As I've written this blog through out the semester Carlos' name has certainly appeared more than once. Always involved in our class (although not always positively), Carlos has entertained, impressed and annoyed. Demonstrating that he is without a doubt one of the more intelligent individuals I have ever taught, Carlos will certainly be missed. Tomorrow is his last day with us as his family has moved and he will be heading to new school. In some ways I'm happy as a teacher to not have to deal with his "mischevious" behaviors or constant need for engagement (god forbid you ask him to chill for a couple minutes!), but in others I am sad to see him go. I know he's going to be an amazing student one day and that his maturity will catch up with his intelligence so it's a bit sad to only see the 14 year old Carols and not the Senior in High School dawning a cap and gown Carlos. Either way, he's one of those kids that you just don't forget.
As a farewell to our audience Carlos has written a brief story for us that occured on March 3.
He writes...
It was 3/3/08 and I cut my finger with a razor blade. I went to the hospital and there were some seanyers (seniors) ther for cretites (credits) they have to do. Somehow we started talking about math and I started telling them about quadratic equations. I got a piece of paper and showed them (-b +/- radical (b^2 - 4ac)) / (2a). They were like "is your school rich or something? We don't learn stuff like that at my school." I made a graph and found a, b, and c and showed them how it worked with the equation y = ax^2 + bx + c. They were amazed and wanted to konw how to do it at their school.
Carlos told me this story the day after it happened and I remember just feeling pride. Here's a freshmen who causes our school all sorts of trouble who opened a conversation about quadratic equations in an ER at the hospital with a group of Seniors from other schools. How do I know Carlos is going to make it? I know because of three things...1.) He loves learning new things 2.) He has amazing retention abilities. and 3.) He loves sharing what he knows and showing his thinking.
Good luck Carlos...we know you're going to do great things.
Thursday, March 20, 2008
Tuesday, March 11, 2008
It's March! (3-10-08) An update
I can't believe the school year is 75% done. It seems like just yesterday I was learning the names of my students and all of the sudden I know about their family dynamics, their learning styles, their likes and dislikes and their abilities with mathematics. What happened? As a teacher you are so buried in the day to day survival that you loose sight of the big picture. Like a caveman constantly searching for food its easy to forget that you've got bigger goals than your next meal or in our case, your next lesson plan. What has happened to the "Power of 17?" Did I loose track of the big goals? Let me get you up to speed on what we're working on and then we'll answer the bigger question.
As individuals the students are working hard (mostly). Currently engaged in an extensive project framed around theoretical and experimental probability the students have been using a lot of originality and creativity to complete their tasks. Creating their own game of chance (i.e. I bet I can roll three 7's before you roll a 3,4, and 5) students were then asked to calculate the theoretical probabilities of both positions in order to choose the most favorable stance. (SIDE NOTE: Probability Theory and that entire field of mathematics originated from wealthy noblemen gambling in the 17th century and asking mathematicians like Pierre de Fermat for help - what an opportunity to bring something fun into learning Algebra) Unfortunately, I wasn't sure how to teach the process behind calculating these figures. I had some ideas, but needed some support form others to finalize my thinking. (what a funny notion, a math teacher who doesn't have all the answers)
Posing the problem to a group of colleagues I myself engaged in some pretty rough mathematical arguing. It wasn't quite a Tyson vs. Holyfield rematch, but it certainly had some sweat and a few jabs. After nearly two hours of discourse between the group of math teachers I decided that our findings weren't conclusive enough to say "hey kids, here's the way to do it." Instead, I decided to pose the two perspectives that we narrowed it down to and let my students decide which approach they wanted to use in calculating their probabilities. The exciting part was sharing the work of the math teachers with my students so they could get a sense of how a person trained in working with mathematics approaches a problem. Here were the two schools of thought that came out of the conversations with the math teachers that the student had to choose from (or make up their own).
THE PASCHAL APPROACH
Although each roll of a set of dice is independent a game involving multiple sums that you're trying to obtain has some dependent factors. For example, if I want to roll a 3,4, and 5 then the probability of my second roll depends on which number I get first.
i.e. The 1st roll has a probability of how ever many outcomes there are for 3, 4, and 5 out of the total outcomes possible for rolling a set of dice.
P(3, 4, and 5) = (2 + 3 + 4) / 36 = 9/36
The 2nd roll then depends on what you've already rolled. So if a 4 came up first than you're now looking for a 3 or 5.
