I taught geometry for teachers course last semester. Towards the end I showed them an animation which depicted a geometry classroom in which students were given a problem to work on, to come up with conjectures and to prove them. The way I went about this was to give my students exactly the same problem before they saw the movie. They came up with conjectures, but didn't have to prove them immediately. Then they saw a movie. I guess I should point out that the students in the movie gave a partial solution, but there were loads of misunderstandings both on the side of students and the teacher. The goals were many, but one of them was for my students to be able to follow what students are saying at a pace at which things happen in reality and to try to determine what is correct, what is incorrect, what they should follow up and what they should leave alone. Things went fairly well, and now I am thinking whether to stick with the same plan or to alter it slightly. One questions is: Is it fair to show them the movie and expect them to be able to follow the action live :) When I saw the movie for the first time, although I have not worked out the problem before hand, I could see where the students were going, and why, and what was causing misunderstandings. Is it fair to put my students into the same position? Should I expect them to be able to do the same? Will the benefit for them be as big as if they had enough time to think about the problem themselves? Why the title above? If teachers have a tendency to assign problems they have not worked out beforehand (I've been guilty of that) then this could potentially show them dangers in doing so. Is that a valuable lesson? On the other hand, if they see the movie first then they loose the perspective of the student trying to solve the problem for themselves. A solution is thrown at them and will they try to come up with something different from what they've already seen? What say you?
P.S. Jonathan, you still didn't tell me why you hate teaching geometry :)
Sunday, March 11, 2007
Friday, March 9, 2007
Back in town
The reason I haven't written in a while should be obvious from the title. Not only did I not write, but I wasn't even following what others wrote (skiing was way too good), so I am trying to catch up. For now, just a link to the third carnival of mathematics. It's not linked here anywhere, so I am doing it now: I loved the second .
Tuesday, February 27, 2007
Mistakes
Algebra class. Teacher writes a list of 5 problems under the heading
Solve and graph each compound inequality
The teachers solves 4 problems, during which time the students are reminded that the compound inequality means that both inequalities need to be satisfied (not in those words), how to graph solutions on the number line, various notation, and so forth. They arrive at problem numbered 4 that reads:
-5 < -5x ≤ -20
While working out the problem the teacher says: "Now listen very carefully. We have to pay attention to our signs". They arrive at the following:
1 > x ≥ 4
and proceed to graph it:
Students make no comment, they copy down the solution and the class proceeds to the next problem. A conversation similar to this happens as the next problem is being solved:
After the class, I brought problem number 4 to the teacher's attention and got the following response:
How and when do we teach our teachers that making a mistake in front of the classroom happens, and is not something we should hide and sweep under the carpet. I remember making mistakes in classes I teach; we all do, sooner or later. I apologize every time I do as if I had wasted their time, and not taught them something of value. We make mistakes, but we need to deal with them in a responsible manner. Go back, fix them, show your students that even when you know something it is not shameful to make a mistake, but it is to hide it. Show your students that we are learning all the time. And that we should not think that we ever learned it all. I think I wrote about this before, but it seems that our teachers think that there is nothing more they need to learn once they have their teaching certificate. They are ready. Are we really?
The teachers solves 4 problems, during which time the students are reminded that the compound inequality means that both inequalities need to be satisfied (not in those words), how to graph solutions on the number line, various notation, and so forth. They arrive at problem numbered 4 that reads:
While working out the problem the teacher says: "Now listen very carefully. We have to pay attention to our signs". They arrive at the following:
and proceed to graph it:
Students make no comment, they copy down the solution and the class proceeds to the next problem. A conversation similar to this happens as the next problem is being solved:
Student: "So, if the bigger one is on the left then they go out, and if it's on the right then they go in?"
Teacher: "That's right"
After the class, I brought problem number 4 to the teacher's attention and got the following response:
Teacher: "I realized it as I was writing it down. I'll fix it by the fifth hour. I didn't want to confuse them."
I, dumbfounded: "?!"
