Mathematics is SO SERIOUS, that we all need to chill out a bit, and stop getting so uptight about decimal points and fractions and parallelograms." (Ian Stewart)
I just received a copy of Ian Stewart's "Cows in the Maze, And Other Mathematical Explorations" to review from Michelle Rafferty at Oxford University Press. I haven't read it all yet. I'm saving that for the long trip back to England this Friday, but I did thumb through it and got distracted repeatedly by interesting topics. The book is from Stewart's articles in Scientific American. I will save a fuller review, but will advise at the onset that it looks like a wonderful gift for a math student graduating from high school this month.
I do want to mention a topic(game) from the first chapter, which is on dice and dice games. I read a lot of math books, and yet I had never seen the game mentioned. I am going to slightly alter the game (Me altering Stewart's games is like telling Ichiro to try a different stance at the plate to improve his hitting, but hey, here I go)
So here is the idea, and a challenge. A point target is determined, say 15, or 21, or any number that sounds good to you; and a single die is rolled (it doesn't matter by whom). The number showing is the starting value for the running total. Each player in turn will give the die a quarter rotation in any of the four possible directions (for example, if a 1 is on the face, a quarter roll might bring up a 2,3,4, or 5, but not the six on the bottom).This number is added to the running total. The first person who goes over the set target value is the loser (alternate game, make him the winnner).
And my question is... given that each player plays intelligently, is there an advantage to one or the other player based on the value of the opening roll?
Wednesday, 2 June 2010
Tuesday, 1 June 2010
Given Two Points????
Each year in the spring my pre-calc kids come to the four brief sections in our text that deal with parametric equations and vectors. (I think of this chapter as the catch-all chapter, anything we might have missed that is in the California or Texas Standards).
And then I throw in about three more weeks of work about vectors that I created because I think it is a)beautiful and b) really important. And I share with my students that it seems incredible to me that they, the best and brightest mathematics students in our school, are nearing the end of their high school education (many are seniors) and they can't do one of the most simple acts of coordinate geometry; that is, "Given two points, write the equation of a line containing the two points" (How does the standard for your school read?).
They look at me in wonder, confusion, and perhaps some doubt of my sanity. After all, we have done that thousands of times. I continue to bemoan their lack of ability until finally someone will challenge me...."But, Mr. B, we CAN do that. We do it all the time."
Ok, We'll see, and I turn and write two points on the board... such as (3,1,2) and (2,4,3).
They are so sure of their ability that they already have their pencils to paper when they realize they have no idea how to begin. I let them talk, explore, suggest ideas, and I wait, and I wait.... I have never had a student come up with an equation. Some will suggest it must be something like z=ax+by+c or something..... but NOT ONE ever hit upon a correct equation of the line in question....
Then we talk... Not about how to, that will come later, they will discover it on their own as a natural generalization, but about why not..
I admit to them that most of the students who graduate from high school (and in fact, many of the math teachers they have studied under) can not do this simple act of writing the equation of a line in the three space dimension that they live in. And I tell them that in the following weeks they will learn to do some of the simple geometry they know in the dimension they live in.
I walk them through a simple vector approach to lines in the coordinate plane. We take y= 3x-1 and rewrite it as (x,y)= (0,-1) + t(1,3). Within minutes every kid in the class can write the equation of a line given two points in this vector form although a few struggle with the seeming reversal of order of the "slope"(in truth, several still mess up regularly when they try to use slope intercept, yet they seem reluctant to adopt the seemingly easier point-slope form).
They are quickly taking two points and talking about "point vectors" and "slope vectors" as if they had used them forever. And each year it startles me anew that after a half-hour of an alternate approach, every student will intuitively generalize the method to produce a three-space equation of a line without any help...and then with a little faltering over the "fourth" variable, they can do the same thing in the barely imaginable four-space.
Later we will write the equations of planes in space given three points and do some simple analytic geometry in three space. Many of them struggle with the idea of projections of lines and minor details, but I at least feel like I have made a small step to preparing them to function mathematically in the three-space they live in. And if the string-theory guys are right, and we really have a ten-dimensional universe.... no big deal, they can extend vectors to any dimension.
But I wonder each year... why are we not introducing this more at an early age (alg I?). I will talk later about some of the advantages I see, and maybe you can tell me what I missing that would make it a bad idea.
And then I throw in about three more weeks of work about vectors that I created because I think it is a)beautiful and b) really important. And I share with my students that it seems incredible to me that they, the best and brightest mathematics students in our school, are nearing the end of their high school education (many are seniors) and they can't do one of the most simple acts of coordinate geometry; that is, "Given two points, write the equation of a line containing the two points" (How does the standard for your school read?).
