One of the services MCTC offers is an ACT preparation class for high school students. I usually teach one or two of these classes each year. There are three two-hour sessions, usually over different nights. In the past, I've spent twenty minutes going over the format of the test and giving general test taking tips, and then spent the rest of the time going over some of the problems on a practice test. I don't review any mathematics in a systematic fashion.

Nobody has complained about the format, and I get good evaluations afterward. However, I never get the feeling that I've done anything to better prepare the students for the test. Does anybody teach and ACT prep class differently? Is there a good format for teaching ACT prep?

## Thursday, October 28, 2010

## Wednesday, October 27, 2010

### Maria Monessori - A Teacher You Should Know

I've been working on a "Mathematician You Should Know" series to give a little credit to some mathematicians that don't get the credit I believe they deserve. Since I've been reading about education reform recently, I though I would expand the series.

I recently read a blog post, written by Sue VanHattum, that listed the qualities of an ideal school. The part that stood out to me is below.

Montessori schools are based on the Montessori Method, developed by Dr. Maria Montessori. Montessori was born in Chiaravalle, Italy in 1870. She attended an all-boy school to prepare to study engineering. She was the first female graduate of the University of Rome La Sapienza Medical School, and became the first woman doctor in Italy. She started teaching at a school for developmentally disabled students in 1896. She was able to have some of the students take the State reading and writing tests with above-average results. Montessori was able to try her methods with developmentally normal students starting in 1907.

Maria Montessori traveled extensively to demonstrate her method. Montessori came to the United States in 1915 to give a lecture at Carnegie Hall. She gave a demonstration of her method at the Panama-Pacific Exposition in San Francisco. During World War II, she went to India at the invitation of the Indian government to teach her method to teachers. She died in the Netherlands in 1952.

Maria Montessori was nominated for the Nobel Peace Prize three times.

Montessori developed her method by observing her students. The core of the Montessori method is a series of activities on which the students can work at their own pace. Children are able to select their own activities. The activities can be worked on different levels of difficulty. The activities are designed to be self-correcting. In his TED Talk, Will Wright credits the self-correcting aspect of the Montessori activities as part of his inspiration for creating Sim City.

Included with the standard academic areas are practical life activities. My favorite of the practical life activities is silver polishing. When my daughter's teacher told me about silver polishing I asked about the wallet stitching activity. She laughed and reminded me that the program was developed in the Victorian Age.

I really support educators going back and dusting off the work of Maria Montessori. There are several misconceptions of her work, but they are quickly dispelled once you see her methods at work. With school reform a popular topic, it is best if educators can have some alternatives to the status quo that we can support.

More information is at The International Montessori Index.

I recently read a blog post, written by Sue VanHattum, that listed the qualities of an ideal school. The part that stood out to me is below.

My ideal schoolThe reason that it stood out is that it described my daughter's school pretty well. My daughter has attended Nativity Montessori School for the past three years, and my son went there for two years. They have a program for three and four-year-olds and Kindergarten. It was my wife's idea to put the kids into the Montessori program. She had learned about Montessori schools in college, where she was studying elementary and special education. I had only a dim idea of Montessori schools.

Is full of resources that draw the kids’ interest

Is staffed with adults who know

That children have their own ways of thinking

That each child moves through learning in their own way

That there must be safety, both physical and emotional

That there must be affection and loving and hugs

Maria Montessori |

Montessori schools are based on the Montessori Method, developed by Dr. Maria Montessori. Montessori was born in Chiaravalle, Italy in 1870. She attended an all-boy school to prepare to study engineering. She was the first female graduate of the University of Rome La Sapienza Medical School, and became the first woman doctor in Italy. She started teaching at a school for developmentally disabled students in 1896. She was able to have some of the students take the State reading and writing tests with above-average results. Montessori was able to try her methods with developmentally normal students starting in 1907.

Maria Montessori traveled extensively to demonstrate her method. Montessori came to the United States in 1915 to give a lecture at Carnegie Hall. She gave a demonstration of her method at the Panama-Pacific Exposition in San Francisco. During World War II, she went to India at the invitation of the Indian government to teach her method to teachers. She died in the Netherlands in 1952.

Maria Montessori was nominated for the Nobel Peace Prize three times.

