So should the same be true of high school chemistry and biology?
The other subjects of science, history, English are all heavy on remembering “facts”. Rote memorization. Water is H2O. Columbus is 1492. Hamlet “To be or not to be” and so on. (Critical reading and interpretation is one of the exceptions to rote memorization.)
On the other hand, genuine understanding of math resists mindless memorization. Yes, kids do use that method to get passing grades but their “knowledge” of math is fragile and built on a house of cards. It crumbles when they see something slightly different.
At the end of Algebra II, you have 2 groups of kids that passed it:
Group 1: The ones that “understood” it
Group 2: The ones that memorized and faked their way to finish it
The Group 2 kids should not be in the Algebra II class at the expense of other quality classes (such as the ones you mentioned).
Here is a very simple question that the Group 2 kids will get wrong:
Which number is bigger? x^2 or x^3 ?
If you hang around enough kids, you start to notice how Group 2 kids supposedly “learn” higher math topics like Algebra. The Group 2 kids see math symbols as “text substitution” instead of a placeholder for underlying mathematical ideas.
Even if you teach them the explicit case that x^2 can be larger than x^3, you can still invent yet another very easy question that will exploit their “text substitution” fake math skills and expose them as pretenders.
You cannot make Group 2 kids stop the reliance on “text substitution” methods. It is part of their coping mechanism to get through the class. And teachers help them with this fakery using FOIL, mnemonics, etc.
Another obvious telltale sign of the text substitution mindset: canceling factors.
Johnny sees that in (x)(y)/(x)(z) the “x” on numerator and denominator are the same and can be “crossed out with a line” to leave behind (y)/(z). He then does the same thing to (x - y)/(x - z) leaving (-y)/(-z). It’s the wrong answer but it “looks right” because that’s what he did with unknown variables that were multiplied instead of subtracted. The unknowns of “x”, “y”, “z” are just blotches of ink to be crossed out with lines and moved from one side of the paper to the other. To them, they are not mathematical ideas.
Now the first reaction is to say, “that’s easy!!! Just teach them that it’s not just chicken scratch symbols and really math ideas!!!”
Easier said than done. Nobody, and I really mean NOBODY has figured out the magic teaching methodology to make this happen for the majority of the population.
What is wrong with teaching someone to the point where they fail, shouldn’t the rant be more about mandatory exit tests than teaching kids to their max potential?
Learning these base ideas is exactly like learning your letters and their sounds…although algebra is repeatable and consistent.
He wants to do the same type of dumbing down that is such a big problem in other parts of STEM education. If you understand WHY things work you understand how they work.
I am not saying there aren’t ways to improve the system but dumping algebra making it about art or music math is exactly like saying “screw words and grammar lets just teach them to memorize Dick and Jane”
Teaching kids “Math” in art is teaching them a parlour trick because you have to teach them ever solution set vs. teaching them the language of mathematics.
Madrasas are pretty damn good at getting kids to memorize the Koran, but the kids have no clue what they are memorizing.
Seriously…what middle school classes do you think are so riveting that they are not so** uninterested students that will forget it**
And you may not remember quadratics but I am sure you remember equivalence or how to move the X across a =.
Do you really remember your US history from the era? Can you remember the theological implications of all the books you read in the 7th grade?
I bet most people remember more from algebra than they think they do.
I would argue that algebra is required for most basic finances - calculating loan interest, taxes and tax deductions, debt, etc. These are practical applications, and if you don’t understand the math behind basic financial instruments you could end up wasting a lot of money.
This section of the original article made me retch a little:
I hope that mathematics departments can also create courses in the history and philosophy of their discipline, as well as its applications in early cultures. Why not mathematics in art and music — even poetry — along with its role in assorted sciences? The aim would be to treat mathematics as a liberal art, making it as accessible and welcoming as sculpture or ballet.
Ugh. I’d rather them just give math a dignified death than to turn it into one of those fuzzy subjects with no right or wrong answers (not that there isn’t a place for math history/anthropology/etc., but that place is *not *as a replacement for basic math education).
I haven’t been following this debate, so I hope this hasn’t been posted up already. It’s an excellent rebuttal. I’ll try and quote the salient points, but recommend reading the full article.
The article contests the fact that maths is as important a factor in kids doing poorly as Hacker claims it is -
It further points out that Math not being useful in ‘real life’ is as true of everything else taught, and the reason a general grounding like this is chosen is because all specific applications of knowledge are not possible to cover.
Higher level algebra can help practice applying simple abstractions to more difficult abstractions, and thus to real life
Math history is awesome (even if half of it boils down to “hey, did you guys know that Isaac Newton was kind of a dick?”), granted it’s not something that should be a core class in High School, an elective at best.
Though I could see an argument for at least having some sort of “applications of curriculum” course which dedicates time to discussing the applications of various fields in the real world. Admittedly it shouldn’t replace math or anything. Maybe just one of those one semester required courses like “exploratory”* or whatever it is they’re doing nowadays.
- In middle school down here, 6th graders had a required course called “exploratory” which rotated a group of students each quarter into different arts. Drama, Visual Arts/Painting, Technical Arts (video, photo, computerized art), and Music to try and find something they’d be interested in. It could be interesting to do something of a similar caliber with the core subjects, except exploring what you can actually DO with said subjects rather than just trying to expose kids to new things.
That sounds almost exactly like me in high school (and dare I say it, now, if I was foolish enough to venture anywhere near algebra these days).
I’m pretty sure I’ve spoken on the boards before about my belief that higher mathematics (ie most stuff more advanced than arithmetic) is not only pointless for most people, it’s actually counterproductive as it makes them despise maths and never want anything to do with it ever again.
Take away the “Intro to the Power of Algebra” and replace it with “tortuous maths that might as well have been in Martian”; throw in English and Science and you’ve got pretty much what I studied at high school in 5th and 6th Form - and I turned out alright (Master’s degree and all). The fact I was able to ditch maths entirely in 6th Form really, really, REALLY helped. I’d have preferred it if I was able to ditch the “tortuous maths” in favour of something like Computer Studies but we didn’t have that as a thing at the time, alas.
Mathematically, statistics and probability have been extremely useful for me. I was fortunate in that I had a cool teacher who decided the best way to teach this stuff was to relate it to card games and gambling (and other more pedestrian real world everyday stuff, too, of course).
In fact, I recall him summarising the entire course with “…And that’s why the best way to win at gambling is to own the casino.”
We did a lot of stuff in Economics on interest rates and so on to which was also extraordinarily useful too, but never once, ever in my life since graduating High School have I needed to solve “X” using the algebra stuff I think I was taught at High School.
Also, the suggestion people who suck at maths are lazy or not trying is a deeply offensive one. There’s a fuckload of mathematics which is simply incomprehensible to people like myself. Not just “a bit hard” or “confusing”, but completely impenetrable. Egyptian Hieroglyphics make more sense.*
I think that’s true of most fields of study, though. The rules of English Grammar are an incomprehensible mystery to a lot of people and countless people deride history as boring and pointless.
As has been mentioned, the key is everything in moderation. It should be pretty obvious by the time you’re 14 or 15 that you’re either good with numbers or you’re not. If you’re not, persisting in trying to make you learn something like algebra really is, IMHO, a waste of time, energy, and educational resources.
*And not just because I have a book on how to read them 
But, that’s a good example of where you would “solve for x.” At least, being able to solve an equation for a variable lets you use the same formula to answer different kinds of questions, like “How much would you have 10 years from now…” or “How much do you have to invest now…” or “How many years would it take…” or “What would the interest rate have to be…” instead of having to have a different formula for each one.
In my experience, people who “suck at math(s)” or do poorly in math classes, may do so for one or more of the following reasons:
(1) They are lazy—or at least, for whatever reason, they don’t put in the time and effort necessary to learn.
(2) Bad teaching, either currently or in a previous class that turned them off to math.