P(3 or 5) = (2 + 4) / 36 = 6 / 36
Assuming that we next get the 3 we've been looking for we now have to hope for the 5.
P(5) = 4/36
Using what we know about the probability of multiple events we simply multiply each probability together to get the overall probability of rolling a 3,4, and 5
9/36 * 6/36 * 4/36 = (216)/(36^3) = .0046 = .46 %
Using this school of thought there are a series of scenarios to consider when determining the overall probability. i.e. 3 then 4 then 5 or 4 then 5 then 3 or .... and all of them have to be considered when finalizing your solution. Perhaps finding the average probability of all scenarios is the way to go or maybe just providing the range of probabilities is a better option. Either way, the Paschal method, named after my colleague here at Manual, uses the notion of dependence and scenarios to drive its calculations.
THE JIMMY METHOD
According to Jimmy, an old friend and colleague who is currently back in school studying mathematics education (he's neither old nor is our friendship, but I've worked with him since nearly the beginning of my career so it seems like an old friendship and since he's fairly wise when it comes to math instruction I have to consider him to be "older") the probabilities are not based on multiple scenarios as Pashcal would suggest. Rather, when we look at the probabilities of each scenario the outcome is always the same.
Consider this table.
Ways to win the game, # of ways to roll each number, Prob. of Mult. Events (*)
(3, 4, 5) , (2, 3, 4) , 2*3*4 = 24
(3, 5, 4) , (2, 4, 3) , 2*4*3 = 24
(4, 3, 5) , (3, 2, 4) , 3*2*4 = 24
(4, 5, 3) , (3, 4, 2) , 3*4*2 = 24
(5, 4, 3) , (4, 3, 2) , 4*3*2 = 24
(5, 3, 4) , (4, 3, 2) , 4*3*2 = 24
According to Jimmy's theory the order of the rolls is unimportant since the properties of multiplication will show that there's always the same total number of ways for your favorable outcome to be produced. In the end, Jimmy's thinking states that there are 6 winning outcomes each of which has 24 ways of happening. Therefore, he concludes the probability to be...
P(3, 4, and 5) = (24 * 6)/(36^3) = 144/(36^3) = .0031 = .31 %
The key to student engagement in the project has been its open-endedness. Moving away from simply right and wrong, the project goes on from these calculations to ask student to construct a set of new dice that will favor their position even greater. Whether that means constructing a die that looks like a loaf of bread (Carlos' idea) or simply creating a lopsided pencil, which is really a hexagonal prism (Latha's idea), the students are thinking, creating and problem solving. That's what it's all about. Sometimes I worry so much about kids solidifying procedural knowledge that I forget what being a mathematician or a scientist is all about. Testing ideas, experimenting, drawing conclusions, sharing your thinking and dealing with counter-arguments. This is the stuff!
As we are currently in the midst of CSAP, our state tests, (something I'll discuss in a later blog) I have a chance to stop searching for food and to remember my larger goals; Creating a collaborative learning community (who can think!) Hey, we're engaged in a project that does just that. Although I need to get back to my die hard effort of creating an effective classroom of collaborative learners at the moment I am pleased with the mere fact that we're engaging in real mathematics. Solving problems using multiple pathways and bringing our own thinking and creativity to the table in order to produce something new. Struggling alongside one another, teacher and student, to answer questions and produce work that neither of us know the outcome to. That's collaboration!
As individuals the students are working hard (mostly). Currently engaged in an extensive project framed around theoretical and experimental probability the students have been using a lot of originality and creativity to complete their tasks. Creating their own game of chance (i.e. I bet I can roll three 7's before you roll a 3,4, and 5) students were then asked to calculate the theoretical probabilities of both positions in order to choose the most favorable stance. (SIDE NOTE: Probability Theory and that entire field of mathematics originated from wealthy noblemen gambling in the 17th century and asking mathematicians like Pierre de Fermat for help - what an opportunity to bring something fun into learning Algebra) Unfortunately, I wasn't sure how to teach the process behind calculating these figures. I had some ideas, but needed some support form others to finalize my thinking. (what a funny notion, a math teacher who doesn't have all the answers)
Posing the problem to a group of colleagues I myself engaged in some pretty rough mathematical arguing. It wasn't quite a Tyson vs. Holyfield rematch, but it certainly had some sweat and a few jabs. After nearly two hours of discourse between the group of math teachers I decided that our findings weren't conclusive enough to say "hey kids, here's the way to do it." Instead, I decided to pose the two perspectives that we narrowed it down to and let my students decide which approach they wanted to use in calculating their probabilities. The exciting part was sharing the work of the math teachers with my students so they could get a sense of how a person trained in working with mathematics approaches a problem. Here were the two schools of thought that came out of the conversations with the math teachers that the student had to choose from (or make up their own).
