A Mathematical Diversion with Cedars-Style Exponents
In the Cedars Christian School 7th-grade math class I teach, someone got confused about 2^3 and thought it meant 2 + 2 + 2 = 6, whereas it really means 2 · 2 · 2 = 8. That got us going as to how we might use exponential-like notation for all the arithmetic operations, not just multiplication. We might say that
Another thing we need to do is define what happens if there is more than one exponent on a number. Someone in a class asked what would happen if we wrote:
Before, we had to make one decision, where to put the 4 different power exponents. It was natural to put the multiplication one at the top right, since that’s the ordinary definition, but we had choices of where to put the others. Now we have to make a second definitional decision: the order of operations. We’d probably want to start with the familiar top right exponent, multiplying 2 by 2, but what happens next? Do we go clockwise, counterclockwise, or criss-cross?
Thus, we define:
We can build up the number progressively, which will teach you the definition better and quicker than reading the equation:
I haven’t heard of this kind of notation or enterprise before. Have any of you, readers?
In my teaching, this idea was one of the gems that are exciting to find along the way, helping the students invent new mathematics and getting them to think about which definitions are good and which are bad.1 I used it to make the point that definitions are not arbitrary: there exist good definitions and bad definitions, meaning useful definitions and awkward definitions. The same goes for notation, the special symbols or patterns we use to depict definitions. Alas, very often these gems are evanescent, ephemeral, transitory, and that is the case with this gem. It worked very well the year it came up, but only because it came up from the class naturally. It would not be the same if I forced it on the class, and I probably would not be able to lead them to it. So many of our best moments are like that . . . .
I like this :))
I majored in mathematics in college, but still was surprised when I first learned about "tetration", which is repeated exponents. Likewise, "Pentation" is repeated tetration, ad infinitum...
My conclusion is that exponents are over-emphasized in the grade school curriculum, as any exponent other than 2 or 3 is almost never used in physics, and only has so much perceived utility because of computer modelling.
Maybe they are overemphasized. Where higher exponents are used is with base 2 and, especially, base 10-- scientific notation. That's worth the price of admission.
They are also good for teaching the idea of an inverse, a very hard idea to get across.