20160404 STICK AND STAKE:

Measuring as counting.

When I mentioned sticks and stakes in MINUS, MINUS, MINUS-MINUS, I knew I had to write an article about them. Mostly, to explain them better, but also to provide an entry point because they are very useful. This said, please, read the old chapter first.

To numeralize space, we tile the distances with a smaller distance called unit of measurement, and we count the tilings. The actual unit can be only one. We can move it forward, applying its beginning to where its end was.

While doing this, we count in one of two ways. First, we can count the units - the measuring sticks - like I did below.

This is what I call the stick model. It works best, if we can view the whole scene of measurement at once. This means, we must be big enough or soar high above to see the beginnings and the ends of the sticks.

Another way to count emerges where the distances are big, and we are small. The first example of it could have been the milestones of Ancient Rome. Numbering the middles of the miles would not be very practical. The milestones marked the beginnings and the ends of the miles. Standing at a milestone, the travelers knew how far away from Rome they were. I imagine a measuring rope with two stakes. The holes left where the stakes were stuck in the ground are points, and we count them. That's what I call the stake model of distance.

I don't supply the picture for the stake model. It's just a number line. I drew plenty of them for MINUS, MINUS, MINUS-MINUS.

On a small scale, the stake model is awfully problematic because we see too much. Counting the distances from zero, kids must exclude the zero point. For those who are not trained in sticks and stakes, zero is a point like any other. Why don't we count it then? Because zero is nothing? OK, but if we count through zero, we count zero point and exclude some other point, at which we begin.

On a line, we always have one extra point. If we join the endpoints of a line segment to make a contour, they become a single point. Think about a clock face. Oh, no, don't think about a clock face. The English AM/PM system is insane.

The intuition behind the sticks is much more solid. Fortunately, the pieces of paper, on which teachers offer their problems, are always smaller than the students, so the stick model is the best. Like probably every parent, I tell my kids to convert stakes to sticks. If in doubt, draw arcs and count them.

I can imagine an overly smart child drawing arcs between the middles of the sticks and counting them. The rule was explained in MINUS, MINUS, MINUS-MINUS: only distances count. It's a metric space.

In the stick model, there is no zero stick. Zero means no sticks. There was no zero in Ancient Rome too, but they did have so-called Golden Milestone, from which every road was supposed to be measured.

A good method to make stakes palatable involves another heresy: a unary operation of increment. Vitruvius described (but not invented) an odometer, a wheel-based mechanism dropping a pebble preloaded in a hole once a mile. In Ancient Rome, they called pebbles calculi, so the distances could be calculated.

Starting from empty (which was exactly 0), the system would increment the number of the calculi in the receiving compartment. One thought of such a machine running along the number line must be enough to answer every question and obliterate any doubt. Anybody knows where to buy a cheap plastic rendition by any chance? OK then, teach to use increment rather than addition. It's even cheaper.

Drawing one-dimensional sticks and stakes, we have an extra dimension for numbers and letters. In two dimensions, sticks become squares, and we can place our marks inside them. Such squares are handy for measuring areas.

Two-dimensional stakes remain points, and we often use grids to highlight their spatial order. The third dimension is not readily available for drawing. This tempts us to draw points (stakes) as squares (sticks).

A chart of addition on this animated page from STEREOLEARNING is made out of squares. The numbers are inside them. The chart looks like a stick model, but it isn't. Invisible stakes are in the centers of the squares. Zero is in place, and a navigator must skip the first number when counting.

Such transformations are frequent and difficult to understand. Many tables have their actual grids passing through the cells and intersecting inside them.

We must specifically train our kids to understand sticks and stakes. Allow me to give you one more reason why: The chart of multiplication available through this link is a stick model. There is no zero, and we count from the first number.

I don't explicitely teach TLG sticks and stakes yet, but I point out the differences and explain them. From time to time I even offer her problems. Occasionally, I still see her making mistakes trying to use a times table as an addition table.