THE PASCHAL APPROACH
Although each roll of a set of dice is independent a game involving multiple sums that you're trying to obtain has some dependent factors. For example, if I want to roll a 3,4, and 5 then the probability of my second roll depends on which number I get first.
i.e. The 1st roll has a probability of how ever many outcomes there are for 3, 4, and 5 out of the total outcomes possible for rolling a set of dice.
P(3, 4, and 5) = (2 + 3 + 4) / 36 = 9/36
The 2nd roll then depends on what you've already rolled. So if a 4 came up first than you're now looking for a 3 or 5.
P(3 or 5) = (2 + 4) / 36 = 6 / 36
Assuming that we next get the 3 we've been looking for we now have to hope for the 5.
P(5) = 4/36
Using what we know about the probability of multiple events we simply multiply each probability together to get the overall probability of rolling a 3,4, and 5
9/36 * 6/36 * 4/36 = (216)/(36^3) = .0046 = .46 %
Using this school of thought there are a series of scenarios to consider when determining the overall probability. i.e. 3 then 4 then 5 or 4 then 5 then 3 or .... and all of them have to be considered when finalizing your solution. Perhaps finding the average probability of all scenarios is the way to go or maybe just providing the range of probabilities is a better option. Either way, the Paschal method, named after my colleague here at Manual, uses the notion of dependence and scenarios to drive its calculations.
THE JIMMY METHOD
According to Jimmy, an old friend and colleague who is currently back in school studying mathematics education (he's neither old nor is our friendship, but I've worked with him since nearly the beginning of my career so it seems like an old friendship and since he's fairly wise when it comes to math instruction I have to consider him to be "older") the probabilities are not based on multiple scenarios as Pashcal would suggest. Rather, when we look at the probabilities of each scenario the outcome is always the same.
Consider this table.
Ways to win the game, # of ways to roll each number, Prob. of Mult. Events (*)
(3, 4, 5) , (2, 3, 4) , 2*3*4 = 24
(3, 5, 4) , (2, 4, 3) , 2*4*3 = 24
(4, 3, 5) , (3, 2, 4) , 3*2*4 = 24
(4, 5, 3) , (3, 4, 2) , 3*4*2 = 24
(5, 4, 3) , (4, 3, 2) , 4*3*2 = 24
(5, 3, 4) , (4, 3, 2) , 4*3*2 = 24
According to Jimmy's theory the order of the rolls is unimportant since the properties of multiplication will show that there's always the same total number of ways for your favorable outcome to be produced. In the end, Jimmy's thinking states that there are 6 winning outcomes each of which has 24 ways of happening. Therefore, he concludes the probability to be...
P(3, 4, and 5) = (24 * 6)/(36^3) = 144/(36^3) = .0031 = .31 %
The key to student engagement in the project has been its open-endedness. Moving away from simply right and wrong, the project goes on from these calculations to ask student to construct a set of new dice that will favor their position even greater. Whether that means constructing a die that looks like a loaf of bread (Carlos' idea) or simply creating a lopsided pencil, which is really a hexagonal prism (Latha's idea), the students are thinking, creating and problem solving. That's what it's all about. Sometimes I worry so much about kids solidifying procedural knowledge that I forget what being a mathematician or a scientist is all about. Testing ideas, experimenting, drawing conclusions, sharing your thinking and dealing with counter-arguments. This is the stuff!
As we are currently in the midst of CSAP, our state tests, (something I'll discuss in a later blog) I have a chance to stop searching for food and to remember my larger goals; Creating a collaborative learning community (who can think!) Hey, we're engaged in a project that does just that. Although I need to get back to my die hard effort of creating an effective classroom of collaborative learners at the moment I am pleased with the mere fact that we're engaging in real mathematics. Solving problems using multiple pathways and bringing our own thinking and creativity to the table in order to produce something new. Struggling alongside one another, teacher and student, to answer questions and produce work that neither of us know the outcome to. That's collaboration!
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