School's mathematics don't add up!
April 10, 2001 7:56 AM   Subscribe

School's mathematics don't add up! PS 234, a primary school in TriBeCa, is at the forefront of the revolution in math instruction being carried out in more than half of New York City's schools. The district's approach to math instruction follows an egalitarian theory called "constructivist math," which is the idea that children shouldn't learn basic techniques for adding, subtracting, dividing and multiplying. Rather, emphasis is placed on "feeling good about numbers" etc. Said one angry parent, "The idea that the home has to be turned into the school because the school is the testing ground for inane programs - that's frightening." And leading university mathematicians have joined parent groups in denouncing the method. All things concidered, is it right for schools to use children as guinea pigs in this manner?
posted by frednorman (53 comments total) 1 user marked this as a favorite

 
Comedic genius Tom Lehrer tackled this subject more than 35 years ago in "New Math."
posted by darren at 8:40 AM on April 10, 2001


*picking my jaw up off of the desk, gasping with stunned outrage, with one thought foremost in my brain*

...and the future of our country is _______. (Fill in the blank.) My word is "uncertain."
posted by NsJen at 8:45 AM on April 10, 2001


You know, that is a very good question, frednorman. I remember our school instituted a new method of teaching algebra based on working in groups and learning with books filled with pictures and word problems. I ended up taking algebra four times until they put me into a regular algebra class with regular textbooks and no one breathing down my shoulder. I don't work well in groups. And I can't imagine that many of my classmates retained any of their hard-won algebraic knowledge, seeing as the workbooks never even got to quadratic equations. We just did a lot of graphing and word problems.

I am surprised that methods like these aren't tested by universities (not on the students, but on "focus groups" or somesuch) or teachers before being tried out in class. Or are they?
posted by annathea at 8:49 AM on April 10, 2001


Or could it be that the Post and the angry parents are misunderstanding, distorting or oversimplifying what constructivist math is about? The idea isn't that students "shouldn't" do these skills, but that they can learn other math skills in preparation for them--like estimating, for example, which is taught to kindergartners as a way of giving them a general feel for quantities and magnitudes before they learn to add. Kindergartners, in my experience, are traditionally not asked to do even that.

As for the "guinea pigs" thing--first of all, this is not much different from the kind of math education Montessori scholls have been doing for nearly a century. Surely there's some data to be found there? (My kids go to a Montessori school, and their math skills, as measured by standardized tests, are fine.) Second, what makes us so sure that the "traditional methods" are successful? They too are founded in psychological theories (constructivism is based on Piaget and Vygotsky) and are subject to empirical validation--what are the numbers on those? Or are we not supposed to ask about that because it's "traditional" and therefore OK and CM is "egalitarian" and therefore suspect?

What's sauce for the goose is sauce for the gander. If you're going to view CM critically (a good idea, I think), then you should view all math teaching methods critically. Otherwise someone might suspect you of really being more concerned about a political agenda than about children. (By "you" here, I don't mean you, Fred, I mean "people".)

See here for some sources on CM.

(I'm not saying CM is good or bad—I'm not equipped to make that judgment. All I'm saying is that the kind of sensationalizing that the NY Post indulges in is not likely to add much light to the debate, whereas paying critical attention to empirical research might--but it's not nearly as easy to score debating points that way.)
posted by rodii at 8:58 AM on April 10, 2001


I'd like to give an opinion on the topic, but this article is so biased, it's impossible.
For instance, clearly the children will learn how to add and subtract. But this article (and especially the post above) implies that they wont.
posted by Doug at 8:59 AM on April 10, 2001


Here's where I'm confused. Near the end of the article, they say the school's math scores are relatively high. This is then said to be misleading because the students are in a position to afford tutors. That's pretty specious reasoning. If the test scores are good, and there's no proof of extensive tutoring, then someone needs to back off their fear of change.

On the other question, I'd need to see more proof that these kids are being used wholesale as guinea pigs. I'd be very surprised if there hadn't been university testing. The article is far too slanted and hysterical to wring any facts from it.
posted by frykitty at 9:04 AM on April 10, 2001


The article you linked didn't give any real details about the program being used, so it's difficult to evaluate it. I would bet, however, that there's a lot of research behind it. I think it's irresponsible to dismiss it out of hand without knowing more about it.

The actual techniques behind math are really not all that complicated. Drilling multiplication tables is not nearly as useful as teaching logic. I remember seeing a documentary about a teacher at a private school who didn't teach arithmetic, etc., because he felt that younger students should concentrate on language, and other subjects that are much easier to learn before puberty. When his students got into the middle school years, they begged to be taught arithmetic, and he brought them up to grade level in six weeks. This could simply be an example of teaching children according to their developmental stages.

