Today I read an article in the New York Times concerning education in the USA. As you might expect it does not present a lot of great news. Our local situation is probably better than what is described in this article but we will all have to live with some of these problems if the author is correct.
Here is the link.
Wednesday, April 23, 2008
Monday, April 21, 2008
Earth Day
Significant and Insignificant
April 22nd is Earth Day. Since there is no practical way to determine otherwise, let's call it Earth's birthday. The best scientific estimate for the age of the earth is about 4.55 billion years, plus or minus 70 million years. This estimate was calculated from the work of an Iowan, Clair Patterson, in 1953. See the Bill Bryson book, “A Short History of Nearly Everything” (p.149-160) for more information.
From what I can tell, what we would call human beings (homo sapiens) have existed on the earth for about the last 250 000 years. Of course this is a rough estimate but it gives the opportunity to do some interesting math. Large numbers are not easily understood by anyone. In one of my earliest posts (October 30th, 2007), I mentioned that one million seconds is about 11.5 days while one billion seconds is about 31.7 years.
Using the above numbers, 4.55 billion years for the age of the earth and 250 000 years for the time human beings (homo sapiens) have been on earth, there are a few ways to show the relative insignificance of man's time on earth compared to the time that earth has been in existence. One way to do this is to compress the entire age of earth into one 24 hour day. Based on that, how long have human beings existed on earth?
The easiest way to solve this would be to use a proportion:
4 550 000 000 years : 250 000 years = 24 hours : X hours
Solving for X we get about .001318681319 hours
Multiply that by 60 min./1 hour and by 60 sec./ 1 min. (which are forms of one)
and we get about 4.747252747 seconds.
Therefore in the 24 hour history of earth, human beings have been on the scene about the last five seconds. Most people would find that pretty stunning.
Another interesting thing to do with the information is more appealing to students on a visual level. Assume that the average sized man has a wing span (arms stretched out wide), from the tip of the fingernail on the middle finger of one hand to the tip of the fingernail on the middle finger of the other hand, of six feet or 72 inches. Assume that length of 72 inches is all of earth's history. What part of that length would represent the length of human beings time on earth? Again set up a proportion:
4 550 000 000 years : 250 000 years = 72 in. : X in.
Solving for X we get about .003956043956 inches which is very close to .004 inches, or about 1/250 of an inch. In the Bryson book (p.337), one of the authors he cites says, “... a single stroke with a medium-grained nail file [would] eradicate all of human history.”
That is a pretty good visual for students to see that the 1/250th of an inch scraped off one of those middle finger nails would represent all of human beings' time on earth.
Another visual way to look at that would be to imagine that the 4.55 billion years of earth be represented by a 100 yard football field. What would the 250 000 years of human beings be on that field?
4 550 000 000 years : 250 000 years = 100 yards : X yards
Solving for X we get about .005494505495 yards. Multiply by 36 in./1 yard you get about .197802197 inches or about 1/5th of an inch. Imagine looking at a football field from one goal line to the other goal line and realizing that the time of human beings would only be about 1/5th of an inch. That is pretty stunning too.
The very bright student may realize that some of these numerical answers can be obtained from the other numerical answers. Time could be spent on that for those interested. For example, the football field (100 yards) is fifty times as long as the wing span of 6 feet. Multiply 1/250th of an inch times fifty and you would get 1/5th of an inch.
I love math!
April 22nd is Earth Day. Since there is no practical way to determine otherwise, let's call it Earth's birthday. The best scientific estimate for the age of the earth is about 4.55 billion years, plus or minus 70 million years. This estimate was calculated from the work of an Iowan, Clair Patterson, in 1953. See the Bill Bryson book, “A Short History of Nearly Everything” (p.149-160) for more information.
From what I can tell, what we would call human beings (homo sapiens) have existed on the earth for about the last 250 000 years. Of course this is a rough estimate but it gives the opportunity to do some interesting math. Large numbers are not easily understood by anyone. In one of my earliest posts (October 30th, 2007), I mentioned that one million seconds is about 11.5 days while one billion seconds is about 31.7 years.
Using the above numbers, 4.55 billion years for the age of the earth and 250 000 years for the time human beings (homo sapiens) have been on earth, there are a few ways to show the relative insignificance of man's time on earth compared to the time that earth has been in existence. One way to do this is to compress the entire age of earth into one 24 hour day. Based on that, how long have human beings existed on earth?
The easiest way to solve this would be to use a proportion:
4 550 000 000 years : 250 000 years = 24 hours : X hours
Solving for X we get about .001318681319 hours
Multiply that by 60 min./1 hour and by 60 sec./ 1 min. (which are forms of one)
and we get about 4.747252747 seconds.
Therefore in the 24 hour history of earth, human beings have been on the scene about the last five seconds. Most people would find that pretty stunning.
