Thursday, December 23, 2010

Toasting Time is Still not Linear

Well, I've finally completed my toaster experiment.  I went through three loaves of bread and four or five days.  My family is happy to have me back.

First, a bit about the toaster.  The settings run from zero to five on the toaster.  Between each number, the dial is divided into four equal parts.  To get the toaster setting for the data, I took the setting on the dial as a fraction and multiplied by four.  That is why the data runs from 1 to 19.5 in the spreadsheet.  I used 19.5 instead of 20 because the dial wouldn't turn that far.
The International Standard Toaster

I made sure to go through each of the settings on the toaster.  The order was scrambled by a Python program.  I did write down one of the numbers incorrectly, so I did four trials on setting four, and two trials on setting fourteen.

I used a stop watch to time the toaster.  I rounded the toasting time to the nearest second.  After each trial, I let the toaster cool to 27 degrees Celsius.  That is to ensure a consistent starting temperature.

For each trial, I weighed the bread before and after toasting.  This is to measure the percent loss of mass during toasting.  My hypothesis is that most of the mass loss is due to the water in the bread evaporating in the toasting process.

The raw data can be downloaded from Google Docs at this link.  The graphs of the data are below.
Time (sec) vs. Toaster Setting
Percent Decrease in Mass vs. Toaster Setting
Percent Decrease in Mass vs. Time (sec)
I haven't applied any regressions to the data yet.  I am planning on doing so in the next couple of days.  I will let you form your own opinions first.  The only conclusion I will draw at the moment is that the toasting time is not a linear function of the toaster setting.

Tuesday, December 21, 2010

Happy New Year!

I know we're ten days from the ball dropping in Times Square.  Today is the winter solstice in the Northern Hemisphere.  This means that today will have the sun visible for the least number of hours this year.

Since calendars are arbitrary, we might as well declare the winter solstice to be the start of the year.  My Mother's family has be celebrating the winter solstice for a few years.  There are different observances in different cultures.  You can find on list on and another on

So, at 6:38 (Eastern Time) countdown to a new year.

Wednesday, December 15, 2010

Video Proof That I'm Not Dead

Yesterday, I finished grading for the semester.  I can now worry about Christmas.

As part of the Title III grant for distance learning, I recorded a short introductory video for my Intermediate Algebra class.  We teach Intermediate Algebra as a hybrid class because it is four credits, which interferes with the scheduling of other classes.  The video is on YouTube, but you can watch it here.  The excellent production was done by Justin Dean.

Thursday, December 9, 2010

Warming up to Python Programming in the Classroom

Last month, I commented on Dan Meyer's blog dy/dan that I wasn't sure about using programming as part of teaching mathematics.  Dan had worked on a project of developing Python programs to be part of the mathematics curriculum.  This was part of a Google project on computational thinking.  I've played around with Python, and I like what I see so far.

I started writing a Python program as part of my toaster data collection.  I wasn't happy with using a die to choose the toaster settings because one setting came up more than the others.  Instead, I wanted to test all of the settings in a random order.

I have programmed enough to be able to implement this program in several different languages.  However, I didn't want to take the time to write it in C or Java.  I started to write a program on my TI-83, but that was too slow.  I remembered that my friend Shawn was working on learning Python, so I thought I would give it a try.  It took me half and hour to download Python, install it, and learn enough to get the result I wanted.  I would like to claim that a short development time was due to my genius.  However, it had to do with the random.shuffle() function which was built into the random module.

Now that I see how fast one can develop a program in Python, I am intrigued by the possibilities.  I am discovering that there are many third-party extensions that could be helpful in the classroom.  I am looking forward to exploring their use over Christmas.

Tuesday, December 7, 2010

Toasting Time is Not Easy

A couple of posts ago, I noted that the data collected by Dan Meyer indicated that a linear model is not the best fit for modeling the time spent toasting vs. the toaster setting.  I've been collecting some data of my own, and it does not look pretty.  The rough data is below.  Remember that the toaster settings were chosen (mostly) randomly.
Raw Toaster Data

From the graph below, it is easy to see there is a lot going on.
The Linear Regression for the Data
The problem with the data is the settings between 2 and 10, which looks linear with a different slope than the settings above 10.