Anyway, if the way that math was taught when we were kids (or our parents are kids) was so superior, a lot more adults would understand math better today.
posted by anapestic at 9:10 AM on April 10, 2001


Or could it be that the Post and the angry parents are misunderstanding, distorting or oversimplifying what constructivist math is about?

Even knowing nothing about CM, I guessed that that's what was going on here. I've noticed a recent rash of MeFi posts from nypost.com--and it's a little disconcerting. This little 200 word article has almost no genuine information at all--it's mostly just the ravings of outraged parents.
posted by jpoulos at 9:10 AM on April 10, 2001


The outrage is pretty simple. New York City public schools (at least in the elementary grades) expect all fundamental education to be done by parents. Teachers teach the frills and administer the tests. Harried parents have to take hours out of their personal and family time to make sure that the kids have the 4R's under their belts.

Where most of the parents are conscientious and themselves educated, like in the wealthy neighborhoods in Manhattan and the middle-class neighborhoods in Queens, Brooklyn, and Staten Island, this works okay, although somewhat exhausting the parents. Where parents lack the interest and/or wherewithall to conduct a daily 6 p.m. to 8 p.m. class in primary subjects, you have absolute chaos and total failure.
posted by MattD at 9:18 AM on April 10, 2001


Once I got beyond the incensed rhetoric, I started to think I'm not sure this is such a bad idea.

Our first language instruction is not to parse sentences into syntactical components, it's to learn how to get basic concepts across and enjoy the results.

Our first art instruction is not to carefully apply brush strokes to mix oil paints into subtle shades of realistic color, it's to squish and squeeze and splash and sketch, delighting in the process of creation.

Why should our first math instruction be to master precise, if simple, calculations and memorize tables of equations?

from NY POST - The idea is that children shouldn't learn basic techniques for adding, subtracting, dividing and multiplying, which are believed to deaden enthusiasm for mathematics.

My entire mathematical education served to progressively deaden my enthusiasm for mathematics. I never had one instructor capable of making 40-problem drill homework assignments compelling.

from NY POST - Rather, emphasis is placed on becoming familiar with math's conceptual underpinnings, and feeling good about numbers. Looking for the correct answer to a math problem is out; a "reasonable" answer is considered acceptable to constructivists.

"Feeling good about numbers" - the reporter's words, the outraged parent, or the curriculm? The article is on such a slant it's impossible to tell.
posted by Tubes at 9:36 AM on April 10, 2001


MattD: I wasn't aware of that policy in the NYC schools. Is that documented anywhere? Because if it's not, I'd almost think it was just more ideology in the guide of fact.
posted by rodii at 9:41 AM on April 10, 2001


(I mean "guise", dammit, "guise")
posted by rodii at 9:44 AM on April 10, 2001


rodii, you didn't hear about NYC's "To Hell With The Little Bastards" policy? You must have been napping.
posted by Skot at 9:49 AM on April 10, 2001


Tubes, I completely agree. I hated math in grammar school and junior high, because it was drills, drills, drills. No thought, just memorization. Then I hit high school, where my teachers mixed things up a bit - small groups of students working together to solve longer problems. Same thing in college, only more advanced - we discovered theorems on our own through guided projects, instead of being told the theorem and then working through the proof. I graduated with a degree in Mathematics. If anyone had told me (or my parents) that I'd grow up to be a math geek based on my K-8 math performance, I would have died of shock.
posted by xsquared-1 at 9:51 AM on April 10, 2001


Rodii -- it isn't the policy, but it is most assuredly the practice. And it works quite simply: teachers regularly, and I mean almost daily, assign homework to their classes which they haven't taught them how to do and expect that homework to be turned in the next day, completed, after having one's parents "help" with the homework.
posted by MattD at 9:54 AM on April 10, 2001


It's the "most assuredly" part I'm not with you on, Matt. It's not that I doubt you, exactly, it's just that it sounds like I'm only hearing one side of something that admits more than one interpretation. It wants a skeptical approach. (Aside: Hey, Matt--did you go to Carleton?)
posted by rodii at 10:05 AM on April 10, 2001


New York City public schools (at least in the elementary grades) expect all fundamental education to be done by parents. Teachers teach the frills and administer the tests. Harried parents have to take hours out of their personal and family time to make sure that the kids have the 4R's under their belts.

