Feedback from peers:
A) Clarity:
1. The structure of the lesson was clear: 4.5
2. Verbal and Visual communication were clear: 4.4
3. The mathematical idea was clear: 4.5
B) Active Learning:
1. I was actively engaged in learning throughout the lesson: 3.8
2. A variety of activities was offered to learners: 3
3. The instructors showed that learns’ active engagement was valued: 4.5
C) Connecting Mathematical ideas:
1. The instructors offered activities that connected to other areas of math: 3.5
2. The lessons connected to other areas of life and culture (history, arts, science, philosophy, etc): 3.5
Things that went well:
- Very clear,
- Methodical discussion
- Good flow
- Good introduction
- Good probing for questions
- Prepared, well organized
- Good example to start with. (Chocolates)
- Good classroom management
- Detailed instructions
- Considered all possible outcomes for the topic
- Great interaction
- Encouraging
- Tried to get everyone involved.
- Good connection with previous math knowledge. (Solving single variable)
These areas need work:
- Time management
- More wait time when questioning for slower thinking students.
- Higher use of terminology
- Could have engaged students to do example on paper as solving the equations.
The overall response from our peers was very constructive. After I reflected a bit, I pretty much had the same comments as my peers on where we could have improved.
One thing I’ve learned through this practice is that having too many teachers teach at the same time is not the greatest idea. It’s like having too many chefs in one kitchen. Although we had a clear plan of how the lesson would go it seemed that time was the only factor that was against us. This was due to a question that I, myself, spent too much time explaining, where as I should have said that Zhi Song’s question would be explained by the examples that Matt wanted to present. I had a split second choice to either answer that question or not to, I chose to answer it, and it took more time than I expected, so it hindered our lesson overall. I wish my co-teachers would have brought my attention to the time and stop me from trying to explain that question.
Another thing that could have helped us was to anticipate how long each part would have taken. I didn’t think that setting up the problem would take that long, but it did. Plus, we should have had the students do the first problem with us. I guess I didn’t think of this, since we had examples for them to work on in the latter half of the lesson.
I wasn’t 100% happy about our lesson, but I guess you can’t always expect things to flow perfectly. But despite it not being up to my standards, we did get some really nice positive feedback from our peers, which reassured us that we are doing at least some things right. :)
Thursday, October 14, 2010
Lesson Plan for System of Linear Equations
For Shannon, Niyaz, and Mathew's lesson on systems of linear equations.
|
| What | How | Who |
| Bridge | There are situations in our daily life where we want to know two (any number) unknown pieces of information at the same time, (I will give you an example) and as long as you can find two (any number) independent relationships between these two (any number) unknowns, you can set up a system of equations and find the exact value of these unknowns. | Explain this, using words, and perhaps giving more examples. | Niyaz |
| Learning Objectives | Students should be able to:
| We’ll mention that we will cover this information and will make sure that they will be able to do this by the end of the lesson. | Niyaz |
| Teaching Objectives | To get students to develop the algorithm for themselves | We’ll give suggestions, but expect them to come up with each step | Shannon |
| Pretest | Asking questions while showing how to go about solving the given example. |
| All of us. |
| Participatory learning |
|
| Do anyone want to print the examples out? If not, we can also write them on the board, I was just thinking of using our time more effectively. |
| Post-Test |
|
| All of us. 2 & 3 will be Matt |
| Summary | Telling the students what they have learned today. |
| Shannon will summarize part way through. Matt again at the end (about the different types of solutions) |
Tuesday, October 12, 2010
Response to Thinking Mathematically: Chapters 2 and 3
I’m not going to lie, my head did hurt after reading these two chapters, not because that it was difficult to read, I just couldn’t give up doing those questions, I had to figure out the answers!
I liked how the author named three steps of which we take to solve problems. I thought they were quite realistic. I was glad that there was a full description of what each step encompassed. Not to mention the endless examples of how they are put into practice.
I also liked that the Attack phase was not the end result of the process. It seemed that the entry phase and reflection phase were of more importance. I thought the review phase was of most importance with my experience of doing math problems.
Most of the times students don’t think of each problem as stepping stones for other problems to come in the future, they always just want the answer so that they can move on. The problem is not really them, it’s just that we as teachers or markers put more emphasis on the final answer and don’t necessarily cultivate this attitude of expanding on a problem and changing the givens to create a new problem. I think it’s necessary for students to come up with their own problems after learning a certain concept. This process of creating a problem will give them a chance to see how they can apply the information they have learned.
I liked how the author named three steps of which we take to solve problems. I thought they were quite realistic. I was glad that there was a full description of what each step encompassed. Not to mention the endless examples of how they are put into practice.
I also liked that the Attack phase was not the end result of the process. It seemed that the entry phase and reflection phase were of more importance. I thought the review phase was of most importance with my experience of doing math problems.
