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. :-)
(no subject)
Date: 2007-01-31 03:47 am (UTC)(no subject)
Date: 2007-01-31 03:57 am (UTC)(no subject)
Date: 2007-01-31 08:04 am (UTC)(no subject)
Date: 2007-01-31 04:35 am (UTC)Good plan! I often do math in my head to keep some small level of skill. :-) I often do a rough tally of groceries in my head, but not always.
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?
(no subject)
Date: 2007-01-31 08:09 am (UTC)Looking at these methods for doing problems, it seems that they're trying to teach intution, sometimes without teaching rules first. (So.. I'm pretty sure that basic addition and subtraction have to be memorized. It seems hard to break 4+7 down into some simpler problem.) The question here that isn't getting mentioned is, how is the teacher teaching it? If they don't understand how it works and can't express that to the students, then it's pretty much equivalent to just teaching them rules, only it's a more complex than necessary set of rules. If it's rules that can generate intution with less requisite experience then they;'re good, but the anecdotal non-evidence suggests that it's hit-or-miss.
Cluster problems... ugh. This sort of backward hole-filling reasoning is really dependent on intuition to go anywhere, and if a particular student can work it, it's good, and if they can't, it's dead. It reminds me of teaching imperative programming proofs when I was a TA--I had no problems coming up with loop invariants, but some of the students just couldn't. The way the professors taught it was the way I thought about it, that the loop invariant was some expression which satisfied a few conditions and enabled proving the conclusion, and as for how you determine what it is...they said nothing. I had no problems with it--stare at a loop, figure what it was supposed to do, think about the shape of the logical hole that needed to be filled... flash! Loop invariant! But several students didn't get it so well, and I'm wishing now that I could have come up with rules for it to tell them until they could get their own intuition. Hole-filling (by which I mean, given the start state and the end state come up with the middle) in some cases may just inherently be a heuristic search problem rather than one susceptible to deterministic rules, which means it's totally intution, which means you just need a lot of experience with seeing all the pieces together. Bleh. It's late, I'm babbling. :)
(no subject)
Date: 2007-01-31 02:28 pm (UTC)Very true, and a likely explanation for some of my college experiences. (You could always tell which professors didn't grok undergrads.)
(no subject)
Date: 2007-01-31 08:10 am (UTC)I've been impressed by (and even jealous of) my daughter's new math education. New math got her ready and eager to take advanced geometry in ninth grade.
My kid and her fourth-and-fifth-grade classmates learned all the techniques lamented in the video. They also learned the standard algorithms, plus visualization techniques (plots, charts, graphs, arrays, maps) that I didn't get to in the old-math regime until ninth-grade algebra. The algorithm-lady sneers at the textbook chapter on visualizations, but visual support for what she was doing with numbers was crucial to my kid's understanding and enjoyment of math.
Example: when Gwen was in fifth grade, she showed me how she'd learned about squared and cubed numbers by drawing me a picture of a "number square" and a "number cube", and showing me how the visualization and annotation were related. Dang, we didn't even get to the *number line* until I was in middle school! Under new math, kids are doing the number line, and learning visually about negative numbers, in third and fourth grade.
Her elementary-school curriculum made math so interesting to my daughter that she routinely shared what she was learning with me. Sometimes she needed help with mechanics, but just as often what she was learning sparked relationships in her mind, and she wanted to talk about them. New math gave me a way to talk to ten-year olds about hypercubes -- fun for them, fun for me, and they've been chomping at the bit for physics ever since.
Compare: in fifth grade, my classmates and I were having our minds numbed with rote memorization and practice of the standard algorithms for multiplication and division. We spent *years* doing nothing but multiplying and dividing columns of numbers. The algorithms are efficient, but what do you have when you're done? Bored, math-loathing fifth-graders who can efficiently multiply and divide long numbers.
Death to old math, I say, and good riddance.
(no subject)
Date: 2007-01-31 09:48 am (UTC)Can you fake being ready for high-school geometry? Is this some new-math trick or lowering of national standards? I'd have to check the statistics, but I think that American students have been doing better and better at math and science over the last twenty years.
(no subject)
Date: 2007-01-31 02:25 pm (UTC)Yes, I also had a knack for math, and I was usually the first person in the class to figure out the untaught "rule" (pattern). I don't think that was most of it. (Interestingly, in most ways my school system seemed pretty bad; it particularly failed high-school students. Maybe they put all their focus on the early years and got that right?)
Oh my, you've put your foot in it...
Date: 2007-01-31 06:03 pm (UTC)Daniel (age 15) uses the Lattice method (Everyday Math) almost exclusively for simple multiplication. Why? Because it's faster and more accurate than the "standard algorithm".
