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.
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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.
I've certainly resorted to this in my own teaching, but I try to come up with an explanation to fill in gaps that students will understand, even if it doesn't tell the whole story.
ReplyDeleteThere are at least a few topics covered in high school where I think the mathematics to prove that what we are teaching is true is actually not really accessible at the high school level.
When I was in high school, I spotted the Cubic Formula and wondered if there was a similar formula for any polynomial equation. My teacher said that such a formula existed for fourth degree polynomials, but not for 5th degree and higher. Instead of trying to justify this statement, she suggested I learn more about it in university and study Galois Theory. I took her advice.
Sometimes, if we aren't able to adequately explain why something is true, we should provide students with some information so they can learn for themselves why it is true.
David,
DeleteI agree. Telling students that "the story will continue" in later courses seems reasonable. And asking them to extend the story themselves is brilliant. Thanks for sharing.
I've caught myself doing that a few times too. I try to give them a "why" though. Sometimes even if I know it's not math they will understand, I'll show them a little bit of what they'll use in the future.
ReplyDeleteFor example, we once spent an entire day on why 0^0 = 1 (and why there is a debate that it could = 0). This was with both my 7th and 8th grade honors classes. It was so much fun! They really got into it.