Montessori developed her method by observing her students. The core of the Montessori method is a series of activities on which the students can work at their own pace. Children are able to select their own activities. The activities can be worked on different levels of difficulty. The activities are designed to be self-correcting. In his TED Talk, Will Wright credits the self-correcting aspect of the Montessori activities as part of his inspiration for creating Sim City.

Included with the standard academic areas are practical life activities. My favorite of the practical life activities is silver polishing. When my daughter's teacher told me about silver polishing I asked about the wallet stitching activity. She laughed and reminded me that the program was developed in the Victorian Age.

I really support educators going back and dusting off the work of Maria Montessori. There are several misconceptions of her work, but they are quickly dispelled once you see her methods at work. With school reform a popular topic, it is best if educators can have some alternatives to the status quo that we can support.

More information is at The International Montessori Index.

## Sunday, October 24, 2010

### Crisis and Opportunity Better Mean the Same Thing, or I'm in Trouble!

Yesterday, I was backing up my teaching files to a flash drive so I can copy them to the new laptop that I am supposed to get soon. For some reason, which escapes me at the moment, I did a "cut and paste" instead of a "copy and paste". Not all of the files copied onto the flash drive and I don't have a backup on my laptop. I managed to lose a large chunk of data.

The sorest part of all of this is that the PowerPoint, LaTeX, and SMART Notebook documents that I use in class are gone. All of them! I don't have a single worksheet or lecture that I've produced in the last three years. This is going to be a real problem on Monday. (To add insult to injury, I do have all of the slides I produced for teaching at UW-Milwaukee. Those slides are for overhead projectors.)

Now that the lectures are gone, I can take this opportunity to get them right. The structure of most of my classes was to summarize the section of the text in a PowerPoint, read the PowerPoint to the students while they copied it into their notebooks, and work a couple of examples. As you would expect, this turned into a race for the students to copy the notes before I grew bored of watching them write. Even my assurances that they could get the notes on-line didn't keep them from writing every word. I did have some activities that broke this mold, but it was hard to produce enough of them to replace every traditional lecture.

I've known for a while that class time lacked interaction with the students. I would only get students engaged in small ways. I realize now that I am going to have to replace "The definition of a linear function is..." with "Read the definition of a linear function on page 143 of your text. Write an example of a function that is not linear and convince your neighbor that it is linear." I am also planning on writing worksheets to go with each lecture. I have used worksheets in the past to lead students, in small groups, step-by-step through some topics. The students prefer the worksheets over lectures.

With due respect to Edward Van Halen, I've fallen down the stairs and I might have landed on my feet.

The sorest part of all of this is that the PowerPoint, LaTeX, and SMART Notebook documents that I use in class are gone. All of them! I don't have a single worksheet or lecture that I've produced in the last three years. This is going to be a real problem on Monday. (To add insult to injury, I do have all of the slides I produced for teaching at UW-Milwaukee. Those slides are for overhead projectors.)

Now that the lectures are gone, I can take this opportunity to get them right. The structure of most of my classes was to summarize the section of the text in a PowerPoint, read the PowerPoint to the students while they copied it into their notebooks, and work a couple of examples. As you would expect, this turned into a race for the students to copy the notes before I grew bored of watching them write. Even my assurances that they could get the notes on-line didn't keep them from writing every word. I did have some activities that broke this mold, but it was hard to produce enough of them to replace every traditional lecture.

I've known for a while that class time lacked interaction with the students. I would only get students engaged in small ways. I realize now that I am going to have to replace "The definition of a linear function is..." with "Read the definition of a linear function on page 143 of your text. Write an example of a function that is not linear and convince your neighbor that it is linear." I am also planning on writing worksheets to go with each lecture. I have used worksheets in the past to lead students, in small groups, step-by-step through some topics. The students prefer the worksheets over lectures.

With due respect to Edward Van Halen, I've fallen down the stairs and I might have landed on my feet.

## Friday, October 22, 2010

### A Brief Conversation with Jack Ditty

I met Jack Ditty on Tuesday. He is the Republican candidate for State Senate in the 18th district. He was in Maysville for a candidate forum that evening. I only had a few moments to talk with him.