The exponential model is listed below.  It is just a bit more accurate, but not noticeably.
The Exponential Regression for the Data

Finally, for your approval is the cubic regression.
The Cubic Regression for the Data

Well, it's back to the toaster.

Friday, December 3, 2010

Carl Sagan in 1995

I stumbled on this video tonight.  It's an Charlie Rose interview of Carl Sagan.  The thought of life off of Earth has been on my mind for a couple of days since this NASA announcement.  Enjoy.

Wednesday, December 1, 2010

If You Were Wondering About the Density of Diet Mnt Dew

We were reviewing fluid forces in Calculus II today.  I decided to give an example using items from the room.  The first fluid I found was my Diet Mnt Dew (the "Mountain" was replaced by "Mnt" a while ago).  The mass density is 901 kilograms per cubic meter.  The weight density is 8830 Newtons per cubic meter.

One of the students asked if the density would change as the carbonation is released.  Our chemistry professor at the college said that the density would change, but not enough to be noticeable.

Monday, November 29, 2010

I'm Making Toast

My last post claimed that the relationship between the settings on a toaster and the time spent toasting is not a linear relationship.  I worked this out using Dan Meyer's data from his blog dy/dan.  Dan responded that one data point changed the distribution.  I'm pretty sure that he is right, but judging from this comment by Dan, neither of us will rest until this is settled.

I've decided to collect my own data.  I'm recording the air temperature inside of the toaster to make sure that I start each trial at 80 degrees Fahrenheit.  I'm also setting the toaster randomly.  I have a super-geeky way to do make the random selections, as you can see below.

This is a slow process, as it takes 10 to 15 minutes for the  toaster to cool.  Hopefully I can finish collecting my data before the sacrificial loaf goes stale.  I'll have to write another NSF grant if it does.

Friday, November 26, 2010

Toasting Time is Not Linear

Dan Meyer just posted some data on his dy/dan blog about toaster settings vs. toasting time.  You can see the video posted below.

Meyer — Toaster Regression from Dan Meyer on Vimeo.

I just ran the toasting times through Microsoft Excel 2010.  The raw data is below.
Raw Data

I graphed the data and used trendlines to analyze the data.  The first graph shows the linear regression.
Linear Regression

The next graph is the exponential regression.
Exponential Regression

Finally, here is the quadratic regression.
Quadratic Regression
Based on the correlation values, the best model is the quadratic.  The exponential is second best.  The linear is the third best.  However, all are good representations.

Also on Dan's blog post, the darkness of the toast is estimated.  The graph, with exponential regression, is below.

Wednesday, November 24, 2010

So, I Used WolframAlpha in Class After All

I gave an example in College Algebra yesterday about the U.S. GDP and Chinese GDP.  We were talking about exponential functions, and I made up some numbers and asked the students to find when the Chinese GDP would be larger than the U.S. GDP.  The year we got with my numbers was 2049.  After I finished the example, I told the students about Hans Rosling's talk on TED about when Asian per capita income would be larger then the U.S. per capita income.  He came up with the year 2048.

I was amazed how close our dates were, even though he is using real data and I just made up numbers.  So, I showed the class WolframAlpha and we were able to find graphs of the Chinese GDP and U.S. GDP over the past sixty years.   Both graphs generated by WolframAlpha looked appropriately exponential.

I was wrong about the starting values for the economies in 2000 (I guessed 4 trillion dollars for the US and 400 billion dollars for China) but the ratio between my guesses and the actual values was correct (10 trillion for the US and 1 trillion for China).  I was confident in my estimates for the growth rate for both economies (4% for the U.S. and 9% for China) because I got them off of the news over the past few years.

In my last post, I discussed overuse of programs like WolframAlpha in teaching.  I felt like a hypocrite when I used WolframAlpha in class.  In my defense, I never did say it was a bad program, just that it shouldn't be the foundation for a mathematics curriculum.  For the record, we did solve the equation that we developed in class by using graphing calculators.