This is, in most cases, false. MattD, you're high. There is no one-method system in New York. It's a school system with extremes: the figure-it-out-on-your-own approach through the micromanaged learning by rote system. It depends on the school, the teacher, the principal, the student, the parents, etc. And in private schools the variation is even greater.
posted by Mo Nickels at 10:08 AM on April 10, 2001


"Looking for the correct answer to a math problem is out; a "reasonable" answer is considered acceptable to constructivists. "

oh well, if the constructivists say a reasonable answer is acceptable.... I had a class like that once. What it taught me was; that if i didn't have enough time to learn the material, I could learn (in much, much less time) how to make it look like i was getting "reasonable" answers. I didn't learn anything. My favorite math instructor from high school was strongly opposed to ideas like this. He claimed that there is no place for "feel good" mathematics. Praising kids for getting the wrong answer does not encourge getting the right answer.
posted by dangel low at 10:09 AM on April 10, 2001


who needs to learn how to add when we have powerful computers that take up less than a room full of space to do it for us?

heh.

anyway, i would think that using focus groups of college aged people wouldn't really work here -- by that time, you already know how to add or subtract (hopefully) so teaching you a new method to go about it wouldn't really work.

although, if i could have learned a better way to do precalculus, i may not have failed it!
posted by sugarfish at 10:16 AM on April 10, 2001


Harried parents have to take hours out of their personal and family time to make sure that the kids have the 4R's under their belts.

I'm sorry if I don't have a lot of sympathy for your position, MattD, but going over schoolwork with kids is a part of family time. A child's education is not something that you entrust entirely to an institution, no matter how good that institution is. I live in a fairly affluent county with a good school system, but when my daughter was in elementary school, she would be in a class with 20 - 22 other kids and one teacher. During the 6 hours or so of class time, the teachers do their best to help all the kids, but there's only so much they can do. And she's had pretty good teachers. I figured it was my job to spend time with my daughter after school to make sure she understood what was going on.

It was time very well spent, too. My daughter is twelve now, and she and I have a very good relationship. And because I helped her when she was younger, she's now much better at doing the work on her own. I'm divorced now, and I only have her two or three nights a week, but whenever she can't understand some part of her homework, she calls me at my office. And I always take the call.
posted by anapestic at 10:27 AM on April 10, 2001


Can I ask an only slightly-off-topic question not meant to generate a flame war? How is it that pedagogical techniques have gotten so politicized? There were all kinds of outcries during the last round of phonics vs. whole language, with people making really vehement stands for one or the other. (IIRC, the breakdown seemed to be liberals for whole language, conservatives for phonics.) I fully appreciate parents being worried that schools are failing to educate their children; I'm just not sure when exactly this became an issue that could be addressed by some sort of larger political worldview. (It was especially odd with the phonics/whole language thing, because schools have gone back and forth on the two for at least a hundred years.)
posted by snarkout at 10:34 AM on April 10, 2001


I think it's dangerous for public schools to try these things in a climate where standardised tests, underwritten by parental hysteria, can render any experiment short-lived. But then again, when I read tabloid pieces like that one, I remember what my tabloid journo friends tell me about how easy it is to cry "outrage", particularly when it's not your kids who suffer from either the teaching or (even worse) the publicity.

Reading that piece, I get the sense that there's a certain amount of "middle-class parent syndrome" going on, whereby parents feel obligated to seek external tuition because that's what other parents do, and they can't be seen to be "letting down" their children.

(I think that partially answers your question, snarkout. And I'd also suggest that "professional" parents are instinctively suspicious of teachers, since they're often perceived as leftie pinko indoctrinists who've chosen to corrupt our children with 60s ideology rather than becoming accountants.)

Praising kids for getting the wrong answer does not encourge getting the right answer.

This isn't high school, though. My three-year-old niece is visiting, and she's at the stage where she's fascinated with numbers -- coins in her little purse, the numbers on the clocks, and so on. She hasn't really got the processing skills yet, but I'd rather encourage that inquisitiveness as she starts school (a year away) than get out the long division books.

(Tangentially: when in NYC a couple of months ago, I walked past a school -- a Catholic school, in fact, all dark blazers and cute hats -- and remember thinking how it was the last thing I'd expect to see in Manhattan.)
posted by holgate at 10:39 AM on April 10, 2001


re: phonics vs. whole language...

in my elementary school we were taught whole language -- there were certain rules in place like "no worksheets ever" and whatnot to make sure that no sneaky teachers taught us a touch of phonics under the table or anything.

it's a wonder i can read at all today. occasionally while reading i'll come across a word i don't know and i realize now that i have more difficulty in this situation than i should -- new words often stop me dead... which is probably one reason why i have a pretty advanced vocabulary, but still.

whole language kids tend to come across a new word, stop, decide on an interpretation (pronunciation, meaning) and then just accept that invented interpretation and go with it. that's bad.

my point is, in kindergarten, 17 years ago, it was decided that i would learn to read whole language and i have not recovered. i think these debates are important and i'm glad they are politicized in such a way that serious adults discuss them rather than just letting a flaky school board or administration decide.
posted by palegirl at 11:20 AM on April 10, 2001


I don't want to turn this into a phonics vs. whole language discussion; I'm just puzzled by why the debate seems to have been politicized in liberal-vs.-conservative terms within the last 5 years (unless I'm imagining it, which people are welcome to say).
posted by snarkout at 11:31 AM on April 10, 2001


It might be illuminating to know that recent brain research seems to have indicated that there are two specialized locations in the brain that handle math, and that they work differently. One part does the arithmetic we all know and love/detest, the other does "guesstimations" using small numbers (three or less, I think), probably something like fuzzy logic. If a stroke knocks one out, the other one may still function. Sounds like this teaching technique may be similarly polarized, to good effect. Remember, even people who use calculators for the details need to be able to recognize when the answer's out of whack.