Most of the times students don’t think of each problem as stepping stones for other problems to come in the future, they always just want the answer so that they can move on. The problem is not really them, it’s just that we as teachers or markers put more emphasis on the final answer and don’t necessarily cultivate this attitude of expanding on a problem and changing the givens to create a new problem. I think it’s necessary for students to come up with their own problems after learning a certain concept. This process of creating a problem will give them a chance to see how they can apply the information they have learned.
Thursday, October 7, 2010
Poem of Zero and Dividing
To divide or not to divide,
They once asked some time ago.
For those who do not understand,
I’m speaking of the famous number, zero.
Let me show you, my dear,
What they thought of it before you were born,
Take pleasure that we can solve this without saying no.
Many minds had to think,
Many minds had to agree,
Finally, everyone was happy,
Even the number itself, Mr. Zero.
The story goes like this, my son.
They asked him, “Why don’t you want to divide?”
He said, “Number one and negative one are my friends,
But they will never understand why I stand on the fence.”
“They both live on the opposite sides of where I stand,
And I must stand here and never move,
So that chaos won’t haunt
The kingdom of rules.
“Why must you stand there? They asked.
“It’s what my mother and father told me to do.”
“Well, who told them first?” they continued.
“I don’t know.” Said Mr. Zero of age 10.
“You should question your existence until the very end.”
“I do as I’m told unlike the rest,
I must stand here all my life and never rest.”
“I do as I’m told, and that’s all there is to it.
I can be added, subtracted or even multiplied,
But sadly I care not much to be divided by.
I am where numbers come to for help
When they have forgotten where to begin.
If I leave now, then who will take my place?”
“Good point, good point.” They replied.
Everyone was pleased with Mr. Zero’s argument.
After all, he was the wisest of all.
However, since Mr. Eight was witty and smart.
He decide to divide in half from the gut,
To take Mr. Zero’s place on the fence for a whole month,
So that Mr. Zero could feel what it was like,
To be divided by.
That is how it was.
Thanks to Mr. Eight.
We, humans, didn’t work it out for them,
The numbers just fell into place as they always do.
They never fail us.
Remember that my little son.
Numbers are our friends,
Including the wise,
Once undividable,
Mr. Zero.
They once asked some time ago.
For those who do not understand,
I’m speaking of the famous number, zero.
Let me show you, my dear,
What they thought of it before you were born,
Take pleasure that we can solve this without saying no.
Many minds had to think,
Many minds had to agree,
Finally, everyone was happy,
Even the number itself, Mr. Zero.
The story goes like this, my son.
They asked him, “Why don’t you want to divide?”
He said, “Number one and negative one are my friends,
But they will never understand why I stand on the fence.”
“They both live on the opposite sides of where I stand,
And I must stand here and never move,
So that chaos won’t haunt
The kingdom of rules.
“Why must you stand there? They asked.
“It’s what my mother and father told me to do.”
“Well, who told them first?” they continued.
“I don’t know.” Said Mr. Zero of age 10.
“You should question your existence until the very end.”
“I do as I’m told unlike the rest,
I must stand here all my life and never rest.”
“I do as I’m told, and that’s all there is to it.
I can be added, subtracted or even multiplied,
But sadly I care not much to be divided by.
I am where numbers come to for help
When they have forgotten where to begin.
If I leave now, then who will take my place?”
“Good point, good point.” They replied.
Everyone was pleased with Mr. Zero’s argument.
After all, he was the wisest of all.
However, since Mr. Eight was witty and smart.
He decide to divide in half from the gut,
To take Mr. Zero’s place on the fence for a whole month,
So that Mr. Zero could feel what it was like,
To be divided by.
That is how it was.
Thanks to Mr. Eight.
We, humans, didn’t work it out for them,
The numbers just fell into place as they always do.
They never fail us.
Remember that my little son.
Numbers are our friends,
Including the wise,
Once undividable,
Mr. Zero.
Wednesday, October 6, 2010
Timed Writing Two
The word: Zero
Zero is when there is nothing in existence, there is nothing, the null space. It is the absence of anything that may be physical. It is a number. It is more than a number at times, I don’t know when it is more than a number, but I’m sure there is, it probably means that someone doesn’t know anything, it could be the base or standard at where we start to consider a problem, whether it is counting or figuring out a social problem, it’s the point where one begins and possibly ends at times when they are zooming into a certain topic, there are no edges near zero. it is where everything begins.
Zero is when there is nothing in existence, there is nothing, the null space. It is the absence of anything that may be physical. It is a number. It is more than a number at times, I don’t know when it is more than a number, but I’m sure there is, it probably means that someone doesn’t know anything, it could be the base or standard at where we start to consider a problem, whether it is counting or figuring out a social problem, it’s the point where one begins and possibly ends at times when they are zooming into a certain topic, there are no edges near zero. it is where everything begins.
Timed Writing One
The word is: Divide
To split, segregation, multiply,
It’s a form of splitting, you can organize things this way, its used when sharing things and there are just ways of doing this so that people are happy. It gives the opportunity for all to get all that they feel is just. They have a sense of belonging to a certain group and there are rules to a certain group. I have no idea what else to say, there is nothing much going on in my head right now about this word, I’m not sure exactly what I should be thinking, but I guess I have to try, since I don’t want to be divided by the rest of the class and seem like I don’t know anything, this is rather hard, I need to try harder.
To split, segregation, multiply,
It’s a form of splitting, you can organize things this way, its used when sharing things and there are just ways of doing this so that people are happy. It gives the opportunity for all to get all that they feel is just. They have a sense of belonging to a certain group and there are rules to a certain group. I have no idea what else to say, there is nothing much going on in my head right now about this word, I’m not sure exactly what I should be thinking, but I guess I have to try, since I don’t want to be divided by the rest of the class and seem like I don’t know anything, this is rather hard, I need to try harder.
Tuesday, October 5, 2010
Response to: Citizenship Education in the context of school mathematics.
I really enjoyed this article. It provided me with a way of looking at mathematics that I have not even considered before. The article outlined it very concisely (as a mathematician should), what it meant to educate for citizenship.
I must say that my thoughts really resonated with this argument. It is true. What I understood of it is that we must value the logical, understanding and reasoning aspects of mathematics and that is what we should, as future high school teachers, focus on teaching the students.
As I was reading the article, I thought to myself that how can I bring this mentality of teaching for citizenship to life in the classroom. A thought occurred to me, that we should teach elementary proofs to enlighten our students about the deeper implications of what math education is really teaching us. (i.e. reasoning and proving by using logic). By elementary proofs I would encourage some proofs from number theory and geometry to be taught to the students, or at least getting the students engaged into math using these demonstrations. The reason of why I thought proofs might be beneficial is it gives the students an impression of why mathematics is so sure of its results. (A question that I always had in my mind when learning mathematics is, “why does it work?”) Understanding of how one goes about to prove something is quite fascinating to observe and learn.
The process of how one constructs a proof and thinking of all the possible conditions that are relevant is a very similar process of what we normally do when we are thinking of an optimal solution to a given daily problem of ours, or at least how we will get to that optimal solution is very similar in terms of how we think of a proof. In addition, trying to proof something often evolves the opportunity to be creative in your approach. Learning that math is creative and not narrow minded, really opens up ones perspective of what math is really about. I only understood this once I started doing math at the university level. Soon you begin to understand that math is not just about a set of rules that one has to follow, in fact at a higher level of math there are no rigid rules, there is plenty of room for being creative, just like any other subject, like English and Art. For example, you could bring up the many unsolved problems in math or at least why there is no existing proof of why certain statements are true to captivate students and letting them know that there are unanswered questions to be solved in math. I feel if we present this creativity aspect of math to students, then possibly we will be able to grab the attention of those many students who think that math is just a computational process of moving numbers around and beginning to give them a taste of the creative reasoning that is behind the whole field of mathematics.
I feel that some may not feel that proofs might be the best way to achieve the goal of teaching for citizenship, I’m sure there are many other ways of bringing this goal to life in the classroom, it’s just a matter of time for someone to try something innovative. Hopefully sooner than later. :)
I must say that my thoughts really resonated with this argument. It is true. What I understood of it is that we must value the logical, understanding and reasoning aspects of mathematics and that is what we should, as future high school teachers, focus on teaching the students.
As I was reading the article, I thought to myself that how can I bring this mentality of teaching for citizenship to life in the classroom. A thought occurred to me, that we should teach elementary proofs to enlighten our students about the deeper implications of what math education is really teaching us. (i.e. reasoning and proving by using logic). By elementary proofs I would encourage some proofs from number theory and geometry to be taught to the students, or at least getting the students engaged into math using these demonstrations. The reason of why I thought proofs might be beneficial is it gives the students an impression of why mathematics is so sure of its results. (A question that I always had in my mind when learning mathematics is, “why does it work?”) Understanding of how one goes about to prove something is quite fascinating to observe and learn.
The process of how one constructs a proof and thinking of all the possible conditions that are relevant is a very similar process of what we normally do when we are thinking of an optimal solution to a given daily problem of ours, or at least how we will get to that optimal solution is very similar in terms of how we think of a proof. In addition, trying to proof something often evolves the opportunity to be creative in your approach. Learning that math is creative and not narrow minded, really opens up ones perspective of what math is really about. I only understood this once I started doing math at the university level. Soon you begin to understand that math is not just about a set of rules that one has to follow, in fact at a higher level of math there are no rigid rules, there is plenty of room for being creative, just like any other subject, like English and Art. For example, you could bring up the many unsolved problems in math or at least why there is no existing proof of why certain statements are true to captivate students and letting them know that there are unanswered questions to be solved in math. I feel if we present this creativity aspect of math to students, then possibly we will be able to grab the attention of those many students who think that math is just a computational process of moving numbers around and beginning to give them a taste of the creative reasoning that is behind the whole field of mathematics.
I feel that some may not feel that proofs might be the best way to achieve the goal of teaching for citizenship, I’m sure there are many other ways of bringing this goal to life in the classroom, it’s just a matter of time for someone to try something innovative. Hopefully sooner than later. :)
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