I found this via Sarah Cannon who blogs at Sarah's Development. The full paper was written by Judah Schwartz and can be found on his old Harvard page. This work came from the Balanced Assessment in Mathematics project.

The sample prompt is really all you need to figure out what's going on here:

If you can't read the questions, they are:

- Which of the rectangles is the "squarest"?
- Arrange the rectangles in order of "square-ness" from most to least square.
- Devise a measure of "square-ness," expressed algebraically, that allows you to order any collection of rectangles in order of "squareness."
- Devise a second measure of "square-ness" and discuss the advantages and disadvantages of each of your measures.
^{1}

I love these -ness problems. There is a ton of high level thinking here and the formulation of measures is the first step in model building. One of my ongoing struggles is helping my students become more precise with their language. What does, ".....works better," mean?

I know the paper says, "Please do not quote," at the top, but I'm going to ignore that:

I know the paper says, "Please do not quote," at the top, but I'm going to ignore that:

Formulating a measure requires one to be explicit about the constituent elements of data that one believes are important in a given situation. (p. 2)Standard measures included rates, ratios and area. Non-standard measures that were listed included crowded-ness, sharp-ness (as in curves), disc-ness (of cylinders) and developing a "size of task" measure given this data set.

On page 10, the author lists the proposed components in formulating a measure.

- Observing
- Comparing
- Ordering
- Making Measurements
- Analyzing Data

At the end of the paper (starting on page 11), the author goes on to differentiate between measures and models (models having the ability to predict in untried instances and being falsifiable).

1: Did you start with (Horizontal/Vertical) with "most square" being closest to 1 as your first measure of square-ness? I did. But Schwartz pointed out that the measure then changes if you rotate the figure 90 degrees. Should this occur for a measure of square-ness? Great great GREAT conversation fodder.

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ReplyDeleteI just spent the last 30 minutes looking through the Balanced assessment in Mathematics project as a result of your post... thanks for the link and the thoughts! It's so challenging to find/create rich problems like these :)

ReplyDeleteI got one more "ness" for you. The logical extension of "squareness" in two dimensions is "cubeyness" in three dimensions. It turns out to be orders of magnitude harder to nail down.

ReplyDelete@Jen - Thanks. And you're local! Say hello if you're going to SF Bay Area Ed Camp or just anytime.

ReplyDelete@Chris - Cool stuff. I spent an hour tonight figuring trying to get a lesson going for having my 8th graders formulate a measure for how much "hella" is. But then I remembered I wasn't a math teacher and could never use it......