(3) Poor preparation: they haven’t mastered the mathematical background necessary as a prerequisite.
(4) A defeatist attitude: They decide ahead of time they’re not going to be able to understand.
(5) Poor attendance: They just don’t come to class.
(6) Poor study skills: They don’t know how to study effectively (either in general, or math in particular), or they don’t practice what they know.
(7) Math phobia or test anxiety.
(8) A fundamental inability to think mathematically: their brains just don’t work that way.
(9) Trouble thinking mathematically for a more correctable reason—e.g. they have trouble concentrating due to ADD or just not getting enough sleep.
And there may be others. And it’s not always easy to tell where the problem lies. When I see someone like you who claims that much of math is completely impenetrable, for example, I can’t help wondering whether that’s a manifestation of #2 or #4 or #8 or…
Yeah, there’s lots of interesting stuff, like Évariste Galois’ death by duel at age 20, who despite that had a profound impact on math. I’d have loved if my school offered STEM history as a substitute for ancient history, for instance. It just doesn’t belong anywhere near math class, not only because it’s a distraction and timesink, but also because it could give the false impression that the sometimes sordid history casts doubt on the truth of the results. It doesn’t, of course.
It’s a tough balance. I generally lean towards teaching basic principles, because they have broad application. Training for specific applications doesn’t necessarily generalize. Still, as an adjunct for the existing math curriculum it might not be a bad thing.
Is it? Really? I would not know.
Because, I did not say that.
In fact, I was very clear that I was NOT referring to math teachers.
Yes, they have. Any number of logic puzzles require math beyond the four basic functions, and a large number of people enjoy doing them.
I’ve had people tell me that sort of thing (Economics) is “Algebra” before and my response to that is while it might be, however I arrive at the answer, it never consciously involves any of that Martian Hieroglyphic nonsense with Xs and Ys and grouped brackets and so on. Which leads me to believe Algebra, in the “pure mathematics” form it’s taught (or was taught to me) is a waste of time for most people and the energy should be directed towards Economics, basic statistical interpretation (with a heavy emphasis on “seeing through bullshit and spin”), and Probability.
At least that way the underlying principles might get taught, in a way that’s useful and hopefully won’t to any long-term damage to students who just aren’t mathematically minded and struggle with equations involving Xs, Ys, Brackets, and so forth.
In my case it’s primarily 7 and 8, with probably some other factors in there. I can’t even understand the Wikipedia page on the subject, for smeg’s sake.
On the other hand, I’m really, really good with words. I always figured most people could write, but… no, they can’t. And I don’t mean in the “I don’t think they can write because I’m a pedantic Grammar Nazi” (I’m not, mostly) kind of way, but because a lot of people cannot effectively communicate in a written form in my personal and professional experience.
Puzzles and games can help in isolated areas but there is no comprehensive system that teaches the entire Algebra I & II curriculum. Nobody has invented it so talking about it seems pie-in-the-sky. In the meantime, while such a breakthrough puzzle teaching system is being invented, kids should pursue some alternative academic topics.
Like I said, there is no curriculum or teaching methodology available that I’m aware of that prevents those 86% of teachers I mentioned from flunking their algebra questions.
We can pat ourselves on the back that we do have a fairly good system for transferring arithmetic skills. Drills work. Adults do retain it. We do not have a good system for transferring genuine algebra skills to those who believe it is a waste of their time. (We do have an ok system for transferring fake robotic algebra skills. I’m baffled why preserving this fake-algebra is so sacred.
These two statements just don’t fit together for me. It is, of course, useful to solve problems like “at 3% interest, how much will a $1000 investment be worth after 20 years”, and use an equation like this:
1000 * 1.03[sup]20[/sup] = 1806
But it’s just as useful to go in another direction, say where you want to know how many years it’ll take to grow the investment to $2000.
1000 * 1.03[sup]x[/sup] = 2000
Suddenly you need algebra! You could, I suppose, memorize every possible permutation of the formula. Or you could learn algebra and only need to know one formula.