How much math does a "normal" human need to know for everyday life? Not calculus, surely. Cooking can require working with proportions, and social life definitely needs various kinds of logic. But I'm a math geek and I haven't had cause to factor an quadratic equation in many years, however pleasurably I might anticipate that feeling of solution. Weird math (such as Boolean algebra, back in the day, or Fourier analysis) is useful mostly in computer applications.

The Tom Lehrer song referred to above isn't quite gratuitous 'cuz it refers to an earlier parent-boggler reorientation in math education, this one in California in the '70s. Suddenly your children were learning set theory in grade school. OTOH, making up groups from intersections and unions of other groups, or selecting elements of a group that meet certain criteria, are things programmers do all the time, and learning them while your brain's still plastic might be a good idea. It'd be interesting to see if we could correlate the rise of Silicon Valley with this.

ps: You never know, this Post story might be a back-skirmish from the Rainbow Curriculum days...
posted by retrofut at 11:39 AM on April 10, 2001


Rodii -- I did not go to Carleton. My sister went to Macalester, though ... and lots of Dundons in Minnesotta and the UP, anyway.

Generally -- I agree that intellectual enrichment is a part of family time, and that parents are responsible, in the final analysis, for the quality of their children's education. I disagree that the day-to-day business of teaching basic concepts necessary to complete one's homework should be a parent's responsibility.

I certainly had attentive and helpful parents, and, at the same time, my homework was for me to do and my teachers to grade, and for my parents to worry about only to the extent that alarm bells were rung at parent-teacher conferences or report-card time. This enabled my parents to focus on talking about what was on the news that night, on what interesting books I was reading, etc., rather than have walk me throgh 45 repitions of long division to assure that I'd mastered the concept.
posted by MattD at 11:40 AM on April 10, 2001


This course is a sick joke, but the concept is hardly new. There has been a steady trend over the last few years to replace objective assessment (tests, exams, memorization (demonized as "rote learning")) with subjective assessment (portfolios, "understanding," "team-learning," feelings about the subject). The recent assault on the SAT is a good illustration. (To the skeptics: yeah, I didn't believe it either until I saw it being done to my own kids).

Part of the reason is simply that teachers are overwhelmed by class size, and it's so much less time-consuming to just express a personal opinion on a student rather than to rate his/her skills and performance objectively vis-a-vis his/her peers. But a larger part of the reason is New Age philosophy that preaches that how you feel about what you're trying to do is more important than how well you do it. So let's not hurt our childrens' fragile self-esteem by actually requiring them to get the correct answers. And never mind the educational damage; we teachers will have retired before anyone notices it, right?
posted by beentheredonethat at 11:58 AM on April 10, 2001


I'm sorry, beentheredonethat, but by saying "New Age philosophy" with a straight face you just lost any credibility you might otherwise have had.

Why are you so convinced that what you call "objective assessment" is a better way of accomplishing whatever it is schools are supposed to do? How do you know the choice of "objective" standard isn't arbitrary? Isn't your idea about what the point of education is a subjective assessment?

-Mars
posted by Mars Saxman at 12:46 PM on April 10, 2001


The Tom Lehrer song referred to above isn't quite gratuitous 'cuz it refers to an earlier parent-boggler reorientation in math education, this one in California in the '70s.

Yeah, it brought the change in the teaching of subtraction and long division that separates my primary school maths from that of my parents. Interestingly, one of the main supporters of the change was J. G. Keremy, the man behind BASIC (another mixed blessing).

Anyway, here's an interesting transcript on the debate:

There are no ads in the paper for people who are highly proficient at long division with big remainders. Employers want employees who can communicate effectively on how they solved the problem.

And I have to shrug and agree.

snarkout: it's the classic ideological division between teachers and other "professionals". I'd imagine that if you're a parent working 12-hour days in the Financial District, you consider teachers as whining lefties with overlong holidays who spout crank "theories" about education instead of teaching; if you're a teacher, you consider parents as overpaid Dilbert clones who haven't a clue about life at the chalkface.

