Saturday, October 17, 2009
Reflection on "Division by Zero"
I think it is a good activity to have students free-write about "divide" for four minutes and then have them do the same for the number zero. I think it is a different approach and students appreciate having something different to do in a math class. Usually they are asked to solve problems, find answers, calculate things, fill out worksheets. . . etc. But this activity stands out. It makes the student reflect about their notions of division and of zero and once you explain that the division by zero is not valid in mathematics they will be ready to talk more and understand better this special suituation.
A poem
Divide by zero and see what you get
the smaller the number the greater result
no business dividing by zero
throw out the idea
separate an apple
among all your friends
separate one unit
but among zero?
not cool
not cool
Zero. four minutes worth of free writing
Zero is a cruel number. When you have zero it means you have nothing. When you put zero effort into something it's as if you had done absolutely nothing. Zero can also make something into nothing. If you multiply by zero ANY QUANTITY it will turn into zero. So zero not a good thing. Unless you have something bad. If you have something bad then you want to have zero of it. Or if you have potential to do something bad you need to put zero effort into it and multiply it and then you get zero of that bad thing. Zero is a round number because other numbers throw them zeros around and they roll. What goes in the middle of zero? Nothing. That is a good way to remember its value. Some numbers like to collect their zeros and put them to the right. On the left of the number it's as if they were too late; they don't get to count or value. Zero: a good thing on the right, not good on left.
Divide 4 minutes worth of free writing
Dividing is no good among people. It's OK in mathematics. It means to separate, to do less. But when you divide by a smaller number than one then you might get a bigger number than what you started with, provided it is a number greater than one. One seems to be a very important number when it comes to dividing. If you divide by one it's as if you had done absolutely nothing. When you divide in the real world you usually end up with less. If somebody asked you to divide your sandwich, you might end up with half of it. Dividing in the real world: not so much a good thing, although sharing is important. In Mathematics it is OK, as long as you don't divide by zero.
Wednesday, October 14, 2009
Reflection on the Micro-Lesson (Assignment 2)
One of the things I enjoyed was listening to my fellow students say things like "Oh..." and the way they said it tells me that there is something going on in their heads. Maybe they saw something they had never thought of before, or saw a thing they had seen before but in a different way.
On the other hand I think there were a few things that could have gone better. I think that the delivery of our lesson was not very well organized. I think that there were times when one of us talked too much and when we interrupted each other. That was because we did not plan it thoroughly before. I think that if we had taken more time to plan the lesson and divide more specifically what each one of us was going to do and say, our presentation would have gone much better.
I liked the way we got the students engaged in learning. I think that manipulatives are a great way to give students something to do. The student feel more comfortable doing math, because they can rely on a concrete image to go further in their mathematical development. The manipulatives do not intimidate the student and facilitate the transition from the concrete to the abstract. In this case it helped us go from operations with areas of rectangles to algebraic expressions. Another thing I noticed is that you, as a teacher, have to let the students play for a little while with the manipulatives once you have given them the manipulatives for the first time. If you don't let them play, they will be playing with them when you are trying to explain something. This is why I think that manipulatives take a lot more time than regular lecturing and worksheets, but in the end the benefits are more fruitful.
On the other hand I think there were a few things that could have gone better. I think that the delivery of our lesson was not very well organized. I think that there were times when one of us talked too much and when we interrupted each other. That was because we did not plan it thoroughly before. I think that if we had taken more time to plan the lesson and divide more specifically what each one of us was going to do and say, our presentation would have gone much better.
I liked the way we got the students engaged in learning. I think that manipulatives are a great way to give students something to do. The student feel more comfortable doing math, because they can rely on a concrete image to go further in their mathematical development. The manipulatives do not intimidate the student and facilitate the transition from the concrete to the abstract. In this case it helped us go from operations with areas of rectangles to algebraic expressions. Another thing I noticed is that you, as a teacher, have to let the students play for a little while with the manipulatives once you have given them the manipulatives for the first time. If you don't let them play, they will be playing with them when you are trying to explain something. This is why I think that manipulatives take a lot more time than regular lecturing and worksheets, but in the end the benefits are more fruitful.
