You can be disappointed in your students, or you can choose another path
When I first started teaching, I worked really hard to plan lessons, making sure that I was prepared and wasn’t leaving anything out. I knew the steps for all the algorithms, and the math that many of the algorithms were based on.
So I was surprised, and honestly disappointed, when my students didn’t do well. I was disappointed in myself for clearly not being a good teacher, but I also found myself becoming disappointed in my students.
I thought they weren’t working hard enough in class. Or putting effort into their homework. Or studying for tests. Why couldn’t they remember a few simple steps? Why did they apply one set of steps when they should have applied another?
One day we were working on… okay, let’s be real.
One day I was showing them how to divide with decimals. I talked about making an estimate and thinking about where the decimal point should go in the quotient so that it made sense, so my class wasn’t totally devoid of reasoning.
But, as I was preparing to deliver the “move the decimal” rules, a brave student said “Hey, if we added the number of decimal places when we multiplied, can we subtract them when we divide?”
I stopped in my tracks.
This question was so intriguing that I had to take it up. Something made me say, “I have no idea, but I think we should try it.”
We did. It worked when we subtracted the number of decimal places in the divisor from the number of places in the dividend. Mind. Blown.
I was amazed that my students had come up with a procedure I had never heard of, and it worked! What other ideas did they have?
I found out pretty soon when we divided with fractions. We had just finished multiplying fractions, which has the easiest procedure known to man. Just multiply the numerators and multiply the denominators.
By the time we started dividing with fractions I was confident enough to ask “What do you think we should do?”
Of course students suggested “Divide the numerators and the denominators?” Oh, the disappointment. You can’t do that. There’s a rule! A process! It’s complicated!
Thinking back to the day we rewrote the book on dividing with decimals, I thought, “Ok, let’s try it. At least we can find out that it doesn’t work and they’ll be a bit disappointed.”
But… and you’re not going to believe this… it works!
Sometimes it’s not pretty, though. 3/4 divided by 1/2 = 3/2 is pretty cool, but 5/4 divided by 2/3 = 5/2 divided by 4/3 is much less cool.
Well, my students weren’t discouraged. We tackled this crazy looking fraction and found out that it has the same value as 5/4 multiplied by 3/2.
I was amazed, again, for two reasons:
1) My students got the experience of trying out their own idea to see if it worked
2) That it worked!
My students were amazed for two reasons:
1) That we would take class time to mess around with an idea that we weren’t sure would lead anywhere
2) That it worked!
Looking back on that early year of teaching, I realized that rather than be disappointed in myself and my students, I could choose to be amazed by them and their crazy ideas.
This also led to my students and me becoming more amazed by math. We weren’t mathematicians, but we had permission to have ideas and try things. We were still constrained by the fact that there is one quotient for 5/4 divided by 2/3, but that constraint helped us decide if we were onto something or not.
I’m lucky that I get to be continually amazed by student thinking. Some of the thinking reveals new conceptual connections for me and some of the thinking sparks my interest about where a wrong answer came from. I credit this to my first groups of students for speaking up and giving me the opportunity to choose to be amazed by them.