Representations are particular kinds of tools, and very interesting ones. The purpose of a representation is not just to show something, but to highlight a particular aspect of it. A weather map shows certain things about the area, usually related to temperature and wind and precipitation. A red-state-blue-state map of the United States focuses on the distribution of Republican and Democratic voters. The weather map doesn’t highlight the political climate, the red-state-blue-state map doesn’t show whether to bring my down coat. Typically a representation, by highlighting certain aspects, suppresses other aspects. This isn’t a failing of a representation, it is an expected consequence.
Some representations are fairly general purpose, others are very special purpose. A subway map of a city may deliberately distort distances to show subway routes and connections more clearly, and as a consequence that subway map may not be useful at all for planning a trip into the city by car. A tourist map of the city may have been targeted towards pedestrians, but be just as useful for bicyclists or motorists, and just as useful for people who live there as for people who are visiting as tourists.
In mathematics, our normal decimal number system is a very general purpose representation of quantity, and so is a number line. Sometimes certain representations are so successful and so widespread that it is hard to imagine there are other representations possible. It is sometimes hard to imagine that there are situations where other representations are vastly superior.
Tally marks are a simple example of a system of representation that not only has survived, but still thrives as a representation in a particular kind of problem situation. No doubt you are familiar with the kinds of situations where you’d use tally marks. In the picture shown, the tally marks represent a certain quantity. That quantity is more generally written as “37”. Whatever it is that you are counting, if a new one shows up, you simply add a tally mark. Adding a tally mark is far less messy than striking through the ‘7’ and replacing it with an ‘8’, and much less error prone than keeping the count in your head.
One of the things that allows tally marks to co-exist with our general number system is that it is fairly easy to convert from the tally mark representation to the general decimal number representation. We did just that when we noted that there were “37” tallly marks.
Representations are usually associated with particular kinds of tasks, and can be judged based on how well they are suited to those tasks. Yet no discussion of representations is complete without looking at how those representations can be converted to (and from) the representations that are in general use. I intend to collect many interesting examples of this in subsequent posts.