Making them think!
I had a very cool lesson today with my sixth graders and and I just had to share. So often we say that our kids don't know how to think on a higher level. Let's share what we do to bring them to that level. Whatever subject you teach, post an idea.
Here's what I did today. It's not much, but the way the kids responded just got me really jazzed.
6th Grade Math:
As the kids came in to class, there were three warm-up questions on the board:
What is the probability of:
1) Tossing heads on a fair coin?
2) Tossing heads twice in a row on a fair coin?
3) Tossing heads three times in a row on a fair coin?
After going over it and discussing concepts like 1/2^3 as the probability for the third question, . I put on a scene from the movie "Rosencrantz and Guildenstern are Dead".
Essentially, the scene begins with Guildenstern finding a coin on the ground. He tosses it and it comes out heads. He does it multiple times and it comes out heads every time. After about 50 sequential heads, the scene continues with this interchange:
GUILDENSTERN: It must be indicative of something other than the redistribution of wealth. He flips a coin to Rosencrantz, who looks at it.
GUILDENSTERN: A weaker man might be moved to reexamine his faith. If for nothing at least in the law of probability. He flips another coin to Rosencrantz.
GUILDENSTERN: Consider. One. Probability is a factor which operates within natural forces. Two. Probability is not operating as a factor. Three. We are now held within un- sub- or super-natural forces. Discuss.
GUILDENSTERN: Look at it this way. If six monkeys … if six monkeys … the law of averages, if I’ve got this right, means that if six monkeys are thrown up in the air long enough, they would land on their tails he throws another coin to Rosencrantz about as often as they would land on their
ROSENCRANTZ (looking at the coin): Heads. Getting a bit of a bore, isn’t it.
GUILDENSTERN: A bore?
ROSENCRANTZ: Well …
GUILDENSTERN: What about the suspense?
ROSENCRANTZ: What suspense?
GUILDENSTERN: It must be the law of diminishing returns. I feel the spell about to be broken. He flips a coin high into the air, catches it, and looks at it. He shakes his head. Well, an even chance.
ROSENCRANTZ: Seventy-Eight in a row. A new record, I imagine.
The scene continues, but you get the gist.
After the end of the scene, I started the discussion.
"Based on today's warm-up, what would be the probability of tossing heads 78 times in a row?"
They came up with 1/(2^78). (Of course, someone asked if that was the same as 78^2, which was a great 2-minute tangent.) We started to show how ridiculously large 2^78 actually was (we made it as far as 2^12 before deciding that it was just a big number) and that 1 over that big number was really small number.
I then asked, "If I've tossed heads 77 times, what's the probability of tossing heads on the next toss?" Some of them felt that it was still 1/(2^78) while some saw that it was an independent event and that the probability would still be 1/2.
Now, I do this activity every year and this was actually the first time that somebody didn't say, "100%" and back it up with an argument about trends and the fact that it must be a bad coin. So...I threw it in there.
Here is what I love about this lesson: They debated. They theorized. They played. It happens this way every year, just on different levels. Today, I had to throw them out of my room at the end of class so that they could get to their next class on time. They discussed it in the halls as they walked to their lockers.
The lesson completely blew their minds. The kids who are the strict "by the book" math students had to listen to the points of view of the global thinkers who bought into the "trend", "chaos", or "diminishing returns" theories that were being thrown around. They had to argue their belief. Sure, some kids just sat back and watched the fireworks, but I didn't see a single kid bored or disinterested. By the way, it's a lot more fun if you play devil's advocate on occasion just to stoke the fire.
At the end of the period, the kids asked me which was correct. I said, "As a math teacher, I have to say 1/2 but in reality, I prefer the chaos/global approach that tails is way overdue."
Do you know that in the past I have had parents call me after that lesson angry at me that I would plant seeds like that? I asked them about the conversation that they had with their child and reveled in every word. It took very little to make them realize that they had a deep, mathematical conversation with their child and that their child was not just thinking in school but continuing that thinking throughout the day.
Anyway, if there is anyone still reading at this point (seriously doubt it), thanks for reading. As I said at the beginning, I was really jazzed about this lesson and just had to get it written. I guess it was therapeutic, if nothing else.