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summary of an 'inflexible knowledge' moment & how I tried to work through it

  • I wanted to calculate the percent decrease in MI students scoring a 4 on their MEAP tests
    question: 22% of MI students used to score 4s; today it's 9%. What is the percent decrease?

  • I started out confidently, thinking I knew how to do the problem. Then I was flummoxed by the fact that the 9 and the 22 were percents, not raw numbers. I was comparing 22 PERCENT to 9 PERCENT, not 22 students to 9 students (or 22,000 students to 9,000 students, etc. My problem wasn't the size of the numbers, but the fact that I was comparing percents).

  • I decided fairly quickly to use two standard strategies:

    1. work backwards from an existing, already solved problem
    2. make it simple

  • work backwards promptly confused me more than it helped, and I dropped it

  • make it simple was by far the better approach

  • once I had make it simple in mind, I saw that I didn't have to find a simple already-worked problem to copy. All I needed to do was take the percent signs away from the numbers 9 and 22. I could figure the percent decrease from 22 STUDENTS down to 9 STUDENTS. That would re-write the problem in the terms with which I was familiar.

  • I also solved an already-worked percent decrease problem in ALGEBRA TO GO to see whether I would get the same answer they did. I did, which told me that my initial confidence was correct. It also told me that my difficulty didn't lie in knowing how to figure a percent decrease or increase, but in recognizing the problem in the first place. In short, I was having the 'Willingham' problem of not knowing when to generalize my knowledge.

  • at that point I started being able to look at my answer and see if it made sense:

    1. I compared my answer to the ALGEBRA TO GO answer
    2. I compared my answer to what it would have been if I'd solved the problem using the incorrect approach a friend once told me to use
    3. I suddenly found myself able to 'see' the problem as a fraction problem, not 'just' a percent problem. When I looked at it as a fraction problem--9 kids out of 22 is around 1/3 of the class--I could start tapping into my 'friendly fraction' knowledge.

tentative conclusions

Basically, everything we read & hear about learning math from cognitive psychologists, mathematicians, & experienced non-constructivist teachers happened here.

  • this was a case of inflexible knowledge, as Daniel Willingham defines it. I knew how to figure a percent increase using raw numbers, but I couldn't tell if percents were a different case requiring a different approach.

  • inflexible knowledge definitely means being tied to the superficial features & specifics of a problem.

  • Therefore, the test (or a test) of mastery is whether a student can generalize what he's learned to a new context or problem

  • I found that although I don't have flexible knowledge of calculating percent increases & decreases, I do have decent procedural knowledge:

    1. I calculated the percent decrease from 22 to 9 quickly and correctly
    2. I calculated the percent decrease in a sample problem quickly and correctly

  • working problems is the magic. I felt very confused as I set out, and became progressively less confused as I worked on the two problems.

  • students need to solve lots and lots of problems of the same type but stated in novel terms

  • observational learning is key. Seeing how ALGEBRA TO GO solved percent increase & decrease problems didn't tell me whether my answer was right. But it did help me see whether my answer was reasonable.

  • make it simple is an incredibly important strategy that we should (probably?) give our kids practice in using

  • Saxon's fragmented 'increments' do, or certainly can, start to cohere as you work unfamiliar problems

  • procedural and conceptual knowledge are linked, as Willingham and many mathematicians say:

    1. Procedural knowledge seems to produce conceptual knowledge as you work more problems, and perform more procedures.
    2. As I worked with various permutations of the problem, I found myself able to look at the numbers '9%' and '22%' and feel that the percentage decrease had to be somewhere in the neighborhood of 2/3, and could not possibly be 13% (as my friend's method would have found)

  • showing kids more than one way to check a problem might be a good idea. I checked my calculations 2 ways (though I don't know whether my answer is right):

    1. I solved a worked-out example to see if I got the right answer
    2. I reformulated the problem, in my mind, as a fraction-of-a-group problem instead of a percent problem

  • telling kids morning noon and night to ask themselves whether an answer is reasonable is a VGI (very good idea)

  • speed is probably an indicator of where you are on the pathway from flexible to inflexible knowledge. When you quickly see that you should try a certain strategy, that's probably a sign that you have enough conceptual knowledge to approach a problem systematically.

My notes are below.

trying to figure out how to do a percent decrease problem in an unfamiliar format

OK, 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

  • I remembered, fuzzily, that first you find the difference between the two numbers; then you divide the difference by the original number.

  • At that point I realized I only knew how to do this with the raw data, not with the percent figures themselves.

  • In other words, if I knew the actual numbers of MI kids who used to score 4s and now score 3s, I would know how to figure the percent decrease.

  • But I don't know how to figure the percent decrease using the percent figures themselves.

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.

Unfortunately--or fortunately--ALGEBRA 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....



It's so fun, here in midlife, being a Rank Amateur at elementary math.

I'll be back.


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.)

At this point it struck me that I should check ALGEBRA TO GO to make sure I'm do have the math correct for raw data-type numbers.

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 div.png 1.24 = 4 percent.

ALGEBRA TO GO gives 4 percent as the answer.

speed with doubt

note: 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 speed-with-doubt is midway to flexible knowledge, since my starting point on problems I have no clue how to do is doubt-with-paralysis.


I'm back.

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.

Looking at 9 and 22, it now strikes me that the formula for finding percent decrease using percents has to be the same as the formula for finding percent decreases using raw data.

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.) struck me that if the percentage decrease from 22 students to 9 students is 59%, the percentage decrease from 22% to 9% also has to be 59%.

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 semi-in-the-realm of being 1/3 of 22....and that 59% is semi-in-the-realm of being 2/3 of 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.

I'm also using a compare-and-contrast approach that I think math books don't give enough attention to.

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 semi-intuit 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!)

So just now I eyeballed this problem to see what the two different answers would be, one answer a simple subtraction of 9% from 22%, the other answer the quotient of the difference divided by the original percent.

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.

I'm going to summarize this in bullet points at the top of the page--

-- CatherineJohnson - 13 Jul 2005


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