It'd be nice to have a teacher-parent job-swap week.
posted by holgate at 1:13 PM on April 10, 2001


I'm a conceptual learner first. If I don't understand the concepts behind something, then the details or implementations of something don't make sense. On the other hand, I've known people who were entirely 'detail level' learners, who couldn't give a hoot about the concepts or theory behind something. If I'd been taught the conceptual basics of math before the adding and subtracting, I would have warmed up to the whole 'number' deal a lot faster. When I went to college as a computer science major (from a blue collar family - a first) I had NO idea why the HELL we'd need to know Calculus (or what it even was/meant) in order to do programming. Still don't. I didn't understand that a lot of what they were teaching was just theoretical math - but had they TOLD me this, then I could have gotten over that hump and down to basics. As it was, at the time I wasn't even sure WHY I was having such difficulty with it - couldn't even put it into words.
posted by thunder at 1:20 PM on April 10, 2001


Get rid of unions. Hand out vouchers. Home teaching for those kids whose parents don't like what is being done. Lower dropout age to 14. Give all students A so that democracy works. More money for field trips and sports. Use public libraries instead of having school libraries. Bus most kids but only one way. Make the political parties pay for using schools for voting at election time. All boys to have crew cuts. No piercing. School unforms for all but only black clothing allowed. Finally, bring back the switch blade and the zip gun so we can have education as it used to be.
posted by Postroad at 3:23 PM on April 10, 2001


Nobody's suggesting that, Postroad. Implying that, is just plain silly. What I am, on the other hand, suggesting, is that we must base education on facts and reality -- and not on "almost-facts" that happen to "feel good".

A does not equal B, after all, even when B is colored pink and resembles -- from a distance -- a panda bear. There is a real world out there. And kids should learn to deal with it by reason, not by "feeling", voting or guessing.
posted by frednorman at 3:54 PM on April 10, 2001


Oh, that's too good an invitation to resist:

"THOMAS GRADGRIND, sir. A man of realities. A man of facts and calculations. A man who proceeds upon the principle that two and two are four, and nothing over, and who is not to be talked into allowing for anything over. Thomas Gradgrind, sir - peremptorily Thomas - Thomas Gradgrind. With a rule and a pair of scales, and the multiplication table always in his pocket, sir, ready to weigh and measure any parcel of human nature, and tell you exactly what it comes to. It is a mere question of figures, a case of simple arithmetic."

Charles Dickens, Hard Times

(warning: all 510k of it, though Chapter II is what you want.)
posted by holgate at 4:08 PM on April 10, 2001


beentheredonethat, in a posting that's remarkably free of any real information, says: Part of the reason is simply that teachers are overwhelmed by class size,

Evidence?

and it's so much less time-consuming to just express a personal opinion on a student rather than to rate his/her skills and performance objectively vis-a-vis his/her peers.

Bullshit. There's nothing more difficult or time-consuming than trying to make a valid holistic assessment of a student's achievement in some area. If we could just give standard, machine-scoreable multiple choice tests on a subject, we could save a lot of time. The problem is, there's often little indication that so-called "objective" tests come up with measures that are valid.

But a larger part of the reason is New Age philosophy that preaches that how you feel about what you're trying to do is more important than how well you do it.

I have to call bullshit on this one too. Show me one place that this idea has been espoused by advocates of CM. This is just sheer, gross, cranky personal opinion.

So let's not hurt our childrens' fragile self-esteem by actually requiring them to get the correct answers. And never mind the educational damage; we teachers will have retired before anyone notices it, right?

And this has what relevance to the actual facts of the case, exactly? I see nothing in this post that makes any reference to anything any actual person has said or suggested regarding CM—just some anonymous random's spleen about some undefined something that was "done to" his kids.

Fred, you're almost as bad. Is CM based on "feeling good" or is this just a caricature? Do you know? Can you cite sources any less biased than the New York Post? Several people have posted here to the effect that the traditional method didn't work for them and that working one's way up to computation via concepts might have worked better. I don't see any response to that, just tired straw men arguments about facts vs. feelings, pink panda bears, etc. Tell me one actual fact you know about CM and we can

Again, I'm not advocating CM. What I'm saying is that it's being used as a whipping boy here because some conservatives have found that to be a useful move in their Kulturkampf. I have seen no actual facts or arguments based on evidence, just generalized cultural grousing and prejudice. I'm not mad, just exasperated at how low the standard of argument on topics like this is.
posted by rodii at 4:24 PM on April 10, 2001


This kind of crap goes on all the time in NYC public schools. There are so many things wrong with our city's school system and the people behind it....... i'm not even going to start talking about it. I'll never be able to stop.

These are the kinds of reasons that my parents refused to put me anywhere near the public schools. Politicians want to give private school vouchers to poor families because public schools aren't good enough, they say we need to make our public schools great, and then they go after new-agey guineau pig stuff like this?

Puh-leese.
posted by tomorama at 4:40 PM on April 10, 2001


There is a real world out there.

Thank god I've never done more than visit there briefly. Horribly boring place, it is. ;-)
posted by thunder at 4:40 PM on April 10, 2001


Thanks for the link, holgate. I'd forgotten just how much I liked Hard Times when I first read it, and it's especially relevant to this discussion. Oh, and for anyone who's interested, it's one of Dickens' shortest books and among the easiest to read.

