*Power and Inverse Operations*

From a memory perspective consider this: If I consider power when thinking about a complex expression my memory load is greatly reduced. The operations are introduced developmentally throughout school. If, each time another set of operations is introduced a big deal is made out of the increased power, memory is barely an issue because this concept will become intuitive through repetition[1]. I learn addition and subtraction when I am very young. They are the least powerful, the least complex operations. They are essentially ‘fast counting by ones’. I then learn division and multiplication as I get older, and understand that these operations are a more sophisticated counting, a skip counting and are therefore ‘fast subtraction’ and ‘fast addition’. They are more powerful than addition and subtraction. Exponents and radicals are even more complex and hold a lot of power as numbers can get very large or very small quickly with these complex operations. Throughout this learning a continued emphasis generally is, and should continue to be, placed on the inverse nature of each power-level of operations. For instance, addition and subtraction: these operations are equally powerful (or in this case, weak!). If I deeply understand the idea that the operations have inverses, I can now approach an expression considering power. This immediately reduces my considerations from six (Exponents, Radicals, Multiplication, Division, Addition, and Subtraction) down to three: strong, medium, weak. When I repeatedly see multiplication and division together as just the inverse of the same medium-powered action, then I am less inclined to think that multiplication has some precedence over division. I see them as equals. As I simplify an expression I will need something to break the tie: Which do I do first? I ‘break the tie’ by addressing these operations as they appear from left to right.

We hope it is becoming clear that the metacognition that is built from classroom conversations rooted in this paradigm are considerably different than the metacognition of a student who is merely pulling up the mnemonic PEMDAS and remembering how to apply it.

[1] Here we treat parentheses simply as the mathematician saying ‘do this first’. There is no greater communication from mathematician to mathematician than an explicit directive such as this! To keep this discussion focused we do not here address that discussion.

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