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summary of an 'inflexible knowledge' moment & how I tried to work through it
tentative conclusionsBasically, everything we read & hear about learning math from cognitive psychologists, mathematicians, & experienced nonconstructivist teachers happened here.
My notes are below. trying to figure out how to do a percent decrease problem in an unfamiliar formatOK, I'm trying to figure out what percent decrease it is when you drop from 22% of Michigan students scoring a 4 (for advanced or exceeds standards) down to 9 percent. sequence of events
This, to me, is an example of inflexible knowledge, because I can't tell whether I can transfer the formula I know (and do have some conceptual understanding of, believe it or not) to this new situation. I can't even tell if this is a new situation. My first impulse was to look it up in ALGEBRA TO GO. Unfortunatelyor fortunatelyALGEBRA TO GO only has examples using the raw data, not the percents. Then it occurred to me that I ought to see if I can work this out myself.....and that the way to do that is....
I know! WORK BACKWARDS!
also: MAKE IT SIMPLE!
It's so fun, here in midlife, being a Rank Amateur at elementary math.
My first thought came pretty quickly, which I think is good, because I think it shows some gains in flexible knowledge. I was sitting here staring at my notebook when it suddenly struck me that since I'm hung up on raw data numbers versus percent numbers, I ought to simply translate the problem into '22 students' versus '9 students.' So I did. I subtracted 9 from 22 & got 13. Then I divided 13 by the original 22 to find out the percent decrease and came up with an answer of 59%.
(I'm sorry, that's big. I may know NOTHING about elementary math, but a 59% decrease is big.)
The problem they give for figuring percent decrease is a drop in gas prices from $1.24 to $1.19. (Pub date on this book is 2000.)
Alright, I'm going to see if I do the calculations right.....
I got it. $1.24  1.19 = .05 .05 1.24 = 4 percent.
ALGEBRA TO GO gives 4 percent as the answer.
speed with doubtnote: I did not feel particularly confident working this problem, but OTOH, I could do it, and I could do it pretty quickly.
I'm assuming that speedwithdoubt is midway to flexible knowledge, since my starting point on problems I have no clue how to do is doubtwithparalysis.
This next part is embarrassing, but what the heck. I looked at my gasoline problem and thought: can I rewrite the two gas prices as percents and see if I get the same answer? I felt I couldn't do that, because a percent 'has to be' between 0 and 100 to be a percent, and both $1.24 and $1.19 are greater than 100. My very next thought, thanks to Saxon Math, was Don't be a moron, of course a percent can be greater than 100!. This is another example of inflexible knowledge. I'm used to seeing percent numbers from 0 to 100, and a percent of 125% throws me. However, that recognition didn't do me much good; I was starting to feel dazed.
So I decided to go back to my 9 and 22, both of which numbers are, indeed, less than 100.
I don't feel confident of this conclusion, but I do think it's right. Assuming it is right, I had another boost from SAXON MATH, which is the various lessons that have you do the four operations on percent numbers. I can't remember the lessons now, but my memory is that they ask the student to perform the operations on percent figures even when it's a bit unwieldy to do so, because they want you to perceive that like can be added to, subtracted from, multplied, or divided by like.
Those lessons made a big impression on me, because apparently I had never been taught this during my own childhood. (Christopher was impatient with these lessons.)
At that point it crossed my mind simply to take a look at the numbers, and see if an answer of 59% made sense. It does.
When I look at the numbers.....I see that 9 is semiintherealm of being 1/3 of 22....and that 59% is semiintherealm of being 2/3 of 22.....in other words, just staring at these numbers I feel that only about 1/3 of the number of kids who used to get 4s are now getting 4s, which means 2/3 of the kids are not, which would be in the neighborhood of a 59% decrease.
Back when I first wanted to learn how to figure percent increases and percent decreases because I wanted to figure out the percent decrease in 4s in Scarsdale schools using TRAILBLAZERS, a friend told me the way to figure it was just to subtract the lower percent figure from the higher percent figure, because 'you can subtract like from like.' I had come far enough in math knowledge to feel that was wrong.....and to semiintuit that there would have to be another division step after the subtraction.
At that point, I looked it up in ALGEBRA TO GO (which I think in that case was the correct response. LEARN HOW TO LEARN, BABY!)
The two answers are very different: 13% versus 59% Eyeballing the problem, 13% seems obviously wrong.
That gives me slightly more confidence that 59% is right.
 CatherineJohnson  13 Jul 2005
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