Friday, May 2, 2014

When It Can't Be Wrecked

We're getting some mileage out of this lately.  Today, I have a new problem to add to the pile of those that foster the process of hypothesis wrecking.

I posed the question with a rubric.

Can a unit fraction always be written as the sum of two unique unit fractions?

5: Precise proof that demonstrates all cases (abstract, general rule)
4: Reasonable argument that demonstrates some cases (numeric, gives examples)
3: Gut level or weak argument
2: Does not present an argument

1: No evidence of understanding

Students played around with a few unit fractions and after a few minutes we had a couple of them.

Shortly, we had a student come up with an hypothesis:

which was soon followed by another student example:

Uh-oh, that doesn't fit the pattern.

"Does this example wreck our hypothesis?"

This led to a nice conversation on whether this new example and our hypothesis can coexist.  It was interesting to see how many students initially thought the hypothesis was wrecked.

We tested a few more examples and shared results--all confirming our hypothesis.

Then I asked, "So where does this put us on the rubric?"

And a student asks, "What has to happen for a 4 to become a 5?"

In other words, when does a numeric (quantitative) argument become abstract [1]?

Had to pause.  This one is worth it.  So we discussed simple example:

I quickly came up with the question and answers 1, 3, and 4.  At lunch I added 2, which really added to the conversation for 6th period.

Which answer provides the stronger argument?  Most saw 4 as the strongest and agreed 1 was the weakest.  But very few saw 2 on the same level as 4.  Then one student says, "I see that 2 and 4 are similar but 4 is just kinda strung out."

Yep, the kid has a feel for brute force vs. elegance.  Love it.

By the end, we agreed that 2 and 4 were more abstract and 3 was more quantitative. What about 1?

Well, 1 was what they would've considered a great answer a few months ago.

[1] This is what prompted my question about SMP 2 on Twitter. 

No comments: