I’ve had a few thousand high school kids come and go in my classes, so I figure I’ve been around long enough to make one or two observations, and lately I’ve been wondering why it is that seemingly bright kids can sometimes have such difficulties in math, and other, seemingly less intelligent and certainly less hard working kids, seem to grasp math so easily. One of the things I notice is that the problems often relate to words that start with the letter I.
Invisible Lines….. One of the first places we seem to lose bright kids is in that point in geometry when we expect them to be able to look at a sketch and see a line that isn’t there, but should be added to make a proof or problem solution easy. I think one of the classics used to show up on SAT tests until it leaked out and became too well known. It gave you a circle with a diameter of 14 and a center at point O, and a rectangle drawn as shown. The question just asked for the length of the diagonal shown.
To the quick student of math, there is another line that makes the answer obvious, the other diagonal is a radius of the circle, and since the diagonals are equal, the requested length must be 14/2 = 7 units. Bright kids who know all the above geometric relations miss problems like this because they seem only to see the lines drawn. It seems that geometry is where this shows up first, but the similar idea shows up repeatedly in math. Mathematical thinkers look for what might be there, or what we would like to know.
The second irritation “I" for the non-mathematical thinker is impossible. Some folks have a difficult time accepting that anything can be proven to be impossible. Their mind can’t get beyond the “maybe you didn’t try the one special way that works yet.” The early Greeks wrestled with the problem of “squaring the circle”, for instance. It is a simple enough idea, take the classic instruments of geometry, a compass and straightedge, and construct, in a finite number of steps, a square whose area is the same as a given circle. We have learned a lot from pursuing the challenge; Hippocrates of Chios (not the medical one from Cos) discovered lots of nice properties about the Lune, but he could never square the circle. Why?...because it is impossible. In 1882 Ferdinand von Linderman proved that pi was transcendental. All that means is that you can’t write a polynomial with rational coefficients (you remember those, things like x2 + 2x + ¾ = 0) which has p as a solution. But every construction with a compass and straightedge can be shown to be expressible as a term of a polynomial with rational coefficients, so no solution, no pi construction, and no pi construction, no pi r-squared. OK, that was over 100 years ago, but college professors are still receiving “proofs” from mathematical armatures who think they have solved the problem, and there is a thread going on one of the geometry news groups where someone is trying to convince those still patient enough to respond that he has conquered the problem. His problem seems to have been an overlooking of the finite part. The problem is more general, because in early geometry, one of the proof methods is called reducto ad absurdum. It is a method by which we assume something we want to prove true, is in fact false. We then have to show that that leads to an impossible consequence, and so we can assume the veracity of our original conjecture. The student who rejects the idea of finitely proven impossibility has forfeited one of the strongest of mathematical tools.
I’ve talked before about the problems students have with the “imaginary” numbers. Ok, everyone in the business thinks it was an unfortunate choice of terms for pedagogical usage, but it is just a name. Still the idea that anything derived from the seemingly impossible (sure… NOW they believe in impossible) square root of a negative number, could prove useful. I suppose that the concept would have to be expected to be difficult for students if only 130 years or so ago, the famous mathematician de Morgan described them as “self contradictory and absurd”
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