I don’t think it’s a coincidence that people who are bad at math think they have no need of it. Every new math concept I’ve learned has opened up new avenues of application. I didn’t know I needed the math until I had it.
Or I could use a scientific calculator, which involves a monumentally reduced amount of frustration and anguish.
The fact so many people like me who are terrible at maths and manage just fine without the advanced concepts suggests we may actually have no need of it. Just like huge numbers of people never read a book again after graduating high school and many of them manage just fine too.
I’m a writer, not a rocket scientist or an accountant or an engineer. I’m fine with arithmetic and I think it’s vitally important and needs to be taught. But I also think that algebra is a waste of time beyond a single introductory class for students who are clearly not of a mathematical bent.
I’m not saying algebra is inherently useless because it’s not. I am saying it’s useless to try and teach it people who can’t (or don’t want to) learn it because it’s not going to achieve anything. They won’t remember any of what they’ve learnt, it won’t help them in later life, and it will (IMHO) put them off maths in general.
Simply plugging the numbers into a calculator means you’ll never understand what’s going on. Among other things, you’ll never develop the intuition needed to detect incorrect answers.
For some definition of “manage just fine”. Yes, illiterates and innumerates can manage not to starve to death in today’s society. They’d still be better off with literacy and math skills.
It’s possible. I just don’t think there’s any evidence that some people are completely incapable of abstract math. The current system for teaching math is subpar at best; it can alienate even those who are good at math and enjoy it. That just means we need to change the teaching method, not eliminate it completely.
Have you looked into the possibility of dyscalculia?
I don’t particularly care that I don’t “understand what’s going on” when I punch equations into the calculator. What I do care about is contuing to develop in my professional field and personal interests, which almost never involve advanced mathematics. I’m not going to waste what precious spare time I have on something I hate and over a decade’s experience has shown I’m not good at.
As The Most Interesting Man In The World once remarked:
I don’t do complicated maths well. Therefore, I don’t do it. And when I do… I use a calculator.
I have thought about that, but the thing is, I’m just fine with arithmetic, general money stuff (working out change, percentages, that sort of thing), dates, chronology, reading analogue clocks, directions, etc. I’m just useless with algebra, calculus, formulae, and that sort of thing. And as I’ve said before, I honestly don’t think my life is any worse off because of it. I live in a professional world of words and ideas, not numbers and equations.
FWIW I’m sure on a maths-themed board my opposite number is saying “I don’t get the whole writing thing; anything more than the basic rules makes no sense to me. But I live in a professional world of numbers and equations, not words and grammatical structure rules. And I have Grad Students to subedit what writing I do need to do, so I’m happy.”
Actually, the math guy over at the other board is probably saying they don’t understand the ART of writing. A very large number of math nerds dabble in formal grammars and by extension a lot take to linguistics (or the computational variety thereof). Many, many math people get grammar, syntax, and structure just fine.
It’s when it comes to making is sound PRETTY, APPEALING, or UNDERSTANDABLE that they fall short.
They had that STEM history course, but all the AI nerds totally freaked everyone out during Ada Lovelace day and they had to cancel it for everybody. That was such a pain to clean up.
Yeah, but I’ll be the first to admit that the course gives a very, very bad impression of what the point of math is. I mean, people kind of grok that calc is used in physics and physics makes airplanes, or whatever, but beyond that I find the “when am I going to use this?” complaints valid. As Feynman noted, textbooks do a piss poor job of giving examples that make a damn lick of sense to anybody actually in the field.
It wasn’t for a good many years that “solving a linear system of equations” actually paid off for me. I think it could be really interesting for kids to see what kind of stuff that can be used for. Computer graphics is awesome and a huge amount of its foundation is in one of the first things you lean in algebra. And since everything in graphics is numbers, even the weirder stuff like Conics can be useful. I mean “did you know that with this knowledge, and a little more, some day you could make Call of Duty?” is a pretty good sell.