When I was a freshman in high school, my home room teacher gave us a very nasty assignment during an after-school detention session - to calculate 35 to the 35th power!
This assignment was particularly cruel and unusual punishment because there were no such things as calculators back in 1967.
What I really needed was something I didn’t know about until college: a closed-form solution.
Finding closed form expressions for partial sums is a standard calculus exercise. The ur-example of this type of problem is the sum of the first n integers, which is easily shown to be n(n+1)/2.
This closed form expression collapses (n-1) operations into three. Because it yields an exact answer, it is not really a predictor, but, in a sense, it is a model of a process.
When this example is done in a calculus class, a typical accompanying story is how young Gauss solved this problem in record time, totally showing up the teacher who had given out the onerous task of adding the first 100 integers. (The version I always heard was that this was a punishment because the students had been particularly noisy that day. The sadistic mathematical punishments of my high school teacher certainly lends credence to this tale.)
Brian Hayes, in his American Scientist Online article Gauss’s Day of Reckoning, questions whether the story is true or myth. Along the way he does prodigious research into Gauss’ writings and those of his biographers, and he reaches some very interesting conclusions that are pertinent to all students and teachers of mathematics. Some excerpts:
...soon I was wondering about the provenance and authenticity of the whole story. Where did it come from, and how was it handed down to us? Do scholars take this anecdote seriously as an event in the life of the mathematician? Or does it belong to the same genre as those stories about Newton and the apple or Archimedes in the bathtub, where literal truth is not the main issue? If we treat the episode as a myth or fable, then what is the moral of the story? After reading all those variations on the story, I still can't answer the fundamental factual question, "Did it really happen that way?" I have nothing new to add to our knowledge of Gauss. But I think I have learned something about the evolution and transmission of such stories, and about their place in the culture of science and mathematics. Finally, I also have some thoughts about how the rest of the kids in the class might have approached their task. This is a subject that's not much discussed in the literature, but for those of us whose talents fall short of Gaussian genius, it may be the most pertinent issue.
Hayes does a great service by pointing out the "fable" may convince some students that they are not capable of doing mathematics. He ends with a "moral of the tale"
The story of Gauss and his conquest of the arithmetic series has a natural appeal to young people. After all, the hero is a child—a child who outwits a "virile brute." For many students, that is surely an inspiration. But I worry a little that the constant repetition of stories like this one may leave the impression that mathematics is a game suited only to those who go through life continually throwing off sparks of brilliance. On first hearing this fable, most students surely want to imagine themselves in the role of Gauss. Sooner or later, however, most of us discover we are one of the less-distinguished classmates; if we eventually get the right answer, it's by hard work rather than native genius. I would hope that the story could be told in a way that encourages those students to keep going. And perhaps it can be balanced by other stories showing there's a place in mathematics for more than one kind of mind.
As I think back on my hichschool detention, I try to picture myself solving the problem instantly, showing it to my teacher (I wish I knew phrases like "virile brute" back then), and jauntily walking out of that classroom hours before anyone else. Of course, I had no idea of what a logarithm was (and how it could be used to do the problem quickly) and had no slide rule, nor could I use one if I did - a corollary of not knowing anything about logarithms. So my solving this problem quickly was never going to happen.
Now, with my trusty TI-84 calculator, I can easily compute 35 to the 35th power (and see the first 10 or so significant digits of the answer.) Ironically, it is easier to do this with a calculator than to find the sum of the first 100 integers.
I want to believe that, even if everyone else in his class had a calculator back in the 18th century, Gauss would still beat the other students (and professor) to the answer. And, while there is a certainly a place in mathematics for "more than one kind of mind" - there is no myth about Gauss’ genius.