I was working with my students today on squares and square roots. We started the lesson with finding the area and perimeter of different rectangles and squares. Then I gave the students squares with a given area and asked for the side length and perimeter. They all did fine until the they got to the square that had area equal to 110 sq meters. Some said 55 and others said 10 point something....maybe 10.5 they said. We then checked a square that was 10.5 by 10.5 and the area was to big. So others suggested 10.4 and the conversation continued.
I had GSP pulled up on the SMARTboard to see if we could make a square that had area 110 and then measure the side lengths. I was going to have the square ready, but I decided to see if they could create a square as a class. The initial response from the students was to just draw a square. I said how and they said with a ruler and pencil. I told them the program didn't have a ruler that we would have to do something different. One person said, "use that line thing and just do one that looks right." This brought up some debate about whether or not our sides were the same or if we had right angles. We used the measure tool and found that we didn't. We decided to erase everything but the line segment. I gave them a hint and drew a circle using the segment on the screen. From there they were able to create the square. They knew that radius would help them and they knew it had to be perpendicular. The most surprising part was that the students were telling me to make another circle. They wanted another circle that had the same radius with the center at the other end of segment.
After creating the square we measured the side lengths and area. We manipulated the square to make it match the squares in the exploration of area and perimeter.
The most amazing part.........these were Freshman taking Pre-Algebra extended. If you don't know what that means....they are the lowest 30 kids of the freshman class. They were able to figure out GSP and some were saying...."do we get to do this." I was shocked. I usually hear "I'm not doing this, it is stupid."
Thursday, March 5, 2009
Monday, March 2, 2009
Supplementary Angles / Colinear Angles
FYI I could not find a definition for supplementary angles that didn't involve the word "pair" or the phrase "two adjacent angles."
I also could not find collinear angles.
So I guess the proof needs revised.
I can see my blog now, I don't know what was wrong.
I also could not find collinear angles.
So I guess the proof needs revised.
I can see my blog now, I don't know what was wrong.
Testing Blog
My blog is not showing up. If you can see my blog, please let me know because I can't see it.
Algebra/Geometry and Frustrations and GO CUBS
Algebra and Geometry
This is my first year teaching algebra and I use geometry mainly by using area models as examples for distributive property. I used area and perimeter both last week when I was working with quadratics. I was giving students a fixed perimeter for a rectangle and asking them to come up with whole number sides that satisfied the perimeter. Then they were to find the area of each rectangle, make a table, and graph. This was one our our introductory graphs to quadratics. When using geometry ideas to teach algebra, I try to stop and ask students some questions about the vocabulary and explore any questions or misinterpretations.
Frustrations
So our corporation is cutting spots, just like every other corporation. We are thinking about combining our two middle schools which is creating a huge fight. As a result of this combining I may be moved back to the Middle School next year. In fact I may be moving back regardless. I said all summer that I didn't want to go to high school, but now I don't want to go back. I am excited about the prospect of teaching high school geometry next year and was planning to use my masters project. Now I might not even be teaching high school....and that's just the beginning. My wife is an art teacher in the same corporation and her job is on the line and she may be cut or only be half day.
GO CUBS
My CUBS are doing awesome 4-1 so far in spring training. I know spring training doesn't mean anything, but it is looking good.
GO CUBBIES
This is my first year teaching algebra and I use geometry mainly by using area models as examples for distributive property. I used area and perimeter both last week when I was working with quadratics. I was giving students a fixed perimeter for a rectangle and asking them to come up with whole number sides that satisfied the perimeter. Then they were to find the area of each rectangle, make a table, and graph. This was one our our introductory graphs to quadratics. When using geometry ideas to teach algebra, I try to stop and ask students some questions about the vocabulary and explore any questions or misinterpretations.
Frustrations
So our corporation is cutting spots, just like every other corporation. We are thinking about combining our two middle schools which is creating a huge fight. As a result of this combining I may be moved back to the Middle School next year. In fact I may be moving back regardless. I said all summer that I didn't want to go to high school, but now I don't want to go back. I am excited about the prospect of teaching high school geometry next year and was planning to use my masters project. Now I might not even be teaching high school....and that's just the beginning. My wife is an art teacher in the same corporation and her job is on the line and she may be cut or only be half day.
GO CUBS
My CUBS are doing awesome 4-1 so far in spring training. I know spring training doesn't mean anything, but it is looking good.
GO CUBBIES
Friday, February 20, 2009
Slavit (Has he worked in a Middle School???)
I thought this was a well written article and enjoyed following the activity. The students did a nice job of using maniplulatives to prove or disprove ideas of congruence and similarity. However, he starts the article by saying "The development of intuitive understandings of geometry is highly important at the middle school level" and then moves on with the article. As I am reading, I try to picture this taking place in a middle school classroom that I have taught in. I am amazed by the conjectures that his students are making and can't believe how they are exploring the ideas presented. Then at the end the of the article he says "even though these activities were used in a college classroom, they are immediately adaptable for use in a middle school classroom with little or not change." Hmmm...... Really. You think so, have you taught middle school before....Ok enough with that.