Friday, November 19, 2010

Whose Computer Can Solve This for Me?

There's an scene in The Simpsons where Edna Krabappel is teaching the students math.  The dialogue is below.
Mrs. Krabappel: Now, whose calculator can tell me what 7 times 8 is?
Milhouse: Oh! Oh! Oh! Low Battery?
Mrs. Krabappel: Whatever.
 I was reminded of this scene when I saw this talk by Conrad Wolfram on the TED website.

I found a link to the talk on this post by Maria Andersen.

The modern version of The Simpsons scene would have Bart's class in a computer lab.  The dialogue would be the following.

Mrs. Krabappel: Whose computer can compute this integral for me?
Milhouse: Oh! Oh! Oh! 404
Mrs. Krabappel: What's the whole answer?
Milhouse: 404 Error: Page not Found
The thesis of the Wolfram's talk is that the mathematics taught in school today does not reflect the real world.  Also, requiring calculations to be done by hand is the bottleneck preventing students working on real world problems.  His solution is to use a program like WolframAlpha, which can solve many mathematics problems automatically, with students.

I do agree with his statement of the problem.  I do have a few points of disagreement with his solution.
  1. Graphing calculator do quite well with teaching mathematics.  At my college, we use graphing calculators at all levels of algebra.  Our College Algebra class is modeling based, so we make heavy use of calculators.  We teach graphing operations, statistical operations, and using the solver application.

    Access to calculators is not a limitation.  The local high schools require the TI-84 calculator, and most of our students still have theirs.  Also, the college has some available for students to borrow.

    The calculators do have limitations, but those are not disadvantages.  The calculator can only process the mathematics that the students put into the calculator.  That means that the students have to develop or find the formula to measure how drunk is someone on their own.  Also, the computers are too fast in presenting the results.  Calculators require the students to slow down and see the fine details.  Finally, calculators cannot get information off of the internet.  Any communication device is also a cheating device.
  2. There is no silver bullet to fix mathematics education.  The problem is that there is a new silver bullet every five years or so.  At some point, people outside of the teaching profession are going to have to realize that people learn in unique ways.  Wolfram is not alone in thinking that he has found the solution.  Some colleges are changing all developmental (below college level) mathematics courses to computer delivered instruction.  This mode of learning only engages two senses.  Three if you count the soreness in your butt from all the sitting.
  3. Computers do not let students feel mathematics instinctively.  Our instincts are not wired for computer simulations or moving sliders with a mouse.  Our instincts are wired for dealing with the tactile world.  To teach surface area, let the students count the tiles on the floor.  Let them see what a square foot looks like.  To teach linear functions, let the students walk down the hallway and make distance and time measurements.  To teach surface area, let the students paint a box and see how much paint they use.
  4. Don't use programming to teach elementary mathematics.  This is adding a layer of difficulty and abstraction on top of a difficult subject.  I know that I would learn well using this technique, but I also know I am rare in this learning style.
  5. It is not true that all calculations have been done by hand except in the last few decades.  My abacus and slide rule want to know what you meant by this, Conrad.
I don't disagree with Wolfram that there is a problem with how mathematics has been taught in the past.  I am concerned by anybody who claims there is a single solution to the problem.  Computers have a part in the mathematics curriculum, but they must not upstage the rest of the valid learning techniques.

Wednesday, November 17, 2010

I'm Done with Twitter

I couldn't find a use for Twitter.  I also haven't had anything to tweet for three days.  If I ever figure out a use for Twitter, I'll post it on the blog.

Sunday, November 14, 2010

Can We Know the Sources of Pseudocontext?

There is a saying that a camel is a horse designed by committee.  I am starting to think that textbooks suffer from the same problem.

On the blog dy/dan, there is a comment by josh g. on pseudocontext.  He is responding to my comment. Here is an excerpt of josh's comment.
I guess that’s part of why I keep coming back to trying to imagine the process in which these kinds of problems get written. We should be able to critique these problems in a way that deconstructs where these things come from, not one that just points the fingers at particular authors or even just specific problems.
I agree with josh completely.  I think that those of us who are worried about pseudocontext are starting to be able to see the boundary between context and pseudocontext.  So, looking for specific examples is not as important now.

