**They're teaching it wrong.**

And I don't just mean teaching the concepts incorrectly (although they do plenty of that), I mean their teaching priorities are

*completely backwards*. Set Theory is really fun. Basic Set Theory can be taught to someone without them needing to know how to

*add or subtract*. We teach kids Venn Diagrams but never teach them all the fun operators that go with them? Why not? You say they won't understand?

**Bullshit**. If we can teach third graders binary, we can teach them set theory. We take forever to get around to teaching algebra to kids, because its considered difficult. If something is a difficult conceptual leap, then you don't want to

*delay*it, you want to introduce the concepts

*as early as possible*. I say start teaching kids algebra once they know basic arithmetic. They don't need to know how to do crazy weird stuff like x * x = x² (they don't even know what ² means), but you can still introduce them to the idea of representing an unknown value with x. Then you can teach them exponentiation and logs and all those other operators first in the context of numbers, and then in the context of unknown variables. Then algebra isn't some scary thing that makes all those people who don't understand math give up, its something you simply grow up with.

In a similar manner, what the hell is with all those trig identities? Nobody memorizes those things! You memorize like, 2 or 3 of them, and almost only ever use sin² + cos² = 1. In a similar fashion, nobody ever uses integral trig identities because if you are using them you should have converted your coordinate system to polar coordinates, and if you can't do that then you can just

*look them up*for crying out loud. Factoring and completing the square can be useful, but forcing students to do these problems over and over when they almost never actually show up in anything other than spoon-fed equations is insane.

Partial Fractions, on the other hand, are awesome and fun and

**why on earth are they only taught in intermediate calculus?!**Kids are

*ALWAYS*trying to pull apart fractions like that, and we always tell them to not do it - why not just teach them the right way to do it? By the time they finally got around to teaching me partial fractions, I was thinking that it would be some horrifically difficult, painful, complex process. It isn't. You just have to follow a few rules and then 0 out some functions. How can that possibly be harder than learning the concept of differentiation? And its

*useful*too!

Lets say we want to teach someone basic calculus. How much do they need to know? They need to know addition, subtraction, division, multiplication, fractions, exponentiation, roots, algebra, limits, and derivatives. You could teach someone calculus without them knowing what sine and cosine even are. You could probably argue that, with proper teaching, calculus would be about as hard, or maybe a little harder, than trigonometry. Trigonometry, by the way, has an inordinate amount of time spent on it. Just tell kids how right triangles work, sine/cosine/tangent, SOHCAHTOA, a few identities, and you're good. You don't need to know scalene and isosceles triangles. Why do we even have special names for them? Who gives a shit if a triangle has sides of the same length? Either its a right triangle and its useful or its not a right triangle and you have to do some crazy sin law shit that usually means your algorithm is just

*wrong*and so the only time you ever actually need to use it you can just look up the formula because it is a

**obtuse edge case**that almost never comes up.

Think about that. We're grading kids by asking them to solve edge cases that never come up in reality and grading how well they are in math based off of that. And then we're

**confused**when they complain about math having no practical application? Well

*duh*. The sheer amount of time spent on useless topics is staggering. Calculus should be taught to high school freshman. Differential equations and complex analysis go to the seniors, and by the time you get into college you're looking at combinatorics and vector analysis, not

*basic calculus*.

I have already seen some heavily flawed arguments against this. Some people say that people aren't interested in math, so this will never work. Since I'm saying that teaching kids advanced concepts early on will make them interested in math, this is a circular argument and invalid. Other people claim that the kids will never understand because of some bullshit about needing logical constructs, which just doesn't make sense because you should

*still introduce the concepts*. Introducing a concept early on and having the student be confused about it is a

*good thing*because it means they'll try to work it out over time. The more time you give them, the more likely it will click. Besides, most students aren't understanding algebra with the current system

*anyway*, so I fail to see the point of that argument. It's not working now so don't try to change it or you'll make it worse? That's just pathetic.

**TL;DR**: Stop teaching kids stupid, pointless math they won't need and maybe they won't rightfully conclude that what they are being taught is useless.

Your article goes on to lament the shortcutting of math and you use a TL;DR? ;)

ReplyDeleteThis comment has been removed by a blog administrator.

ReplyDeleteMy comment got deleted so fuck it.

ReplyDeleteI agree with you, don't know the difference between trig and geometry, blah blah blah, fuck the system.

Double that last part because commenting on here is a broken piece of shit for anyone not on here. Hooray!

Hooray for denigrating trig identities, which I always had trouble with, yet I got my doctorate.

