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Tuesday, December 15, 2009

The hills are alive...with the sound of healthy confusion

Thinking about the analogy of marching learners into confusion, and then marching them out, we embarked on this problem from the Algebra Connections series from CPM:

First, most of my 8th graders "got" the veiled references to "Sound of Music". They also "got" why xylyphones and yodelers (x and y).

Oh, but what a storm of confusion upon writing the equations related to this problem.

They initially want to write: x = 2y .

That is perfectly understandable because it is mimicking, at least in order, how the English language lays it out for them.

A few students, though, saw a problem in this. Debate ensued. No agreements easily reached.

I suggest that one way to determine whether this equation would work would be to assign values to the variables. They all understood that there are twice the number of yodelers as xylophones. So to speak, two yodelers carry one xylophone onto the gondola. I assigned the value of "1 to x, but that was too hard to conceive. So I upped the value of x to 2, and some students started to get that y would have to be 1.

This means that for every 2 xylophones, there would be one yodeler. But the problems says it is the other way around. Still, many students were not convinced.

So I assigned a value of 1 to y, meaning 1 yodeler. The value for x would be 2. Still not coherent to the problem.

A student suggested we switch around the variables to write y = 2x . Many students were unsure about this, but I overtly supported the idea, so they started to pay attention. By assigned a value of 1 to x (meaning one xylophone) we find there are two yodelers: matching the problem as stated.

While I thought the case to be convincing, several students still questioned the logic. I tried assigning new values to the variables showing them how it would work.

In the end, I would say that the conversation was rich. The problem really elicited the difficulty of translating oral or written language into algebraic form. I do not believe this was a moment of "confusion in vain", but rather, a deep discussion of the meaning of variable in algebra. This is just one of many different discussions we have had and will continue to have as we move forward in this journey out of confusion.


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