Another interesting thing to do with the information is more appealing to students on a visual level. Assume that the average sized man has a wing span (arms stretched out wide), from the tip of the fingernail on the middle finger of one hand to the tip of the fingernail on the middle finger of the other hand, of six feet or 72 inches. Assume that length of 72 inches is all of earth's history. What part of that length would represent the length of human beings time on earth? Again set up a proportion:
4 550 000 000 years : 250 000 years = 72 in. : X in.
Solving for X we get about .003956043956 inches which is very close to .004 inches, or about 1/250 of an inch. In the Bryson book (p.337), one of the authors he cites says, “... a single stroke with a medium-grained nail file [would] eradicate all of human history.”
That is a pretty good visual for students to see that the 1/250th of an inch scraped off one of those middle finger nails would represent all of human beings' time on earth.
Another visual way to look at that would be to imagine that the 4.55 billion years of earth be represented by a 100 yard football field. What would the 250 000 years of human beings be on that field?
4 550 000 000 years : 250 000 years = 100 yards : X yards
Solving for X we get about .005494505495 yards. Multiply by 36 in./1 yard you get about .197802197 inches or about 1/5th of an inch. Imagine looking at a football field from one goal line to the other goal line and realizing that the time of human beings would only be about 1/5th of an inch. That is pretty stunning too.
The very bright student may realize that some of these numerical answers can be obtained from the other numerical answers. Time could be spent on that for those interested. For example, the football field (100 yards) is fifty times as long as the wing span of 6 feet. Multiply 1/250th of an inch times fifty and you would get 1/5th of an inch.
I love math!
Wednesday, April 16, 2008
Number Tricks
I have been gone from posting anything for awhile. One of my former students sent me an e-mail saying, "patiently waiting for another blog" so I thought I better get back to posting and that maybe somebody was actually reading them. If you want to comment on a posting directly to my e-mail, send it to me at crxos@yahoo.com.
NUMBER TRICKS
All students enjoy what many people call "Number Tricks." Number tricks are
in two basic types. The first type involves the student starting with their own
chosen number, performing several operations on it, and then ending up with
the same number they started with. The second type involves the student,
again, starting with their own number, performing several operations, but
then everyone in the group ending up with the same number, instead of
their starting number. The students find the first type pretty interesting but
are more fascinated by the second type. At first they are somewhat mystified
that these work. That opens the door for the Algebra teacher to demonstrate
algebraically why they do work and then challenge the students to develop
their own number tricks. I eventually show them why each trick works by showing
three columns that have (1) the direction written out, (2) a numerical example,
and (3) the algebraic expression for that step. I require students to develop an
example of both types of tricks and each trick must have at least six steps.
They show their number tricks just like I show mine to them. The first column is
where they write the direction for that step. In the second column, they write a
numerical example of that step. In the third column they write an algebraic
expression for that step. I encourage them to be as creative as possible. I give
them an example of "dressing up" their directions to allow more creativity. Of
course you cannot do project this until they possess the algebraic skills needed.
This has proven to be an enjoyable project for my students involving creativity
and algebraic knowledge.
(Please excuse the fact that the three columns I wanted to show, separating the direction, the number, and the algebraic expression didn't come out well. The auto editor crams them together so they are a little hard to read. If needed I can e-mail you my document copy which is easier to read.)
Here is an example of the first type of number trick in which everyone ends up
with the same number they started with:
Pick any number: 32, X
Multiply by 20: 640, 20X
Add 1000: 1640, 20X + 1000
Divide by 2: 820, 10X + 500
Subtract 300: 520, 10X + 200
Divide by 5: 104, 2X + 40
Subtract 40: 64, 2X
Divide by 2: 32, X
The teacher should do this trick at least twice (before showing the algebraic expressions)
so the students can see that it works regardless of what number they choose at the beginning.
You might ask them if it matters if they had started with a negative number or a decimal.
If they understand the algebraic expressions they will be able to answer that question.
Here is an example of a number trick in which no matter what number the student chooses
to begin with, they will all end up with the same number - in this case the current year.
Pick any number: 3.14, X
Multiply by 100: 314, 100X
Add 250: 564, 100X + 250
Multiply by 4: 2256, 400X + 1000
Add 1000: 3256, 400X + 2000
Subtract 400 times your
original number: 2000, 2000
Add 8: 2008, 2008
I would have several number tricks of each type ready on that first day in order
to heighten their interest. Make them as fun and interesting as possible. Many times
students would ask how old I was. This was my way of telling them.
An example of "dressing up" the above trick would change the directions to:
Pick any number
Multiply by the number of years in a century
Add the number of pennies in $2.50
Multiply by the number of sides in a quadrilateral.
Add the number of meters in a kilometer.
Subtract twenty squared times your orginal number
Add the value of two to the third power
The steps to any number trick can be "dressed up" in a variety of ways depending
on the age, maturity level, and mathematical knowledge of the class. They could
involve other school subjects also. (Add the number of milliliters on a liter or subtract
the year in which the Declaration of Independence was signed.)