He asked me about the amount of developmental math students enrolled at the college. There has been some concern about the number of students who are enrolling into community colleges, and four-year colleges as well, who are placed into developmental classes. It is the belief of some that students who are placed into developmental classes are there because they weren't successful in high school math classes. I got the feeling that was Dr. Ditty's opinion as well.

The problem that I have with that interpretation of the increasing number of students in developmental mathematics is that it assumes that developmental students at the community college are recent high school graduates. I know from teaching a few developmental courses, and extensive time in the advising center, that developmental students are there for many reasons. It is common to see student in advising has been out of school for ten or more years, and has tested out of developmental reading and writing. It is the math skills that these students lose the fastest. Also, one of my prealgebra students this semester was home-schooled. It is important to remember that the community college population is very diverse.

One suggestion that Dr. Ditty made to reduce the number of students in developmental classes is to allow students to take classes at the community college during their last two years of high school and graduate high school with both a high school diploma and an associates degree. Again, I don't see this as a solution to this perceived problem.

I have a problem with students taking too many college classes in high school. First of all, colleges are not designed to accommodate high school students. I teach a section of College Algebra to the students of Mason County High School. We are very careful to limit the number of regular college students in that section. If we didn't the section would fill up during spring enrollment, and there would no room for the students. That happened last year, and we had 42 students at the first class meeting in a room with 30 seats. Also, the students coming from Mason County have already taken precalculus, so I have to teach college algebra with a different emphasis than I would for students who are coming out of intermediate algebra. The high school students quickly get bored, so classroom management is an issue.

With more students, community colleges will have to hire more faculty. At our college, all of the full-time math faculty are teaching overloads. The adjunct instructor pool is small for rural communities like ours, so adding additional sections can only be done with great pains. Hiring faculty is difficult because of budget cuts and competition from industry for people with advanced degrees in math. Adding more students from the high schools will only add to the problem.

High schools will suffer when students are siphoned to the community colleges. When I was in high school, there were students who were bussed to the local vocational school. Being high school students, we picked on those students. By moving the "better" students to the community college, the students who are left in high school will be the second class students. Student moral is already low enough without adding such a clear distinction between students.

I am able to articulate these points over the internet after a few days to collect my thoughts. My only response to Dr. Ditty at the time was, "Nice to meet you."

In the interest of equal time, Dr. Ditty's opponent is Robin Webb. I am unable to find a campaign website for Ms. Webb.

He asked me about the amount of developmental math students enrolled at the college. There has been some concern about the number of students who are enrolling into community colleges, and four-year colleges as well, who are placed into developmental classes. It is the belief of some that students who are placed into developmental classes are there because they weren't successful in high school math classes. I got the feeling that was Dr. Ditty's opinion as well.

The problem that I have with that interpretation of the increasing number of students in developmental mathematics is that it assumes that developmental students at the community college are recent high school graduates. I know from teaching a few developmental courses, and extensive time in the advising center, that developmental students are there for many reasons. It is common to see student in advising has been out of school for ten or more years, and has tested out of developmental reading and writing. It is the math skills that these students lose the fastest. Also, one of my prealgebra students this semester was home-schooled. It is important to remember that the community college population is very diverse.

One suggestion that Dr. Ditty made to reduce the number of students in developmental classes is to allow students to take classes at the community college during their last two years of high school and graduate high school with both a high school diploma and an associates degree. Again, I don't see this as a solution to this perceived problem.

I have a problem with students taking too many college classes in high school. First of all, colleges are not designed to accommodate high school students. I teach a section of College Algebra to the students of Mason County High School. We are very careful to limit the number of regular college students in that section. If we didn't the section would fill up during spring enrollment, and there would no room for the students. That happened last year, and we had 42 students at the first class meeting in a room with 30 seats. Also, the students coming from Mason County have already taken precalculus, so I have to teach college algebra with a different emphasis than I would for students who are coming out of intermediate algebra. The high school students quickly get bored, so classroom management is an issue.

With more students, community colleges will have to hire more faculty. At our college, all of the full-time math faculty are teaching overloads. The adjunct instructor pool is small for rural communities like ours, so adding additional sections can only be done with great pains. Hiring faculty is difficult because of budget cuts and competition from industry for people with advanced degrees in math. Adding more students from the high schools will only add to the problem.

