aid at the start of every class so that he wouldn’t have to answer questions. The tutorials with fifty students spread across two non-adjacent rooms while one TA ran back and forth were also a nice touch. Constant typos through the practise questions so that you could never be sure if your “wrong” answer was actually incorrect, too.

I still haven’t the faintest idea how to do calculus.

Even if we assume that I got perfect marks for every question I attempted in the final exam (which I didn’t, because I had no idea what I was doing), it was not possible for my end-of-subject score to be higher than 35%, due to how poorly I’d done in the midterm quizzes.

And yet I still passed the subject. Turns out that the entire class had failed so badly that the grade-curve normalisation gave me a twenty percentage point boost.

So, yeah; it’s both. Mathematics is innately difficult for some people [1], but it is also badly taught to a heroic degree. I spent over a decade as a university student, and I have never seen teaching incompetence and malpractice come anywhere close to that calculus class [2].

[1] I have a near-perfect memory for things that interest me, and no memory at all for things that don’t. And, possibly relatedly, I’m also universally incompetent at non-English languages (French, German, Ancient Greek and Latin in High School; can’t speak a word of any of 'em).

[2] And, no, it wasn’t just one bad teacher; we had multiple lecturers, and they were all appalling.


#17

well, I feel like this could have helped me as a humanities nerd. however, I think they tried this with the Cemrel math I got first-through-partway-into-third-grades until i moved. perhaps the problem was I was removed from the program before they integrated the “fundamental language-of-math” stage with the “actually doing equations” stage. I went from conceptual math to traditional arithmetic over the course of a couple days when I was eight years old. I often wonder if my understanding of math would have been better had I been able to complete the program. as it was, it was pretty unfair that they didn’t have an exit program for kids whose families moved, which is a pretty common externality that they completely ignored.

maybe simplistically, I feel like the base problem is that STEM people intuitively grasp math concepts. when something is intuitive, explaining it becomes hard–it just exists for you, “getting it” is fundamental, not a path that you can “give directions” to someone else. A humanities person might could explain the journey better to a fellow humanities person, but that’s assuming they can arrive at the full understanding. Surely all types of people are capable of reaching top-level math understanding, but if you don’t have a knack for math to start with, developing the interest to pursue it is probably truncated. I’m aware that I’m oversimplifying/feeling sorry for myself.

High school geometry was a breeze, though, because it was visual and concrete rather than abstract. It was very strange to see the “good” math students struggle that year. I thought algebra was interesting and still do, but I was just not good at doing it. I took trig and algebra II because I was told I’d need it for college but only passed with heavy tutoring and none of it stuck. I’ve always felt that I was a logical person, but I really do feel like the math teaching was created by-and-for people who just had no idea how I perceived the concepts. but, of course , there’s certainly the possibility that even with some sort of tailor-made curriculum i’d still suck at math, so I’m not trying to make excuses, either.


#18

wisdom


#19

I can describe math pretty well to people who aren’t very interested/good-at/whatever math but I can do so because I have empathy, not because I know math (I guess it’s at least partly because I know math).

This is also partly because in a lot of programs they let in more students than they really want to graduate and then they give the job of winnowing the field to the math department. I know math several people who have been stuck in this position. Some business program is going to have a failure rate of about 1/3, so their stats class is going to have a failure rate of about 1/3. It’s just letting math be the bad guy since people already think it is.


#20

Meh. For anyone who thinks they know the right way to teach math, this is a rare growth area in academics, and NSF grants in Math Ed are typically many times the size of grants in Math proper. I encourage you to bring your ideas to that table.

It is pretty easy to just blame the teachers. Everyone who did well in math in school, everyone who did poorly, and everyone who did well in some classes and poorly in others - basically, everyone - thinks they have a useful opinion about math teaching. Why would any sane person want to be a K-12 math teacher when everyone on the street is slamming them all the damn time?

As for “the way we teach math,” there are several competing curricula (which includes mode of instruction) in the US alone, and some non-US curricula (eg, “Singapore math”) are all the rage here. Most of them come complete with “evidence” that they are better than the competition.

While I don’t think you’re alone in finding this Riemannian definition of limit natural, it is of course relatively recent by math standards (though the basic idea probably goes back to the Eudoxus derivation of the area of a circle), and its adoption in curricula was contested in the late 19th century by people who thought that the infinitesimal definition (the limit is L provided whenever x is infinitesimally close to a, f(x) is infinitesimally close to L) was more natural. I think one is simply likely to favor the first “non vague” definition they see of any math concept.