The students may very well surprise you with their creativity in this situation.
NUMBER TRICKS
All students enjoy what many people call "Number Tricks." Number tricks are
in two basic types. The first type involves the student starting with their own
chosen number, performing several operations on it, and then ending up with
the same number they started with. The second type involves the student,
again, starting with their own number, performing several operations, but
then everyone in the group ending up with the same number, instead of
their starting number. The students find the first type pretty interesting but
are more fascinated by the second type. At first they are somewhat mystified
that these work. That opens the door for the Algebra teacher to demonstrate
algebraically why they do work and then challenge the students to develop
their own number tricks. I eventually show them why each trick works by showing
three columns that have (1) the direction written out, (2) a numerical example,
and (3) the algebraic expression for that step. I require students to develop an
example of both types of tricks and each trick must have at least six steps.
They show their number tricks just like I show mine to them. The first column is
where they write the direction for that step. In the second column, they write a
numerical example of that step. In the third column they write an algebraic
expression for that step. I encourage them to be as creative as possible. I give
them an example of "dressing up" their directions to allow more creativity. Of
course you cannot do project this until they possess the algebraic skills needed.
This has proven to be an enjoyable project for my students involving creativity
and algebraic knowledge.
(Please excuse the fact that the three columns I wanted to show, separating the direction, the number, and the algebraic expression didn't come out well. The auto editor crams them together so they are a little hard to read. If needed I can e-mail you my document copy which is easier to read.)
Here is an example of the first type of number trick in which everyone ends up
with the same number they started with:
Pick any number: 32, X
Multiply by 20: 640, 20X
Add 1000: 1640, 20X + 1000
Divide by 2: 820, 10X + 500
Subtract 300: 520, 10X + 200
Divide by 5: 104, 2X + 40
Subtract 40: 64, 2X
Divide by 2: 32, X
The teacher should do this trick at least twice (before showing the algebraic expressions)
so the students can see that it works regardless of what number they choose at the beginning.
You might ask them if it matters if they had started with a negative number or a decimal.
If they understand the algebraic expressions they will be able to answer that question.
Here is an example of a number trick in which no matter what number the student chooses
to begin with, they will all end up with the same number - in this case the current year.
Pick any number: 3.14, X
Multiply by 100: 314, 100X
Add 250: 564, 100X + 250
Multiply by 4: 2256, 400X + 1000
Add 1000: 3256, 400X + 2000
Subtract 400 times your
original number: 2000, 2000
Add 8: 2008, 2008
I would have several number tricks of each type ready on that first day in order
to heighten their interest. Make them as fun and interesting as possible. Many times
students would ask how old I was. This was my way of telling them.
An example of "dressing up" the above trick would change the directions to:
Pick any number
Multiply by the number of years in a century
Add the number of pennies in $2.50
Multiply by the number of sides in a quadrilateral.
Add the number of meters in a kilometer.
Subtract twenty squared times your orginal number
Add the value of two to the third power
The steps to any number trick can be "dressed up" in a variety of ways depending
on the age, maturity level, and mathematical knowledge of the class. They could
involve other school subjects also. (Add the number of milliliters on a liter or subtract
the year in which the Declaration of Independence was signed.)
The students may very well surprise you with their creativity in this situation.
Friday, March 21, 2008
Looking into the Past - Really!
For awhile now I have been reading an absolutely fascinating book. It is, "A Short History of Nearly Everything." It was written by Bill Bryson in 2003. This book is full of extremely interesting information related to science, astronomy, chemistry, biology, geology, anthropology, physics, paleontology, taxonomy, history, mathematics, genetics, and the people who were and are involved in discovering some of the mysteries of the universe. I will eventually post a great deal of information about this book but for now I want to focus on one thing: gamma ray bursts. These are among the biggest and most violent events that take place in the universe.
Today in the New York Times, I read about a gamma ray burst that was briefly visible to the naked eye on Wednesday morning (if you knew where to look). Of course when you look at something like that, you are actually seeing something that happened a long time ago - in this case about seven billion years ago which is about 2.5 billion years before the earth had even formed! Right now the scientific best estimate for the age of the universe is 14 billion years old.
I have been fascinated for a long time by the fact that when we look at the sky, you are seeing light that left the stars, moon, or sun sometime in the past, and sometimes in the ancient, to say the least, past. Actually if you think about the speed of light, whenever we look at anything, we are seeing the light that left that object at some point before we actually experience the sight. In most cases the amount of time is so brief that we simply cannot comprehend it.
However I digress. Here is the link to the Times article on the gamma ray burst.
Here is a link to an opinion article written by Arthur C. Clarke for the Times in 1994. In the article he makes reference to a novel he wrote in 1973, and in a quote from that article, the date of September 11th is used for a monumental event in the history of the planet.