They look at me in wonder, confusion, and perhaps some doubt of my sanity. After all, we have done that thousands of times. I continue to bemoan their lack of ability until finally someone will challenge me...."But, Mr. B, we CAN do that. We do it all the time."
Ok, We'll see, and I turn and write two points on the board... such as (3,1,2) and (2,4,3).
They are so sure of their ability that they already have their pencils to paper when they realize they have no idea how to begin. I let them talk, explore, suggest ideas, and I wait, and I wait.... I have never had a student come up with an equation. Some will suggest it must be something like z=ax+by+c or something..... but NOT ONE ever hit upon a correct equation of the line in question....
Then we talk... Not about how to, that will come later, they will discover it on their own as a natural generalization, but about why not..
I admit to them that most of the students who graduate from high school (and in fact, many of the math teachers they have studied under) can not do this simple act of writing the equation of a line in the three space dimension that they live in. And I tell them that in the following weeks they will learn to do some of the simple geometry they know in the dimension they live in.
I walk them through a simple vector approach to lines in the coordinate plane. We take y= 3x-1 and rewrite it as (x,y)= (0,-1) + t(1,3). Within minutes every kid in the class can write the equation of a line given two points in this vector form although a few struggle with the seeming reversal of order of the "slope"(in truth, several still mess up regularly when they try to use slope intercept, yet they seem reluctant to adopt the seemingly easier point-slope form).
They are quickly taking two points and talking about "point vectors" and "slope vectors" as if they had used them forever. And each year it startles me anew that after a half-hour of an alternate approach, every student will intuitively generalize the method to produce a three-space equation of a line without any help...and then with a little faltering over the "fourth" variable, they can do the same thing in the barely imaginable four-space.
Later we will write the equations of planes in space given three points and do some simple analytic geometry in three space. Many of them struggle with the idea of projections of lines and minor details, but I at least feel like I have made a small step to preparing them to function mathematically in the three-space they live in. And if the string-theory guys are right, and we really have a ten-dimensional universe.... no big deal, they can extend vectors to any dimension.
But I wonder each year... why are we not introducing this more at an early age (alg I?). I will talk later about some of the advantages I see, and maybe you can tell me what I missing that would make it a bad idea.
Wednesday, 26 May 2010
RIP Martin Gardner
More than any classroom teacher I ever had, Martin Gardner shaped my mathematical interests. "For 35 years, he wrote Scientific American's Mathematical Games column, educating and entertaining minds and launching the careers of generations of mathematicians"
I learned that he died on Tuesday. Only two days before, I stood in the front yard of my Mother's home in Fort Worth and told Alex, my sister's grandson, aged 12, that if he wanted to nurture his curiosity for math and science he should find anything in the library by Martin Gardner and read it every year for the next ten years of his life, and each year, I promised, he would find something new in the reading.
I can not do justice to the life of a man who was the mathematical Pied-Piper of mathematics for a generation of us; so here is link to the article in Scientific American.
I learned that he died on Tuesday. Only two days before, I stood in the front yard of my Mother's home in Fort Worth and told Alex, my sister's grandson, aged 12, that if he wanted to nurture his curiosity for math and science he should find anything in the library by Martin Gardner and read it every year for the next ten years of his life, and each year, I promised, he would find something new in the reading.
I can not do justice to the life of a man who was the mathematical Pied-Piper of mathematics for a generation of us; so here is link to the article in Scientific American.
Thursday, 13 May 2010
The rumors of my death have been greatly exaggerated?
After several inquiries, I thought I should admit that I am alive and well, but not finding much time to blog. I am attending to my beautiful sweetheart while she has a minor surgery and along the way trying to teach my two AP classes some extended "after the exam" topics from five time zones away.... go ahead, try to get up and teach at 3am.... I am not at my best at that time...and I really need to have them see my hands wave....
My stats kids are doing some stuff on game theory, and the calc kids are exploring Topology...
Doing all this over a well-secured but not always compliant military connection has required extreme patience on the part of my students, my very capable substitute teacher, and occasionally even I have had to monitor my temper bursts...
I promise to roar back soon and blast away with my trivia/mathematica...
In the meantime, if you haven't seen Dan Meyer's talk on Ted.... check it out...he is one of my favorite bloggers, and the video will show why...
My stats kids are doing some stuff on game theory, and the calc kids are exploring Topology...
Doing all this over a well-secured but not always compliant military connection has required extreme patience on the part of my students, my very capable substitute teacher, and occasionally even I have had to monitor my temper bursts...
I promise to roar back soon and blast away with my trivia/mathematica...
In the meantime, if you haven't seen Dan Meyer's talk on Ted.... check it out...he is one of my favorite bloggers, and the video will show why...
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