Bad Idea: Exponents are about a set of rules
First off, it is worthwhile to point out that algebra, philosophically speaking, really is a "set of rules", when dramatically oversimplified. It gives us symbolic representations of various notions, and predictable ways for manipulating those representations. This act alone is good practice for the mind, and development of skills along these lines is certainly worthwhile. Bearing this in mind, however, it is important to note that algebra as a rule-set is an incomplete picture.In a good enough pre-algebra classroom, students sometimes see the following definition:
22 = 2 x 2
Or something equivalent. This is a very good starting point. Unfortunately, the context often gets dropped from here on in. Next, we present to students the rules that go along with exponents. They learn that when multiplying powers of the same base, the powers add:
22 x 23 = 25
They are given no explanation for this, (or perhaps only a very brief one), and are simply told to memorize it as a rule. "Algebra is about rules", they are told, "follow them, and pass the course," is the promise.
A similar approach is taken to dealing with composition of exponents:
(23)2 = 26
This is again presented as a rule that algebra students simply need to memorize, more than likely because to spend time investigating the content of this rule seems wasteful to teachers who know such an investigation won't lead to more correct answers on assessments for their students.
Then, of course, are the remaining exponent rules, which again are presented to students to be memorized:
20 = 1
21 = 2
2-1 = 1 / 2
21 = 2
2-1 = 1 / 2
And so on. The way it appears in an algebra and pre-algebra classroom, each of these facts is a separate entity, and collectively, they all give us the rules for how exponents work. Algebra, after all, is only about rules.
Good Idea: Exponents are repeated multiplication, nothing more
The original presentation of exponents, i.e:
22 = 2 x 2
is in fact, totally correct. By definition, when one puts an exponent on a number, it literally means "multiply that number by itself that many times". While this may be the initial presentation of the idea, many math teachers ignore the fact that all of the exponent rules and properties inherit directly from this one fact.
Consider how to deal with multiplication of powers of the same base:
22 x 23 = ?
(2 x 2) x (2 x 2 x 2) = ?
(2 x 2 x 2 x 2 x 2) = 25
22 = 2 x 2
23 = 2 x 2 x 2
24 = 2 x 2 x 2 x 2
25 = 2 x 2 x 2 x 2 x 2
As can clearly be seen, using the original definition of exponents as repeated multiplication makes this "rule" not only obvious, but necessary. There is absolutely no other way to position this "rule", and upon viewing this elementary algebraic "proof", (a dirty word in our high school math classrooms for absolutely no reason, in the author's opinion) the fact is inescapable. The inevitability of this result, and triviality of its justification will actually carry with it the benefit that students are more likely to have better recall of it.
A very similar justification for the composition of powers can also be demonstrated:
(23)2 = 23 x 23 = (2 x 2 x 2) x (2 x 2 x 2) = 26
Both of the above proofs can also lead to a discussion of the associative property and commutative property of multiplication, if the teacher so chooses. (Again, though, since there is only a small chance that there's a standardized test question on these properties, it won't happen).
Zero and negative exponents can be a bit trickier, but not for a skilled teacher who truly has a grasp of the underlying content. Please note the following:
22 = 2 x 2
23 = 2 x 2 x 2
24 = 2 x 2 x 2 x 2
25 = 2 x 2 x 2 x 2 x 2
And so on. It is instructive to notice that as the power of 2 goes up, an extra "x 2" is added to the right-hand side. Clearly to advance down this chart means to multiply by 2. Students can then develop some mathematical inquiry by being guided to ask the question "What happens if we go backward?"
It is easy to see that if going down the chart means multiply by 2, then going up the chart means divide by 2. Once this is realized, since 4 / 2 = 2, 21 simply must be equaled to 2. There is no other possibility, nothing to get it confused with. In context, this is not a rule, but another inevitable fact about exponents. 20 similarly must be equaled to 1, since 2 / 2 = 1. Negative exponents then clearly yield fractions, as we divide 1 by 2, and so on. If we keep dividing by 2, we clearly get the reciprocals we would expect.
It is extraordinarily important that exponents be taught in this way. Students will see that there is a rule to understand, and that algebra is a rule-governed system, but it is important that they then discover for themselves all the inherent necessary formulas which inherit directly from that initial definition. This is not only important to nurture early-developed mathematical inquiry and curiosity, but it will also instill these algebraic notions into the minds of students for years to come, making the furthering of their mathematics education possible.
It is extraordinarily important that exponents be taught in this way. Students will see that there is a rule to understand, and that algebra is a rule-governed system, but it is important that they then discover for themselves all the inherent necessary formulas which inherit directly from that initial definition. This is not only important to nurture early-developed mathematical inquiry and curiosity, but it will also instill these algebraic notions into the minds of students for years to come, making the furthering of their mathematics education possible.