Today in the New York Times, I read about a gamma ray burst that was briefly visible to the naked eye on Wednesday morning (if you knew where to look). Of course when you look at something like that, you are actually seeing something that happened a long time ago - in this case about seven billion years ago which is about 2.5 billion years before the earth had even formed! Right now the scientific best estimate for the age of the universe is 14 billion years old.
I have been fascinated for a long time by the fact that when we look at the sky, you are seeing light that left the stars, moon, or sun sometime in the past, and sometimes in the ancient, to say the least, past. Actually if you think about the speed of light, whenever we look at anything, we are seeing the light that left that object at some point before we actually experience the sight. In most cases the amount of time is so brief that we simply cannot comprehend it.
However I digress. Here is the link to the Times article on the gamma ray burst.
Here is a link to an opinion article written by Arthur C. Clarke for the Times in 1994. In the article he makes reference to a novel he wrote in 1973, and in a quote from that article, the date of September 11th is used for a monumental event in the history of the planet.
Wednesday, March 19, 2008
Arthur C. Clarke has died.
One of the most influential artistic experiences I have ever had in my life was in the spring of 1968. I was in college at UNI at the time. A few college friends and I drove down to Des Moines to see a movie. Yes, there were plenty of movie theaters in the Waterloo-Cedar Falls area so why go to Des Moines? Well the movie we wanted to see was "2001: A Space Odyssey" and it was being shown at the River Hills Theater in Des Moines which, at the time, was the only wide screen movie theater in Iowa and it also had the best (surround) sound system of any movie theater in the state.
I honestly don't remember how I had heard about the movie but probably there had been an article in the newspaper. I do remember hearing enough that I was very curious and felt that it would not be just another movie. I was also looking forward to seeing a movie in the new wide screen (180 degrees) format with the, at the time, new "surround sound."
I convinced some other guys to go down there with me. On the two hour trip to Des Moines I read a very long magazine interview with the director of the movie, Stanley Kubrick. Of course he talked a lot about the making of the movie and I remember the interviewer asking him about the meaning of some of the scenes in the movie. Kubrick's answer was interesting. He basically said to not try to figure it out, but rather just "experience" the movie. In later years Kubrick would say that he himself didn't really know what some of those scenes, and the movie itself, were trying to say. Think about that. Kubrick was the director of the movie and more than any other human being responsible for what people saw and heard on the movie screen. Since then I have read or heard other artists say similar things about their own work. Bob Dylan is a good example. He has often said that his songs "come through him." That implies that they started somewhere else ( location unknowable to the artist), went through him and then emerged into the public consciousness. That makes it a lot easier to understand when an artist claims to not know the meaning of their own work. They aren't trying to be mysterious - they really don't know.
Where does Arthur C. Clarke come into this? He was a science fiction writer among many other things. In 1948 he wrote, "The Sentinel" which in some way was the forerunner to "2001: A Space Odyssey." I won't go into the details but Kubrick and Clarke worked together to bring the movie to the screen. Arthur C. Clarke died on Mar.19, 2008 at the age of 90. He is most remembered for his role in creating the movie but he did many other things. In 1945 he wrote a scientific article that proposed the theory of satellites in geosynchronous orbit around the earth and how valuable that could be. Of course he was way ahead of his time in his thinking. Where would we be today without those communication satellites that almost the entire world relies on these days? This also reminds me of one of Clarke's three famous laws. Law #3 said, "Any sufficiently advanced technology is indistinguishable from magic." I think about that many times when I consider some of the new technology that comes into public usage. How else could you explain using a cell phone to make a call from the middle of a corn field in Iowa to someone in the middle of an oil field in Saudi Arabia? That has to be magic! Here is a link to a Wikipedia article on Clarke if you are curious.
Now back to the trip. We got to the theater and there wasn't a real big crowd. We sat right in the middle of the theater and I was agog at the size of the screen. You had to physically move your head to see from the far left of the screen to the far right. I could see the speakers on the side and behind the audience. To use an expression not known at that time, I was pumped! The lights went down, the sound came up and the movie started. I remember gripping the armrests on each side of me as I saw the first images come on the screen. I felt like I was going to fly off my seat!
I was enthralled the entire time at what I was seeing. The special effects were way ahead of their time ( I didn't see anything comparable until Star Wars in 1978). There were long passages of time in the movie in which there was no dialog, and that was unusual. One scene early in the movie was an encounter between two groups of early humans. The image on the screen was where we saw one of the groups and we knew the other group was behind the camera. We heard that group, behind the camera, from the speakers in the theater behind the audience. That was amazing to me.
The use of classical music was a brilliant touch. With the surround sound and the wide screen it was like the movie had opened its arms and invited the audience into its embrace.
I won't go into detail about the movie. Some people hated it. Some critics hated it. I loved it. Some critics loved it. You might want to do a search for Roger Ebert's review of the movie. The last half hour, in particular the last few minutes, are unforgettable to me. It is pure cinema and I still have no idea what it means (Kubrick always said he didn't know and Clarke never, to my knowledge, gave any explanation) but I "experienced" it and the last scene just gave me such a hopeful feeling about the human race.