What I did learn about textbook writing this week is that there are many different people who are involved in the process.  There is the author, project manager (I met two at the dinner), the editor, the supplemental materials authors, and the reviewers.  I guess that a textbook is the camel designed by committee.

I asked several different people about who controls the content of the text book.  Everybody answered that the authors have a lot of freedom in the content of the text.  Every problem is written by the authors.  However, one person mentioned that the reviews influence the number and type of exercises in each section.  There are nineteen reviewers for the textbook from which I took the example from the post.  I've seen texts with a full page of reviewers' names.

So, imagine that you are a textbook author and you just finished your masterpiece of a text.  The reviews come in, and the editor says that you need to include three more word problem in the section on systems of linear equation, and fifty new word problems in the entire text.  You have to get everything done next week so that the publisher can get the book to print, incorporate your problems into their flagship online homework system, and add them to the solutions manual.  You need to do this on top of your teaching load.

I don't know first hand about publishing a textbook.  If my scenario is far fetched, then please correct me (politely) in the comments.  However, I can see how well intentioned and intelligent authors get stuck with problems that they don't like in their texts.

There are two ways that I can think of to get involved in improving the quality of our textbooks.  Both options require a time commitment on your part.
  1. Become an author.  Textbook companies are looking for authors.  If you have the next great textbook in your head, get it on paper.  You can also self-publish, but be sure to cover all your bases before going that route.
  2. Review textbooks for publishers.  I've been asked once to review textbooks, and that's after only four years of teaching.  I declined because I had six preps that semester.  You could go looking for opportunities.  There is a small amount of money to be earned

AMATYC Boston Day 3

It was my third and final day in Boston.  The conference runs until tomorrow morning, but we usually have to leave early.

Awards Breakfast

The awards breakfast was the only event that I attended today.  The winners of the Student Math League were announced.  Los Angeles City College won the team competition this year.  Congratulations to them.

The keynote address was Infinity Bottles of Beer on the Wall by Lew Lefton.  He told mathematical jokes for forty minutes or so.  I was laughing too hard to keep track of time.  I had heard some of the jokes before.  I wrote one down because it relates to the pseudocontext that we've been talking about.
If you put six white balls and nine black balls in a bag and draw five of them out, then there is a 85% chance you are in a word problem.
After the awards breakfast, I went souvenir shopping and checked out of the hotel.


I'm home now, and my head is swimming.  I've absorbed a lot of information, and I need to distill it.  My ears are still plugged from the flight. 

Friday, November 12, 2010

AMATYC Boston Day 2

It's been a good day today in Boston.  I had a good talk with Gary Rockswold and Terry Krieger and a productive committee meeting with the ITLC.

Sessions for the Day
I started the day with the Innovative Teaching and Learning Techniques Themed Session.  I sat in the following fifteen minute talks.
  1. Promote Active Learning Using Real-World Applications - Frank C. Wilson
    This talk is along the same interests of mine.  The activity was about the dice game Pig and computing associated probabilities.  I think this is a good project, but I don't know if I would consider it "real world".
  2. Digital Learning Projects - Maria Andersen
    There was a lot of information to process.  She had good ideas about students using social media to reflect on and share about their learning.  She also had a good idea about using data visualization.  I am going to look into doing these projects in college algebra and calculus.
  3. Symbolic Processors: Wave of the Future? - Fred Felton
    This talk was about using Wolfram Alpha in the classroom.  I learned later that I misinterpreted Fred's use of this program in his classes.  I am worried about this program because it does all of the work for a student in solving a problem.  I don't want student to believe that technology can replace thinking, and Wolfram Alpha is close to being able to make the replacement.  In addition, I really, really don't like Stephen Wolfram's views on science and mathematics.  I tried to read A New Kind of Science, but could only get through 21 of the 1300 pages before getting too angry to read.
  4. Beyond Tables - Introductory Statistics - Dianna Cichocki
    This was a good talk about using JAVA applets to replace tables of probabilities for normal distributions.  She asked during the talk if we still use tables of trigonometric values in class.  I raised my hand because I still show how to use them in my trigonometry class.  I want students to be aware that there are ways of doing things that don't involve batteries and plugs.  I would never require students to use trig tables instead of calculators.  Dianna is making effective use of the applets in her classroom.
Lunch With Gary Rockswold and Terry Krieger
Pearson, the textbook publisher, arranged a lunch with a few instructors with the authors Gary Rockswold and Terry Krieger.  I thank them for the lunch.
I was fascinated to get the views of the authors and the publisher reps.  We have been discussing pseudocontext on my blog and at dy/dan.  I tried to bring it up with the authors, but I couldn't make myself clearly understood.  I didn't press because I didn't want to be a jerk.

