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The move to middle school has really allowed me to consider not only why we do what we do, but why students do what they do. I had one such opportunity the other day during my introduction to quadratics.
I gave my students the “magic number” worksheet where they choose a number and then select integer pairs whose sum is the magic number and place them in a table. The third column of the table is the product of the integers.
Filling out the table took all of 30 seconds because my students are very good at following directions.
Plotting the relationship between columns 1 & 3 went very smoothly as well. So smoothly that students were very nonchcalont about the symmetry of the resulting parabolas.
Things got interesting when I asked them to give me an equation that represented the relationship between column 1 and column 3. It was interesting to me when I asked them what was happening in column 2.
Most students saw column 2 through the lense of addition. For example, if the magic number was 12 and the number in column 1 was 4, students would mentally figure,
“4 + what = 12?”
The answer’s 8. Obvious. And they got this answer as if it were a thoughtless reflex.
As long as column 1 contained an integer, there was no problem. However, this was no help when it came to generalizing an equation. As a result, many students struggled with going from the concrete to the abstract.
“What’s happening in column 2?”
“Addition.”
“Look again. Is it really addition?”
“No, subtraction. You’re subtracting in column 2.”
“OK, so what happens if we write column 2 like this…
…and then let column 1 be x?”
“Ooooh, column 2 is 12-x.”
Recognizing what was happening in column 2 made finding the equation simple. The real problem was that there was a disconnect between what they thought they were doing and what they actually were doing.
This led to a nice discussion on what I like to call “taking thoughts captive.” Teachers call it metacognition. Students call it…well, they don’t call it anything, because they don’t usually realize they’re doing it.
Somewhere in there is the reason we get a shoulder shrug and an “I don’t know” when we ask questions.
But once they realize that they ask themselves questions hundreds (maybe thousands) of times each day, it’s impossible for them to not think about it.
About this time last year, I didn’t know much about blogs, Twitter or PLN’s. I did, however, have questions about why we do what we do and how to make it matter. This blog was born because there are a lot of great teachers having great conversations and I thought 14 years of going it alone was enough. It continues to boggle my mind that many of you have a better idea of what takes place in my classroom than many of the teachers in my own school; it matters to you what is happening in classrooms of others. I hope that in the next year, the questions asked here will be more honest and continue to challenge.
I am humbled and honored that Sam and Kate found this blog worthy of a Best New Blog nomination. If you feel so inclined, you may cast your vote here.
Triangle Centers Lab
Published December 10, 2009 Geometry , Lessons , Triangles , Uncategorized 3 CommentsThe other day I made up a triangle centers lab for my 8th graders.
Here is how it went:
Day 0: Homework for tonight is to make a triangle larger than your hand out of some material heavier than paper. Cardstock or cardboard are ideal.
Day 1: Open GeoGebra and get to work. Kids got after it. Some slowed themselves down by not reading directions very well. The nice thing about GeoGebra is that it’s easy to erase.
Day 2: Most finished the lab and went onto extension activity. Those who didn’t finish had a difficult time managing time. They could do the work, but staying focused was the issue.
Extension: Now that you know how to find the circumcenter and incenter, construct an inscribed and circumscribed circle using only compass and straight edge. These students haven’t done anything with a compass, so I offered a 6th point (assignment was worth 5) for those who could figure out how to do the constructions on their own. If they chose to look up the “how to” of constructions, they then would have to prove that the method of angle bisecting works.
David came up with his own extension. He asked, “why does the centroid allow you to balance the triangle?”
“Nice question. Now go away and come back with an answer.” He’s figured out that the three medians divide the triangles into six smaller triangles with equal area and that would account for equal weight distribution from the centroid. He can see it in GeoGebra, but is working on a formal proof. Brandon tried to backdoor me with a proof by contradiction, “well the six triangles have to have equivalent areas because if they weren’t, the large triangle wouldn’t balance.”
Don’t try to beat me at my own game, son.
Chris’ reflection: “Hey Mr. Cox, you just kinda gave us a test without giving us instructions.”
“Yeeeeaah kinda, huh?
For Next Time: Stamp each page after students have demonstrated the correct constructions. Then allow them to go to the next page. Take a little more time discussing the difference between “drawing” and “constructing.”
For the past two years I have had a classroom Voicethread account. It has been really difficult to set up the accounts for the students, though. So much so, that I have put it on the back-burner until today. I saw that I could import multiple accounts as long as I had a Name, UserName, Email and Password in a .csv I could import.
Piece of cake. I set up a Google Form, embedded it on our wiki and had the kids fill it out. Done. Now I’m gonna eat lunch.
Today, my 7th graders were dealing with a problem that eventually they will solve by setting up a system of equations. Right now, they don’t have that in the tool kit, so guess-and-check would be the strategy of choice. Here is the problem and our first step:
One student was able to come up with the equation: 1.50 a + 2.00 d = 360, but another was quick to point out that there would be many different solutions, so we couldn’t use it…yet. So the first guess is 20 and we recognize that the total is too high. Usually that indicates that the first guess is too high, so we normally go with a lower second guess. However, this time, Aaron pointed out that it didn’t make sense to decrease the guess on advanced tickets because they are cheaper. In fact we want to increase that one. This is right about the time I almost lost my poker face:
Aaron: “So if we add to the advanced tickets our total will actually go down.”