The reality is that the group that made this slick video (and other related groups) are faux advocacy groups that appear to be focused primarily on the idea that "because it's not what I learned, it's bad". This doesn't reflect the fact that different kids learn differently.
From Valorie (in an email sent on the discussion we had at work about this):
Another thing to remember is that one of the reasons we are so pathetic in math these days is that so many of the *parents* believe they are awful in math. Parents believe themselves competent to help their child write/edit a paper but many parents (in my experience) believed the math was too hard by third grade. Once students start two digit multiplication or long division, parents started to drift away. Fractions typically got NO support from home, and don't get me started on units of measure! If the parents convey to the kids that "Math is hard. I don't understand it so I don't expect you to understand it," then kids believe they can't learn math. I had some kids doing fourth grade work by the end of second grade, but by the middle of the third grade standard math curriculum, they'd decided that math was too hard even though the math was easier than what they'd been doing in second grade.
This is why I had to start teaching the first graders "algebra" - their parents needed to feel competent to help their child and every parent I've met thus far doesn't think first grade math is too hard. Once I started to get the parents feeling competent, the number of kids who struggled with math in 4-6th grades decreased dramatically. Of course, this is relatively anecdotal and a very small sample set, but it makes sense to me. How many times do you hear someone say something along the lines of "Let me get out my calculator so I can figure out the tip. I'm lousy at math"? A lot fewer times than I hear someone say something along the lines of "You read it to me. I'm a lousy reader."
Having a curriculum that changes the way students think about math makes this innumeracy worse, but most of these progressive math curriculums support a lot of parent teaching. Everyday Math includes materials for school-wide math nights, games parents can play with their kids and letters to send home which include "How-to" examples of the methods covered in class. It isn't great, but parents have to be willing to ask questions if they want their kids to be able to ask questions.
Re: Oh my, you've put your foot in it...
Date: 2007-01-31 06:42 pm (UTC)Re: Oh my, you've put your foot in it...
Date: 2007-02-01 02:58 am (UTC)Yeah, the discussion has been fascinating (and educational).
teachers and administrations have a part in this
Date: 2007-01-31 10:23 pm (UTC)Monica, your school experiences are pretty close to what the "new math" curriculums are trying to create. Teach kids in a variety of different ways, a variety of different concepts and give them the opportunity to think through what they are learning. For instance, rather than giving kids endless multiplication worksheets to complete, the next unit could be area and perimeter which has the kids visualizing arrays so they can see visually that 6*8 means 6 rows of 8 or 8 rows of 6 and that all totaled, 6*8 is 48. From the mathematical standpoint, kids are just practicing their multiplication tables in a different way while they learn other concepts. From a student's perspective, they are just learning something new. It's not rote and because it is presented in a different topic, it isn't quite as boring.
I grew up with the more rote-style curriculum, but after seeing the newer style curriculums in action for the last eight years, I've become a convert. I am pathetic at arithmetic (lousy handwriting and a worse memory!) but math was always easy for me. The new curriculums, in the right hands, make the transition from arithmetic to math a smooth transition. Kids are exposed to thinking algebraically much earlier and to thinking about why rules are true and, more importantly, when they aren't true. Watching the kids transition through and succeed in algebra at a much higher rate than previously has made me a believer.
Now I'm REALLY going to put my foot in it...
Having said that, the curriculum is only as good as the teachers who teach it and the administration who demands certain requirements be met. When I student taught last year (who'da thunk I'd become a teacher?!?), the 6th grade kids had just completed the area and perimeter unit. When it came time for the test, only 2 kids out of 23 passed with a C or better. The official teacher went off to her move-up meeting with the local junior high teachers and was told that it doesn't matter; just keep going in the curriculum. So I got to start teaching the students the next unit (Volume and Surface Area) when the kids don't understand area and perimeter... As Larry said, my nickname is the Math Fairy but even I knew that you couldn't build a house over a pothole. I had 12 class days to complete the volume and surface area unit because we were "behind schedule" in our math curriculum. I had one kid flunk the final test that I made up (which included perimeter, area, volume and surface area) but he was allowed a make-up test (which was the "required" end of unit test) on which he recieved a 95%.
What changed? These were about as "creme-of-the-crop" kids as you could get. No one was hungry, homeless or on drugs, and all of them had two parent, high-income homes. The difference (I think) was that I firmly believed all the kids could grasp the mathematics and the official teacher still struggled with the concepts. I attended the district training on the new version of the fractions unit and while I was there with 12 or so other 6th grade teachers, the two at my table declared that they didn't understand fractions and would not cover this unit in their class. When I asked why, their responses were that the kids were poor and wouldn't be able to understand the concepts. As we worked through the problem sets during the training, it became quite clear that these two teachers didn't understand how to add, subtract, multiply or divide fractions and were mostly unwilling to learn.
