Rediscovering i

No, this is not going to be a post about me reacquainting myself with my inner child.  This is going to be a post about imaginary numbers!  Yay!

Yesterday, during a conversation with friends, we somehow got onto the topic of i.  I know what you’re thinking, how can you NOT have a conversation with friends and have the topic turn to imaginary numbers?  Am I right?  You remember i, right?  i = \sqrt{-1}?  Of course you do!  We were trying to remember what i^2 equaled.  Having worked with imaginary friends much more recently than I’ve worked with imaginary numbers, I was leaning towards i^2 = 1.  A friend, who happened to be a math major in another life, was leaning towards i^2 = -1.  Neither of us were confident in our answer, though, so I did what any smartphone wielding dork would do and looked it up.  Yeah, i^2 = -1.  Never doubt a math major.  About math.  Doubt them about everything else because they thought it was a good idea to be a math major.

This whole exercise, of course, got me thinking about one of my favorite subjects; how we teach our children.  Given the way we learn about squares and square roots, it would be perfectly natural for someone to think that i^2 = 1.  After all, we have it drilled into our heads that the square of any number results in a positive number.  I had that in mind when I was leaning towards i^2 = 1.  I remembered enough about all the stuff that we were taught were rules that turned out to be more like generalizations to be unsure that I was right, but someone who was never exposed to imaginary numbers would likely encounter a cognitive dissonance that would be difficult to overcome after years and years of believing one thing.

So why don’t we teach children about imaginary numbers immediately?  We don’t need to get into the nitty-gritty of how to figure out that i^k = i^{k\mod{4}}, but it sure would be nice to stick something into the back of those sponge-like minds that there are these things called imaginary numbers that they may run into someday that don’t fit the mold that is being taught.

The big problem, of course, is that we teach children individual things and we require them to understand that one individual thing as quickly as possible.  If you can’t add two numbers together after second grade, you are a failure.  If you can’t multiply two numbers together after third grade, you are a failure.  Our standardized tests show it to be true.  Nonsense, I say.  I cannot count the number of times I attempted to learn some math subject only to have it make complete sense after something in another subject made the pieces all fit together and I would wonder why these discrete pieces weren’t taught together as part of a whole.  We need to stop teaching children answers and start teaching them solutions.

Today, we teach kids that 123 + 321 = 444.  Add the ones and carry the remainder, then add the tens and carry the remainder, then add the hundreds and carry the remainder.  It works, but it’s not getting kids intimately familiar with numbers like they need to be if they want to succeed in more difficult subjects.  Kids need to learn that numbers can be split apart and replaced and moved around using a set of very simple rules.  Yes, 123 + 321 = 444, but it also equals 100 + 20 + 3 + 300 + 20 + 1 = x + 300 + 20 + 1 = x + y = 444.  I don’t think things like this are too difficult for the second grade mind to be able to grasp even if it takes them years more to fully grasp the implications.