Monday, April 7, 2014
Day 134: How Can They Be So Different??
After being energized at #EdCampPGH to continue my work on critical thinking and problem solving skills over content, I had two activities in mind for today. One is something I've been thinking about for a while but haven't had a place to implement and the other was inspired by Dylan Kane's post on low-floor problems.
In geometry, the students were asked to work in groups to find as many Pythagorean Triples as they could. The task was then to see if they could find a pattern for determining other primitive triples. They worked very well through the period and I started noticing a drastic difference between students.
The groups with a better sense of numeracy were able to identify patterns, or spot differences that allowed them to look at the problems differently. Those with less numeracy spent much of their time grinding through the calculations.
Since I didn't want the calculation, but rather the problem solving, I gave them a long list of primitive triples and asked them to come up with more, or at least identify patterns. One students started with this:
If you can't tell what's happening here, she wrote the triples in pink and the difference between the second and third numbers in blue. After looking for a pattern there and finding none, I asked her if it might be beneficial to set them into groups by that difference.
When we came back together as a group, I asked groups to describe the formula they discovered and said they could name them. Almost everyone got a formula for triples with a single difference between sides b and c. ("3, 4, 5", "5, 12, 13", "7, 40, 41" etc.) As a class we decided that if we took any odd number, we could find a Pythagorean triple by squaring that number, dividing the answer by 2, then subtracting .5 to get b and adding .5 to get c.
3^2 = 9, 9/2 = 4.5, 4.5 -.5 = 4 and 4.5 +.5 = 5.
So the Pythagorean Triple is 3, 4, 5. We tested this with any off number that the students wanted and found that it worked.
Fewer groups came up with a way to find triples with a difference between b and c of 2. ("8, 15, 17", "12, 35, 37", "16, 63, 65.") but after we began discussing it (a student led discussion) we agreed on a rule that seemed to work.
I asked them if we could make a general form for it and which numbers it applied to. We decided that it worked for values of a that were divisible by 4 and came up with:
I think that tomorrow, I'm going to have them work on proving that these formula work by plugging them into the Pythagorean theorem.
But I'll make them determine HOW we're going to prove it.
I clued them in at the end of the period that they had been working on both algebra 2 concepts AND number theory.
"HOW CAN YOU NOT BE SUPER EXCITED ABOUT THIS??"
They were, but were too cool to show it. In their hearts, however, I know they were bouncing around the room with the beauty of mathematical discovery.
Pre-algebra didn't go as well. The lesson, I thought, was a pretty darn good one. The task set before them was:
Describe (as in write down) how you would calculate 358 + 453 + 556 if the "5" button on your calculator was broken.
As I expected, several groups just did the column addition by hand. When they did, I said "That's great! Now what if you couldn't use a 5 at all? How would you do it?"
We had a discussion about what the number 5 means and represents. If it means you have "5 things" then how else could you say that without using the word or number "5"?
The talk went into various ways to represent different kinds of numbers and I asked them to do a few basic problem in their heads, then describe what they did. For the most part, I got answers like I did last week.
Me: What do you get when you add 7 and 6?
S: **without hesitation** 13.
Me: Great! How did you do that so quickly?
S: I took 3 from 6 to make 7 into 10. Then I had 3 left over so I went up another 3 to 13.
THIS is the kind of numeracy that speaks to fluency. THIS is what we destroy with the column method, not because it's wrong, but because it uses tricks that makes students think that their brains are wrong.
I fear, however, that the entire lesson was lost because of lack of consistent attention. This is about as far as we got because I gave up.
I don't know how to ask students to stop talking, singing, yelling, dancing, etc. without them taking it personally and claiming that I'm attacking them.
I don't know how to say "Could you please stop talking over me?" and have that be the end of the interaction.
I don't know how to adequately convey that having a conversation about shoes while I'm trying to teach is rude and unacceptable. When I speak to individual students, the response I get, consistently, is "Other people were louder."
The conversation goes as follows (and went exactly this way with the 5 students I spoke with today):
Me: I'm having a lot of difficulty trying to conduct my class with the volume that's coming from this area of the room.
S: I was paying attention! I was answering questions!
Me: I know you were, and I very much appreciate that. I'm not worried about you paying attention because I know you are. When I ask you questions, you're on the ball! I love it! What I'm having an issue with is the volume when you're talking to (other student).
S: But other people are louder than me.
Me: That may be true and I'll speak to them, but you're in charge of you. I need you to be more aware of your volume and more considerate to the rest of the class.
S: But I'm paying attention!
Me: I know. I said that. That's not the problem. The volume is the problem.
S: But if other people are louder, why are you talking to me?
Me: Because you control you. I will speak to them, just as I am speaking to you.
S: But I'm paying attention!
And so on, ad infinitum...
I don't have any idea how to end this cycle. I don't know how to convince some of my students that they are not alone in the room.
It is worse this year than in previous years and I think that it's because in the past, I was much more strict about the rules. I was very traditional (and angry) and students were afraid of me.
This year, with me trying to be less angry and less fear-inspiring, I didn't take into account that I have no idea how to do class management without it.
Up until Christmas, I think the students were still getting used to a type of classroom that they had not experienced before. Their hesitation to be rude/disruptive was more about trepidation than respect.
I fear that how smoothly my class goes has become dependent on the emotions of teenagers rather than the expectations of the teacher and I don't know how to fix that.
This slide from Michelle King's talk this weekend has me thinking. I know where I want the kids to be, but I don't know how to get them there...