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Friday, December 10, 2010
Sunday, November 28, 2010
Microteaching Reflections
This microteaching exercise ran into several roadbloacks that I didn't see coming. There was a great deal of miscommunication between the group members and also between us and the students. We were meant to have a bunch of construction paper cut to represent 2x4s in order to demonstrate the construction project. However, due to some miscommuniation between us, the construction paper was provided as full sheets which led the students to believe that it was expected to be treated as a piece of plywood. Therefore, when I asked them to figure out which angle to cut the wood at in order to build the ramp, everyone was confused. Seeing the paper as representing a plywood sheet, they didn't see why any cutting was necessary at all and instead told me the angle it should be laid at.
It comes down to a failure of planning, I suppose. We should have had the correct materials on hand so that I could have performed a visual demonstration. I also should have communicated better with the students to recognize their misconception sooner. Minutes were wasted as we talked past each other because we interpreted the question differently. I finally did manage to relay my intended lesson, but it happened after much back-and-forth and confusion.
It's a lesson in the perils of team teaching, I suppose. I think we each had a different plan in mind for the lesson, and they didn't precisely line up. Vincent's section, Meghan's section, and my section didn't flow well into each other and the prolem with the supplies pushed it over the top. I think this exercise has convinced me that a lesson should have only a single teacher. If two are involved, either one plays a supporting role or the lesson is split into two mini lessons.
I'm still going to hold onto this lesson for my class, though. I like it and I think that if I had a second shot, it would work much better, especially now that I've seen some of the potential hurdles.
It comes down to a failure of planning, I suppose. We should have had the correct materials on hand so that I could have performed a visual demonstration. I also should have communicated better with the students to recognize their misconception sooner. Minutes were wasted as we talked past each other because we interpreted the question differently. I finally did manage to relay my intended lesson, but it happened after much back-and-forth and confusion.
It's a lesson in the perils of team teaching, I suppose. I think we each had a different plan in mind for the lesson, and they didn't precisely line up. Vincent's section, Meghan's section, and my section didn't flow well into each other and the prolem with the supplies pushed it over the top. I think this exercise has convinced me that a lesson should have only a single teacher. If two are involved, either one plays a supporting role or the lesson is split into two mini lessons.
I'm still going to hold onto this lesson for my class, though. I like it and I think that if I had a second shot, it would work much better, especially now that I've seen some of the potential hurdles.
Microteaching Lesson Plan
Group Micro-Teaching: Triangles
Meghan Bentley, Paul Spoering, Vincent Kwong
Grade 10 Apprenticeship and Workplace Mathematics
Geometry:
B4: Demonstrate an understanding of primary trig ratios (sin, cos, tan) by:
* applying similarity to right angle triangles
* generalizing patterns from similar right triangles
* applying the primary trig ratios
Activity: See learning objective
Materials:
* Construction Paper
* Calculators
* Rulers
* Protractors
* Measuring Tape
* Scissors
* Large Paper Cutter
--------------
(30 sec) Bridge: Models are used all the time as a draft of a construction project, and in architecture. By creating a smaller compact replica of your project, you can test out logistics, on cheaper materials. In this lesson we will review the concepts of similar and right triangles, to create a model of a ramp that will sit flush to the floor and wall.
Learning objectives:
Application of properties of right angle triangles, similarity and trig ratios to construct a ramp that is flush to the wall and to the floor.
Teaching Objectives:
Review of concepts of right angle triangles, similarity and trig ratios
Guide students in applying these concepts to a hands-on activity which involves designing, calculating and constructing a ramp with the correct angles to fit flush against the wall and against the floor.
Try to mimic this to make it as real as possible
( 1-2 min) Pre-test:
Who can remember the properties of similar triangles?
Who can remember the properties of right angle triangles?
Who can remember the properties of trig ratios?
(9 min) Participatory Learning:
Measure height of wall and length of floor (Use measuring tape)
Use similar triangles to scale down these measurements for the construction (Use calculator)
Use trig ratios to determine the angles of the ramp. (Use calculator)
Use trig ratios to determine the length of the ramp. (Use calculator)
if we have time, ask them to compare Pythagoras Thm results to this method
Sketch out the ramp on construction paper (Use ruler, protractor, pencil)
Cut out the ramp (Scissors or larger paper cutter if available)
Place against wall and see if angles of ramp are flush with the wall.
