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An empty rectangular tank, 60 cm long by 50 cm wide, contains 3 metal cubes of edge 10 cm. The tank is being filled with water flowing from a tap at a rate of 10 liters per minute. If it takes 6 minutes to fill up the tank, find the height of the tank.
How many Scarsdale kids could do this problem?
Few. (I'll ask my friend the next time I see her.)
Here's another:
A car left Town A at 10:00 a.m. and travelled towards Town B at the average speed of 70 km/h. At the same time a truck left Town B and travelled towards Town A over the same road at an average speed of 50 km/h. If the distance between Town A and Town B is 420 km, at what time would the car and the truck pass each other?
I talked to the Admissions Director at the Masters School, which admits kids from Asian countries, about their knowledge of mathematics. She basically said: no comparison. (She said it politely. But that's what she said.) The kids they see, some of them, started algebra in 5th grade.
They're good at it, too.
Elite private schools in Manhattan, some of them, start algebra in 5th grade, too. Here in Westchester, however, not only do public schools not start algebra in the 5th grade, they are adopting constructivist curricula that progress even more slowly than the curricula they replaed. When C. went through grade school here he learned long division in 4th grade. Now the school, using Math Trailblazers, is teaching long division in 5th, and they're mostly not teaching long division at all. They're teaching forgiving division.
Fareed Zakaria needs to get a clue.
the deathless meme of the high-performing school
9 comments:
Okay, that forgiving division is neat. Granted, I'm all grown up and know how to do long division so I can afford to say that but still...
I think I'll show that to my daughter tommorrow and see if she can get it.
I don't think she'll like it though. She has this love affair with long division...
I once tried to do a super long forgiving division & got completely tangled up!
What if the cubes are hollow and they float? That was my first thought.
All of this points to my position that this is all about high versus low expectations, not direct instruction versus constructivism. After years of Everyday Math and seeing the problems my son did in class and for homework, I saw very little real constructivism and very low expectations. Schools want to talk about conceptual understanding and how the brain works, but what they are hiding are low expectations.
"They're teaching forgiving division."
They like forgiving division because it's forgiving about not knowing the times table. You are not required to be able to figure out (in your head) how many times 17 goes into 96 without going over, such as with:
9645 / 17
How many immediately start multiplying 4 * 17 = 40 + 28 and then 5 * 17 = 50 + 35 in their heads?
With forgiving division you can guess and it will all work out in the end. They claim that they do this because it provides more understanding. That's rubbish.
Coincidentally, I just posted a comparison of 2nd grade Everyday Math and Singapore Math problems for OILF's problems of the week. Even that early on in the curriculum, huge gaps emerge (I'm specifically looking at place value and regrouping).
Fareed Zakaria! Last thing I read of his was "why they hate us", in Time, after 9/11. He was a college classmate of mine, and has always struck me as quite bright. Unfortunately, like so many reporters, he's apparently quite sloppy when it comes to education!
I thought forgiving division was the standard method for mental calculations? I don't know how to do long division in my head by any way other than simply breaking the number up into sane parts.
Anyway I have to say that little did I know when I took the PSLE that five years later I would be basically be doing the same thing, only with derivatives. I must say that the calculus mindset messed me up when I tried to mentally solve the problem though (I tried making height a piecewise function of volume) before I realised the more straightforward solution.
I tried forgiving division with my daughter. I really liked it, she didn't. I can see how it would be really useful for mental math and it actually pointed out a bit of a weak spot in my daughter's math.
It seems to me the two methods have different strengths and possible applications.
lolll, Dawn --- yeah, I find that one's kids are never quite as enthusiastic over a cool new math-thingie as we are....
I guessed that the cubes were completely covered and I won (21cm). I divided the volume (60l + the cubes = 63000cm^3) by the floor area (50x60=3000cm^2). But If the water didn't reach 10cm, I would have had to work it out by dividing just the straight water (60000cm^3) by the reduced area (50x60cm^2 - 300cm^2). Some of those problems can have some blocks submerged and others sticking out!
We need to make sure that middle school kids can convert 420km to 3.5 hours (without a calculator) given the "conversion factor" of 120km/hr (the rate the vehicles are converging) ; they need to multiply by hr/km so the km's cancel and they are left with hours. The sneaky part is to remember 10am + 3.5hr = 1:30pm.
You can always use conversion factors when your relationship is a straight line that goes thru zero!
OTOH, the blocks in the water were a bit of a "challenger"
--rocky
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