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Dear Mitch,

I'm an elementary school teacher in a school just outside Sydney, Australia, and I am a team-teacher for a section of third-fourth graders, in a program for gifted children. Most of the students are remarkably strong in at least two subjects, but some show talent across the curriculum. In any case, for our end-of-year field trip, the students were given a selection of three local activities or tours that interested them, and they voted for a day at a nearby ceramic and pottery studio where each child would get to paint a ceramic piece of his/her choice. Afterwards, the pieces would be fired in a kiln. Some students may be given the opportunity to make their own items from clay, which they'll paint and have fired as well, but basically the day will consist of a pottery painting workshop.

My question is this: My hope was to find a way to subtly incorporate many of the topics we studied this year, from science (the heat of the kiln, for example, and how it affects the temperature of the adjoining rooms), to the social sciences and the historical importance of pottery in the ancient world for survival. HOWEVER, I have yet to come up with an idea I like that I like for a MATHEMATICAL perspective, and I'm only looking for two or three thought questions to present, rather than try to force a whole year's curriculum in. I figured you'd surely either know of some way in to the subject that I would like, or you'd come up with one. Could you do that? Would you?

Before I end this note, I just wanted to say how much the other teachers in my school and I appreciate your ideas. You've done more for our mathematics program than you can imagine!


Stuck in clay and I feel like it's kiln me...


Dear Get-Unstuck,

In most of the pottery painting workshops I've experienced, there tends to be one or two ideal moments to present a few questions regarding symmetry: A round ceramic plate, for example; has lines of symmetry. The real question is how many such lines can students identify and count.

Well, before the plate is painted and decorated, the piece has an infinite number of these lines. All circles do, which is related to the fact that all circles have an infinite array of diameters, and, of course, each of these diameters is the same length (or, to be official in our geometric terminology, these diameters are all "congruent". And with that fact comes the idea that each circle has an infinite supply of radii, and each of these is congruent.

But once the ceramic plate is decorated, there may be one line of symmetry, or two, or more, but most likely none (given most children's passion for spontaneous Jackson-Pollack like bursts of energy!). A ceramic bowl also has the same issues of symmetry, unless the bowl is oval, diamond shaped, square, star-shaped, or has some other form that is a regular polygon. In those cases, there could be one, two, or more lines of symmetry, but often there are none. Rectangles, students are often surprised to discover, have vertical lines of symmetry and horizontal lines of symmetry, but not diagonal ones unless that rectangle happens to be the "special" rectangle we call a square.

If the students observe a potter's wheel at the studio, they can actually watch as geometry comes to life; they will see that the center point of the circle, or vertex, or whatever it's currently the fashion to call it, must be in the true center of the wheel, or the potter will have trouble getting its shape to be round. (If that is indeed the result desired.)

It should also be easy to incorporate the mathematical concept of "more than"/"less than", particularly with regard to estimation: Do the pieces probably weigh more before they are fired or less? Many students will associate weight with hardness or brittleness, as in a glass bottle verse a plastic, disposable one, but clay presents a different case. Why? Because in the course of firing, the clay's water is expelled, and water is the part that ranks high on the heaviness scale. You might want to ask students which is heavier, ice or water? They are likely to say ice is heavier because it is 'harder' and more solid and more brittle. Then ask: But which floats in which? Does ice rise to the top of a soft drink, or does ice sit at the bottom of the beverage? Typically, ice cubes rise. Why? Because they are lighter... But how can they be lighter? They are less dense or more spread out. And that is the reason, of course, why when water freezes and becomes ice it often puts cracks into the surface of highways. How????? Water gets into tiny, barely visible cracks, then freeze, and as it freezes it expands with such might that the pavement buckles and crumbles apart. Concrete is tough stuff, for sure, but it is certainly no match for the eternal power of mathematics!

As all kinds of shapes and patterns present themselves (such as spheres, cylinders, circles, and the whole range of polygons) there arises the topic of surface area. For example, if two students start off with the same-sized lump of clay, will their finished products necessarily end up with same surface area? NO! Same volume? That depends upon whether or not you consider empty spaces like open cups to have more volume than the original ball of clay...

Then there is the very real-world issues of "positive" and "negative" value. If a design is scratched into a piece using a stylus, then it is a negative addition; if the student then changes his mind, and he or she decides to remove that etched-in design, he is taking away a negative, which is the equivalent of adding a positive of the same absolute value, and taking away a negative causes an increase. The inscribed line brought up to "surface level".

On the other hand, thickly applied paint or decorations are 'positive' in that they go above or beyond the neutral point of zero elevation, and then if one takes that way, well, the eventual removal of a "positive" is neutralizing or adding the negative equivalent.

Final question for the class:

"Which is a better shape for a cup, one whose bottom is wider than its top or one whose top is wider than its bottom?"

Answer: That depends on who happens to be answering the question: A bigger bottom is often better for the consumer, as it is less likely to tip over. But the narrower bottom enables the store to stack many glasses and save money on shipping...

The very same idea, by the way, is, of course, true for trash cans. On a windy night, when your trash can goes over, if you are like most people, you probably aren't thinking that much about the store, and how nice it is for them to be able to stack those cans thirty high, correct?

I hope this helps. Either way, it's an end-of-year field trip, so have fun!