( 1-2 min) Post-test
Reiterate the concepts of right angle triangles, similarity and trig ratios
Open discussion about any issues that may have come up (eg ramp angles were not flush)
Ask for other examples where right angle triangles, similarity and trig ratios could be used or would be useful.
(30 sec) Summary
In this lesson we created a model of ramp that needed to be flush to the wall. This activity relied on applying our past knowledge of similar and right triangles.
Meghan Bentley, Paul Spoering, Vincent Kwong
Grade 10 Apprenticeship and Workplace Mathematics
Geometry:
B4: Demonstrate an understanding of primary trig ratios (sin, cos, tan) by:
* applying similarity to right angle triangles
* generalizing patterns from similar right triangles
* applying the primary trig ratios
Activity: See learning objective
Materials:
* Construction Paper
* Calculators
* Rulers
* Protractors
* Measuring Tape
* Scissors
* Large Paper Cutter
--------------
(30 sec) Bridge: Models are used all the time as a draft of a construction project, and in architecture. By creating a smaller compact replica of your project, you can test out logistics, on cheaper materials. In this lesson we will review the concepts of similar and right triangles, to create a model of a ramp that will sit flush to the floor and wall.
Learning objectives:
Application of properties of right angle triangles, similarity and trig ratios to construct a ramp that is flush to the wall and to the floor.
Teaching Objectives:
Review of concepts of right angle triangles, similarity and trig ratios
Guide students in applying these concepts to a hands-on activity which involves designing, calculating and constructing a ramp with the correct angles to fit flush against the wall and against the floor.
Try to mimic this to make it as real as possible
( 1-2 min) Pre-test:
Who can remember the properties of similar triangles?
Who can remember the properties of right angle triangles?
Who can remember the properties of trig ratios?
(9 min) Participatory Learning:
Measure height of wall and length of floor (Use measuring tape)
Use similar triangles to scale down these measurements for the construction (Use calculator)
Use trig ratios to determine the angles of the ramp. (Use calculator)
Use trig ratios to determine the length of the ramp. (Use calculator)
if we have time, ask them to compare Pythagoras Thm results to this method
Sketch out the ramp on construction paper (Use ruler, protractor, pencil)
Cut out the ramp (Scissors or larger paper cutter if available)
Place against wall and see if angles of ramp are flush with the wall.
( 1-2 min) Post-test
Reiterate the concepts of right angle triangles, similarity and trig ratios
Open discussion about any issues that may have come up (eg ramp angles were not flush)
Ask for other examples where right angle triangles, similarity and trig ratios could be used or would be useful.
(30 sec) Summary
In this lesson we created a model of ramp that needed to be flush to the wall. This activity relied on applying our past knowledge of similar and right triangles.
Friday, November 12, 2010
Commentary on Adaptive Expertise
I like the story about Ferit's strategy. I think this is an interesting problem that extends far beyond mathematics. We all have our preferred way of doing things, and tend to stick with them longer than we should.
This happens everywhere, not just in math. For example, in rugby the need occasionally arises to kick the ball through the goalposts. We all have a slightly different style for doing this, and most of our approaches are not ideal. The goal is to line up your kick, then take a series of practiced steps to place yourself at such a distance from the ball that you can perform the same kicking motion every time. It's the same thing that American football kickers do. But many of us take such unusual steps. Instead of the simplest "2 back, 2 sideways" that the NFL kickers perform, many of us have such strange sequences. I've seen people move 4 back, 2 forward, 3 sideways...in such a convoluted pattern. This is surely not the ideal strategy, but it works for them!
And this is really the crux of the matter. Most of us don't do things in the simplest and most reliable way. But we do things the way we're used to. We do things the way we've practiced them, and after so much practice, doing it the hard way is more reliable for us than doing it the simplest way!
This raises an interesting question. Once a person has a considerable amount of practice in doing something the hard way, is it still worthwhile to teach him a new and simpler way? At what point do the gains of learning a simpler method no longer outweigh the advantages of being highly practiced?
