Have you ever been reading a book where things are getting tense and you wonder how the hero is going to get out of the crisis he finds himself in, only to be surprised by some solution that seems to come out of nowhere? This is a writing device called, deus ex machina (god from the machine), and I find it to be a very dissatisfying way to address plot problems. Unfortunately, this same method is used frequently in math lessons.
|From Science Cartoons Plus|
I have written before how I like to view teaching as storytelling. What I have not shared is the story I used to tell early on in my career to my middle school classes whenever a student asked, "Why?" It goes something like this:
Do you know what the largest animal on Earth is? <brief pause, but not long enough for anyone to answer> It's the blue whale. The thing is huge! But its throat is less than a foot wide. Do you know why? <another brief pause> Because that's the way it is. And that's why we invert and multiply <or some other mathematical rule>, because that's the way it is.
There you have it - my own version of deus ex machina in my teaching. In retrospect, I am sure it was as dissatisfying to my students as it has been to me in my reading, but I needed to move the plot onward and did not want to get bogged down in details.
I am not proud of this chapter in my teaching history, but like so many others I taught the way I was taught. Consider this a cautionary tale. So the next time a student asks why we subtract one from the exponent in the explicit formula of a geometric sequence but not in the exponential function, do not just respond with, "Because that's the way it is." Help the students to come up with a better reason. If you do, they are more likely to stick with the story of math instead of discarding it because it doesn't make sense.