The entire time I was an undergraduate at MIT, no one expected me to memorize facts. Homework was always problems sets. Exams were always problems. Except for my foreign language courses, there was never a multiple-choice or true/false or fill-in-the-blank test. Never. During the course of doing all those problems (and writing papers for my literature courses), I learned certain facts, which made solving the problems faster, but that wasn't the point. And frankly, I've forgotten almost all of those facts. But the problem-solving skills remain.

In the real world, it's frequently useful to know facts. But it's more important to be able to apply them. And, after all, you can usually look facts up. Problems in adult life are almost always open book.
posted by anapestic at 4:48 PM on April 10, 2001


rodii: Can you cite sources any less biased than the New York Post?

Less biased? Hardly. Biased the other way, though, politically? Here's from the New York Times'* take on the issue:
Indeed, the manifesto of constructivist mathematicians, the 1989 standards of the National Council of Teachers of Mathematics, urges teachers not to demand too much accuracy too early. Math should be "flexible," the standards say, and "reasonable" answers should be valued over a single right answer.
Now this is horrible enough in itself. But it is only the result of something far more fundamental, i.e. the methodology's philosophical foundation:

According to Prof. George E. Hein (and others), constructivism promotes the idea that learners construct knowledge for themselves; that each learner individually (and socially) constructs meaning as he or she learns, and that this construction of meaning indeed is learning; that there is no other kind.

This, in turn, of course implies that there can be no knowledge independent of the meaning attributed to experience (constructed) by the learner (or community of learners). Which, again, implies relativism. And there's my problem with constructivist math for you, alright.

(Now, let's just hope that MetaFilter's HTML parser isn't based on constructivist HTML, because from the preview I'm getting here, it sure looks like it :-)

* Free online registration required for access to nytimes.com
posted by frednorman at 6:25 PM on April 10, 2001


(I know, the preview thing throws me off too.)

OK, thanks. I think the key thing in your NYT quote is "too early"--the idea is that learners converge on the right answer, and converge on the right method. So, for instance, when I show a pile of marbles to a bunch of four year olds, and say "how many are there?", it's more important that they get the right order of magnitude than that they get the right answer--because they're becoming accustomed to the relative sizes of things, not learning to count.

Now, I agree, the idea of "close counts" with, say, multiplication seems kind of odd--and I think there must be more to it than that. But I can't offer any more details. However, check out the chart in that very New York Times article and I think you'll note that, although the traditional and constructivist methods are different, they come up with exactly the same answers. So maybe the CMers aren't throwing out the whole idea of correctness after all.

There's a lot in that NYT article worth commenting on, but I've already gotten awfully long-winded, so I'll just say it's worth a careful read. I appreciate your finding it.

* * *

As for the Hein thing, all I can say is that those ideas seem absolutely, trivially, obviously true to me, because it views knowledge as, if you will, a particular configuration of neurons in a learner's brain. That is, the fact, say, that hydrogen has an atomic number of 1 may be "out there" objectively in the world somewhere, but the knowledge of that is something in the learner's head.

How did it get there? The learner constructed it out of a huge flow of sensory data (including classroom instruction) and previous knowledge. This is just basic cognitive psychology, or even earlier, Gestalt psychology, which is fairly non-controversial even with the right.

How is it socially constructed? As social beings (among other social practices, as people who share ideas and information), we decide which ideas to value and pass along, which to treat as givens or articles of faith, which to treat as merely questions of semantics, and so on. The acquisition of knowledge in fact comes from millions of social encounters (as well as the occasional experiment or brush with physical reality). Much of what we "know" we know as the result of countless stories told about reality by teachers, parents, friends, etc.

None of this denies the existence of objective reality or necessitates relativism. It's about how knowledge gets into the head of a learner, not about what is or isn't true.

Constructivism in the philosophical sense (say, the late Wittgenstein) or mathematical sense (the intuitionism of Leopold Kronecker* or L.E.J. Brouwer) has some pretty dangerous suggestions to make about some of these issues. But the constructivism we're talking about here is, I think, much more innocuous.

*Kronecker is famous for saying, "God made the integers; the rest is the work of man," which is a genuinely extreme viewpoint. But Kronecker wasn't a new-ager or a deconstructionist or any of the other right-wing boogey-men—he was a great 19th-century mathematician. It seems to me we should at least try to understand what he's saying before we condemn it for undermining our stubbornly held realist convictions.
posted by rodii at 7:04 PM on April 10, 2001


Constructivism in the philosophical sense [...] or mathematical sense [...] has some pretty dangerous suggestions to make about some of these issues. But the constructivism we're talking about here is, I think, much more innocuous.

I hope you're right, rodii, although I'm not convinced you are.