Both the authors, and Gary's daughter Jessica are nice people.  I feel bad for using one of their examples on the dy/dan blog.
Dana's Talk
I went to the talk of one of my coworkers.  It was on grant writing.  I went to show support for Dana.  I don't have much interest in the topic right now.
ITLC Meeting
The Innovative Teaching and Learning Committee had its meeting in the afternoon.  It was a good meeting.  Most of the committee is in favor of proctored exams for online classes.  I was under the impression that the opposite was true.  Proctored tests are very important to me.  The committee is looking into a position paper on online educational resources (OER), with which I have volunteered to help.

I'm much more alert tonight than I was last night.  I am planning on going for a walk around the hotel and the nearby shops to burn off some energy and calories before bed.  I went through Barnes & Noble too quickly last night.

Thursday, November 11, 2010

I Caved on Twitter

I just posted that I was almost convinced to try Twitter.  In a moment of weakness, I signed up.  It was quick and painless.  I promise to never Tweet what I have for breakfast.

AMATYC Boston Day 1

I've had a busy day in Boston today.  It's the first day of the AMATYC Conference.  This is my third year attending the conference.  I'm starting to recognize people from one year to the next.

Morning Sessions

Here is a summary of the sessions I attended.
  1. Collaboration is the Key! - Vicki Gearhart and Honey Kirk
    I thought this was going to be a talk about student collaboration.  I was wrong, but I'm glad I went to the talk anyway.  It was about curriculum alignment in Texas.  The collaboration is between the high schools, community colleges, and four-year colleges.  The point is that the alignment is supposed to be a bottom-up approach, which I like.  Kentucky is going through a similar thing right now, but they are taking a top-down approach.
  2. On the Use of Social Media - Mike Martin, Maria Anderson, Fred Feldon, and Mary Beth Orrange
    You can tell I have an interest in social media because you are reading my blog.  You also can tell that a talk is good if it almost convinces me to sign up for Twitter.  I did say almost. 

    The biggest thing I took away from this talk is that employers are looking to social media in the hiring process.  It would be a good idea to encourage students to leverage their on-line presence to reflect their strengths.

    Maria Andersen showed us Imagination Cubed, a website for sharing drawing.  It can be used for displaying writing on a tablet PC to online students.  It caused a buzz of excitement in the audience.
  3. The Power of Google Docs for Effective Online Course Management - George M. Alexander and Calvin Williamson
    I was hoping to learn more about the mechanics of Google Documents.  I've been using Google Documents as a file server for this blog.  They had good ideas on how to use Google Documents to replace a course management system like Blackboard.  It looks like a good alternative.

    The problem I have with intirely online classes is that you can never know who is really doing the work.  If there is one class that would make an otherwise honest student cheat, it's math.
  4. I Can't Teach Calculus and It's Not My Fault! - Philip Cheifetz
    This talk did a good job highlighting the difficulty with falling standards.  The main point was that students are passing Precalculus and Calculus I with such weak skills that they can't do the work for Calculus II.  Dr. Cheifetez made a good case for mastery learning.
  5. Second Life in Higher Education - Fred Felton
    Fred Felton really enjoys Second Life.  I don't see the point.  I think that Second Life is one delivery method for distance learning courses, but not the best.  My concern with Second Life is that it can be so addicting for some people that it would negate any educational benefit.  I joked in the talk that I wish I could make MyMathLab so addicting.