As I said earlier, I think the activity is a great idea and if I ever get the chance to teach geometry again I will try to use parts of this activity to discover the ideas behind triangle congruence. I also agree with Slavit's approach to teaching. He believes that our students should be "doing math" not just "hearing or seeing mathematics." This a difficult law to live by though. It takes a lot of preparation and you have to be willing to face defeat. Not all activities work as well as this one. Despite the difficulty of this type of teaching, if we believe it is best for students, we must persevere.
SPRING BREAK
March 23-27.
I am pretty sure that we are planning on going.....that's right no where. Unless I get a big fat stimulus check in the mail......lots of sarcasm.
As I said earlier, I think the activity is a great idea and if I ever get the chance to teach geometry again I will try to use parts of this activity to discover the ideas behind triangle congruence. I also agree with Slavit's approach to teaching. He believes that our students should be "doing math" not just "hearing or seeing mathematics." This a difficult law to live by though. It takes a lot of preparation and you have to be willing to face defeat. Not all activities work as well as this one. Despite the difficulty of this type of teaching, if we believe it is best for students, we must persevere.
SPRING BREAK
March 23-27.
I am pretty sure that we are planning on going.....that's right no where. Unless I get a big fat stimulus check in the mail......lots of sarcasm.
Saturday, February 14, 2009
Learning and Masters Project
These two things seem to be connected. I would really like to learn more about the proof process and how I can use GSP and other activities to help students connect to the ideas. Some of my early thoughts have been about setting up a template for a digital dictionary for geometry definitions. I have thought about what format I would use and what things might be with each word. Ideas....
1.) Picture/Sketch
2.) Definition in own words
3.) Formal Definition
4.) Examples and even counter examples
I also would like to build activities that would allow students to discover some of the definitions themselves. I am a little concerned about how many things I can use because many of the activities are already in the GSP activity book. Another concern that I have is availability of a computer lab and the programs that I would be able to use.
Even if I can't make a digital dictionary I would definitely like to find a unique way for students to keep track of the things they have discovered/learned during the year. I would also like to compile and create activities that engage students in geometric thought.
I have been researching different geometry texts this week and I have been looking at the topics covered in geometry. I have never taught geometry so I have been talking with teachers in my building and in Muncie to see what is covered and how they are currently covering the material.
1.) Picture/Sketch
2.) Definition in own words
3.) Formal Definition
4.) Examples and even counter examples
I also would like to build activities that would allow students to discover some of the definitions themselves. I am a little concerned about how many things I can use because many of the activities are already in the GSP activity book. Another concern that I have is availability of a computer lab and the programs that I would be able to use.
Even if I can't make a digital dictionary I would definitely like to find a unique way for students to keep track of the things they have discovered/learned during the year. I would also like to compile and create activities that engage students in geometric thought.
I have been researching different geometry texts this week and I have been looking at the topics covered in geometry. I have never taught geometry so I have been talking with teachers in my building and in Muncie to see what is covered and how they are currently covering the material.
Friday, February 6, 2009
Memory is the residue of Thought!!!!
Response to the Jane Keiser article and an Article from the American Teacher by Daniel T. Willingham.
It seems that the main point from Jane's article is that students need to produce their own definitions first. The concise definitions that we have for terms may not and in most cases do not register with the students. It is very possible that there are words in the definition itself that they don't understand. However, I think the idea goes beyond just definitions. The key is that the students are working with and idea and making discoveries. They aren't being feed definitions and examples. My experience has been that if students discover something themselves they are much more likely to remember the definition, concept, or procedure in the future.
"What Will Improve a Student's Memory?" In this article by Willingham there is a statement the supports the thought of students struggling with an idea, making their own conjectures, and then revising. His statement has nothing to do with mathematics, but definitely applies.
"..., the first principle for students is that memories are formed as the residue of thought." If are students are just given a definition and are not asked to think about the concept or make their own definition there most likely won't be a memory. If a student creates a definition and then defends the definition he or she has definitely devoted thought to the definition. Then as the created definition receives praise or is revised the student is engaging in thought.
Therefore, (sounds like a proof) it make sense that we give students opportunities to think about mathematics through activities and investigations.
It seems that the main point from Jane's article is that students need to produce their own definitions first. The concise definitions that we have for terms may not and in most cases do not register with the students. It is very possible that there are words in the definition itself that they don't understand. However, I think the idea goes beyond just definitions. The key is that the students are working with and idea and making discoveries. They aren't being feed definitions and examples. My experience has been that if students discover something themselves they are much more likely to remember the definition, concept, or procedure in the future.
"What Will Improve a Student's Memory?" In this article by Willingham there is a statement the supports the thought of students struggling with an idea, making their own conjectures, and then revising. His statement has nothing to do with mathematics, but definitely applies.
"..., the first principle for students is that memories are formed as the residue of thought." If are students are just given a definition and are not asked to think about the concept or make their own definition there most likely won't be a memory. If a student creates a definition and then defends the definition he or she has definitely devoted thought to the definition. Then as the created definition receives praise or is revised the student is engaging in thought.
Therefore, (sounds like a proof) it make sense that we give students opportunities to think about mathematics through activities and investigations.
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