I have been interested recently in the mathematics lying just under the surface of middle school math problems (see also “How many ways to roll a 10?” ). I enlisted a former student, Ben Harste, to help me write an article on such an investigation. The premise is below; until and unless it is accepted for publication, you can read the full article by downloading it from my Papers page.
In the Connected Mathematics unit Filling and Wrapping, students find all of the rectangular prisms that can be made using whole number side lengths and 24 unit cubes. Then they find the surface areas of these prisms and notice that surface area can vary greatly for prisms with fixed volumes.
Commonly, teachers have their students conjecture how to tell which of two prisms will have a smaller surface area based on appearances alone, and a frequent observation is that the closer a prism is to being a cube, the smaller its surface area will be (in comparison to other prisms of the same surface area). This property could be called cubeyness.
It is easy to judge the relative cubeyness of a 1 by 1 by 24 prism against a 2 by 3 by 4 prism. But what about when two prisms appear similarly cubey? How can we measure cubeyness without calculating the surface area? And furthermore, how can we compare the cubeyness of two prisms that have different volumes (so that, for example, a 2 by 3 by 4 prism would be just as cubey as a 4 by 6 by 8 prism, or so that the cubeyness of a prism does not depend on the units we use to measure it?)
These turn out to be much harder questions than they appear to be on the surface, and answering them calls into question a lot of what many teachers know about measurement, statistics and geometry.
Overthinking my teaching 
As an addendum, the “cubeyness” paper was rejected. It will remain on my Papers page as an unpublished paper.
Christopher, this is a post I meant to respond to a while ago but I am more inspired now by this concept of a measure in “cubeyness.” I am sorry to hear the paper was rejected but in light of this type of measure what would be a proper definition of “quantity” ? It seems with that might be part of the discussion as I read the paper today. I certainly see a lot of potential for mathematics in comparisons without needing to “quantify” the cubeyness. I will read the paper and be sure to get back to you.
I think of “quantify” rather simplistically, I suppose. I mean “assign a number to”. As it turns out, the number we assign has unit=1/inch (not especially meaningful). I’m curious about your ideas on “non-quantified comparisons”. Here’s how I see the issue that I think you are discussing: The question at hand is “which of two prisms with equal volume has larger surface area and is it possible to know without computing the surface area directly?” In lots of cases, we can see which is more cubey, and therefore which has smaller surface area. But sometimes it’s close, or we have non-cubey prisms where it is not obvious visually which one will have smaller surface area. That’s where it would be nice to quantify-to have some way to make a measurement to tell the difference.
I don’t think we are out to complicate the meaning of “quantity” (at least I don’t think), but we are using something really similar to what you said: Quantities as objects (or phenomena) that are measurable. We ran into some disagreement on that definition recently so we are scoping it out. In the MeasureUP curriculum we are working from “Non-Quantified” comparisons are what open the door to bigger concept thinking in young elementary students. First graders that work relatively fluently with comparisons like A A/C and then discuss the relative sizes of B and C. I am just impressed by the amount of mathematics, some quite abstract, that young minds can take on without introducing the more specific case with actual values. I am imagining the “cubeyness” question as a an awesome extension of a similar problem we just played with about “Who’s farm is more square?” There were pretty cool strategies used without values, just two drawings. Ultimately adding value to the problem is what we want of course. I am wondering if this “cubyness” activity could be adapted for younger students I suppose; I think it can and could be really fun.
Sorry Christopher, that last post of mine was a copy and paste nightmare. The primary fix is A/B=D compared to A/C=D and talking about the relative sizes of B and C without the use of value. There were a few transpositions of text too but I think the gist is there. Thanks for letting me voice ideas! It really helps solidify my thoughts.
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