ReplyDeleteI would have enjoyed math more if my teachers/instructors would have shown me practical applications such as, the 3/4/5 rule to ensure the big damn hole you dug in your yard is square (before beginning digging), or how many cedar planks you need minus the 2" of space between the boards you forgot to calculate (realized once you have an expensive stack of cedar firewood next to the house, for a firepit you don't own), and the difference between a sale price and the regular price is still more expensive than ordering it from Amazon.com (with shipping).

ReplyDeleteVery good points. I've always thought the way math was taught made it appear harder than it had to be.

ReplyDeleteOne of my gripes is that proofs in textbooks generally have MISSING STEPS. Arrgh! Instead of zooming through the logic, you spend hours trying to fill in the gaps, or give up.

It's like speed bumps on the highway of learning!

That's the biggest problem in schools these days, overall - we're consistently underestimating the capacity of kids. They're like sponges - give them some information, they'll absorb it. Will they understand it? Maybe. Maths is only one example.

ReplyDeleteNumbers in first grade? Great. Addition, subtraction, division, multiplication, fractions... you could at least suggest at that in first grade, and definitely teach all of it by second grade. Advance the program. Partial fractions? You can learn that as soon as you understand factorisation of polynomials. Complex numbers? Don't leave them until last year of high school - the basic concept of complex numbers was outlined to us using a picture book for Montessori levels! They're fun, they're different... they're probably not useful if you're not an engineer, but at least they're different!

I was casually trying to explain basic molecules (like O²) to a 2nd grader back when I was in primary school. Now I'll also add they hadn't even tried to teach chemistry *at all* in primary school. I was talking to a 2nd grader, and I was in 7th, and she did have some clue what I was talking about. She wouldn't have understood the whole idea, but something was clicking. If I'd been taught better at the time, I could've drawn some diagrams, and she probably would've gotten it - instant high schooler! (at current level of education)

Children are *far* more intelligent than people assume, and we wouldn't have these terrible rates of children actually staying through to the end of high school, and the following shortages of college graduates, or skills shortages, if we actually kept kids stimulated. Kids aren't staying in school? They're bored! Delinquency rates? They have nothing interesting to think about!

@alankdkd I'm not sure the missing steps would matter if we were moving at a more realistic pace. The steps that are skipped are usually something you spend ages and ages on two topics before... about a month ago!

Teaching basic set theory, compact sets etc. was tried in the 1960's, for elementary and middle school. It was called "New Math". I had it in the 1970's in the south.

ReplyDeleteI never understood why it was taught until I audited real analysis in graduate school. I think the math teachers had not studied enough math to understand why it was important and how it was used. Also, at least my high school math teachers presented theorems as the key useful tools for application, rather than the methods for proving the theorems.

The other problem is the disproportionate attention to standardized testing over some of those useless tasks. That not only holds them up, but gives a worthless benchmark against which they are measured. Furthermore, the teachers are under so much pressure to teach to the test, there's no way for even the creative teachers to branch out!

ReplyDeleteI homeschool, and I never make my kids do exercises if they already understand the concepts. I honestly think that's what kills the joy of math early on. Pages and pages of arithmetic problems. Kids either understand how basic functions work, or they do not and they need more teaching, not more practice.

ReplyDeleteMuch radical change has been made to the way children learn language; Math should follow suit. Children are receptive to wonder-ful things and math contains wonders aplenty. Learning math;however, can be a painful and dreary process.

ReplyDeletePS

Can you tutor my 12-year-old? :)

The teachers are only teaching the wrong way due to being pressured by standardized testing. Take away standardized testing, give teachers a little more slack, and maybe they'll be able to come up with better methods of instructing students.

ReplyDeleteBTW Love the point - "We're grading kids by asking them to solve edge cases that never come up in reality and grading how well they are in math based off of that" - Entirely true!!!

I graduated high school never going past algebra. I took pre-cal in college, never went further than that. I'd like to understand math, but it frustrated and angered me, so I don't get it. As Erik said, the teaching is the main reason why. There's a load of more stuff I could do with programming, automotive applications, and firearm design if I understood math any more than 2+2=4, but all during my education my time was wasted learning how to do stupid triangles and whatever the hell those things are where you have (x + 1 + 5)(x+5) or some fucking useless bullshit like that.

ReplyDeleteConcept quantity. Have you paged through a 4th grade or 9th grade math book? The shear number of topics is overwhelming. Every week is a new topic with sometimes fragmentary association with prior topics. Bam, bam, bam with no application, no point to the topics with regards to real life. It's just this glut of content the teacher HAS to get through in the school year - dictated by the school district/state/fed.

ReplyDeleteTeachers are great - the school system sucks.

Hooray - exactly right, although I'd make two additions:

ReplyDeleteTrig identities are another example, not a exception to your point: sin² + cos² = 1 and Euler's identity provide all of the standard trig identities with a little algebra and no memorization. Understanding trig in terms of geometry on the unit circle also gives a whole different understanding, and brings some fun back into trigonometry.

