Patrick Honner has several blog articles about poor problems (sometimes outright mistakes) on some of the exams that New York state requires for students. Since I don’t live in NY I’m not super familiar with these exams, though Mr. Honner’s posts actually make me happy that I’m not. I get the feeling that the more I knew about them the more they’d drive me crazy.
In any case, his latest article is here:
and the problem he discusses is:
“The medians of a triangle intersect at a point. Which measurements could represent the segments of one of the medians?
(a) 2 and 3
(b) 3 and 6
(c) 3 and 4.5
(d) 3 and 9”
Unlike some of the prior problems that he has written about the problem itself doesn’t have any mathematical flaws. Instead this problem is simply testing if you know a single math fact. Really no deep understanding of geometry is required to solve it at all if you know this fact. I think that Mr. Honner’s concern – which is essentially that if state exam questions end up looking similar to this one, math education is just going to turn into something equivalent to prepping for a night on Jeopardy – is spot on.
This critique hit me for another reason, though. Three years ago when I decided that I wanted to start making fun little math videos for kids, I thought that I should practice a little and see if I actually had any ability to explain math. At that point it had been more than 10 years since I’d been in front of a classroom. Oh, and 10 years in finance doesn’t exactly sharpen your explanation skills.
What I decided to do was grab my copy of Geometry Revisited off the shelf and pretend I was doing a few lectures from the book. I went through the first two chapters or so, but two of the early sections are relevant here. The second “lecture” was about Ceva’s theorem, which is a beautiful theorem with a fascinating and incredibly instructive proof (keep in mind that I’d not talked about math for a long time in this video, so it isn’t the best. Also my older son is watching for some reason I don’t remember):
So we get a beautiful theorem with a really instructive proof right in the second section of Geometry Revisited! Also, I saw that Steve Leinwand gave a lecture at a conference for teachers last week and said that ratios were one of the most important pieces of early math. Ratios play a surprising role in this proof of Ceva’s theorem, so it may have even more educational value than I realized the first time around. Finally, the result we can see pretty easily from Ceva’s theorem that is relevant to the question on the NY state exam is that the medians of a triangle intersect in a single point.
With a few fun ideas about cevians in hand, you might be interested in learning even more, and Geometry Revisited doesn’t disappoint. The next section shows a couple of nice results about the medians that also have incredibly instructive proofs (the part about the medians starts at 3:20):
I remember vividly how going through these early sections in Geometry Revisited reminded me how much I loved math. With just two short sections on Ceva’s theorem and medians we have a couple of beautiful results and several really instructive proofs for students to see. I understand that no all kids are going to find these ideas to be as fascinating as I do, but I think that lots of kids will. It seems like such a shame to me to reduce it all to the question posed above, and frankly even worse to essentially reduce it all to something like –
Answer: They trisect each other.
Question: What do the medians in a triangle do?