math education

[cellio]

Being of a certain age, I learned arithmetic the conventional way and neatly dodged New Math. I knew things had changed since then -- at least in the ability of high-school graduates to do arithmetic unassisted -- but I didn't realize just how strange things had gotten. amergina posted a link to this (longish) news story broadcast: math education: an inconvenient truth. Sigh.

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. :-)

Comments

teachers and administrations have a part in this

[(Anonymous)]

Hi Monica! Larry passed on a pointer to your blog. - Valorie

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

[(Anonymous)]

I never have been a brief writer!

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

[cellio]

Hi Valorie!

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?