    During the talk, Fred talked about Booland, where you can do virtual bungee jumping and hang gliding.  It seems a little odd to me that you would want to do those things virtually.  I thought the whole point was the adrenaline rush from risking your life.  How would you get that from your basement?
Opening General Session

The Opening General Session had the presentation of the Mathematics Excellence Award.  The two winners were Sadie Bragg and Ed Laughbaum.  Both winners have impressive experience in teaching mathematics and are worthy winners.  The keynote address was The Treasure of Polynomials  by Javier Gomez-Calderon.

AMATYC Exhibits

There are many companies that make their money off of students, and they are all selling their wares here.  The exhibits are popular because of the freebies.  I did grab a few things for the kids, but I don't go Christmas shopping.  The exhibits that I especially noticed were
  1. The MAA - They were giving out pi temporary tattoos.  I grabbed two for the kids.
  2. Easy Worksheet - They have worksheets for free download.  I'll be checking it out soon.
  3. AMSER - The Applied Math and Science Education Repository.  They have free materials for teachers.  Free is my favorite word.
  4. Minitab - I grabbed a t-shirt from them.  I didn't pack enough.
  5. Pearson Publishing - I grabbed a couple of foam puzzles for the kids.
After visiting the exhibition hall, I was wiped out.  I left my coworkers and headed out for dinner by myself.  I got pizza at the food court at the mall across the street.  I bought an abacus at Barnes and Noble,and two pairs of socks.  Now, I am watching Big Bang Theory.  I love it when Wil Wheaton is on.

It's going to be an early night for me.  I'll be meeting my coworkers for breakfast at 7:30 for breakfast in the lobby.

Monday, November 8, 2010

AMATYC Student Math League - Round 1

Maysville Community and Technical College participated in the Student Math League organized by AMATYC.  This semester, we had 19 students participate.  That is a record by a factor of two.  Our team score was 60 points, another record.  Our top scorer was a high school student in my Calculus II class.  He managed 18.5 points out of 40.  That is another record.

I'm pleased with this year's turn out.  In 2007, my first year, we managed a total of 6.5 team points for both rounds.  That tied for last place in the nation for schools with positive scores.  I would have been less humiliated if we had just scored 0 for the year.

I'm hoping that we will beat one of the other KCTCS schools this year.  Watch out, Madisonville.

Thank you to Susan Strickland for organizing the SML.  I promise to get my results in on time this year.

Update:  It looks like MCTC is the only KCTCS college in the SML competition this year.  We win!!!!!!!

Sunday, November 7, 2010

On Pseudocontext

On the blog dy/dan there has been a discussion about pseudocontext.  Pseudocontext is when a word problem is asking the students to use math that has little to do with the word problem.  Think of the infamous problems with trains leaving stations and moving at different speeds.  I'm having a discussion with another teacher about what qualifies a problem as pseudocontext.  You can check it out here.

I'm trying to finagle a dinner with Gary Rockswold at AMATYC this week in Boston.  I want to get an author's point of view on the subject of pseudocontext.

By the way, I want credit for the word "homocontextmorphism".

Saturday, November 6, 2010

ACT Prep (part 3)

I lead my ACT prep class this morning at Bracken County High School.  It was weird to teach in a high school.  I've taught high school students before, but the classes were at the college.  I was on their home field.  To add to the pressure, there was one of the high school teachers observing me.  The high school is going to be holding ACT Prep classes after school.

It went better today than before.  I think that holding the session in the morning helped.  Usually we have the sessions at night on weekdays and the students are tired after a day of school.  I changed my methods a bit, and actually reviewed the topics like I was teaching them.  The students were more engaged.

I followed the same format as before.  I gave a fifteen minute presentation on the format of the test and a few test taking hints.  Then, I let the students take a practice test for twenty minutes.  Finally, we reviewed the questions for the rest of the two hours.

If you are interested, my prepared notes can be downloaded here.  The handwritten notes during the review are here.