Here were teachers, taking the time to attend district-sponsored training on concepts they didn't understand, who refused to ask for help to understand the concepts so they could teach their kids. There is no curriculum in the world that can help that problem.
When "my" class got to the unit, the official teacher sat in the back while I taught the unit. She admitted that she while she knew the rote rules for fraction math, she didn't understand why any of it was necessary or why it worked. When the unit was done, she thanked me for explaining the methodologies to her. It doesn't make a difference what curriculum that teacher teaches, she works to make her skills better which has to make her students learn just a bit better, too.
Re: teachers and administrations have a part in this
Date: 2007-01-31 10:25 pm (UTC)I haven't yet found a curriculum I think is perfect, but I've found an awful lot that are good. In my classwork to become a teacher, we're told that it takes the average teacher 4 years before they teach the curriculum as it is supposed to be taught, and another four years before accurate results of the curriculum can be determined because no curriculum works in a vacuum. For instance, Connected Math works very well with science curriculums that encourage exploration rather than rote experiments. When the science and math curriculums help reinforce the concepts of the other, students really start to excel at both.
Ok, this is starting to ramble a bit, but this is a subject near and dear to my heart. When your local school district is thinking about changing curriculums, think about whether you are willing to wait eight years to determine if its better or whether you might want to spend that money training teachers to use the curriculums they already have.
Re: teachers and administrations have a part in this
Date: 2007-02-01 03:10 am (UTC)To clarify, my school experience involved a lot of rote memorization and timed drills. But there was other stuff on top of it; we did do area, and I remember the number line pretty early, and of course word problems (don't know if that's "new" or "old"). I don't remember precisely how I learned that, say, multiplication is commutative, but I have a sense of it being "show, not tell". I don't remember any unconventional notation (like lattices).
If parents and, especially, teachers and administrators don't do their jobs, none of the rest of it matters. I guess I was lucky there; while I don't remember my father the math fan teaching me (and he explicitly refused to teach me "tricks" or rules, because I should learn those on my own in due time), he was certainly available to review homework and answer questions. (My mother is not so mathematically inclined, but she indirectly supported my math education: when we went grocery shopping it was my job to tell her which package to buy for the best price, from a pretty early age, long before stores started posting the price per ounce.)
I'm rambling; sorry. :-)
Your story about teachers unable or unwilling to learn the material saddens me, and you're right that in that case the curriculum doesn't matter. I suppose I have an unsupportes sense that if that's the situation in your school, you're probably still better off sticking with the basics just because it's the most common approach and, maybe, some future teacher will be able to fix your students (and that that would be harder if you used techniquest that future teacher might not know). But talk about playing to the lowest common denominator... :-(
How practical is it to get parents (= voters) to go along with an eight-year experiment?
(no subject)
Date: 2007-02-01 02:08 am (UTC)Usually the program requires some fundamental changes (like, let's make sure the kids learn this smaller set of topics than crash through this huge set of topics) which are only half-heartedly supported by the school (the state's standardized tests demand you cover all of these things by January). So the program is kind of implemented half-assedly, and parents complain to the school board, which is made up of elected people with no educational training whatsoever, who deride these newfangled ideas as stupid hogwash and wind up vilified on TV like above, and then everybody says "let's go back to basics" and all the research that the scientists did that might one day lead the way to some gender equity in mathematics is thrown down the drain for another ten years.
educating parents
Date: 2007-02-01 06:10 am (UTC)As to Monica's question about whether parents will wait the eight years or not, the answer depends on whatever is happening in your state's required annual testing. Since it can't ever be the kid's fault they aren't meeting the standards, there have to be other reasons why the kids are failing. The teacher unions pretty much make sure it can't be the teacher's fault. The administrations just push paper. Curriculum is the only possible reason the kids are failing. Just as with any other important issue, an educated electorate is the best case scenario. It's easy to read a simple article (or watch a video :-) )talking about how a curriculum scars children for life but it's much harder to do the work of actually learning about the curriculum and how it works.
Ultimately, I think our children get out of school what we parents put into it, but then again, I'm an over-involved parent.
Valorie
Re: educating parents
Date: 2007-02-04 12:25 am (UTC)Also, a child's home life is the strongest predictor of academic performance, and the one thing a parent has the least influence over.
Re: educating parents
Date: 2007-02-05 02:06 am (UTC)Re: educating parents
Date: 2007-02-05 05:54 am (UTC)Those parents aren't going away and I'm not necessarily saying they are a bad thing --- sometimes the teacher isn't very good --- but the point is that promoting changes in education requires a lot of P.R. and political work; you cannot unfortunately expect that new methods take hold based on their merit.