I have seen the movie a few times since then on television and on DVD but it isn't even close to the experience I had in 1968 at the River Hills Theater. Watching that movie on a normal TV is like watching a football game on a postage stamp.
I remember walking out of the movie that night thinking I might have experienced something very special in my life. Now, 40 years later I know I did. I have seen many great movies since then, but no other movie has affected me in the way "2001: A Space Odyssey" did. I also remember thinking that one of my goals in life was to see that movie again in the year 2001 at the same theater. I realized I would be very old at that point but I wanted to compare the movie to the reality of 2001. I was so disappointed that Stanley Kubrick died in 1999. Then the River Hills theater was torn down to make way for Wells Fargo Arena. When I went to Wells Fargo Arena in 2006 and 2007 to watch the Spirit Lake girls play basketball in the state tournament, there was a slight tinge of sadness that they were shooting baskets where I had once been sitting and watching and hearing and experiencing my all time favorite movie.
One other thing that relates to parents and teachers. If or when you encounter a young person that is different, and I mean really different, like Arthur C. Clarke or Stanley Kubrick must have been as children, don't dismiss them right away. They can be a nightmare at times because they don't seem to follow the normal rules of life. It's almost like they are aliens from another planet. They might talk about things that seem impossible to a normal way of thinking. Our world badly needs these type of thinkers and dreamers. Some of those dreams may never come to pass, but some of them may end up affecting deeply the people in the world that experience them.
Arthur C. Clarke R.I.P.
I honestly don't remember how I had heard about the movie but probably there had been an article in the newspaper. I do remember hearing enough that I was very curious and felt that it would not be just another movie. I was also looking forward to seeing a movie in the new wide screen (180 degrees) format with the, at the time, new "surround sound."
I convinced some other guys to go down there with me. On the two hour trip to Des Moines I read a very long magazine interview with the director of the movie, Stanley Kubrick. Of course he talked a lot about the making of the movie and I remember the interviewer asking him about the meaning of some of the scenes in the movie. Kubrick's answer was interesting. He basically said to not try to figure it out, but rather just "experience" the movie. In later years Kubrick would say that he himself didn't really know what some of those scenes, and the movie itself, were trying to say. Think about that. Kubrick was the director of the movie and more than any other human being responsible for what people saw and heard on the movie screen. Since then I have read or heard other artists say similar things about their own work. Bob Dylan is a good example. He has often said that his songs "come through him." That implies that they started somewhere else ( location unknowable to the artist), went through him and then emerged into the public consciousness. That makes it a lot easier to understand when an artist claims to not know the meaning of their own work. They aren't trying to be mysterious - they really don't know.
Where does Arthur C. Clarke come into this? He was a science fiction writer among many other things. In 1948 he wrote, "The Sentinel" which in some way was the forerunner to "2001: A Space Odyssey." I won't go into the details but Kubrick and Clarke worked together to bring the movie to the screen. Arthur C. Clarke died on Mar.19, 2008 at the age of 90. He is most remembered for his role in creating the movie but he did many other things. In 1945 he wrote a scientific article that proposed the theory of satellites in geosynchronous orbit around the earth and how valuable that could be. Of course he was way ahead of his time in his thinking. Where would we be today without those communication satellites that almost the entire world relies on these days? This also reminds me of one of Clarke's three famous laws. Law #3 said, "Any sufficiently advanced technology is indistinguishable from magic." I think about that many times when I consider some of the new technology that comes into public usage. How else could you explain using a cell phone to make a call from the middle of a corn field in Iowa to someone in the middle of an oil field in Saudi Arabia? That has to be magic! Here is a link to a Wikipedia article on Clarke if you are curious.
Now back to the trip. We got to the theater and there wasn't a real big crowd. We sat right in the middle of the theater and I was agog at the size of the screen. You had to physically move your head to see from the far left of the screen to the far right. I could see the speakers on the side and behind the audience. To use an expression not known at that time, I was pumped! The lights went down, the sound came up and the movie started. I remember gripping the armrests on each side of me as I saw the first images come on the screen. I felt like I was going to fly off my seat!
I was enthralled the entire time at what I was seeing. The special effects were way ahead of their time ( I didn't see anything comparable until Star Wars in 1978). There were long passages of time in the movie in which there was no dialog, and that was unusual. One scene early in the movie was an encounter between two groups of early humans. The image on the screen was where we saw one of the groups and we knew the other group was behind the camera. We heard that group, behind the camera, from the speakers in the theater behind the audience. That was amazing to me.
The use of classical music was a brilliant touch. With the surround sound and the wide screen it was like the movie had opened its arms and invited the audience into its embrace.