Lesson Plan - Factoring
Bridge: By introducing students to the manipulative of algebra tiles in an associative style, we will demonstrate the idea and some simple mechanics behind factoring before actually addressing it in a purely mathematical sense.
Learning Outcomes: Students will gain a hands on association to the mechanics of factoring, which will supply them with an insightful experience to relate to the larger problem of factoring quadratic equations.
Teaching Outcomes: To have the students actively participate in a creative group effort. Expand students experience with mathematics. Learn from the students experiences with the manipulatives.
Pretest: Ask students what kind of simple geometric shapes can be made with the manipulatives. What methods are there to finding the area of those shapes.
Participation: Students will work in groups to form various rectangles with the manipulatives, noting different configurations of their classmates. By introduction the notion of variables as lengths of some manipulatives, students will be asked various ways to find the area of their rectangles.
Post Test: Inquire to students what kind of rectangles were created, and how the areas related to the length and width of the rectangles. Stress the use of symbolic equations in the final steps of their exploration.
Summary: Have students note the findings of their group, the relation between the length and width, and the total area of the rectangle. From this demonstrate that through symbolic equations what they've been doing is factoring simple quadratics.
Learning Outcomes: Students will gain a hands on association to the mechanics of factoring, which will supply them with an insightful experience to relate to the larger problem of factoring quadratic equations.
Teaching Outcomes: To have the students actively participate in a creative group effort. Expand students experience with mathematics. Learn from the students experiences with the manipulatives.
Pretest: Ask students what kind of simple geometric shapes can be made with the manipulatives. What methods are there to finding the area of those shapes.
Participation: Students will work in groups to form various rectangles with the manipulatives, noting different configurations of their classmates. By introduction the notion of variables as lengths of some manipulatives, students will be asked various ways to find the area of their rectangles.
Post Test: Inquire to students what kind of rectangles were created, and how the areas related to the length and width of the rectangles. Stress the use of symbolic equations in the final steps of their exploration.
Summary: Have students note the findings of their group, the relation between the length and width, and the total area of the rectangle. From this demonstrate that through symbolic equations what they've been doing is factoring simple quadratics.
Tuesday, October 13, 2009
Citizenship Education in the Context of School Mathematics
How will I make my class a place where students learn to be a democratic citizen?
I think that, as teachers, we usually lose focus on what our real goal in the classroom is. We are not there to throw knowledge to the students and then assign them grades according to how much they get. Our job is to help them understand the world around them, and how they as students can affect that world. In other words, how they can use what they learn in school to go out and affect their families, their own and other communities, their country and the global community.
In my class I will make sure to have discussions taking place. I think that students need to be able to explain their understandings to the rest of the class so that they learn to make, and at the same time understand, political statements and arguments.
I will also try to incorporate current events into the classroom and show them with a mathematics perspective. For example, we can discuss the current unemployment rate and make arguments on what does this mean and make predictions as to what would happen if it grows or decreases.
I will try to encourage students to build arguments, but in a respectful manner. They need to be able to listen to other students’ opinions and enrich and form one of their own. I will value critical thinking, more so than just getting the "right answer." I will focus on the "how" and the "why." I will emphasize that mathematics is not about totality where an answer is either wrong or right because that would translate into their own minds as if they were able to think right or wrong.
I think that, as teachers, we usually lose focus on what our real goal in the classroom is. We are not there to throw knowledge to the students and then assign them grades according to how much they get. Our job is to help them understand the world around them, and how they as students can affect that world. In other words, how they can use what they learn in school to go out and affect their families, their own and other communities, their country and the global community.
In my class I will make sure to have discussions taking place. I think that students need to be able to explain their understandings to the rest of the class so that they learn to make, and at the same time understand, political statements and arguments.
I will also try to incorporate current events into the classroom and show them with a mathematics perspective. For example, we can discuss the current unemployment rate and make arguments on what does this mean and make predictions as to what would happen if it grows or decreases.
