math education
Jan. 30th, 2007 09:54 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
cellio![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif){kind=small}
amergina posted a link to this (longish) To summarize, some (apparently-big-name) published curricula are now skipping conventional methods to teach new ways of doing arithmetic. Some are different ways of breaking down the problems; others are primarily notational differences. All of them seem, on average, slower and more error-prone.
Now granted, I sometimes do arithmetic by the "reason through it" process the reporter dislikes (what did they call that, clusters?), but it's kind of specialized. For example, a 15% tip reduces to a 10% tip and half again; that's fast and easy. If I'm multiplying by a number ending in 9 or 1, it's often easier to reduce to another problem and then deal with the leftovers. If I need the square root of 4862 (I just pulled that number out of thin air), I can't tell you exactly what it is but I know it's a bit less than 70. Sometimes I think in patterns like that. I think this is a fine thing to teach people after they have mastered conventional write-it-down-and-work-it-out methods. Not before, and certainly not instead of. (And I think it's better if you can give them an educational environment in which they figure out these "tricks" for themselves, like I did.)
I assume these new teaching methods (which include "use calculators") are largely responsible for many people being unable to get order of magnitude right. Those of the previous generation undoubtedly said that about the move away from slide-rules, but I never used a slide-rule (except as a novelty) and I can approximate... I once had a calculator-armed teenage clerk at a produce stand insist that my bag of vegetables came to over $200. Even if he had no instincts about what vegetables cost, he should have been able to tell that the price codes he'd read off the list didn't add up to that and maybe he'd mistyped something.
(When shopping I tend to keep rough a mental tally, so when I get to the check-out I know approximately what the total should be. I gather that this is unusual. It's just the way I learned to shop, probably from a time when you had to make sure you didn't exceed cash on hand. Now I use plastic for everything, but the habit remained.)
Well, I guess I can take comfort in one thing: if what they say about mental exercise is correct, I should be pretty close to immune to Alzheimer's. :-)
I work for one of those companies...
Date: 2007-01-31 05:23 am (UTC)Oops. Looks like someone's premise is incorrect.
Gary Larson drew a Far Side about "math terror" before either of those curricula existed; people have been complaining about the mathematical inadequacy of high school graduates long before either of those curricula existed. The practice of teaching grade-school "math" as the memorization of arithmetic answers and rote methods -- and the complete avoidance of reasoning skills until pre-algebra -- is precisely why for generations, students hit the wall in 7th and 8th grades when they first encountered, you know, actual math. These things aren't the product of "new math", they're the precipitator of it.
For generations, it was socially OK for nobody but a small elite to make it past (or even through) algebra. Unless you were going into a specialist field, all you needed to get by was your sums and your times tables. And then came WWII, and then the Russians put a man in space, and then the Internet happened. As a society, we think it's no longer OK for people to stop with sums and multiplication tables. We no longer think it's adequate to have a small elite of "numerate" people.
So various people who studied how kids learn looked at just why so many people opt out of math around the 7th grade. And what they found was that the method that that pretty, well-spoken meterologist advocated, whereby one is drilled in a synthetic method and the memorization of facts was pretty much directly analogous to teaching reading by having kids memorize vocabulary for six or seven years without ever telling them what the words meant or showing them how they were used, and then beginning teaching reading comprehension. As one might expect, some kids managed to figure out how math worked either on their own or with the help of parents or other supports... and other kids didn't. Rather a lot didn't.
And I remember vividly tutoring my peers in the 7th grade. And let me be clear, by peers I mean the advanced (honors) math track. I remember how many, even most, of my classmates did not know that multiplication was repeted addition; that AxB==BxA (oh, they'd memorized such a rule, and could recite it, but if you posed them a problem which required using that fact, they wouldn't know if "swapping them was allowed"); that a fraction represented division, and vice versa; etc. But they were very good at sums; that's how they got in the advanced class, after all. I remember thinking, incredulously, "I think someone needs to teach these kids arithmetic all over again, because they don't understand anything about it."
New math is an attempt to do just that. It's an attempt to bring the sort of mathematical reasoning that permeats real math down into the lower grades, so that mathematical reasoning is learned young; it is an attempt to teach math comprehension instead of just math memorization, the same way we demand reading comprehension instead of just vocabulary memorization.
Does it succeed? That's a good question. A lot of people think so, especially people who are specialists in studying how children learn. Let me be clear: the product isn't cheap. It's never anyone's lowest bidder. Nobody gets on this bandwagon for any reason other than they think it does the job really well. And the reason there's so much noise about it is that so many people -- teachers, schools, districts -- are opting for it... and other people really hate it. And make things like that video.
Re: I work for one of those companies...