High schools will suffer when students are siphoned to the community colleges. When I was in high school, there were students who were bussed to the local vocational school. Being high school students, we picked on those students. By moving the "better" students to the community college, the students who are left in high school will be the second class students. Student moral is already low enough without adding such a clear distinction between students.

I am able to articulate these points over the internet after a few days to collect my thoughts. My only response to Dr. Ditty at the time was, "Nice to meet you."

In the interest of equal time, Dr. Ditty's opponent is Robin Webb. I am unable to find a campaign website for Ms. Webb.

## Wednesday, October 20, 2010

### My First Attempt at Astrophotography

Here is a picture of Jupiter and the Moon. The alignment was striking last night.

This was a 4 second exposure with a f-stop of f/3.5. I am using a Cannon PowerShot S2 IS. It's a nice camera for its size.

This was a 4 second exposure with a f-stop of f/3.5. I am using a Cannon PowerShot S2 IS. It's a nice camera for its size.

## Tuesday, October 19, 2010

### Benoît Mandelbrot - A Mathematician You Should Know

Yesterday, I heard of the passing of Benoît Mandelbrot on October 14th. Mandelbrot is most famous for coining the word fractal, and one of the most famous fractals is named after him. A picture is below.

To generate this image, you need to look at the one-parameter family of functions

with complex variable z and parameter c. The process that one performs with this function is

One of the properties for which fractals are famous is that you can zoom in on one part of the fractal and that part will look like the whole fractal. This is the property of self-similarity. You can see a close-up of part of the mandelbrot set below.

Benoît Mandelbrot was born in Warsaw, Poland in 1924. His family moved to Paris in 1936. He studied in France, and briefly at Cal Tech. He moved to the United States in 1958. He spent 35 years at IBM's Watson Research Center, and then taught at Yale Univerity, retiring in 2005.

Mandlebrot is also famous for his book

I first encountered Mandelbrot while writing a paper on fractals in high school. I read about him in James Gleik's book

Here is Mandelbrot giving a talk on the TED website.

The Mandelbrot Set |

with complex variable z and parameter c. The process that one performs with this function is

__iteration__of some starting value. That means that the function is evaluated for the starting value, and then the function is repeatedly evaluated at the output from the last step of the iteration. This iteration process can continue forever. The Mandelbrot set is the set of all parameters c for which iteration of a starting value of 0 does not become infinite. This is the black region in the picture above.One of the properties for which fractals are famous is that you can zoom in on one part of the fractal and that part will look like the whole fractal. This is the property of self-similarity. You can see a close-up of part of the mandelbrot set below.

A close-up of part of the Mandelbrot Set |

Benoît Mandelbrot was born in Warsaw, Poland in 1924. His family moved to Paris in 1936. He studied in France, and briefly at Cal Tech. He moved to the United States in 1958. He spent 35 years at IBM's Watson Research Center, and then taught at Yale Univerity, retiring in 2005.

Mandlebrot is also famous for his book

*The Fractal Geometry of Nature*, which describes how nature produces objects with self-similarity. Tree branches, blood vessels, and romanesco broccoli are a few examples.I first encountered Mandelbrot while writing a paper on fractals in high school. I read about him in James Gleik's book

*Chaos: Making of a New Science*. I learned more about his work on fractals in graduate school, where I was studying dynamical systems. I certainly count him as one of my influences in mathematics.Here is Mandelbrot giving a talk on the TED website.

## Sunday, October 17, 2010

### Cooling Water - A New Media Project

I just finished a PowerPoint of a new media project. You can get the file here on Google Documents. I show a tea pot of cooling water over an hour, with photos taken every six minutes. The photo at 24 minutes is blurry, which is an accident. However, you can challenge your students to find the missing value.

This is the third attempt at this project. In the past, I shot photos of water going from room temperature to boiling. The problem is that the steam interfered with the temperature readings. This time, I started with boiling water and let it cool. I got better results in the long run.

The large thermometer in the picture is an indoor/outdoor thermometer I have at the house. I placed the outdoor sensor about a meter from the tea pot to get a good read on the room temperature. That is why I used the outdoor temperatures when computing the difference in temperature.