How and when do we teach our teachers that making a mistake in front of the classroom happens, and is not something we should hide and sweep under the carpet. I remember making mistakes in classes I teach; we all do, sooner or later. I apologize every time I do as if I had wasted their time, and not taught them something of value. We make mistakes, but we need to deal with them in a responsible manner. Go back, fix them, show your students that even when you know something it is not shameful to make a mistake, but it is to hide it. Show your students that we are learning all the time. And that we should not think that we ever learned it all. I think I wrote about this before, but it seems that our teachers think that there is nothing more they need to learn once they have their teaching certificate. They are ready. Are we really?
Wednesday, February 21, 2007
Mental?
I am impressed by lots of people who manage to write regularly. I supposed I havne't quite learned how to manage my time. That's why my posts are short. Not that that's a bad thing.
I am still observing. Yesterday and today I saw something that seems recurrent, so I have to comment/ask. In a process of solving a problem my teacher gets to:
She writes
Or, let's have even simpler example:
She proceeds to multiply 6 and 50, and then divide by 3. I saw identical process in the first classroom that I observed. In fact, kids (9 grade, I believe, possibly even 10) there couldn't do things like -4-2 or 7*3 without a calculator (the latter we figured out, and then two minutes later we had 8*3 in a problem and the student couldn't do it even after I reminded her that she knew 7*3). The teacher there said "Their mental math skills are terrible"?! I wouldn't call that mental math. Should I? Regardless of what we may call it though, I was terrified. I still am.
Back to my examples, I can understand using the long process when teaching it, but this was not an introductory lesson. Why do the teachers not use the "shortcuts"? Is it out of some consideration for the students? Do they think the students can't handle it? Or they'd be confused? Or is there some other explanation?
I am still observing. Yesterday and today I saw something that seems recurrent, so I have to comment/ask. In a process of solving a problem my teacher gets to:
 |
She writes
 |
Or, let's have even simpler example:
 |
She proceeds to multiply 6 and 50, and then divide by 3. I saw identical process in the first classroom that I observed. In fact, kids (9 grade, I believe, possibly even 10) there couldn't do things like -4-2 or 7*3 without a calculator (the latter we figured out, and then two minutes later we had 8*3 in a problem and the student couldn't do it even after I reminded her that she knew 7*3). The teacher there said "Their mental math skills are terrible"?! I wouldn't call that mental math. Should I? Regardless of what we may call it though, I was terrified. I still am.
Back to my examples, I can understand using the long process when teaching it, but this was not an introductory lesson. Why do the teachers not use the "shortcuts"? Is it out of some consideration for the students? Do they think the students can't handle it? Or they'd be confused? Or is there some other explanation?
Sunday, February 18, 2007
Mathematics and lyrics
I just saw this on Darren's blog, and just in case there are people here who read my blog, I had to put it here. I would have anyways, because it's awesome. And, yes, I guess I'm a nerd. For this kids out there who may think that's bad: It's not really! Enjoy:
I have to say it all sounds sligthly less impressive when you know that the leader was a math graduate student , who in the mean time finished his PhD, but still. Pretty neat. I have to go continue investigation, and enjoy my nerdom.
Tuesday, February 6, 2007
Geometry teachers out there?
Here is a problem:
Prove that the midpoint quadrilateral (a quadrialteral obtained by connecting consecutive midpoints of sides) of an isosceles trapezoid is a rhombus.
I know of three different proofs. Would love to know if there are more.
Update Here is one remaining proof that I know of. Well, sort of proof :) more of an idea.
Prove that the midpoint quadrilateral (a quadrialteral obtained by connecting consecutive midpoints of sides) of an isosceles trapezoid is a rhombus.
I know of three different proofs. Would love to know if there are more.
Update Here is one remaining proof that I know of. Well, sort of proof :) more of an idea.
Funding for what?
I've been reading a lot about NCLB lately. This morning it was Washington Post's turn to enlighten me further:
This reminded me of somebody's comment about testing agencies and how they might be profiting from NCLB (can't find the actual comment now) that I read while following the discussion on Dan Meyer's blog about NCLB.
The budget would add about $1 billion for the education law, most of it directed toward high schools to help pay for a proposed expansion of testing and other programs.
This reminded me of somebody's comment about testing agencies and how they might be profiting from NCLB (can't find the actual comment now) that I read while following the discussion on Dan Meyer's blog about NCLB.
Subscribe to:
Posts (Atom)
