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Archive for April, 2008

Clever HansMental calculators of yesteryear were usually described in magazines, newspapers and books in ways that can be startling in our more cynical age. But even today newspaper articles, documentaries and television features on modern lightning calculators appear almost regularly, often with a “hook” such as diminished capabilities in other areas (the “Einstein” effect). Surely there must be some reports that try to be objective, but I haven’t found them. At best they are naively written by people with little mathematical background; at worst they use considerable license (deception, really, if only by omission) to present a better story. This part of the essay is not directly related to the historical art of mental calculation itself, but I think it serves as a cautionary tale in evaluating articles on it.


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Brain Scan Comparision of Rudiger Gamm and OthersThe types of calculations performed by lightning calculators were historically quite limited, notable mainly for the size of the numbers and the speed at which they were manipulated. But remember that the questioner had to verify every calculation by hand, making higher powers and roots (particularly inexact roots) much less feasible. The dawn of calculators and computers propelled some of these tasks into hitherto uncharted territories such as 13th or 23rd roots, deep roots of inexact powers, and so forth, much of it supported by more sophisticated mathematics. Here we will review the methods of calculation used in the past, many of them not commonly known, as well as other techniques that are relatively new.


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Bogdanov-Belsky (1895)Individuals with preternatural abilities to calculate arithmetic results without pen, paper or other instruments, and to do so at astonishing speed, are the stuff of mathematical and psychological lore. These “lightning calculators” were sometimes of limited mental ability, sometimes illiterate but of average intelligence, and sometimes exceptionally bright, this despite the popular notion of the idiot savant. The techniques used by these people are not generally well known. In fact, despite claims by educators that acquiring a mental facility with arithmetic operations is essential to a student’s mathematics education, I see little in the textbooks other than simple estimations based on rounding values, surely the most basic and least interesting mental task. The field of mental calculation may not be a lost art per se, but in this digital age it most certainly is a neglected one.

Part I of this essay attempts to take a fresh look at both historical and modern lightning calculators. Part II describes classic and modern methods of mental calculation. And finally, Part III demonstrates as a cautionary tale the shallow and deceptive nature of most media coverage of lightning calculators, an important consideration in analyzing reports on them.


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