“Why?”
“Because for every $1.50 we add, we lose $2.00. So every time we change the tickets by one, we drop $.50. Since we need to drop $30, we need to sell 60 more tickets in advance.”
At this point, I almost blew it. Nearly jumped right in and said something stupid like, “That’s right Aaron. You guys get it?” To which they would have all nodded “uh-huh” and we would have moved on. But I caught myself, gave him the “eh, I am not sure about that” and removed myself from the conversation. So Aaron had to try to re-explain to his classmates what he was saying. I could tell that a couple of them got it, but many were still perplexed. After a couple of minutes I asked Jose if he could explain what Aaron was saying. Jose nailed it and a bunch of kids have an “a-ha” moment. Good stuff.
Later I asked Jose why he didn’t speak up a little sooner. He said that he didn’t figure that he needed to ask any questions because he understood it. I asked for a show of hands on how many understood after Jose’s explanation and that’s when about 15 hands shoot up.
We have been talking a lot in my classes about how important it is to join the conversation. Some kids still think that it’s just about them. They don’t realize that if they offer something to the conversation, not only do they benefit from explaining something they already understand, but there is no telling how many other kids benefit too.
Today, I think Jose gets it.
To tell you the truth, I don’t really have a problem with my state’s math standards (here and here). I do, however, have a serious problem when the standards become the target rather than the scope through which we aim at the target. So what’s the target? What should be the point of math education today? It has become very clear to me that it has never been easier to find correct answers to anything rooted in computation. With WolframAlpha, GeoGebra and all the other resources available, there probably isn’t a question we could ask a student where they couldn’t quickly look up an answer. When I first started teaching, the big question was whether or not we should let our algebra students use a calculator. A lot of the “veteran” teachers were dead set against it because they “gotta add, subtract, multiply and divide, for cryin’ out loud.” But the calculator would allow a student to speed up all the calculations (read arithmetic) and get to the “math.” Well can’t the same be said for WolframAlpha or GeoGebra? Don’t these tools give students access to certain problems where previously they would have been bogged down by calculations they couldn’t complete?
I remember when basic skills were being able to perform the four operations over the Real Number system. What’s a basic skill look like today? Is algebra the arithmetic of the 21st century?
I’ve been getting real tired of having to sift through my students chicken scratch to find the few nuggets of information they have hidden in their “work.” It kind of came to a head the other night as I was grading our most recent geometry test. It became clear that many of these students really have not been taught what it means to show work in a clear and organized manner. I have modeled it many times, pointing out how one should line up the equal signs, how arithemetic isn’t necessarily showing work, etc. However, I think that when I’ve tried to explain this, all they hear is Charlie Brown’s teacher. “Wha wha wha…wha wha whaa.”
I decided to give them a taste of their own medicine. I took the warm up I had planned and dumped it into Wordle (I figured Wordle must be good for something). Yesterday’s warm up:
”But Mr. Cox, what are we supposed to do?”
“The directions are all there. It’s worth 100 pts. you know, so ya better make it snappy.”
“It’s hard to understand.”
“Ok, you don’t like this one; how about this?”
“That’s no better.”
“No?”
They’re on to me by now.
“Yeah, of course I’m proving a point; but what is it?”
The conversation went something like this:
“You think this is confusing? Well that’s what you do to me whenever you show your work. Why is it that in your Language Arts classes you all understand that we start writing in the top left, we work left to right and top down, but in your math class, you seem to think that starting right in the center of your workspace and then going any which way is a good idea? How in the world am I supposed to understand what you’re telling me?”
So I threw up some examples of what to do… 
and what not to do…
and we discussed what made one student’s work acceptable and the other’s unacceptable. We also discussed the difference between “showing your work” or “showing your steps” and that which belongs on scratch paper.
Oh, we also agreed that starting in the upper left hand corner of the workspace is acceptable in a math class as well.
A couple of weeks ago, I posted a question regarding how I should handle my advanced 8th grade class. I got a bunch of great responses which cemented my opinion that these PLN’s are no joke. In a matter of hours, I got a responses from Kate Nowak (telling me to pull my head out and quit encouraging this “jump through hoops” mentality), Darren Kuropatwa (detailing some great extension lessons as well as some online resources), a tweet from Jackie Ballarini (suggesting that I go with an analytical approach to geometry) and an email from Alison Blank(offering a problem based analytical geometry curriculum). By the end of the weekend, I was in conversation with Alison and Jim Wysocki regarding the geometry curriculum.
This problem based geometry is good stuff. It is rigorous, but the skills the kids need are all review. We may only get through 10-12 problems every day or two, but they are really causing them to think. That’s good, right?
Anyway, this allows me to try to implement something I have been working on for a while. Chris Lehmann called it “inverting the classroom.” I like the idea because it allows me to have kids work on the skills review outside of class while we spend the class time discussing the meaty stuff; the problems that make our heads explode only to find out that if I would have just stepped back and taken a different look at the problem, there is a simple yet elegant way to solve it.
I can see two camps evolving in the class. Those who embrace the problem solving and those who feel like they can’t do it. I’ll keep you posted on how it goes. But, so far, so good.
Thanks again.
David Cox
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