Jamie Stroud
Table of contents
Even though tallying is considered unary (base-1), most widely spread methods of tallying essentially function in congruence with quinary (base-5), while another widely spread method essentially functions in congruence with decimal (base-10). There's probably a biological reason why tallying in congruence with base-5 became so popular. Evidence strongly supports that most people can subitize up to 4 symbols but not beyond. (Trick & Plyshyn, 1994) That is to say, recognize that amount of symbols immediately and accurately without having to consciously count them. While it might seem that tallying up to 5 is going 1 past this limit and thereby would slow work down, it should be noted that if you can utilize a distinct symbol then higher digits are possible to be subitized. For example "9" is a distinct symbol (excluding the quotation marks) which most people subitize, meaning they immediately and accurately recognize the symbol 9 to represent 9 digits (or 9 things). The symbol "9" is somewhat arbitrary and in another place or another time could be a different symbol yet still represent the same thing.
North Americans, Australians, some parts of Africa, and some parts of Europe tend to tally up to 4 separate but the same symbols, yet by the 5th tally the 4 symbols become connected with a single diagonal line, thus making it something which can be learned to represent a distinct symbol, and thus all these tallies can be subitized. South Americans, some parts of Africa, and some parts of Europe tend to utilize a tally system with symbols that are composed of all connected lines that end up forming a square with a diagonal line (forming hypotenuses) drawn across it. These tallies can all be learned to represent a distinct symbol respectively, and thus all these tallies can be subitized. Asians tend to utilize a tally system with symbols that are composed of singularly increasing lines that end up forming 正 which is generally an already learned and recognized symbol in their language. These tallies can all be learned to represent a distinct symbol respectively, and thus all these tallies can be subitized.
The important information to take away from these examples is that drawings can generally be learned to represent a particular concept (numbers, words, ideas, etcetera) so long as they're distinct and not too complex, and thus they can become there own meaningful symbol. We can immediately and accurately process up to 4 symbols at a time without having to consciously count them.
These methods of tallying are just as fast yet even more accurate than the current widely spread methods of tallying. By putting the tallies in units that are congruent with one's preferred numeral system (octal (base-8), decimal (base-10), or duodecimal (base-12)), both the speed and accuracy of rechecking will be increased. In other words, the tallying system would be in 2 units of half of the numeral system followed by a recognizable amount of space and then repeated. While the decimal and duodecimal methods might seem to break the rule of not having more than 4 symbols so that subitization can occur, in practice this isn't much of an issue. Generally when one makes the symbol for the halfway mark (5 in decimal and 6 in duodecimal) it's possible to consciously remember that you did so, and thus only have to look at the next 4 symbols which then become a single symbol at the next number. On the chance you don't consciously remember what number you left off at, because the symbol for the halfway mark is distinct from the other tallies, it's still faster to process than, for example, using all the same straight-line tally up to the end of the base number. It's also faster than creating another system that doesn't break this "no more than four" symbols rule, and that's because bottom-left to top-right diagonal strokes and top-to-bottom straight line strokes are physiologically the fastest strokes for most people. These methods are optimized for those that write right-handed and for those that use a left-to-right (x) then top-to-bottom (y) writing system. For those that write left-handedly and/or use a different writing system, these methods would need to be modified in order to be similar in speed.
These methods of tallying use just as little space while also being faster than the current widely spread method of space efficient tallying. However, they're unlikely to be anywhere near as fast as the time efficient tallying methods. For decimal and duodecimal, for increased accuracy, it's best to draw the half-sized lines starting from the center of the square.
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