Thinking Outside the Box -- or Maybe Just About the Box
The Box problem is so timeless that it even goes back almost a hundred years to the classic Granville and Smith (1911) text. Except for the fact that it is made out of tin, doesn't this look familiar?
Figure 2. The box problem on page 115 in Granville and Smith (1911)
However, this instance was not the first appearance of the box problem in either an academic publication or in popular culture. A derivation of the box problem first appeared in popular culture in the 1903 Henry Dudeny's puzzle column in the Weekly Dispatch and again in 1908, it was rephrased for his puzzle column in Cassell's Magazine. Below is the graphic and phrasing of the problem contained in Cassell's Magazine:
Figure 3. Illustration from Cassell's Magazine.
But just in case one might think that the problem is only a century old, a Treatise on Differential Calculus by Todhun
ter (1855) contains the following related problem:
Figure 4. The box problem on page 213 of Todhunter (1855).
Consequently, this problem has appeared in calculus texts for at least the last one and a half centuries. It has become so entrenched in the calculus curriculum and articles are still being written about it. For instance, an article by Miller and Shaw (2007) breathed new life into this problem for it served as the backdrop to explore a conventional problem in unconventional ways or an article by Fredrickson (2003) reconceptualizing the constraints on the ''cutouts'' while striving for a container of maximum volume. But what is it about this problem that makes it so universal and mathematics teachers keep coming back to it? Does it really motivate student learning of essential calculus concepts? We concur with Underwood Dudley (2002) when he pointed out in his talk on Calculus Books, "when have you encountered a cardboard box constructed in such a way?" and the problem is ''quite silly'' according to Friedlander and Wilker (1980), Dundas (1984) and Pirich (1996) since the corners are wasted.
As previously identified, this particular calculus problem predates modern use of cardboard packaging. But, most would agree that constructing a cardboard box in this manner is quite unusual. So, why do calculus teachers keep including it as part of their repertoire of examples other than the fact that it is straightforward and easily comprehensible? Doesn't the inclusion make students begin to wonder about the ability of calculus to model reality? In fact, if one attempts to rip apart the classic open box used for the typical shirt box that one receives at Christmas, it is not constructed in this manner.
It is interesting to note that both the Granville and Smith text and Dudeney's puzzle used tin, a little more practical material than cardboard for such a construction. A box of cardboard might use the squares as tabs to connect the sides together and increase stability and strength. In fact, we have often asked students to construct boxes from index cards we gave them and they tend to do use these tabs naturally to make the connections with tape, glue or a staple. The students can typically answer the mathematical questions of the corner-removed box problem but at the same time they recognize the lack of realism when the corners are removed since boxes just are not constructed that way.
Meel, David and Thomas Hern, "Thinking Outside the Box -- or Maybe Just About the Box," Loci (August 2009), DOI: 10.4169/loci003321
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