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## A Response to a Response to....

Two weeks ago, A Mathematician's Lament was suggested as good thought-provoking reading material. Last week, D.B.'s contrary reflections were printed. This week, two more responses are provided. Dialogue is great and I appreciate the willingness of teachers to sahre their thoughts.....Thanks.

First, M.H. (Portland, OR) focuses on D.B.'s claim that: "Another of his [Lockharts] math problems leads me into a world of research of why does a negative exponent mean a reciprocal .."

M.H. writes: Hmmmm... A simple sequence produces the basis for the logical necessity of the negative exponent, to say nothing of the elegance of the notation and resulting simplicity of computation:
16 * 1/2 = 8 = 2 ^ 3
8 * 1/2 = 4 = 2 ^ 2
4 * 1/2 = 2 = 2 ^ 1
2 * 1/2 = 1 = 2 ^ 0
1 * 1/2 = 1/2 = 2 ^-1 = 1 / (2 ^ 1)
1/2 * 1/2 = 1/4 = 2 ^-2 = 1 / (2 ^ 2)
1/4 * 1/2 = 1/8 = 2 ^-3 = 1 / (2 ^ 3)
......

Wouldn't familiarity with some pattern similar to the one above be a necessary prerequisite for guiding students to an understanding of operations with exponents, as opposed to simple memorization of the rules of computation?

(As an aside, doesn't the sequence also point to a great entry point for a discussion about the meaning of zero as an exponent?)

Then, J.S. (Whatcom County, WA) writes: ...I am in wide and general agreement with Mr. Lockhart. I was often frustrated in the classroom by what I saw as profound apathy from the students in general. I had hoped, over the years to teach a math course which included all sorts of "weird stuff".....It never came to fruition and, sadly, I have to report that things have actually taken a turn for the worse in the very district where I taught. A long time ago, I noticed that quite a few students were reading Jurassic Park. At first, I assumed that it was an assignment from an English class. I subsequently found out that it was given to them by the biology teacher. Also, some of them asked me about the "mysterious squiggles" at the beginning of each chapter that said, "First iteration, second iteration, etc." I went to the biology teacher and expressed a willingness to give a class on iterations and fractals to his group. The class period was a 1.5 hr block and I went through some calculator discovery of iterative processes with them for a half an hour, then [a colelague] spent a half an hour with them showing how they could generate the fractal with Sketchpad, and then I took them again and we went further with some more calculator based processes. We got a lot of positive feedback from the teacher and students and I've always felt excited by what happened. Unfortunately, it only happened that one time. Scheduling conflicts and numerous other problems always got in the way. However, what bothers me most is that recently I was told that in that district, it most likely will not happen again because it would interfere with curriculum constraints. They have to follow the curriculum and there is not likely time enough for embellishments.

I have always been a proponent of problem solving and the belief that mathematics is the science of patterns. I think they go together and are inseparable. I'm pretty sure I read somewhere that in Hungary, elementary school teachers do not have to spend as much time with many rote memory tasks like times tables because the students are given many problems to solve from the beginning of their mathematical experience. The theory is that once they are armed with problem solving skills, then all they need is a basic understanding of how the times tables work, a few examples, and off they go. Whether Hungarian schools really do that or not, it certainly seems to make sense.

One area where I may have a slight disagreement with Lockhart is that I do believe that at a certain level, mathematics is a language. Having said that, I think I would agree that for most k-12 students I would avoid teaching it as such.

In closing, I think that it would be interesting to discuss the structure he would envision for mathematics in the k-12 world.

Thanks also for taking the time to read this.