I would use this in my Intermediate Algebra, College Algebra, or Precalculus class. I took some photos of the temperature every thirty seconds for the first five minutes. I would use the second set for Differential Equations or Calc I to attempt to derive the differential equation for cooling.

If you download the file (successfully, this is my first attempt with Google Documents), feel free to use it in class and make your own modifications. Just be sure to give me credit for the photos.

Edit:

I buried the link to the document in the text. Here it is again.

This is the third attempt at this project. In the past, I shot photos of water going from room temperature to boiling. The problem is that the steam interfered with the temperature readings. This time, I started with boiling water and let it cool. I got better results in the long run.

The large thermometer in the picture is an indoor/outdoor thermometer I have at the house. I placed the outdoor sensor about a meter from the tea pot to get a good read on the room temperature. That is why I used the outdoor temperatures when computing the difference in temperature.

I would use this in my Intermediate Algebra, College Algebra, or Precalculus class. I took some photos of the temperature every thirty seconds for the first five minutes. I would use the second set for Differential Equations or Calc I to attempt to derive the differential equation for cooling.

If you download the file (successfully, this is my first attempt with Google Documents), feel free to use it in class and make your own modifications. Just be sure to give me credit for the photos.

Edit:

I buried the link to the document in the text. Here it is again.

## Saturday, October 16, 2010

### Why Educators Need to Get Along

This afternoon, my family was waiting for a parent/teacher conference at my daughters preschool. While waiting, we overheard some drama from the meeting before us. One of the parents was yelling at the teachers loudly enough to be heard through the closed door. I was only able to hear one side of the argument, so I don't know what set the parent off. We were able to pick up enough of the conversation to know that the parent was also a teacher. This is the aspect of the argument that bothered me the most.

The reason I am concerned about cohesion in the teaching profession, at all levels, is that we have enough enemies outside of the profession. In the public schools, teachers are already bearing the blame for failing

schools. School reformers, of whom Michelle Rhee of Washington D.C. is one of the more famous, are looking to clear the "dead wood" from the classrooms. Rhee resigned yesterday as Chancellor of the Washington D.C. Schools, but it looks like her successor, Kayla Henderson, will continue with her reforms. I haven't seen the movie Waiting for Superman yet, but by all accounts it should add fuel to the reform fire.

In higher education, especially in community colleges, budget limitations are straining faculty. We are expected to teach more students with fewer resources. In my own division, we've been asked to add more math classes even though all the faculty are already teaching overloads. It is difficult to get adjunct instructors because very few people have advanced degrees in mathematics in rural Kentucky.

For the record, I am not saying that all teachers must be in complete agreement at all times. We just need to work out our disagreements quietly in private. It is better for a department to say, "This is our position on this topic" than to squabble about it publicly. I do not always agree with some of the policies of the other math faculty, but I do follow the department policies. Being consistent helps when grade disputes occur. Students will have less room to argue about grades if every class has a uniform policy.

If your department has a teacher that is under-performing, it best to work it out with the teacher before the administration is looking to replace that teacher. An experienced teacher is rarely replaced with a more experienced teacher. If your colleagues tell you that you are under-performing, it is a good idea to listen. I've been there, and it hurts. I realize that the criticism I received was accurate, and I trusted the other faculty enough to use their help to improve. Just remember that you chose to go into the profession to facilitate, not prevent, student learning.

Finally, remember that not everybody has the same teaching style. Some teachers use direct instruction, some like discovery learning, other prefer self-paced instruction. As professionals, we need to respect the differences of other teachers and trust that they will develop their style to what work best for them. This is why the angry parent got to me so much. It's very easy to think that your way of teaching is the only way to teach. This can turn you into the same type of parent that you dread in the parent/teacher conferences.

Teaching is like parenting, the only way to understand it is to do it. Educators need to appreciate the people in other classrooms. If we want respect from people outside of the profession, we need to respect each other first.

P.S. The preschool in question is the best in town. My son was very well prepared for public school, and my daughter is doing well in kindergarten at the preschool.