I won't go into detail about the movie. Some people hated it. Some critics hated it. I loved it. Some critics loved it. You might want to do a search for Roger Ebert's review of the movie. The last half hour, in particular the last few minutes, are unforgettable to me. It is pure cinema and I still have no idea what it means (Kubrick always said he didn't know and Clarke never, to my knowledge, gave any explanation) but I "experienced" it and the last scene just gave me such a hopeful feeling about the human race.
I have seen the movie a few times since then on television and on DVD but it isn't even close to the experience I had in 1968 at the River Hills Theater. Watching that movie on a normal TV is like watching a football game on a postage stamp.
I remember walking out of the movie that night thinking I might have experienced something very special in my life. Now, 40 years later I know I did. I have seen many great movies since then, but no other movie has affected me in the way "2001: A Space Odyssey" did. I also remember thinking that one of my goals in life was to see that movie again in the year 2001 at the same theater. I realized I would be very old at that point but I wanted to compare the movie to the reality of 2001. I was so disappointed that Stanley Kubrick died in 1999. Then the River Hills theater was torn down to make way for Wells Fargo Arena. When I went to Wells Fargo Arena in 2006 and 2007 to watch the Spirit Lake girls play basketball in the state tournament, there was a slight tinge of sadness that they were shooting baskets where I had once been sitting and watching and hearing and experiencing my all time favorite movie.
One other thing that relates to parents and teachers. If or when you encounter a young person that is different, and I mean really different, like Arthur C. Clarke or Stanley Kubrick must have been as children, don't dismiss them right away. They can be a nightmare at times because they don't seem to follow the normal rules of life. It's almost like they are aliens from another planet. They might talk about things that seem impossible to a normal way of thinking. Our world badly needs these type of thinkers and dreamers. Some of those dreams may never come to pass, but some of them may end up affecting deeply the people in the world that experience them.
Arthur C. Clarke R.I.P.
Sunday, March 16, 2008
NCAA Tourney and Math
In the past 10-15 years the NCAA Basketball tournament has become a popular item of interest to students, particularly when their favorite team is in the tournament. I found that even kids who weren't normally interested in basketball were talking about it and some of them even would join their classmates (or parents)in picking the outcomes of the games. I wanted to take advantage of this natural interest and come up with some math questions related to the whole thing.
What I ended up doing was pretty simple but turned out to work very well and certainly was interesting to the students. I made a transparency of the full set of brackets with all the teams listed. I showed that to the students and explained the basic idea that a team continued to play as long as they won. (The NCAA actually invites 65 teams into the tournament. Two teams play an early "play-in" game and that narrows the field to 64 teams.) I ask the students to tell me how many total games will then be played, after the "play-in" game, in the tournament to determine a winner. It's the type of question that most, if not all, kids don't already know. Neither do many adults.
When I first thought of this question, I solved it by doing what most people do when they hear this question. I looked at the bracket sheet and counted the first round games -quickly realizing that there had to be 32 games in the first round because there are 64 teams (after the "play-in" game). The next round would be 16 games (because there were 32 winners n the first round), then the third round would have 8 games and the fourth round (regional final) would have four games. Those four games produce the "Final Four." Of course then you have two games in the national semi-finals and then the final game for the championship.
32+16+8+4+2+1 = 63. (I prefer not to count or sometimes even mention the "play-in" game because it isn't usually shown in the brackets that kids see. That way they can count all of the games one by one if they don't see any faster way.) When I first did this question, many years ago and saw the answer "63" I quickly realized that I had not used the most efficient method. When you have 64 teams in a single elimination tournament, 63 teams have to be eliminated in order for the champion to be determined. Each game eliminates one team so 63 games need to be played to eliminate 63 teams. An even better way to look at it is that all teams have to be eliminated except one, so 64-1=63.
Most kids and adults can easily figure out by the original counting method, that the tournament with 64 teams requires 63 games to determine the winner. They usually don't notice or think about the shortcut. Therefore when I present the original question I also present an additional question. Almost every year there are people who say that all of the NCAA (Division 1) teams should participate, regardless of record, just like they do in high school. In a given year I wouldn't always know exactly how many teams were in Division 1 so I would tell the students to assume that there would be 200 teams and to figure out how many games would need to be played. Assuming they didn't know the shortcut mentioned above, they would normally try to solve it just like they had the original question. With 200 teams there would be 100 games, then the second round would have 50 games, the third round 25 games and then they would get stuck. Very few, if any, students or adults, would be able to figure it out from there. What do you do with an odd number of teams?
So the next day when I asked them how they solved the original 64 team question (after they passed their papers in) most would have the correct answer but would not have seen the shortcut. I can't remember any of my 8th graders getting the 200 team question right because they always got stuck with the odd number of teams. At this point, if I had the time, I could suggest two problem solving strategies - solving a simpler problem and finding a pattern along with making a list.