I will try to encourage students to build arguments, but in a respectful manner. They need to be able to listen to other students’ opinions and enrich and form one of their own. I will value critical thinking, more so than just getting the "right answer." I will focus on the "how" and the "why." I will emphasize that mathematics is not about totality where an answer is either wrong or right because that would translate into their own minds as if they were able to think right or wrong.
Thursday, October 8, 2009
The Art of Problem Posing
"What if Not"
I think that the advantage of using the "What if Not" (WIN) strategies is that the teacher can come up with interesting questions. Students are very used to the obvious questions. If the topic being covered is right triangles, then they expect to work around the same type of problems, such as "using the Pythagorean theorem, find the missing lenghts... find the missing angles, find the area..." and they all sound the same, are predictable, and no wonder the students end up being bored. If on the other hand you present the student with a question that seems completely different from what they had heard before you run the risk of having the student engaged into learning.
The way I would use WIN strategy would be when I am coming up with "good" questions when I am lesson planning. A good question has multiple approaches and can lead to rich conversations with many different topics, and at the same time, can serve for the purpose of assessing student progress. I could also use WIN strategies when coming up with a class project. One question might be challenging enough for students to take time, and, why not?, group effort to solve it and then present it to the rest of the class. I think that if I could frame a WIN question with the right real world story behind it, the students might feel motivated to work the problem ahead of them with diligence, enthusiasm, excitement and curiosity as to how the problem is solved.
I think that one weakness of this strategy is that if not thoght thoroughly, when a teacher uses WIN strategies the questions that might arise might not be meaningful or interesting enough. So I think that is why the author even warns the reader on coming up with this questions in a mechanical or automatic way.
I think that the advantage of using the "What if Not" (WIN) strategies is that the teacher can come up with interesting questions. Students are very used to the obvious questions. If the topic being covered is right triangles, then they expect to work around the same type of problems, such as "using the Pythagorean theorem, find the missing lenghts... find the missing angles, find the area..." and they all sound the same, are predictable, and no wonder the students end up being bored. If on the other hand you present the student with a question that seems completely different from what they had heard before you run the risk of having the student engaged into learning.
The way I would use WIN strategy would be when I am coming up with "good" questions when I am lesson planning. A good question has multiple approaches and can lead to rich conversations with many different topics, and at the same time, can serve for the purpose of assessing student progress. I could also use WIN strategies when coming up with a class project. One question might be challenging enough for students to take time, and, why not?, group effort to solve it and then present it to the rest of the class. I think that if I could frame a WIN question with the right real world story behind it, the students might feel motivated to work the problem ahead of them with diligence, enthusiasm, excitement and curiosity as to how the problem is solved.
I think that one weakness of this strategy is that if not thoght thoroughly, when a teacher uses WIN strategies the questions that might arise might not be meaningful or interesting enough. So I think that is why the author even warns the reader on coming up with this questions in a mechanical or automatic way.
Sunday, October 4, 2009
Comments, Questions on "The Art of Problem Posing"
1. Asking students to pose problems has really changed the way I perceive teaching mathematics. It has opened unlimited possibilites for studentsto show their mathematical abilities. I was teaching English in Korea for a few weeks, and teaching English was, at that time, something new for me. My wife is an English teacher and so I borrowed many of her strategies for teaching English. One of those strategies was to ask students to write a "K W L" for each piece of literature as they read it. The "K" stands for Know and so you would write what you already know. The "W" stands for Wonder, and so you would write what you wonder about, and it could be in a form of a question. The "L" stands for what you learned. After asking the students to come up with questions for the "W" column about what they were reading I realized what a powerful tool this is. I was not asking students, on that column, what they knew or prove that you know something, which would have made them feel obligated to bring up something that would be seen as wrong or right. But by coming up with a question, usually about a story we were reading, the student was implicitly showing me that they understood the story.
2. When you ask a student to pose a problem, how do you evaluate him or her? I would think pass or fail. But it is complicated because the teacher is going to go through each problem posed by the kid, think about it and determine whether the student really understood or only paraphrased a problem from the textbook.
