My son is in 2nd grade. His teacher gave the class a quiz, and one question was this:

If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have?

My first reaction was "0" or "undefined". But my son wrote "$\infty$" which I think is a reasonable answer. However, it was marked wrong with the comment, "the answer is 1".

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Is there an accepted correct answer in geometry?

edit: I ran into this teacher recently and mentioned this quiz problem. She said she thought my son had written "8." She didn"t know that a sideways "8" means infinity.

geometry education circles
edited Jan 20 "18 at 17:34
asked Apr 8 "11 at 19:26

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The answer depends on the definition of the word "side." I think this is a terrible question (edit: to put on a quiz) and is the kind of thing that will make children hate "Side" is a term that should really be reserved for polygons.

edited May 20 "11 at 10:30
answered Apr 8 "11 at 19:30

Qiaochu YuanQiaochu Yuan
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My third-grade son came home a few weeks ago with similar homework questions:

How many faces, edges and vertices do the following have?

cube cylinder cone sphere

Like most bsci-ch.orgematicians, my first reaction was that forthe latter objects the question would need a precisedefinition of face, edge and vertex, and isn"t reallysensible without such definitions.

But after talking about the problem with numerous people, conducting a kind of social/bsci-ch.orgematical experiment, I observed something intriguing. What I observed was thatnone of my non-bsci-ch.orgematical friends and acquaintances hadany problem with using an intuitive geometric concept here,and they all agreed completely that the answers should be

cube: 6 faces, 12 edges, 8 verticescylinder: 3 faces, 2 edges, 0 verticescone: 2 faces, 1 edge, 1 vertexsphere: 1 face, 0 edges, 0 vertices

Indeed, these were also the answers desired by myson"s teacher (who is a truly outstanding teacher). Meanwhile, all of my bsci-ch.orgematicalcolleagues hemmed and hawed about how we can"t reallyanswer, and what does "face" mean in this context anyway,and so on; most of them wanted ultimately to say that asphere has infinitely many faces and infinitely manyvertices and so on. For the homework, my son wrote an explanation giving the answers above, but also explaining that there was a sense in which some of the answers were infinite, depending on what was meant.

At a party this past weekend full ofbsci-ch.orgematicians and philosophers, it was a fun game to firstask a bsci-ch.orgematician the question, who invariably made various objections and refusals and and said it made no sense and so on, and then thenon-bsci-ch.orgematical spouse would forthrightly give a completely clearaccount. There were many friendly disputes about it that evening.

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So it seems, evidently, that our extensive bsci-ch.orgematical training hasinterfered with our ability to grasp easily what children andnon-bsci-ch.orgematicians find to be a clear and distinctgeometrical concept.

(My actual view, however, is that it is our training that has taught us that the concepts are not so clear and distinct, as witnessed by numerous borderline and counterexample cases in the historical struggle to find the right definitions for the $V-E+F$ and other theorems.)