The chart you pointed to, on the other hand, looks eatable. Simplistic, sure, but exact and correct nevertheless. And that should always be the objective. Which was my main point all along.
posted by frednorman at 7:39 PM on April 10, 2001


(eatable?)
posted by rodii at 7:49 PM on April 10, 2001


Experimental education is already the defacto standard in most classes. In my experience, the most effective teachers all supplemented or entirely deviated from the supplied course materials as they saw necessary, sanctioned or not. I think one thing we all can agree on is the quality and value of astronomically priced textbooks that schools are limited to sucks. Beware those magic bullet claims that always make the soundbites, as teaching has always been as much art and experimentation as science. Teachers mature, ripen, and sometimes go bad. We need to concentrate on keeping thoroughly educated and experienced professionals leading all our kids' classrooms. Micromanagement from on high is ineffective and obscures the big picture that we really need to focus on.
In public school we were all tutored in addition and subtraction by the class ahead of us, which was effective as we all were exited to make friends with the bigger kids back then. I learned multiplication and division by brute force. Geometry and algebra teachers freewheeled it throwing in crazy contests, puzzles and computer labs to coax us through the thick daunting books that weighed down our packs. I went to a small private school my senior year and learned calculus from and experimental textbook my teacher got from the U of MD. I raised my eyebrows skeptically at the mass of photocopies cheaply bound, but my whole class learned calculus thoroughly enough to perform well on tests and solve problems in physics.
But it boils down to this - the success of teaching the whole class depended far more on the quality and personal investment of the teacher than the prescribed lesson plan.
posted by roboto at 8:21 PM on April 10, 2001


Not sure about the whole math thing, although surely it was taught in a mostly boring fashion in high school and most American students don't do well in math. What the heck's wrong then with at least a modicum of experimentation?

What worries me more here is that it came from the NY Post. That means it's a conservative slating article. And the article fits with what I've seen coming out of the conservative rags and mags lately: utter hostility, as opposed to honorable criticism, to anything remotely newish in education, a taking of an opposite side even when one isn't at all necessary.

For a typical example see George Will's column from Sunday on folklorist Bill Ferris, director of the NEH . Note that one of the former persons to win the NEH honor here was author Walker Percy, who wrote entire essays or at least pages of books about movies, the Incredible Hulk, psychology-via-Johnny Carson, the Donahue Show (with a special appearance by Cotton Mather), radio evangelists, etc. and who was generally obsessed with suburban/media culture as it intersected with older America. Yet George believes that studying similar things would be out of bounds for a federal agency in 2001. What a goofball.
posted by raysmj at 9:36 PM on April 10, 2001


Oh, and George W. thought Walker was one of the winners whose work presented quite a contrast to the work of the NEH today. Wrong.
posted by raysmj at 9:50 PM on April 10, 2001


And I always wondered about the rationale behind "add a 0" in long multiplication... but it reminds me that ultimately, maths is a techné, a set of techniques, that sometimes have no "factual" basis other than that they work. And if you can teach not only the components, but the techniques of combination, you're onto a winner. Dickens' point was that emphasising facts alone stagnated thought, so I was doing frednorman a disservice by quoting Gradgrind, since he was talking of "facts and reason". (Apologies.)

(I should really read Russell and Whitehead's vast logical proof of 1 + 1 = 2 in Principia Mathematica some day. When I have time. Or can't sleep.)

As for George Will: jealousy, thy name is George. Opinion journalism is thankfully transient, unlike, say, the work of a folklorist such as James Lileks who deserves public funding for the amount of research he's done. (I'd certainly use his site as a teaching aid, if I were at a school in the Twin Cities.)
posted by holgate at 9:53 PM on April 10, 2001


Topic drift...

Speaking of bias and Newspapers, every publication has it's own amount of bias. It might reflect the owner's views or the reporter's views, but it's usually there in some form. Personally, I never read the NY Post. Out of the 3 major newspapers here, I consider it the worst. The writing reflects that of a HS freshman english class, and the post itself never met a cop it didn't like. The NY Times, although one of the better papers in New York and a great source of information, is sometimes way to liberal for it's own good. For instance, they did a story on the lack of black police officers in the NYPD. In itself, it's a worthwhile topic for a newspaper to bring to light, but the Times decided it would be ok to illustrate their point by using a photo of a troop of all-white cops.... taken at the St Patricks day parade. That annoyed me a bit...
posted by tomorama at 9:59 PM on April 10, 2001


What worries me more here is that it came from the NY Post.

There have been a number of comments in this thread slamming this article in its entirety because it came from the Post. Yet MeFites post articles from the Guardian on the front page almost every day and consider them to be quite legitimate.