ACT Prep (part 2)

I'm finishing my preparations for an ACT prep class tomorrow at Bracken County High School this morning.  I was searching on-line for hints on leading an ACT prep class, and I came across this article about a report that too much time spent on ACT preparation in class can hurt scores.  I remarked in a previous post that I didn't feel like I was helping the students over the course of two hours.  Perhaps two hours is enough after all.

Wednesday, November 3, 2010

Robin Webb Defeats Jack Ditty

A couple of week ago, I remarked on a conversation I had with Jack Ditty.  Last night, Jack Ditty lost his race for Kentucky's 18th Senate District.  This was the only race last night that I was worried about, as the other races had results that I expected.  Ms. Webb has been on the campus before for several events, and has taken the time to interact with the students.  She seems to genuinely care about our college.  Our representative to the Kentucky House, Mike Denham, ran unopposed.  Mike Denham was one of the first students at the college, and has also been a supporter of the college.

Monday, November 1, 2010

Halloween, My Car, and the Night Sky (part 2)

In response to my last post, Sue VanHattum (the only person who reads this blog :-)) asked why the Pleiades are called the Seven Sisters while there are six stars in the logo for Subaru.  I was going to respond in the comments, but I realized that I had too much to say.

The Pleiades from Greek Mythology where seven daughters of Atlas.  According to the myth, Zeus transformed the Pleiades into stars to protect them from the hunter Orion.  Due to the rotation of the Earth, the constellation Orion appears to follow the Pleiades in the night sky.

The Subaru logo represents the five companies that merged to form Subaru's parent company Fuji Heavy Industries.  Apparently the larger star represents the merged companies.  It looks like there is one star (Celaeno) on the right of the picture that is not part of the Subaru logo.  The Greeks must have seen it differently than the Japanese.

I finally remembered where I heard about the date of Halloween.  I saw it on a Jack Horkhimer: Star Gazer segment last year.  This year's version is below.

I just learned that Jack died this year.

Sue also mentioned that the pagans celebrate the periods halfway between the equinoxes and the solstices.  The astronomy professor at my college told me that Groundhog's Day is at the halfway date between the Winter Solstice and the Vernal Equinox.  I was saving that tidbit for February.

Halloween, My Car, and the Night Sky

Out of all of the superstitions that people have come up with, astrology makes the most sense to me.  I can imagine an ancient person watching the sky and being amazed by how the change of seasons coincide with the change in the stars' position from one night to the next.  It's not far of a leap to think that the position of the stars affected the seasons.

Most people do not plan their lives around the stars anymore, but there are some artifacts of astrology that are still observed.  One of those artifacts is the date of Halloween.  The date of Halloween is October 31st because that used to be the date that the Pleiades would be at their highest at midnight.  This date was set several thousand years ago, and due to the movement of the galaxy, the Pleiades reach their highest point at midnight in mid-November.

The belief was that a bridge between heaven and Earth was formed on Halloween and the dead were able to move back and forth.  Most of our traditions were formed out of this belief.

The Pleiades, or the seven sisters is an open cluster of stars located in the constellation Taurus.  It is one of the Messier objects (M45), and is visible to the naked eye.  A picture I took of the Pleiades is below.  The Earth's rotation and the length of the exposure causes the double images of the stars.
My Photo of the Pleiades
Here is a professional photo from Wikipedia.
A Professional Photo
You may be wondering what is the connection between Halloween and my car.  The Japanese name for the Pleiades is Subaru.  Look at the picture of the logo from my car and compare with the pictures above.

Thursday, October 28, 2010

What is a Good Way to Run an ACT Prep Class?

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?

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 , that listed the qualities of an ideal school.  The part that stood out to me is below.
My ideal school
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
The 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.

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.

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.

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.

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.
The Mandelbrot Set
 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 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.


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.

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.

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 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.


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 "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
This is a very easy algorithm to program into a computer or even a graphing calculator.  Computers have been used to test the first billion billion (10^18) or so natural numbers, and each returns to 1 eventually.  This would be enough for most people, but mathematicians need to know that there is not some unimaginably large number that will not go to 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.


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.
 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)
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.

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