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.
For example:
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...
Thank you for the great post Justin! I have the same struggles when I give my students similar problems. They seems to slowly lose interest and I helplessly try and bring them all back together, each time less successful.
ReplyDeleteMaybe it is the fact that I am uncomfortable with a classroom that is noisy and energetic. But I believe students learn best in that setting, I just wish their energy was focused more on the math! I continue to pursue my passion in presenting students with a variety of problems that push them to their limits in critical thinking. It sounds like you are doing the same which is inspiring for me.
I am comfortable with a noisy classroom, provided I'm not trying to give instructions or answer questions. Also, I'm not sure they are slowly losing interest as much as refusing to engage and refusing to acknowledge that others might want to.
DeleteIt's a level of self-absorption that I don't know how to break through.
The Geometry kids are getting the critical thinking skills, but the rest? I don't think so. They are lacking some basic skills and when I try to help them develop those, they, rather than blow off the class try to blow it up.
I don't think they are being mean or malicious by wrecking the educational environment. I think they simply can't grasp that other people might want to take advantage of the class. I hear frequently "I'm not talking to them. Don't worry about me" with no idea that screaming across the room isn't just about them.
Yup. My Period Six manifested all these symptoms today. I also have a hard time not blaming myself for not having a perfect lesson. Or not being able to focus on the positives (the students that aren't off-task and behaving unacceptably). But ... keep trying, that's it.
DeleteMark, the problem I'm having is that I think these are really good, engaging lessons, but I can't even get into them. I can't get the hook set because the fish are busy scaring away all of the other fish.
DeleteSometimes I think that these great, project-based lessons are just not going to work for middle school kids. There's just too much other stuff they're tuned in to: the light fixture is blinking, the kid next to them just burped, their sweater is scratchy, the girls they like smiled at them at lunch, the teacher's socks don't match, there's a weird smell in the room, the science teacher said there was homework but they lost their books, their pencil broke and the teacher doesn't like them to get up to sharpen it, their calculator was stolen, they have to use the bathroom, someone said something really funny at lunch, sports try-outs are after school, that quiet kid in the corner just made a face at them........... Blah blah blah blah blah blah. And, oh, someone's trying to tach them something by having them do seemingly random things with numbers. Weird time of life, by the time high school gets underway, they're so much more focused and able to handle inquiry.
ReplyDeleteI think to a certain extent, you're right, but I don't know if that means we have to do plug and chug for all of the middle grades. There is a maturity aspect to it, but I also think there is a cultural aspect and a egocentric world-view that's difficult for students to grasp.
DeleteJustin -- thanks for the shout-out! Love that you used that calculator problem...it stumped a lot of my 8th graders when I gave it to them. A few surprised me with some gems, however, including one who just cut out the 5s putting 0s in their place, then adding 1 over and over again to make up the difference. I thought that was some pretty cool thinking.
ReplyDeleteReally like the Pythagorean triple question. I spent awhile trying to figure out the pattern for the a, n, n+1 Pythagorean triples awhile back. I might steal that for my classroom next year....maybe try and build something in Desmos to make it more concrete/easier to guess?
Low-floor problems are a puzzle. I've really struggled to figure out what will consistently engage my students this year, and the best answer I have right now is anything that looks like a puzzle is decently reliable, and the rest is a crapshoot. That's got to be a big key with low-floor problems. They're accessible, but only if the students try to do them, and there's nothing more frustrating than asking a question you chose so that every student could access and answer it, only to have them not read it or take it seriously and waste the opportunity.
What I forgot to put in was for at least 6 kids, when I said "see if you can do it without the 5" they did 38+43+6. They just pulled the 5s off!
DeleteI don't even know where to go with that!