The Trachtenberg System for mental arithmetic
en.wikipedia.org
en.wikipedia.org
I was kind of disappointed to discover it wasn't that Trachtenberg.
In Texas high schools there is -- or perhaps was -- a mental math competition called "Number Sense." Eighty problems in ten minutes, ranging from 12 + 72 = ____ to geometry, trig, and calculus problems. Training requires lots of tricks, memorization, and practice. My coach had accumulated a file of close to a hundred practice tests, including the tests for most of the official state, regional, and district competitions going back over ten years. One of the best Number Sense competitors I knew actually missed some meets because she failed Spanish -- she spent all day taking practice tests during her other classes.
(Caveat: I graduated fifteen years ago and have no idea how things have changed since then.)
There are (or were) a couple of other math competitions, and both also focused on speed and rewarded mental arithmetic skills, though to a lesser extent. One was calculator-based, and was a little silly since doing well depended quite a lot on your ability to punch buttons quickly and accurately. (RPN was a must, and very, very few competitors used anything other than an HP.) You had thirty minutes to do eighty problems, of which sixty were boilerplate button-punching problems called "crankout." (You could save a few seconds here and there by doing some mental math, and a few of the crankout problems on each test were series that you could recognize as, say, e^1.26, and you could save yourself a few seconds by just punching 1.26 e^ on your calculator.) You'd do the "crankout" perfectly (hopefully) in 10-15 minutes and spend the rest of the time on the word and geometry problems. Having a fast, accurate "crankout" was the key to success, which made the calculator competition by far the stupidest of the competitions, though it was also the most exciting from a sporting perspective, since you could tell by the sound of the keys who was fastest and when each competitor finished the "crankout." The competition was also superbly paced; usually, the competition ended while each competitor was trying desperately to figure out a problem that was neither routine nor unattainable for him or her. As with Number Sense, training for Calculator meant lots of tricks and practice, and my coach had dozens of practice tests she had collected over the years.
The last competition was simply called "Math" and gave you forty minutes for only sixty problems, which seemed luxurious. Typically, if you were good, you would blaze through 50-55 problems in 25-30 minutes and then tackle the harder ones that remained. None of the problems required math beyond simple calculus, but on every test there were problems that would confound the average "A" honors calculus student.
This all sounds terribly silly, and it was. I considered it a sport for my mind, with all the benefits of a sport: camaraderie (such as it was,) fitness (of the mental kind,) and an opportunity to excel and attract positive attention to myself. It might seem irrelevant to my academic and intellectual development, and in an ideal world it would have been just a distraction. In reality it was a godsend. High schools do a terrible job, pretty much a nonexistent job, of challenging talented math students with advanced or difficult material. You're rarely going to find a high school teacher who learned anything beyond calculus, let alone remembers it, let alone was any good at it. Letting us train the hell out of less-challenging math was the only way to compensate. It rewarded kids who studied and learned more, which was nice, since even honors classes didn't differentiate much between a good student and an excellent one. By the time I took calculus, I already knew a lot of calculus from training for competition. It didn't matter that my honors calculus teacher (who had an engineering degree from a local university) didn't seem to get the basic concepts of calculus and probably never did. (To be fair to him, he didn't volunteer to teach calculus, and to be fair to the system, he was the worst calculus teacher the school had had in the past decade, and they found somebody else to teach calculus only a few years later.)
Although the competitions didn't contain any advanced material, they did provide some stimulating puzzles. The puzzle of optimizing one's approach to the competitions -- optimizing problem solutions a la Trachtenberg, optimizing test-taking skills by tackling the problems in the optimal order, and optimizing practice time by figuring out what tricks to learn and which tables to memorize -- was the closest thing to engineering that I did until I got my first software job. The test problems sometimes brought new ideas to my attention and pointed out areas that I had overlooked in my classroom study. Obviously, this is a pretty stupid way of educating kids, but it existed where I grew up, and it was better than anything else that I could share with other people. For everything else, it was just me alone with a book, with no one to talk to when I felt proud or frustrated.
Anyway, I thought I would describe this curiosity since the subject of mental math came up. FWIW, the Trachtenberg system as described in the link seems primitive compared to the bag of tricks that I learned from the yellowing self-published pamphlet that my coach kept locked in a cabinet, but it is more practical for people who don't want to "train" as they would for a sport. The best Number Sense competitors dropped a lot of the shortcuts that they relied on when they were beginners as they got better at raw computation. It was common to memorize the squares up to 30x30, and a few memorized the multiplication tables up to 20x20 (or so they claimed.) Memorizing the squares up to 25x25 and the digits of the fractions over 6, 7, 8, and 16 were fundamental skills taught to freshman. Logarithm tricks were less common, but the best competitors used them. Who would do that stuff nowadays except to excel at a "sport?"