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Notes -
I got hit by the tail end of the new math. The way I was taught to do subtraction is definitely closer to Tom Lehrer's second method than the first; we would say that the two "borrowed" a one from the four, so the four got crossed out and replaced by a three while the two became twelve, and twelve minus three is nine.
I... like it? It makes a lot more sense than the first method. We didn't get taught that the four is actually four tens, since it's in the tens place, and that you are substracting ten from the forty and adding them to the two, but hearing him say it makes it obvious in retrospect that's why the algorithm works.
Tom Leher says "the important thing is to understand what you're doing rather than to get the right answer" like it's a joke, but I actually agree with that. As my calculus teacher said "your only advantage over the machine is your ability to think. Once you lose that, I prefer the machine. Calculators are faster and make no mistakes".
Anyway, we also did simple sets in elementary school. No non-decimal bases, though; I learned that on my own reading about computers, because binary and hexadecimal.
“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”
Your ability to think matters because it enables you to get the right answer. The only problem with students who don’t understand is that they won’t be able to get the right answer in more general situations. An athlete doesn’t need to understand the physics of his sport or the biology behind his movements.
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