The reason I am concerned about cohesion in the teaching profession, at all levels, is that we have enough enemies outside of the profession. In the public schools, teachers are already bearing the blame for failing

schools. School reformers, of whom Michelle Rhee of Washington D.C. is one of the more famous, are looking to clear the "dead wood" from the classrooms. Rhee resigned yesterday as Chancellor of the Washington D.C. Schools, but it looks like her successor, Kayla Henderson, will continue with her reforms. I haven't seen the movie Waiting for Superman yet, but by all accounts it should add fuel to the reform fire.

In higher education, especially in community colleges, budget limitations are straining faculty. We are expected to teach more students with fewer resources. In my own division, we've been asked to add more math classes even though all the faculty are already teaching overloads. It is difficult to get adjunct instructors because very few people have advanced degrees in mathematics in rural Kentucky.

For the record, I am not saying that all teachers must be in complete agreement at all times. We just need to work out our disagreements quietly in private. It is better for a department to say, "This is our position on this topic" than to squabble about it publicly. I do not always agree with some of the policies of the other math faculty, but I do follow the department policies. Being consistent helps when grade disputes occur. Students will have less room to argue about grades if every class has a uniform policy.

If your department has a teacher that is under-performing, it best to work it out with the teacher before the administration is looking to replace that teacher. An experienced teacher is rarely replaced with a more experienced teacher. If your colleagues tell you that you are under-performing, it is a good idea to listen. I've been there, and it hurts. I realize that the criticism I received was accurate, and I trusted the other faculty enough to use their help to improve. Just remember that you chose to go into the profession to facilitate, not prevent, student learning.

Finally, remember that not everybody has the same teaching style. Some teachers use direct instruction, some like discovery learning, other prefer self-paced instruction. As professionals, we need to respect the differences of other teachers and trust that they will develop their style to what work best for them. This is why the angry parent got to me so much. It's very easy to think that your way of teaching is the only way to teach. This can turn you into the same type of parent that you dread in the parent/teacher conferences.

Teaching is like parenting, the only way to understand it is to do it. Educators need to appreciate the people in other classrooms. If we want respect from people outside of the profession, we need to respect each other first.

P.S. The preschool in question is the best in town. My son was very well prepared for public school, and my daughter is doing well in kindergarten at the preschool.

## Thursday, October 14, 2010

### My Latest Video is on YouTube

I've been working on a screencast on using LaTeX (pronounced "La tech") to write notes for a SMART Board. You can find the video below.

If you are interested in learning LaTeX, a good reference is First Steps in LaTeX by George Grätzer.

You can download the software from the video at the following sites.

Pre-production of the second part of the trilogy will start tomorrow.

If you are interested in learning LaTeX, a good reference is First Steps in LaTeX by George Grätzer.

You can download the software from the video at the following sites.

Pre-production of the second part of the trilogy will start tomorrow.

## Wednesday, October 13, 2010

### Claude Shannon - A Mathematician You Should Know

If you asked a random person on the street to name a famous mathematician of the twentieth century, you would only get a couple of answers with high probability. The the highest probability answer would be Albert Einstein. The next highest would be "the guy Russell Crowe played in

Clearly, von Neumann's contributions to mathematics were the farthest reaching. He did work in quantum mechanics, functional analysis, economics and game theory, computer science and also participated in the Manhattan Project. However, I believe that Shannon's work had the larger impact on the lives of people after the Cold War.

Claude Shannon was born in 1916 in the northern part of the Lower Peninsula of Michigan. He studied electrical engineering and mathematics at the Univeristy of Michigan. At MIT he wrote his master's thesis, "A Symbolic Analysis of Relay and Switching Circuits,". Since you are reading this on a computer, you are using relays and switching circuits.

During World War II, Shannon worked at Bell Labs on cryptography. There he wrote a classified memo in 1945 which would be declassified as the 1949 paper "Communication Theory of Secrecy Systems", which gave one of the first mathematical descriptions of cryptography. Cryptography is used in secure communications, which allows for relatively safe commerce on the internet.

In 1948, Shannon published his most famous paper "A Mathematical Theory of Communication" where he lays the foundations of information theory. The information content of a message source is measured by entropy, which means the average number of bits needed to encode a symbol. We use data compression, like .zip or .mp3 files, to reduce the number of symbols required to encode a computer file to the minimum.

