A prior post on tally marks looked at the ease of using tally marks for counting, and its survival as a niche representational system for quantity that coexists peacefully with the decimal number system. An interesting issue I want to pursue in this post is: if we didn’t have the decimal number system, and had to rely on tally marks alone as a number system, how would we likely extend it so that it would properly support operations other than counting?

Before diving into this question, and the concomitant question of what is missing in the tally mark system that makes it a more widely useful number system, let’s take another look at what works really well about tally marks.

The tally mark system is based on a direct relationship between the number of strokes and the number of things counted. If I’m counting cars, each stroke stands for a car, if I’m counting bales of hay, each stroke stands for a bale of hay. In a sense, you might say I haven’t done any counting yet, I’ve merely moved the things to be counted from the field of hay to a piece of paper on which I’ve made marks. Now I need to count the marks on the piece of paper. But in the tally mark system, I haven’t merely put marks on paper. I have grouped them by fives as I went along. The five-groups are easily distinguished and separated from each other. Each group of five huddles together, separate from the others, like a group of teenagers at recess – but unlike cliques of teenagers, each clique consists of exactly five marks.

What’s the importance in the groupings of five? The first answer that comes to mind is that it is helpful in the conversion, later, to a decimal number. We can count off the groups while skip-counting by fives: five, ten, fifteen, twenty, twenty-five. Or perhaps you prefer to count off pairs of groups, and skip-count by tens: ten, twenty. If you skip-counted by five, you complete the conversion by counting up by ones for the remaining tally marks: twenty-six, twenty-seven, twenty-eight. If you skip-counted by ten, there is 1 five-group left, and three single tally marks, so you count up by five, then by one, by one and by one, also ending up at twenty-eight.

But our first answer can’t be the complete answer. If conversion back to the decimal system was front and center, surely we’d be tallying ten-groups, not five-groups. From a sufficiently abstract perspective, it makes no difference how large the tally groups are, as long as we are consistent. If we stuck to six-groups, we’d better get good at skip-counting by six, but otherwise they’d work just as well. One important reason for sticking with five-groups is that almost everybody can identify a group of five without having to actually count to five. You could even argue that all that’s needed to be a good tally marker is to be able to identify a group of four marks, because it is the four-group that triggers the diagonal mark that completes a five-group. After suitable training, a group of four just “looks” like a group of four, different from a group of three, and different from a group of five – and the four-ness of the group can be seen and apprehended without needing to count.

So if recognizing five-groups is the strength of the tally mark system, the weakness is quite clearly the lack of organization of the resulting five-groups. If the quantity I’m representing with tally marks is 2009, there are too many five-groups (about 400 of them) to be just spread out on a piece of paper without further internal organization. An obvious idea is to group five-groups in groups of five (how to mark a group of five five-groups is an issue we will come back to). If there was a way to see a group of five five-groups as a distinct entity, off in its own corner, marked and dealt with, then we would be better able to compare large numbers and see which one is bigger. To be able to deal with something as a single thing, the group of five five-groups needs a name of its own. Let’s call it a five-five-group.

Let’s also deal with how the five-five-group is marked. A simple way is to arrange tally marks in a two-dimensional display, as follows:

In this arrangement, we can quickly see 3 completed five-five groups, and on the right we see an incomplete one. The right-most column shows 4 five groups and 2 single tally marks. And as long as we’re working wholly inside of the tally mark representation (that is, we’re not yet in the business of conversion to and from decimal numbers) – we’re done. The five-five-group is easily apprehended as a single entity, a column, separate from the other tally marks. Deciding when to start a new column only requires us to see at a single glance that we’ve completed five five-groups.

We might even be tempted to come up with symbols to stand for a five-five-group, a five-group and a single tally mark. If we were to settle on “C” for a column of five five-groups, and on “X” for a five-group and on “I” for a single tally, we could write the number represented in the figure above as CCCXXXXII. Short of numbers that get so big that there were many many columns, we would have the basis for a reasonably well-functioning system. However, it is not clear that the system we sketched here scales very well for bigger numbers. We will come back to this in a future post.

Tally marks I can get, collatz…..

Sharon

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Sharon,

There are several new posts on tally marks… I hope you’ll enjoy them.

As to the posts on the Collatz problem: this is material I’ve been working on with a ninth grader – though not a typical ninth grader by any measure.

I do mean the thing it says on top, about math as a garden, friendly and always new. Feel free to read what you like and skip whatever you don’t like. Some people like the roses, some the crocuses and some head straight for the oregano. It’s all fine.