This happens everywhere, not just in math. For example, in rugby the need occasionally arises to kick the ball through the goalposts. We all have a slightly different style for doing this, and most of our approaches are not ideal. The goal is to line up your kick, then take a series of practiced steps to place yourself at such a distance from the ball that you can perform the same kicking motion every time. It's the same thing that American football kickers do. But many of us take such unusual steps. Instead of the simplest "2 back, 2 sideways" that the NFL kickers perform, many of us have such strange sequences. I've seen people move 4 back, 2 forward, 3 sideways...in such a convoluted pattern. This is surely not the ideal strategy, but it works for them!
And this is really the crux of the matter. Most of us don't do things in the simplest and most reliable way. But we do things the way we're used to. We do things the way we've practiced them, and after so much practice, doing it the hard way is more reliable for us than doing it the simplest way!
This raises an interesting question. Once a person has a considerable amount of practice in doing something the hard way, is it still worthwhile to teach him a new and simpler way? At what point do the gains of learning a simpler method no longer outweigh the advantages of being highly practiced?
Word Problem
A student at St. F. X. decided to become his own employer by using his car as a taxi for the summer. It costs the student $693.00 to insure his car for the 4 months of summer. He spends $452.00 per month on gas. If he lives at home and has no other expenses for the 4 months of summer and charges an average of $7.00 per fare, how many fares will he have to get to be able to pay his tuition of $3280.00?
Solution: # of fares = (3280 + 4*452 + 693) / 7
Commentary: This is a problem that actually has a certain practical application. Not because anybody is going to run his own taxi business during the summer (nobody does that.) Not because it's about a university student. But because it forces the student to think about expenses. It's actually good practice in accounting because it makes the student think about whether expenses should be added or subtracted from his tuition total, and why. This is the crux of the problem and I imagine that most wrong answers will be because the student subtracts, rather than adds, the expenses. I actually quite like this word problem for this very reason. Most word problems I think are just arithmetic in code. But this one actually contains an important lesson about income and expenses.
Solution: # of fares = (3280 + 4*452 + 693) / 7
Commentary: This is a problem that actually has a certain practical application. Not because anybody is going to run his own taxi business during the summer (nobody does that.) Not because it's about a university student. But because it forces the student to think about expenses. It's actually good practice in accounting because it makes the student think about whether expenses should be added or subtracted from his tuition total, and why. This is the crux of the problem and I imagine that most wrong answers will be because the student subtracts, rather than adds, the expenses. I actually quite like this word problem for this very reason. Most word problems I think are just arithmetic in code. But this one actually contains an important lesson about income and expenses.
Wednesday, November 10, 2010
Quad Cut Triangles
1) Simplest model. Lines have to touch vertices of original triangle.
2) I can make 2 sides and the middle quadrilateral, but thet the third side is a pentagon
3) I can make all 3 sides quadrilaterals, but then the middle is a triangle
4) How can I divide up the middle so that it is no longer a triangle? Am I allowed to have lines starting from the midpoint of my inner triangle?
5) Assuming my lines don't have to start at the vertices of my inner triangle, this is the simplest solution. If they do, then this is cheating and there's no solution.
Monday, November 1, 2010
Practicum Stories
I did my practicum at the ideal minischool. Har har har, puns!
But this school really does deserve the name and after spending two weeks there, I wish I could work there and send my kids there. It's an amazing school. Not because it has much to offer (it's resources are very limited) but because it operates on the human scale that we all crave.
I don't really have much to write about the first lesson I taught. It was a quick and awkward experience. I learned a few things, of course, and I got comfortable enough to teach another lesson. The first lesson wasn't a disaster, but it wasn't a success. It just was. It deserves no further comment.
But the second lesson was incredible. This is what I want teaching to be like. I got to teach physics to a small class (only 11 students). The lion's share of the class was taken up by a big group activity and the whole class was interested. They got out of their seats to gather closer to the center of the action and spontaneous discussion sprang up about the nature of physics. It was easy, it was natural, it felt like a gathering of philosophers in the park.
