inflexible knowledge redux

Another example of inflexible knowledge.

C. and his friend J. were assigned a "digit word problem" for homework. Example:

The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number.

source: purplemath

Neither of them had a clue how to do it. (I think digit problems had been taught in class that day for the first time. They'd both seen 1 or 2 worked examples.)

I tried to teach it by writing the number 23 and then, below that, writing a variable for each digit, then asking them something wildly imprecise like, "What would you have to do to these variables to get them to equal 23?"

Actually, as I think about it now, I don't know what words a person educated in mathematics would use to express this, so maybe you all can fill me in.

In any case, they did get the idea they were to let x equal 2 and y equal 3, and they were to write an equation using these values for x and y that would equal 23.



Looking at this arrangement of numerals and variables didn't help.

J. said, "x + y?"

"That's 5," I said.

"x times y?"

"That's 6."

They were stumped.

This is a classic case of inflexible thinking, for two reasons.

First of all, they've seen "plus" and "times" more often than they've seen 10x + y. Confronting a novel problem, their brains went straight to the more familiar instead of the less familiar.

Of course, that's what a brain ought to do: When you hear hoof beats on the bridge don't think of zebras.

The problem was that, pace Willingham, once they figured out the hoof beats weren't made by horses they didn't think of zebras next.

Both boys know how to set up word problems using letter variables; both boys are proficient (or close to) at solving number and consecutive integer problems. They can set up and solve coin problems, distance problems, and age problems; they can also set up and solve simple linear function problems.

They've been writing algebraic expressions for at least a couple of years.

And they were stopped cold by a simple digit problem.

I wish I'd taken notes on what I did next. I think I said something like, "What is the 2?" (That's imprecise, too, right? What's the correct language there? "What value does 2 represent"?)

I don't think I had to go as far as to remind them of the base-10 place system, but, on the other hand, maybe I did and I'm repressing it.

In any case, the moment it dawned on them that the key was recognizing the 2 as 20 they could both set up and solve digit problems rapidly and efficiently.

Knowledge transfer and generalization are core, unsolved problems in education.

update: Looking at this now, I think I should have had them start with the "3" and write an equation using y alone, then had them write an equation using x and 2 (or 20). If I wasn't simply going to show them a worked example myself (should I have? I don't know) I needed to break this down into the smallest, simplest possible steps. And I should have started with the ones digit, not the tens.

I think.


cumulative practice and problem solving

Speaking of which, I keep promising to write a post about one of the three most valuable papers I've ever read on the subject of teaching math:

The Effects of Cumulative Practice on Mathematics Problem Solving
Kristin H. Mayfield and Philip N. Chase

Here is the introduction:

Over 35 years of international comparisons of mathematics achievement have indicated problems with the performance of students from the United States. According to the latest international study, the average score of U.S. students was below the international average, and the top 10% of U.S. students performed at the level of the average student in Singapore, the world leader (Wingert, 1996). In addition, recent tests administered by the U.S. National Assessment of Educational Progress revealed that 70% of fourth graders could not do arithmetic with whole numbers and solve problems that required one manipulation. Moreover, 79% of eighth graders and 40% of 12th graders could not compute with decimals, fractions,
and percentages, could not recognize geometric figures, and could not solve simple equations; and 93% of 12th graders failed to perform basic algebra manipulations and solve problems that required multiple manipulations (Campbell, Voelkl, & Donahue, 1997).

These statistics reveal students’ deficits in the fundamental skills of mathematics as well as mathematical reasoning and problem solving. Indeed, poor problem-solving skills have been targeted by the National Council of Teachers of Mathematics (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980; Carpenter, Kepner, Corbitt, Lindquist, & Reys, 1980; Kouba et al., 1988; National Council of Teachers of Mathematics, 1989, 2000). Thus, it seems appropriate that current behavior-analytic research in mathematics education should address problem-solving skills as well as basic mathematics skills (e.g., Wood, Frank, & Wacker, 1998).

I'm hoping I can get to this later on today.

But first -- must write chickens chapter!