Re: educating parents
Date: 2007-02-05 05:50 am (UTC)parentteacher has the least influence over."(no subject)
Date: 2007-02-02 04:07 pm (UTC)Well, I'm here to tell you that set formulas are, or can be, death. No, really, I mean that literally. Because you have to understand the logic of what you're calculating. We got zero credit for "decimal point errors" in pharmacy school, thank goodness! "Sorry: your patient is dead."
An Illustrative Anecdote
Every semester, in one class or another, we got a version of the same problem. You get a prescription for one pint of potassium permanganate solution, such that, when the patient takes a tablespoonful and
dilutes it to a gallon, he gets a 1:5000 concentration. How much potassium permanganate do you need? It's the sort of problem most students figure you "need to know for the test" becasue you'll never see it in real life.
Guess again.
When I worked at the hospital, my director was an incompetent Captain Queeg from "the Caine Mutiny". And one day, we got this problem. Everyone recognized it, and Queeg shut down the department while he set everyone working on it. I mean, really, c'mon, it's just a basic ratio problem but NOOOoooo, it was that ratio problem! Work on it and show your work.
Well, what I figured was this. I needed to know how much KMnO4 had to be in that tablespoon. And I needed to know how many tablespoons I had to make. So I told him the answer, and when he asked for my work I told him what I just said. Wouldn't buy it. Where's your work, where's your formula? I just told you. Nope, unacceptable. And how come you didn't use the metric system, you know we use the metric system here.
That's where dependence on formulas can lead you. What you need to realize is that disciplines reduce to simpler disciplines. Medicine is biology. Biology is chemistry. Chemistry is physics. Physics is math. And math is logic. What kids need to be taught is how to solve math problems with logic. And from what I see, that isn't being done any more.
(no subject)
Date: 2007-02-05 02:09 am (UTC)I think everyone needs to be taught logic. For some, alas, it won't stick, and I'd rather everyone have some established techniques to fall back on.
(no subject)
Date: 2007-02-05 06:07 am (UTC)Mathematics is about problem solving. When you have solved a problem, and you just give me the answer in a vacuum, why should I believe that the answer is correct? Hopefully you can convince me by presenting an argument and walking me through your reasoning. This is sort of what school calls "show your work" but most classes this means "write out the steps in the formula we gave you". Classes where students actually have to reason the problem out and present their reasoning look like "kids are struggling and teacher won't show them how to do the problem" and students/parents/sometimes teachers get frustrated and don't see the point.
Anyway, if "the right answer" is the goal of every math problem, and reasoning/understanding is just a means to an end that the class is not interested in, then there is no need for a concept like partial credit, because your answer is either right or wrong. Hell, it would be easier to grade.
On the other hand, if having some sense of what is going on is important, and maybe the student doesn't arrive at the right answer because of a mistake in computation, but she did write out what she was doing, maybe she should get more credit than the student who left the answer space blank?
(no subject)
Date: 2007-02-04 08:44 pm (UTC)However, what you say is certainly correct: you must know how to do the basics before any of these techniques becomes useful. If you do not know your multiplication tables, you do not stand a chance of getting a correct answer. And letting kids use calculators is boneheaded. One may as well allow them dictionaries for spelling tests, or Clif notes for book reports.
Where the methods of estimation are really useful, though, is for people who have some skills but cannot come to appreciate mathematics because it was drilled into them that it was hard, boring, and useless. I may not be able to come up with 20% of $69.34 easily, but 20% of $70 is a snap. And I know enough about transitivity that I could come up with a first-order error correction (say, adjusting my answer by 20% of $.65).
I am told that I learned math in the age of "new math". I don't think so, because, like you, I had to come up with those tricks myself. And that's a useful learning experience in and of itself. But many people are unable to discover those tricks.
(no subject)
Date: 2007-02-05 02:21 am (UTC)I have sometimes taught people who only have the basic methods to see a particular class of problems logically. Once I demonstrate I can see the light bulb go off. I think that only happens because they can see why it must be true. Multiplying by 9 seems hard, but multiplying by 10 and then subtracting the number once makes sense once you point it out. Multiplication is just specialized addition, so you're free to break it up into managable chunks.
(no subject)
Date: 2007-02-06 03:30 am (UTC)It is easy for others, though, who "suffered and suffered" through rote memorization to see these tricks and wonder why they weren't taught them to begin with. Imagine all the wasted time! But the problem comes because they simply want to teach the tricks, not the process. There is a big difference between telling someone to divide by six to get a tip and explaining that 1/6 ~= 16%, describing commutativity, and someone really grasping what is going on.
One way to put it is that the joy is in the synthesis of knowledge into these tricks, not just the knowledge of the tricks. And that is what we have to learn how to teach: that learning is enlightening and interesting, not just something to be suffered through so you can get a bigger paycheck. Such a shift in beliefs seems unlikely to me, though, when there is the panacea to be offered by "new new new math".