How To Use A Counting Board Abacus
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The first of two DVDs that together are: Ancient Computers, Part II: Video Users' Manual.
These are the companion videos to the book: Ancient Computers, Part I: Rediscovery.
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His ideas are somewhat speculative about the details, but what I like about it is that (a) he takes ancient technologies seriously instead of brushing them aside as obsolete relics [nearly every modern description of Roman numerals aimed at a popular audience talks about how terribly cumbersome they are, without bothering to explain their context or purpose, and mostly use them as a straw-man foil for Hindu–Arabic numerals which are presented as clearly superior], (b) he tries to reconstruct/invent working solutions to problems that would come up when attempting to do real concrete computations, approaching it as an engineer rather than a historian.
In the context of primary/secondary schools, I think it would be great to teach this kind of arithmetic (as a supplement to standard Hindu–Arabic arithmetic, the Japanese Abacus, a slide rule, etc.) for a few reasons:
0) It requires almost nothing in the way of supplies. Just a pile of pebbles and some lines drawn on paper (or lines drawn in the dirt with a stick).
1) There is practically no rote memorization involved, but lots of pattern matching, discovery, and thinking about the concrete meaning of numbers.
2) Unlike a fixed-frame abacus, the counting board allows for unreduced numbers to sit on the board, and shows very clearly and explicitly the relation between reduced and unreduced numbers. The method of reducing a number to a standard form can be handled using tiny obvious steps in many possible orders. Other arithmetic operations can also be done in a variety of possible paths, as long as each step is valid. This is the core of algebraic thinking.
2a) The counting board can make use of “balanced” positional numeration, i.e. the use of negative numbers within a particular place value instead of only complete integers being positive/negative, and there is obvious symmetry between positive/negative versions – just flip everything across the divider line; I think the loss of this idea in Hindu–Arabic arithmetic as usually practiced and on fixed-frame abacuses is a real step back for conceptual understanding of more sophisticated later mathematics.
3) The counting board is a fabulous piece of our cultural history as humans, and it’s a shame that a mere 4–5 centuries of disuse (after millennia of use) have been enough to almost completely wipe it out of mainstream awareness (except as backgammon boards, and in the etymology of various words/phrases). Moreover, understanding the counting board makes it easy to understand what Roman numerals are for and how they function. Namely, they are the written record of finished computations, not a tool for actively performing arithmetic.
4) In Steve’s version, there is an explicit treatment of an exponent as an integer, instead of as just a movable position of a decimal point. This is a good stepping stone to scientific notation, the floating point arithmetic used by computers, and logarithms.
5) The counting board can be fluidly/easily adopted to many different number bases for different purposes, unlike Hindu–Arabic numerals. Students figuring out how to properly change the “rules” of the board to deal with new number bases will have to think deeply about the relationships involved and the fundamental nature of positional numeration.
6) In a modern era when electronic calculators can handle the rote performance of computation, it’s more important than ever to focus thinking on the *meaning* of numerical relationships, and the *design* of algorithms. The counting board does this very well. Pen-and-paper arithmetic might turn out to be more efficient if you need to multiply 1000 pairs of 4-digit numbers, but these days nobody needs to do that.
Readers should treat this book as a speculative but plausible description of counting board methods and a practical guide for using a counting board in an effective manner, *not* as a work of precisely factual scholarship. Even if some of the proposals here turned out to be entirely different than Babylonians’ and Romans’ actual historical methods, they would still be worthy of consideration by primary and secondary school mathematics students.