Formulating Measures

Back to assessment! Not technically research so it's not getting the edresearch tag.

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:

1. Which of the rectangles is the "squarest"?
2. Arrange the rectangles in order of "square-ness" from most to least square.
3. Devise a measure of "square-ness," expressed algebraically, that allows you to order any collection of rectangles in order of "squareness."
4. 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I know the paper says, "Please do not quote," at the top, but I'm going to ignore that:

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.

1. Observing
2. Comparing
3. Ordering
4. Making Measurements
5. 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). 

Other than the very obvious ones (formulating a measure for speed or for density) I fully admit to slacking on this.




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.