Last week, attempting to show how times (e.g. goals, content focus, etc.) have changed in mathematics education, I provided a copy of questions from the "1983 Oregon Invitational Mathematics Tournament." Again, I do not know who produced, coordinated, or scored the exam.
One reader of this website (T.R., Bellingham) pointed out that one of the problems had no answer. It was:
[Q#10] How many solutions does the equation cos x = x/(3π) have?
(a) 2 (b) 4 (c) 6 (d) 8 (e) infinitely many
So, the new question or problem for this week is...what is the correct answer to this question?
Plus, any ideas as to why this answer was omitted in phrasing the question in 1983?
Also, I wonder how many students taking this competitive exam obtained the correct answer and paniced when it was not included as an option?
Thanks, T.R. for catching this error...and noting that the error is connected in spirit with a story by Richard Feynman, physicist extraordinaire. (See solution commentary next week as to why!)
Hint: T.R. writes: "It is an error I see pretty frequently, most often in the alternate version that if given the graphs of x and xsinx, students assume that there is a local extremum at each point of intersection since it is a point of tangency."
Now, how does this help?
Solution Commentary: T.R. writes: "There is an interesting mistake in one of the problems in that Oregon math exam--problem #10 about the number of solutions to cos(x) = x/3Pi. The correct answer is 7, which is not one of the options given. I suspect the difficulty is at - 3Pi where there is one obvious root (- 3Pi !) and the other root is close enough that if you don't realize that it must be there because the line is not tangent to cos at - 3Pi, then you could easily miss it. Even if you look at a default graph of the entire interval it is not obvious, though of course if you zoom in, you see it."
T.R. continues: "And an afterthought--you shouldn't need graphing technology to know there are two zeros near -3Pi--do you know the Feynman story about the time in his freshman year at MIT in a drawing class when he told the class that french curves were specially designed so that when you held it on its side and put a ruler tangent to the curve at its lowest point, the ruler would be horizontal no matter how you turned the french curve? Then he makes one of his cutting remarks about how most people don't really know what they know."
The full story from Feynman's book Surely You're Joking, Mr. Feynman!: Adventures of a Curious Character (1985):
"I often liked to play tricks on people when I was at MIT. One time, in mechanical drawing class, some joker picked up a French curve (a piece of plastic for drawing smooth curves--a curly, funny-looking thing) and said, "I wonder if the curves on this thing have some special formula?"
I thought for a moment and said, "Sure they do. The curves are very special curves. Lemme show ya," and I picked up my French curve and began to turn it slowly. "The French curve is made so that at the lowest point on each curve, no matter how you turn it, the tangent is horizontal."
All the guys in the class were holding their French curve up at different angles, holding their pencil up to it at the lowest point and laying it along, and discovering that, sure enough, the tangent is horizontal. They were all excited by this "discovery"--even though they had already gone through a certain amount of calculus and had already "learned" that the derivative (tangent) of the minimum (lowest point) of any curve is horizontal. They didn't put two and two together. They didn't even know what they "knew."
I don't know what's the matter with people: they don't learn by understanding; they learn by some other way--by rote, or something. Their knowledge is so fragile!"