3. Are some observations made on a simple statement as "x^2 + y^2 = z^2" more valuable than others?
4. When one comes up with a question, the nature of the question should reflect the level of undarstanding, but it is difficult to judge that.
5. How do the two meanings of undertanding, relational understanding, play into the "Art of Problem Posing"?
6. Is the author aware of the difference between "relational" and "instrumental" understanding?
7. Why does he think that Problem Posing has not taken off in the curriculum and therefore in the majority of the classrooms in North America?
8. Are we going to find great problems in this book?
9. I appreciate the fact that the authors take their time to explain what the book is about, how to read it and who is it directed to (audience).
10. I am really enjoying reading this book, although at some point it can get too hard to understand.
2. When you ask a student to pose a problem, how do you evaluate him or her? I would think pass or fail. But it is complicated because the teacher is going to go through each problem posed by the kid, think about it and determine whether the student really understood or only paraphrased a problem from the textbook.
3. Are some observations made on a simple statement as "x^2 + y^2 = z^2" more valuable than others?
4. When one comes up with a question, the nature of the question should reflect the level of undarstanding, but it is difficult to judge that.
5. How do the two meanings of undertanding, relational understanding, play into the "Art of Problem Posing"?
6. Is the author aware of the difference between "relational" and "instrumental" understanding?
7. Why does he think that Problem Posing has not taken off in the curriculum and therefore in the majority of the classrooms in North America?
8. Are we going to find great problems in this book?
9. I appreciate the fact that the authors take their time to explain what the book is about, how to read it and who is it directed to (audience).
10. I am really enjoying reading this book, although at some point it can get too hard to understand.
Friday, October 2, 2009
Hey you!
Yo,
I did not appreciate being in your class. I was late most of the time and you gave me a total hard time about it. I did not learn nothing. So thanks a lot for wasting my time!
You only cared about math and not about me. I am not good when it comes to math, ok? but I am good for other things and, contrary to what you might think, not everything in life is about math!
I think there was not a single day that I said, "Gees, I'm glad I went into Mr. R's class 'cause I learned something." No. It was always: "What a drag this class is! And what a waste of time!!" 'cause that's exactly what it was.
Anyway, I am out of your class and happy about it. I hope I don't cross paths with you, or better said, if I ever see you on the street, beware! 'cause there's gonna be me, with a bat, and ready to smash your "brilliant" head.
Sincerely,
Me, your EX-student.
I did not appreciate being in your class. I was late most of the time and you gave me a total hard time about it. I did not learn nothing. So thanks a lot for wasting my time!
You only cared about math and not about me. I am not good when it comes to math, ok? but I am good for other things and, contrary to what you might think, not everything in life is about math!
I think there was not a single day that I said, "Gees, I'm glad I went into Mr. R's class 'cause I learned something." No. It was always: "What a drag this class is! And what a waste of time!!" 'cause that's exactly what it was.
Anyway, I am out of your class and happy about it. I hope I don't cross paths with you, or better said, if I ever see you on the street, beware! 'cause there's gonna be me, with a bat, and ready to smash your "brilliant" head.
Sincerely,
Me, your EX-student.
Your student, Bart
Hi,
I was your student about 4 years ago, in 2015. I just want to say that I really enjoyed your class and at the same time I learned a lot. I enjoyed your class because it was really interesting. Even if the topic being covered was not very interesting, you found a way in presenting it that made us want to learn about it and want to learn more about it.
I think I learned a lot because now that I am in college I think I am doing a good job in my calculus classes and I have a good foundation that has supported me.
Anyway, I just wanted to say thanks for your help and hope that you are all right.
Your student,
Bart
I was your student about 4 years ago, in 2015. I just want to say that I really enjoyed your class and at the same time I learned a lot. I enjoyed your class because it was really interesting. Even if the topic being covered was not very interesting, you found a way in presenting it that made us want to learn about it and want to learn more about it.
I think I learned a lot because now that I am in college I think I am doing a good job in my calculus classes and I have a good foundation that has supported me.
Anyway, I just wanted to say thanks for your help and hope that you are all right.
Your student,
Bart
Subscribe to:
Posts (Atom)