Deeper Arithmetic

I have always believed the mathematical truths that my teachers told me growing up: multiplication is nothing more than repeated addition and exponents are shorthand for repeated multiplication. Of course, I knew there were flaws in this- how does really mean taking the square root of a number? Eventually, I could make sense of this idea and I reasoned through it by thinking of multiplying a number by half of itself would mean that if you multiplied two halves together you should get back the original number, which followed logically.


A bigger question I had was about . Somewhere in high school, I decided that this was a function that I needed to graph and explore. To the right of the y-axis, everything made sense; the graph was a smooth continuous line. To the left, I knew there were places where the function was defined and yet, I saw nothing. It got me thinking: if x was a simplified rational number and the denominator was even, then there would be a hole. If the denominator was odd, the function would be defined. But what about the irrational numbers? Where was their place in this? If all of the defined points were graphed and looked at from afar, would there still be a "smooth" pattern?


Somewhere, this hit a disconnect for me. A process as simple as repeated multiplication seemed to tell me that the graphed solutions should be continuous. Something was wrong with exponentiation that was further illuminated when I later studied Taylor series and was able to produce Euler's Formula: . (Sorry, I can't seem to make the exponent look correct. The exponent should be: .) While this formula blew me away mathematically, what really gave me pause was rewriting it to read: . This meant that e, a positive number, when multiplied by itself some mathematical amount of times would yield a negative number. True, there was an i in the exponent, but I could not reconcile this equation with the idea that exponents were just repeated multiplication.


Keith Devlin explained the idea of how multiplication and exponentiation cannot be simplified forms of other operations in his articles "It Ain't No Repeated Addition" and "It's Still Not Repeated Addition." It is amazing to me that although I had evidence that exponents were not multiplication, I still could not help but teach my students that same fallacy. (Although after my own experiences in high school, I have always had my BC Calculus students explore Euler's Formula to help them start to ponder what exponents really mean.)


Devlin leaves us K-12 math educators to determine how to rectify this situation. Part of me wants to say that working with whole number exponents gives students a chance to explore the basic rules and make sense of them before applying them in abstract situations. Begin by talking about multiplication as repeated addition and addition as counting items in a set. But I know the type of student that arrives in my Algebra I class that still does not know how to add and subtract signed numbers. They have to work to memorize the rules for multiplying and often fail. How are these students being served correctly by being told that math is simple and based on concrete representations when it is not? Not everyone can naturally abstract on their own and reconcile untruths.


I'm not sure I have a good solution to Devlin's challenge yet, although I will try to expand the minds of those students who come to me already believing an oversimplification of an abstract system.