Date: 2007-01-31 05:34 am (UTC)meta
Date: 2007-02-01 02:48 am (UTC)I need to think more about this. During the first 30 seconds of the video I pegged it as propaganda, yet by the end I accepted the argument. What was it that caused that effect? And is it relevant that probably 95% of my intake of intentionally-published news, opinion, and commentary is in written form? (Is it about the argument or the medium?)
Re: I work for one of those companies...
Date: 2007-01-31 07:55 am (UTC)"...in the new approach, as you know, the important thing is to understand what you are doing, rather than to get the right answer." (Emphasis mine, but Lehrer -- a math teacher himself -- made very clear in interviews that the New Math methods of the time really truly did NOT care whether the student got the answer right, which he obviously found quite snarkworthy.)
A better approach, which I hope the newer methods are using, is to both get the right answer *and* understand why you are getting it right. This is the approach I used when working with
I don't pretend to be up to date on the more recent methods; I just hope that they are far more integrative than *any* of the methods (Old *or* New) that were around when I was in grade school.
Re: I work for one of those companies...
Date: 2007-01-31 02:15 pm (UTC)I agree that students need to learn math comprehension. When I was in school that was done by teaching the standard methods and then discussing why they worked. If kids are only getting the standard methods without that explanation, that would certainly be a problem -- but so would teaching only a method that is long or complex enough that some folks will just saw "aw, screw it" and pull out a calculator.
I guess my math education was better than I thought it was; I thought the kind of reasoning skills I learned were par for the course. (Not knowing whether you could swap operands? Gah!)
Re: I work for one of those companies...
Date: 2007-01-31 11:52 pm (UTC)A question for you about your math background: were you in any sort of special classes, or was that instruction given to everybody in your school? One apparently common pattern is that the advanced kids get more "theory", but the other kids just get drill.
At the grade school I (and many of those 7th grade peers) attended, they mouthed all the right words about needing to understand. And they did things which, I suppose looking back, were meant to enhance understanding -- but they were just different forms of drill and rote learning (e.g. memorizing the communicative and distributive laws). In the end, it all boiled down to memorizing answers or methods.
Re: I work for one of those companies...
Date: 2007-02-01 02:45 am (UTC)With two possible exceptions, I got the same math education everyone else did. The first curriculum split was in 8th grade, when some students took algebra and the rest took something called "math 8". (One of my frustrations with my school was, in fact, that I thought they didn't do enough for me. So I found ways to fill the time on my own.)
Possible exception 1: I had this tutor from the blind association for one class period a week. We did assorted stuff. We didn't do anything math-like that I can remember until either fifth or sixth grade, when we did some algebra. We did do logic puzzles and stuff like that a lot earlier; to the extent there is a correlation between logic and arithmetic, I was getting more there. On the other hand, I read (and solved) books of logic puzzles for fun all on my own, so the tutor might not have been relevant there.
Possible exception 2: my father enjoys math. I don't recall him tutoring me, and he explicitly refused to teach me "tricks", but he was all too happy to talk with me about the tricks and patterns I'd discovered on my own, and either help me validate them or show me counter-examples.
My education in arithmetic involved a lot of rote memorization -- timed multiplication drills particularly stand out in my memory. This is why I thought I hadn't been much affected by "new math", but I don't know very much about that approach (beyond Tom Lehrer :-) ) and it appears I was wrong about that. I'm not sure; I could attempt to describe what I remember in more detail if that would help.
Re: I work for one of those companies...
Date: 2007-01-31 03:53 pm (UTC)Anyway, somewhere along the line (probably around Algebra II), I figured out that what we had been learning was not math, but arithmetic. And that math is quite interesting. So in high school, I learned probability and statistics, and taught myself calculus, and read a lot of books about the philosophical and logical foundations of math.
But it cuts both ways. In college, I took the grad-prep sequence of math courses, with the consequence that I got rather good at proving theorems and understanding the structure of modern mathematics, but, as a general rule, I can't compute anything much. Fortunately, I became a computer scientist, which depends more strongly and the ability to estimate and reason than it does on the ability to compute (after all, that's what the computer is there for).
I guess I just throw that out there to say that I'm not a big fan of the old math, and, in my case it least, I think it was largely irrelevant to my later mathematical development. Perhaps one of the newer methods would have suited me better.
Re: I work for one of those companies...
Date: 2007-02-01 12:30 am (UTC)These somewhat rarified abstract questions have some amazingly concrete ramifications. I am very good at math, and have always been -- both computational tasks and mathematical reasoning. Is it useful for people who aren't good at math to be instructed in how to think the way I think? Or is there some difference in how my mind works, such that my methods exploit strengths of mine which others lack, and they'd learn math better with a method which draws on their intellectual strengths? Or would teaching to their strengths amount to neglecting their weaknesses? Is the way I (and other highly mathematically adept people) reason about math required for reasoning about math, such that it's necessary? If I sit down and tutor someone in math (as I have done) is it useful for me to use my own reasoning process as a guide for what good mathematical reasoning looks like? What if that way of thinking is really alien to the student?