- The extension I get is that the purpose of mathematics is logical thinking and solving puzzles *not* memorization. If math is being taught "by formula" it is a complete miss, not of interest to anyone except standardized test takers and over-competitive academics. If we taught by puzzle and by problem there would be more love in the topic, and more thinking in the rest of the academic world.

You're missing the biggest reason why it's taught poorly: Until children reach high school or thereabouts, they're being taught math by completely innumerate dumbasses, who, despite having years and years of repetition of the subject matter, have no better (and in many cases worse) understanding of the material than the kids they're teaching.

ReplyDeleteThese "educators" introduce hatred and fear of mathematics because of their own inadequacies, and -- as is typical of those suffering from Imposter Syndrome -- become aggressive and punitive when challenged about it. My undergraduate degree was in mathematics, and in the honorary math fraternity meetings, my friends and I shared many stories of completely incompetent grade school "math teachers."

I had several run-ins; my favorite was a sixth grade teacher who INSISTED that 6/0 = 0! When in exasperation at her failure to understand the basic reasoning behind it, I showed her my older brother's algebra book -- which explained in detail why x/0 makes absolutely no sense -- she patronizingly informed me that, "That's only true in algebra. It isn't true with numbers. In this class it's 0."

If you honestly think you're going to be able to get people with this level of conceptualization to teach anything that involves mathematical abstraction, you're living in a dream world.

While you are reforming topics and sequence, please provide a foundation in descriptive and inferential statistics before leaving high school.

ReplyDeleteArguably, there is no post-arithmetic math more essential to citizenship.

Erik, thank you for this topic. I find it useful.

ReplyDeleteAt least we could correct the school system by some in-family activity...

But I should note that it is not so funny to find coarse words as "bul...it" in the topic related to math and to kids. Is it common in your country? (Some comments use even more ugly words...) :(((((

This was a rant. The only place I actually shared it was on HN where engineer profanity-laced rants are commonplace. It is not designed to be read by children, although any kid old enough to comment (13+) should know what's appropriate and what isn't (and are probably swearing out of earshot of their parents anyway).

ReplyDeleteI'm reminded of the claim I've seen a few times that although Sudoku is a number puzzle game it "doesn't need you to do any maths". But of course it does! It's just that it's all set theory and logic, and no arithmetic.

ReplyDeleteThe thing is that that claim is generally offered as a positive aspect of Sudoku, which supports your idea that people who find arithmetic tedious can be perfectly happy with set theory.

I agree with the premise of teaching these things earlier, but I think you drastically overestimate the difference it will make in how much children enjoy it. You can dismiss my argument all you want, but a lot of people simply aren't interested in math, just like a lot of people aren't interested in history or grammar or science or literature or music theory or sports or needlepoint. People like different things.

ReplyDeleteWill teaching it differently help them understand it better? Most likely. But in the words of Homer Simpson, "Just because I don't care doesn't mean I don't understand."

We can argue about whether or not this will work all day long and it won't mean anything until we try it and see what happens.

ReplyDeleteI think you're on the right track with most of this, but I'm going to question 'Introducing a concept early on and having the student be confused about it is a good thing' - it at least calls for qualification, I think. Many students get turned off maths by the feeling they're never going to understand things - it's not *just* a question of kids not getting why they'd ever want to understand it! So you do need to be careful about what concepts you introduce when, and how...

ReplyDeleteYou might enjoy my blog

ReplyDeleteoldpoliticaljunkie.com

I am a retired mathematician and a veteran of the "math wars"

According to me mathematics is all about practice.If students have fear of mathematics ,it is just because of they don't practice it.If someone try to teach them in a good way,they can also score very good in this subject.The parents and education boards should promote this thing that mathematics is one of the most important subject of their syllabus and appoint good teachers for it or they can provide sample papers to them like science sample paper of cbse board.

ReplyDeleteI strongly disagree! I'm a high school freshman and the reason I abhor math is because of the constant problems and practice sheets teachers assign. I have never feared the subject but I don't want to do it. I don't want to sit and do 30 freaking problems on something I probably will never use unless I become a scientist or mathematician. They need to come out and show us why it matters how much distance Tom covered when he walked to and from the store.

DeleteBecause they're teaching it wrong – most likely! It’s a given that Math could be tough and complicated. I think the flaw here is with teachers implementing a single and general method of teaching to all of their students, without considering the learning preference and difference may vary. Some students will be ready to take Algebra lessons as soon as they learn arithmetic, while it may take more time and practice for some. Sarah @ GradePower Learning

ReplyDelete