NavList:

Brad, you wrote:
"How can we stand on the shoulders of giants like Pythagoras, Euclid or Maskelyne without speaking their language?"

Well, in a way, you answered your own question. It is highly doubtful that you speak Classical Greek and probably only one or two NavList members can read more than a few sentences in Classical Greek. But we can comprehend their words through translation. And to some extent at least, the same is true of their mathematical culture. For example, if I can work a historical logarithmic problem once on paper, I really don't need to do it again. I can translate that solution into a spreadsheet equivalent. I know that, if necessary, I can dig deeper into the old mechanics of historical calculations if the need should arise. But my understanding of historical navigation material is not significantly impaired if I bypass the details after an initial study.

You also wrote:
"Some young associate will state the answer came from his calculator, therefore it 'must' be correct. No concept of the order of magnitude of the proper solution or understanding of the problem formulation. "

This is an excellent example. And I think this is where both sides of this debate are on the same page. Just the other day, I was reading something that noted that you can't easily figure out the value of "17 times 24" off the top of your head. You have to do it on paper or using a calculator. I thought to myself, "it's about 400... probably 408". There are certain algorithms that led me to that answer (it's nearly 20*20 and the last digit has to be 8). But the traditional algorithm for working a multi-digit multiplication didn't enter into it. I could have gotten out a piece of paper, stacked 24 over 17... then worked 7 times 24, dropped down a line written 24 one column to the left... and then added. OR I could have grabbed a calculator and tapped six keys. Either of those approaches might have confirmed my initial guess. But it doesn't matter. By FAR the most important computational algorithm here is the ability to estimate the final result in order of magnitude and preferably with one significant digit. That is, ANY student past a certain age should be able to state that the answer is nearly 400.

This past weekend, I had another opportunity to watch students wrestle with angular arithmetic. How do you calculate 90°-72°45' on paper? All of them "remembered," at least vaguely, the idea that we can "borrow one" from the degree column and convert into sixty minutes in the minutes of arc column making 90°00' into 89°60'. Those kinds of manipulations are essential tools. But do students really need to learn degrees and minutes and "sexagesimal" calculations in detail? Do they need that more than they need to learn "FF" and "00A" and other "hexadecimal" calculations? It's hard to say. Briefly when I was nine, we were forced to study base-eight arithmetic because, in the future, we would all be programming computers on our daily trips to the Moon. The parents and teachers revolted and the books were pulled after a few weeks, but even today, that exposure to "new math", ill-conceived as it was, definitely helped me even though it drove the adults crazy.

I come down squarely on the fence on this debate. We need more conceptual math and less old-fashioned computational algorithms. And we also need more "old-fashioned" computational algorithms but geared to managing the modern tools we have available. Now quick: what's 180*60/pi? ;)

-FER

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