To be clear, here, I am waaaaaaaaaaaaay off my employers' page here. I think (I don't work for the math department!) that their premise is that taking the way adept math students think about math and trying to teach less adept kids those ways of thinking is precisely the way to go. Indeed, that the premise is to teach little kids, even little kids who are bad at math, how to think about math the way adult mathematicians (and engineers and scientists) think about math.
But I think it's an interesting and open question. MB testing suggests 75% of Americans prefer concrete to abstract reasoning, and are more adept with concrete operations than abstract ones. Let us assume that is true, and that therefore memorizing and rote learning is easier and more familiar to the majority of American kids in elementary school (and their parents!) than is generalizing from cases and finding patterns. For such students, traditional drill-and-kill arithmetic lessons teach to their strength -- or allow them to coast on their strength -- at the complete neglect of their weakness; when they suddenly are expected to reason deductively, to invent or pick approaches to solve problems, or, heaven help them, prepare an equation to solve a word problem, they are completely unprepared. So the question is then how to teach that 75% to reason abstractly all along. The particular challenge is that any situation that they can turn into one of memorization or method, they will, because they prefer it. These are students that, (when you tutor them) when asked to reason out a problem, say crossly, "Just show me how to do it!" So on one hand, you want to exploit their strengths, but on the other hand, you have to fight against the students' attempt to reduce all problems to right-answer-getting.
But then, that leaves the question of what happens to the other 25%! What about the kids who are good at abstraction and weak on concrete operations? I'm one of those, and from your description, I'd guess you are, too. If you are teaching to the other 75%, how does that serve the kids like we were? Does it neglect our weaknesses? Or would it be a mercy not to be required to work exclusively from our weakness?
I don't know the answers to these questions; I find them fascinating. And this doesn't even get into the other learning-styles questions. There's lots more questions where these come from.
Re: I work for one of those companies...
Date: 2007-02-01 02:50 am (UTC)Here's a (weak and naive) analogy: in elementary school, they teach you how to use the library. Rather than have you read every book, they teach you about card catalogs and where the reference section is and so forth. Supposedly, then you can go and find the knowledge you need when you need it. But teachers are resources like libraries, and so is your own learning style. I wonder if there's a way to teach some sort of meta-learning skill.
I sometimes wonder what college would be like if I went back now. On the one hand, my brain is probably less flexible, but on the other hand, I'm a lot more mature and have a lot more experience teaching myself...
Re: I work for one of those companies...
Date: 2007-02-01 03:21 am (UTC)Heh. http://siderea.livejournal.com/440902.html
I sometimes wonder what college would be like if I went back now.
Hehheh. I'm running that experiment now. I've returned, at the ripe age of 35, to college to finally get a degree (this time). Things are much better this time, for a whole bunch of reasons, but not least because I've gotten over my past schooling and become an autodidact. Apparently, this is normal in "adult learners", according to the returning student orientation materials the school gave me -- and backed up by my experience in all-"adult-learners" classes. The grown-ups are voracious and keep the profs on their toes. They'll show up with their own learning (and other) agendas and hold the profs accountable for meeting them. They're no little bit cynical and know how the world works, and have an attitude of "I'm paying for this, so you'd better live up to my standards." They butt into lectures to demand clarifications. They raise contradictory examples to challenge professors' examples. They have no problem saying, point blank, that school has to take a back seat to work/kids/etc. when they feel it does. They're intensely involved in the classroom and with the material, just like professors always are saying they want. As one prof put it, "The up-side is you guys bring an incredible wealth of experience and engagement to the classroom. The down-side is, damn, ain't nobody can tell you guys nothing!"
tangent
Date: 2007-02-01 02:53 am (UTC)This is, of course, a problem that goes well beyond math, and gets to the whole question of when it's ok to cater to the majority and what you owe the minority. I felt largely cheated by my school system because they had no accommodations for people near the tails of the bell curve; it was very much a "one size fits all" approach. I don't know what the answer is; even if it were practical for every student to have an individual, tailored experience, I don't think that would be ideal either. Somewhere in there you also have to learn about compromise and getting along in community, too.
Re: tangent
Date: 2007-02-01 03:38 am (UTC)If we are educating kids to a certain standard.... why do we have (or try to have) all kids stay in school until 12th grade? And why do we then throw kids out at age 18, whether or not they've met the standard?
Or are we providing an educational opportunity from ages X to Y, during which kids are entitled to make the most of themselves, whatever that is?
Are we trying to prepare a mete workforce for the exigencies of the 21st century? Are we trying to unleash human potential? Are we trying to educate an electorate?
What is the purpose of the enterprise? And how will we know if/when we're achieving it?