Of course, I found ways to make it awkward by having a novice lesson plan and generally being a n00b. There were many things that went poorly and need to be fixed for the next time. But the core of the lesson, the spontaneous emergence of interested discussion and the look in their eyes when they had that "aha!" moment were truly magical and reassured me that I'm doing the right thing with my life.
Then there was Churchill. Compared to Ideal, Churchill is a big noisy impersonal city. The character of the students was noticeably different between the two schools. The group at ideal was very diverse, in ethnicity and character. Many students had their hair dyed, many wore extravagant clothing. There was an incredible diversity of styles and interests, and every student brought something special that made the school a better place.
At Churchill, it was a melting pot. I saw no dyed hair, I saw no extravagant dress. I saw nary a peep of a halloween costume, few and far between. Nearly every student at Ideal wore a costume. The children at Ideal were loud, involved, and passionate. The children at Churchill were quiet and subdued. They were passive receptors, while the children at Ideal were full of life. I have no idea what happens at Churchill to produce such a stifling atmosphere, but it made Ideal all the more vibrant by comparison.
I observed a gym class at Churchill and it was shocking. The students were told to stretch, and they stretched. They were told to warm up, and they warmed up. They were told to sit and wait, and they sat and waited. They were told to take their turn, and they took their turn. Never in my life have I seen such an obedient gym class. They weren't chasing each other in circles, they weren't goofing off, they weren't fighting and hardly even competing! I was not the only person to notice this, another student teacher commented on their unusual passivity. What I saw really worries me. Why are these kids not acting like kids? I saw nothing of the sort at Ideal. At Ideal, each student was energetic and full of life. Perhaps the small environment helps them develop as individuals. I'll be looking into this in my long practicum.
But this school really does deserve the name and after spending two weeks there, I wish I could work there and send my kids there. It's an amazing school. Not because it has much to offer (it's resources are very limited) but because it operates on the human scale that we all crave.
I don't really have much to write about the first lesson I taught. It was a quick and awkward experience. I learned a few things, of course, and I got comfortable enough to teach another lesson. The first lesson wasn't a disaster, but it wasn't a success. It just was. It deserves no further comment.
But the second lesson was incredible. This is what I want teaching to be like. I got to teach physics to a small class (only 11 students). The lion's share of the class was taken up by a big group activity and the whole class was interested. They got out of their seats to gather closer to the center of the action and spontaneous discussion sprang up about the nature of physics. It was easy, it was natural, it felt like a gathering of philosophers in the park.
Of course, I found ways to make it awkward by having a novice lesson plan and generally being a n00b. There were many things that went poorly and need to be fixed for the next time. But the core of the lesson, the spontaneous emergence of interested discussion and the look in their eyes when they had that "aha!" moment were truly magical and reassured me that I'm doing the right thing with my life.
Then there was Churchill. Compared to Ideal, Churchill is a big noisy impersonal city. The character of the students was noticeably different between the two schools. The group at ideal was very diverse, in ethnicity and character. Many students had their hair dyed, many wore extravagant clothing. There was an incredible diversity of styles and interests, and every student brought something special that made the school a better place.
At Churchill, it was a melting pot. I saw no dyed hair, I saw no extravagant dress. I saw nary a peep of a halloween costume, few and far between. Nearly every student at Ideal wore a costume. The children at Ideal were loud, involved, and passionate. The children at Churchill were quiet and subdued. They were passive receptors, while the children at Ideal were full of life. I have no idea what happens at Churchill to produce such a stifling atmosphere, but it made Ideal all the more vibrant by comparison.
I observed a gym class at Churchill and it was shocking. The students were told to stretch, and they stretched. They were told to warm up, and they warmed up. They were told to sit and wait, and they sat and waited. They were told to take their turn, and they took their turn. Never in my life have I seen such an obedient gym class. They weren't chasing each other in circles, they weren't goofing off, they weren't fighting and hardly even competing! I was not the only person to notice this, another student teacher commented on their unusual passivity. What I saw really worries me. Why are these kids not acting like kids? I saw nothing of the sort at Ideal. At Ideal, each student was energetic and full of life. Perhaps the small environment helps them develop as individuals. I'll be looking into this in my long practicum.
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