Attacking the statements in an article based on where the article comes from is a meaningless logical fallacy, and makes the complainants look petty.
posted by aaron at 10:58 PM on April 10, 2001



Aaron: The reason I worried that it came from the Post had to do with what it reflected about conservatism at the moment (to me it sounds like straining too hard to make everything fit an ideological mold, but that's another story entirely). The NY Post is a well-known conservative paper.
posted by raysmj at 11:45 PM on April 10, 2001


Apples and oranges, aaron. While there's always going to be some blurring, broadsheets at least attempt to maintain the distinction between reportage and commentary. The NYPost is a Rupert Murdoch tabloid; its politics are as set in stone as Fox News or The Sun in the UK:

"I think that's one reason why Murdoch has returned the favor and gone after Clinton even more energetically in the NYPost and the Times of London. Murdoch is the only big player with the clout to fight these guys, and I hope he can pull this one off."

My source for that quote? A Freeper.
posted by holgate at 12:07 AM on April 11, 2001


(I guess I've rendered the actual subject of this thread so utterly boring that we can return to our usual low-level background noise.)
posted by rodii at 5:35 AM on April 11, 2001


rodii: No, if anything it's odd that a NY Post story sparked such an intelligent discussion. It's not only a conservative paper, but biased to the point of being comical, biased in what has seemed to me to be a very self-aware fashion. The fact that the article came from the paper was an absolutely legitimate point of discussion, even if such a debate gets tiresome after a while.
posted by raysmj at 7:07 AM on April 11, 2001


I'm perfectly happy to discuss a story from the Post, just as Aaron's perfectly happy to discuss a story from Guardian, and both of us are quite right to point out that each publication has its ideological bias. Basing one's entire opinion on something from a slanted summary is foolish (and the bias was more apparent in this particular article than in some).

To wrench the thread back slightly more on topic, that chart was interesting. I was a math major in college, so I clearly enjoyed the stuff, but I have a distinct memory of being freaked out by watching a substitute teacher do a multiplication problem in a marginally different way (I think involving not writing out all the values to be summed) when I was nine or so, because I hadn't gotten the concept so much as the procedure. All other things being equal -- and I'm willing to accept that they may not be with constructivist math; I spent some time last night reading some reasonably convincing academic criticism of it* -- I think the method that exposes the most of the underlying mechanism is preferable. Some of my math teachers in high school were very good at that teaching technique, and I think it helped me, at least, learn better. (New math exposed lots of the underlying method, but not in a terribly useful way -- bases just aren't that important for applications -- and it almost certainly didn't teach children as well as alternatives.)

It's just very weird to me to see Lynn Cheney weighing in on teaching methodologies; from what I could read from Fred and BeenThere's statements, it seems that the politicalization comes from a feeling that public school education in rigorous basics has been hijacked by a touchy-feely "New Age" agenda. Fred, am I stating your position correctly? (Why this is being posited as a conservative/liberal distinction is a question best left for another thread.)

* No links at the moment, but apparently one of the studies that forms the basis for constructivist philosophy was done on working-class women who could calculate the best buys in a shopping scenario more accurately than the exact same problem expressed as a paper-and-pencil quiz; the criticism I read asked whether how meaningful this was.
posted by snarkout at 8:37 AM on April 11, 2001


Regarding the vast logical proof of 1 + 1 = 2 in Principia Mathematica (for anyone who's interested):
In normal math (starting with number theory, i.e., arithmetic) no proof is needed for 2 = 1 + 1, simply because 1 + 1 is the definition of 2. In normal math, we define addition, and introduce axioms such as:
- associativity: (a+b)+c = a+(b+c)
- commutativity: a+b = b+a
- identity: 0 is the only number for which a+0 = 0+a = a
Principia Mathematica isn't normal math. Principia doesn't start by assuming the above axioms about addition. In fact, it doesn't even assume the existence of addition, or even the existence of numbers. Instead, it assumes the basic axioms of symbolic logic and set theory, and builds numbers and the rest of arithmetic out of them. Because Principia builds numbers and addition (rather than assuming their existence), in order to prove that the constructions are valid, the axioms of number theory must be proven to work on the constructions.
If you're really interested, here is a massive hyperlinked and cross-referenced set of proofs similar to Principia.
Included are proofs of such things as the existence of the natural numbers, and 2+2=4, a "very advanced" proof.
Note that the steps in proving 2+2=4 includes such basic things as "4 is a real number" and "addition is associative" which in this context, require long proofs.
What good is this? Well, in mathematics, we like to keep our set of assumptions to a minimum. (Note Euclidean geometry, built out of a mere five axioms.) We use the axioms of set theory, number theory, and symbolic logic quite extensively (and of course, theorems proven from these axioms). Principia shows that we need not assume the axioms of number theory, since we can build them out of set theory and logic.
Of course, most mathematicians (myself included) don't need to build numbers out of sets; we just need to use numbers. I am aware of the existence of Principia Mathematica, and I think its kinda neat, but I don't actually think of numbers that way.
(Some mathematicians hardly use numbers at all, but that's a different story.)
posted by CrunchyFrog at 11:28 AM on April 12, 2001


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