Also in "Theory of Communication", Shannon studied information moving though a channel, which transmits a message from a source to a receiver. The capacity of a channel is measured in bits per second. We worry about the channel capacity of our internet connections when we are downloading large files or streaming video.

In 1949, Shannon published "Communication in the presence of noise", in which he proved a sampling theorem. The theorem states that it is possible to encode an analog signal into a digital signal and back. This process allows a CD to store music digitally, and then play the music back as a analog (sound) signal.

Claude Shannon was an all around interesting person. He enjoyed juggling, unicycle riding, and chess. He was one of the first people to consider using a computer to play chess. He also built several devices. One of the more famous ones is the "Ultimate Machine", which you can watch in action below.

Hopefully, I've piqued your interest in the work of Claude Shannon. Many topics in information theory are easily accessible to students. The book by John Pierce, linked below, is a readable introduction to information theory. If you can read a book, thank a teacher. If you can hear an audio book, thank Claude Shannon.

References:

*A Beautiful Mind*". I'm going to guess that a distant third would be "the guy from Num3rs". Personally, my first two answers would be Jon von Neumann and Claude Shannon.Clearly, von Neumann's contributions to mathematics were the farthest reaching. He did work in quantum mechanics, functional analysis, economics and game theory, computer science and also participated in the Manhattan Project. However, I believe that Shannon's work had the larger impact on the lives of people after the Cold War.

Claude Shannon was born in 1916 in the northern part of the Lower Peninsula of Michigan. He studied electrical engineering and mathematics at the Univeristy of Michigan. At MIT he wrote his master's thesis, "A Symbolic Analysis of Relay and Switching Circuits,". Since you are reading this on a computer, you are using relays and switching circuits.

During World War II, Shannon worked at Bell Labs on cryptography. There he wrote a classified memo in 1945 which would be declassified as the 1949 paper "Communication Theory of Secrecy Systems", which gave one of the first mathematical descriptions of cryptography. Cryptography is used in secure communications, which allows for relatively safe commerce on the internet.

In 1948, Shannon published his most famous paper "A Mathematical Theory of Communication" where he lays the foundations of information theory. The information content of a message source is measured by entropy, which means the average number of bits needed to encode a symbol. We use data compression, like .zip or .mp3 files, to reduce the number of symbols required to encode a computer file to the minimum.

Also in "Theory of Communication", Shannon studied information moving though a channel, which transmits a message from a source to a receiver. The capacity of a channel is measured in bits per second. We worry about the channel capacity of our internet connections when we are downloading large files or streaming video.

In 1949, Shannon published "Communication in the presence of noise", in which he proved a sampling theorem. The theorem states that it is possible to encode an analog signal into a digital signal and back. This process allows a CD to store music digitally, and then play the music back as a analog (sound) signal.

Claude Shannon was an all around interesting person. He enjoyed juggling, unicycle riding, and chess. He was one of the first people to consider using a computer to play chess. He also built several devices. One of the more famous ones is the "Ultimate Machine", which you can watch in action below.

Hopefully, I've piqued your interest in the work of Claude Shannon. Many topics in information theory are easily accessible to students. The book by John Pierce, linked below, is a readable introduction to information theory. If you can read a book, thank a teacher. If you can hear an audio book, thank Claude Shannon.

References:

## Monday, October 11, 2010

### Unsolved Problem - The Collatz Conjecture

I was introduced to the Collatz Conjecture, also called the 3n+1 problem, in graduate school. This is one of the many problems in mathematics that are easy to state, can be understood by middle school students, but is difficult to prove.

The set-up of the problem is simple. Define the function below on the natural numbers.

The problem is to prove that if you start with any natural number and repeatedly apply this function, you will eventually get back to 1. For example:

Many people have tried to prove the Collatz Conjecture, and none have yet succeeded. For me, the draw is that I can feel that there is a solution on the tip of my tongue. It is the feeling that comes when you know there is a solution that you will see when you look at the problem in the correct way. Despite that feeling, I have yet to find the solution.

References:

The set-up of the problem is simple. Define the function below on the natural numbers.