I start with a tournament that has only one team which I know is ridiculous but it serves the strategy. Obviously with only one team, zero games need be played. If there are two teams, you need one game. When you get to three teams, you show them that the only way to do it is to have two teams play and the winner of that game plays the other team for the championship. So three teams requires two games. Four teams (like the "Final Four") needs three games. As this list is being made some kids will see the pattern that every time you add a team you add a game. That is true but if that is all they see then a tournament with 200 teams is going to require making a long list. Eventually somebody usually does see the most efficient method - just subtract one from the number of teams. They can then instantly determine that if the tournament had 200 teams that 199 games would be needed. If the tournament had 35 teams, then 34 games are played.
If you have any other additional time (highly unlikely) you can add in the concept of powers of two. When you have a single elimination tournament, you have to have enough "play-in" games to cut the field down to a number of teams that is a power of two. So a tournament with 35 teams would require three "play-in" games to cut the field to 32 teams. In fact you could ask the students at the beginning why the NCAA field (after the play-in game)is 64 teams. Why not 60 teams or 80 teams or 50 teams?
This was a great problem solving situation that took advantage of the natural interest that students and adults have in the NCAA tournament. One of the things I used to always say to my students is that, "A good mathematician is a lazy one." This particular situation is a perfect example of that.
What I ended up doing was pretty simple but turned out to work very well and certainly was interesting to the students. I made a transparency of the full set of brackets with all the teams listed. I showed that to the students and explained the basic idea that a team continued to play as long as they won. (The NCAA actually invites 65 teams into the tournament. Two teams play an early "play-in" game and that narrows the field to 64 teams.) I ask the students to tell me how many total games will then be played, after the "play-in" game, in the tournament to determine a winner. It's the type of question that most, if not all, kids don't already know. Neither do many adults.
When I first thought of this question, I solved it by doing what most people do when they hear this question. I looked at the bracket sheet and counted the first round games -quickly realizing that there had to be 32 games in the first round because there are 64 teams (after the "play-in" game). The next round would be 16 games (because there were 32 winners n the first round), then the third round would have 8 games and the fourth round (regional final) would have four games. Those four games produce the "Final Four." Of course then you have two games in the national semi-finals and then the final game for the championship.
32+16+8+4+2+1 = 63. (I prefer not to count or sometimes even mention the "play-in" game because it isn't usually shown in the brackets that kids see. That way they can count all of the games one by one if they don't see any faster way.) When I first did this question, many years ago and saw the answer "63" I quickly realized that I had not used the most efficient method. When you have 64 teams in a single elimination tournament, 63 teams have to be eliminated in order for the champion to be determined. Each game eliminates one team so 63 games need to be played to eliminate 63 teams. An even better way to look at it is that all teams have to be eliminated except one, so 64-1=63.
Most kids and adults can easily figure out by the original counting method, that the tournament with 64 teams requires 63 games to determine the winner. They usually don't notice or think about the shortcut. Therefore when I present the original question I also present an additional question. Almost every year there are people who say that all of the NCAA (Division 1) teams should participate, regardless of record, just like they do in high school. In a given year I wouldn't always know exactly how many teams were in Division 1 so I would tell the students to assume that there would be 200 teams and to figure out how many games would need to be played. Assuming they didn't know the shortcut mentioned above, they would normally try to solve it just like they had the original question. With 200 teams there would be 100 games, then the second round would have 50 games, the third round 25 games and then they would get stuck. Very few, if any, students or adults, would be able to figure it out from there. What do you do with an odd number of teams?
So the next day when I asked them how they solved the original 64 team question (after they passed their papers in) most would have the correct answer but would not have seen the shortcut. I can't remember any of my 8th graders getting the 200 team question right because they always got stuck with the odd number of teams. At this point, if I had the time, I could suggest two problem solving strategies - solving a simpler problem and finding a pattern along with making a list.
I start with a tournament that has only one team which I know is ridiculous but it serves the strategy. Obviously with only one team, zero games need be played. If there are two teams, you need one game. When you get to three teams, you show them that the only way to do it is to have two teams play and the winner of that game plays the other team for the championship. So three teams requires two games. Four teams (like the "Final Four") needs three games. As this list is being made some kids will see the pattern that every time you add a team you add a game. That is true but if that is all they see then a tournament with 200 teams is going to require making a long list. Eventually somebody usually does see the most efficient method - just subtract one from the number of teams. They can then instantly determine that if the tournament had 200 teams that 199 games would be needed. If the tournament had 35 teams, then 34 games are played.
If you have any other additional time (highly unlikely) you can add in the concept of powers of two. When you have a single elimination tournament, you have to have enough "play-in" games to cut the field down to a number of teams that is a power of two. So a tournament with 35 teams would require three "play-in" games to cut the field to 32 teams. In fact you could ask the students at the beginning why the NCAA field (after the play-in game)is 64 teams. Why not 60 teams or 80 teams or 50 teams?
This was a great problem solving situation that took advantage of the natural interest that students and adults have in the NCAA tournament. One of the things I used to always say to my students is that, "A good mathematician is a lazy one." This particular situation is a perfect example of that.
