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Peace through Constructions

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Illustration from Euclids Elements.pngCompass and straightedge constructions have been in the geometry curriculum for a loooonng time. These constructions were around before Euclid included them in his textbook, The Elements, 2300 years ago. The painting shown here is an illumination from a translation of that book 1600 years later. I had to do constructions in junior high; I don’t remember being quite as unhappy about the endeavor as these monks or the woman teaching them. And now constructions are in the Common Core, too. Why are we eternally requiring students to construct things with these arcane and arbitrary tools?

Here is one answer: Plutarch tells a story of Plato and his buddy Simmias of Thebes coming back from a trip to Egypt when they encountered a delegation from the isle of Delos. The famous oracle at their temple had told them that the Delians and the other Greeks could put an end to their current troubles if they could figure out how to double the temple’s altar, which was a cube. In other words, they had to make a new cubical altar that was twice the volume. They told Plato that they had tried to double all the dimensions, but they ended up with an altar that was 8 times the volume, not double the volume.

Plato told them that this was not a problem for estimation but for precise construction. (It can be solved by finding two lengths in mean proportion, in other words, construct a length of the cube root of 2.) Although a few Athenians knew how to do it, the point was that the Delians should figure it out themselves. The oracle’s motive was that the Greeks would only end their troubles when they focused on cultivating the art of mathematical problem solving, including making logical arguments for their constructions. This would give them the ability to sort out their differences through reason rather than force. 

This classic problem of doubling the cube turns out to be impossible with only a compass and straightedge, but you can do it with other Greek tools (and you can do it with paper folding, too!). The work that occurred on this problem led to entirely new fields of mathematical study in ancient Greece.

I think maybe the reason we keep doing constructions is that it allows us to engage in at least two of the Common Core’s Standards for Mathematical Practice (SMP 1: Make sense of problems and persevere in solving them and SMP 3: Construct viable arguments and critique the reasoning of others). By forcing ourselves to figure out how to do something with a limited set of tools, we refine our problem-solving skills. By forcing ourselves to justify why we know our constructions will always work, we improve our ability to argue rationally, to show how ideas logically rely on other ideas.

Maybe the oracle of Delos was onto something. . . . 

Rob Ely, MTMS Blogger Rob Ely is an associate professor in the Department of Mathematics at the University of Idaho.  He likes researching student reasoning and how it relates to historical reasoning, particularly with the infinite, infinitesimal, and algebra.  He likes playing hammered dulcimer, bridge, and fetch. 





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