The "phi" funcion |

The problem is to prove that if you start with any natural number and repeatedly apply this function, you will eventually get back to 1. For example:

- 4 -> 2 -> 1
- 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

Many people have tried to prove the Collatz Conjecture, and none have yet succeeded. For me, the draw is that I can feel that there is a solution on the tip of my tongue. It is the feeling that comes when you know there is a solution that you will see when you look at the problem in the correct way. Despite that feeling, I have yet to find the solution.

References:

## Friday, October 8, 2010

### What is an application?

A couple of years ago I was reading a problem in our college algebra text, and it made me laugh out loud. The first sentence is below.

What made me laugh is the thought that Canadian buffalo is just as abstract for my students as the formula. This is a word problem and it does give students the opportunity to practice skills for translating between the word problem and the computations. However, I would not call this an application because it is not relevant to the lives of the students.

With the multimedia technology of today, it is possible to find applications in the real world and bring them into the classroom. Dan Meyer gave an amazing talk about this topic at TEDxNYED. The video is linked below.

I have been trying to incorporate more applications in my classes. My focus is working on phenomenon that are difficult to see with the naked eye due to time frames that are too long or too short for use in the classroom. To date, I've created three PowerPoint files that track the motion of a ball in the air, a weight oscillating on a spring, and water boiling. The spring and the ball are videos that were reduced to individual frames so measurements can be taken at intervals of 1/30 of a second. The boiling water is a twenty minute experiment with photographs taken every minute. Some stills are below.

These videos are useful in every mathematics class. I use them to teach the students to build a mathematical model and then use the model to find more information that we can check using the media. The complexity of the model changes from one class to the next.

I encourage you to come up with your own media applications for your classes. I feel that the impact on the students is stronger when they know that you produced the video. It would be best to let the students make the media themselves, which would have to be done outside of class due to time constraints.

If you have an application that you want to share, please share in the comments.

Waterton Lakes National Park of Canada, where the Great Plains dramatically meet the Rocky Mountains in Alberta, has a migratory buffalo (bison) hear that spends falls and winters in the park.The problem then gives the logistic growth formula for the herd, and then asks the student to compute the number of buffalo in the herd in 2002.

What made me laugh is the thought that Canadian buffalo is just as abstract for my students as the formula. This is a word problem and it does give students the opportunity to practice skills for translating between the word problem and the computations. However, I would not call this an application because it is not relevant to the lives of the students.

With the multimedia technology of today, it is possible to find applications in the real world and bring them into the classroom. Dan Meyer gave an amazing talk about this topic at TEDxNYED. The video is linked below.

I have been trying to incorporate more applications in my classes. My focus is working on phenomenon that are difficult to see with the naked eye due to time frames that are too long or too short for use in the classroom. To date, I've created three PowerPoint files that track the motion of a ball in the air, a weight oscillating on a spring, and water boiling. The spring and the ball are videos that were reduced to individual frames so measurements can be taken at intervals of 1/30 of a second. The boiling water is a twenty minute experiment with photographs taken every minute. Some stills are below.

Ball Video (Spring 2007) |

Boiling Water (Spring 2010) |

Spring Video (Spring 2010) |

I encourage you to come up with your own media applications for your classes. I feel that the impact on the students is stronger when they know that you produced the video. It would be best to let the students make the media themselves, which would have to be done outside of class due to time constraints.

If you have an application that you want to share, please share in the comments.

## Wednesday, October 6, 2010

### Oh, That Darn "+C"

We were working on integration using trigonometric substitutions today in Calculus II, and the last problem of the day was the one below.

The integrals looked close, but there is a factor of 1/4 in the answer that we worked by hand that seemed to disappear when the calculator computed the integral. I was stumped to see how they where the same.

When I got back to my office one minute later, I sat at my desk and the answer hit me. The factor of 1/4 was absorbed by the arbitrary constant as follows.

The integral |

I was chatting with the students in the last couple of minutes of class, and I showed them the TI-92 calculator emulator that I have on my laptop. When I did the integral on the calculator, this is what I got.

Output from VTI |

When I got back to my office one minute later, I sat at my desk and the answer hit me. The factor of 1/4 was absorbed by the arbitrary constant as follows.

Now I see! |

So, the arbitrary constant comes back to get us again. That is why math teachers are so picky about it.

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