Friday, March 14, 2008
Iowa Model Core Mathematics Curriculum and Motivation
The Iowa Department of Education has come out with two documents detailing what it calls the Model Core Mathematics Curriculum. Prior to this Iowa was the only state in the USA that had not established state standards in subjects like mathematics. I will provide a link to the introductory article. In that article is another link to the downloadable curriculum document for high school mathematics. Teachers and parents can read it and draw their own conclusions.
One complaint about mathematics education in the USA has been that the curriculum is "a mile wide and an inch deep," meaning that it has attempted to cover way too many topics and therefore has not been able to cover them in the depth necessary for true understanding and mastery. I believe I first heard that at least as far back as 1983 when the "A Nation at Risk" document came out. I don't disagree with that comment but, if people think that simply decreasing the number of topics addressed in the course of study will solve all of the problems, they will end up being disappointed.
I think it is a first step but what is more important, perhaps even decisive, is the need to find ways to motivate students to do the hard work necessary to learn at a high level. We should work for a higher percentage of parents who put the proper emphasis on education but that is a segment of the population that educators have less control over. What I believe is that math teachers must work hard to determine the most effective ways to teach their students (people other than teachers determine what is to be taught). The effectiveness of a teaching method includes the particular way of having students learn the skill or concept but also must also include motivational ideas that will cause the student to want to spend the time necessary to learn the skill and overcome the obstacles involved in learning difficult things.
Students today have a large number of demands on their time and, getting them to choose learning academic topics over extracurricular activities or jobs or video games or TV or text messaging or phone calls or anything else, requires special attention. I am not saying that students can't do some of those things, some of the time. The ideal thing would be that the student, himself or herself, chooses to put their emphasis on academics and then work in the other activities as their available time permits. It is necessary but not sufficient to just tell them or show them that some particular topic is important to their future. Motivation, motivation, motivation can overcome many obstacles. Motivate daily, motivate weekly, motivate monthly, motivate yearly, and motivate creatively. Lesson planning (daily, unit or chapter) should include motivation.
Success is an excellent motivator but that requires teaching that is correct in how it teaches students to think about a skill or concept so that the success is real and not artificial. When students find that the brain that they have really does work, they get pretty excited and want to do more. Having fun is important to students, but it has to come in the context of learning, otherwise students get the idea that learning is necessarily separate from fun. Having fun is good for teachers too since they may be teaching the same lesson several times a day and/or many times over the course of their career.
So my prescription for the USA to do better in mathematics education (and all education for that matter) is to work to get more positive involvement from parents, find the most effective ways to teach the required content (requires a lot of time and effort on the part of the teacher), and then motivate, motivate, motivate. If you can do only one of those three things, MOTIVATE.
One complaint about mathematics education in the USA has been that the curriculum is "a mile wide and an inch deep," meaning that it has attempted to cover way too many topics and therefore has not been able to cover them in the depth necessary for true understanding and mastery. I believe I first heard that at least as far back as 1983 when the "A Nation at Risk" document came out. I don't disagree with that comment but, if people think that simply decreasing the number of topics addressed in the course of study will solve all of the problems, they will end up being disappointed.
I think it is a first step but what is more important, perhaps even decisive, is the need to find ways to motivate students to do the hard work necessary to learn at a high level. We should work for a higher percentage of parents who put the proper emphasis on education but that is a segment of the population that educators have less control over. What I believe is that math teachers must work hard to determine the most effective ways to teach their students (people other than teachers determine what is to be taught). The effectiveness of a teaching method includes the particular way of having students learn the skill or concept but also must also include motivational ideas that will cause the student to want to spend the time necessary to learn the skill and overcome the obstacles involved in learning difficult things.
Students today have a large number of demands on their time and, getting them to choose learning academic topics over extracurricular activities or jobs or video games or TV or text messaging or phone calls or anything else, requires special attention. I am not saying that students can't do some of those things, some of the time. The ideal thing would be that the student, himself or herself, chooses to put their emphasis on academics and then work in the other activities as their available time permits. It is necessary but not sufficient to just tell them or show them that some particular topic is important to their future. Motivation, motivation, motivation can overcome many obstacles. Motivate daily, motivate weekly, motivate monthly, motivate yearly, and motivate creatively. Lesson planning (daily, unit or chapter) should include motivation.
Success is an excellent motivator but that requires teaching that is correct in how it teaches students to think about a skill or concept so that the success is real and not artificial. When students find that the brain that they have really does work, they get pretty excited and want to do more. Having fun is important to students, but it has to come in the context of learning, otherwise students get the idea that learning is necessarily separate from fun. Having fun is good for teachers too since they may be teaching the same lesson several times a day and/or many times over the course of their career.
So my prescription for the USA to do better in mathematics education (and all education for that matter) is to work to get more positive involvement from parents, find the most effective ways to teach the required content (requires a lot of time and effort on the part of the teacher), and then motivate, motivate, motivate. If you can do only one of those three things, MOTIVATE.
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