Series: About Early Math

Beyond Base-10: Three Unusual Counting Systems

Counting is one of the first mathematical skills that children develop. While counting to ten is a major milestone, it is also a temporary roadblock because children run out of fingers to count on! From there, we see that each decade, or series of ten, is a mini-milestone for young students to conquer. Note how this student pauses and adds emphasis every time he crosses into a new decade.

It is clear that counting in sets of ten, also called a base-10 or decimal numeral system, is a skill that we develop and internalize at a very young age. However, surprisingly, there are dozens of different counting systems, many still in use today. Understanding and exploring other counting systems can be helpful to children further down the line when attempting to measure an object that does not cleanly add up to 10 units. It is also helpful when children begin to tackle “the dreaded F word,” fractions.

  1. Shepherd’s Counting System

    Older students might enjoy trying to wrap their heads around this counting system. Years ago, shepherds in Scotland, England, and Wales counted their sheep using this set of rules, and some of these counting words still pop up colloquially in rural regions. There are regional variations, but most deal exclusively with numbers up to 20. They often have a sub-base of 5 between 10 and 20, so it may be an avenue for exploring a base-5 system. Written in the Arabic numerals that we know, an example of a Shepherd’s counting system might look and sound like this:

    Shepard Counting System

    This chart looks like it goes up to 40, but 20 is the actual quantity reached. We see the numeral 40 written, because when using place value in a base-5 system, we would reach a new decade after every fifth number.

    This example is fun, but unless you plan on traveling to rural Scotland, you might not get the chance to use it very often. In an early education classroom, delving into this different number system, maybe even with a little European history and cultural spin to the lesson, can be a useful exploration.

  2. Binary (base-two)

    There is a non-decimal number system you most likely use every day and are actually using right now reading this article. Computers and electronics use this number system to do the bulk of their processing. As it sounds, it uses only two different digits, 0 and 1. It looks like this:

    Binary Counting System

    The year 2014 would look like this: 11111011110. It’s pretty hard to read. Why would computers use only two digits? Think of another electric device you use daily: a light switch. You have two choices: flip it up to turn it on; flip it down to turn it off. It’s very easy to determine whether the lights are on or off; you’re either sitting in the dark or you aren’t! This “on or off