Homer Simpson disproves Fermat's Last Theorem

Simpsons fans know that Lisa seems to know a lot of math for someone her age but Homer can also surprise us, and higher math often sneaks its way into the show. Two university professors give some references to some of the math that has been used on the show in their site Simpsonsmath.com, and they also link to another of their pages on the quite significant mathematical backgrounds of the Simpsons' writers.

I recently learned from another article that discusses math on the Simpsons that the writers once sneaked into an episode a counterexample to Fermat's Last Theorem. This theorem is a famous problem in the history of mathematics and states that no three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n > 2. The theorem was conjectured in 1637 and not proven until 1995, yet Homer writes on a blackboard a counterexample 3987¹² + 4365¹² = 4472¹².

Try Homer's counterexample on your calculator or spreadsheet. Read Simon Singh's article for more on the counterexample.

Women in STEM on television

Other than female physicians, there are not a lot of women on television drama and sitcoms in science and technology.

I don't know how popular the television show "The Big Bang Theory" is among young women - probably not very - and while that show does have a pretty but somewhat dumb blonde, it also has several actresses portraying women with doctorates in science.

Melissa Rauch portrays Howard's wife Bernadette, a doctorate level microbiologist with a well-paying job.

Mayim Bialik, who portrayed Blossom in the 90's, portrays neurobiologist Amy Farrah Fowler. In real life Mayim has a Ph.D. in neuroscience from UCLA. She speaks on a variety of topics besides acting, including to scientific and mathematical groups.

Sara Gilbert portrays the very sharp-tongued Dr. Leslie Winkle, a physicist.

Christine Baranaski, currently a regular on "The Good Wife", portrays Dr. Beverly Hofstadter, Leonard's mother. She plays a neuroscientist and a psychiatrist. I think she is hilarious.

These women are all brilliant, witty and funny, and are not the social misfits that the male characters are. So I think these are pretty good role models for women in STEM (Science, Technology, Engineering and Math).

MOOCs

OK, we've all heard about MOOCs: Massive Open Online Courses. Very few of them are for college credit at the moment. But for those of us reading this blog - we probably don't need three more credits.



So let's enroll in one of these MOOCs, for the fun of it. I chose Stanford University's EDUC115N: How to Learn Math, which starts July 15, 2013.

I'm expecting an exceptional teacher, from a university I could never get into, zero one-to-one interaction with this teacher, an enormous number of fellow students, but a group of fellow students who will try to answer each other's questions. And I'm expecting to learn something I probably wouldn't learn somewhere else.

Why not join me, and see what a MOOC is like?

Copyright and math teaching

Lately I have been intrigued with the subject of copyright and math teaching.

I have done some reading and concluded that the copyright law is intentionally vague to provide users with flexibility, and that there are two schools of thought on permissible use of copyrighted materials in teaching.

The first school of thought is to take a very conservative approach and not do anything that could potentially trigger a lawsuit. This school of thought adopts the Section 107 Fair Use statute literally, and adopts the so-called Guidelines as rules to follow. This school would say that Section 107 does not apply to for-profit schools, period. It would also say under the Guidelines that you had better obey the brevity, spontaneity and cumulative effect guidelines.

The second school of thought is to take a more liberal approach, and that Section 107 requires an individual assessment of all four factors. For example, if a copyrighted work has no economic value, then the owner is not going to lose money if I use the work in a for-profit school. As another example, if I show before and after steroid use photos of a famous ballplayer in the context of measuring the probability of his post-steroids results, then this is a transformative use of these copyrighted photos from their original purpose, which is permissible.

I'd love some comments to help me think through this.

New tech / old tech

Hi.

So many people are saying teachers ought to be including more web 2.0 technologies in their classroom, so I dipped my toe in the water. I created several five minute videos (my college age son liked the idea of five minutes) to help my students visualize and save for later use some calculations. I wrote a script, and I used screenr.com (free) to record my screen. I asked my class to be gentle in their comments, and I said I don't think George Clooney needs to be worried about me taking his job.

So what happened? I got no student responses on the videos. But what I did get a favorable response on, is a handwritten diagram that I scanned and posted. So sometimes maybe old tech is the best.

Mr. Finch and pi

March 14, Pi Day, is coming up. Is there any more to be said about $\pi$ that hasn’t been said?



In the January 3, 2013 episode of the TV show “Person of Interest” (“you are being watched ... ”), computer genius Mr. Finch says that since $\pi$ is an infinite non-repeating decimal, “contained within this string of decimals is every single other number. Your birth date, combination to your locker, your social security number ...”

Really? I didn’t think that is necessarily so. Mr. Finch doesn’t explain why.

There is a site that lets you search for a specific string of digits within the first 200 million digits of $\pi$, http://www.angio.net/pi/piquery, but of course just because your number doesn’t appear, doesn’t prove anything. 200 million is a long way from infinity.

Are the digits in $\pi$ truly random? There is something called a normal number, which is a real number whose infinite sequence of digits is distributed uniformly. See http://en.wikipedia.org/wiki/Normal_number which says it is believed that $\pi$ is normal, but this has not been proven.

So we think Mr. Fitch was right, but we’re not 100% sure.

Counting with my fingers

We just did matrix multiplication, and I shocked my class by telling them that I think that is the perfect place to count with your fingers. I put my left pointer finger on the first element of the appropriate row, I put my right pointer finger on the first element of the appropriate column, do the multiplication, and move one element right and one element down to the next multiplication. Of course you need to mentally, or some other way, add these products, so that c_ij = ∑_k a_ik * b_kj. The other time I count with my fingers is when I do modular arithmetic, such as determining what is 7 months after October. I don't teach physics or three dimensional vectors, but I guess another application is the Right Hand Rule.

The class seemed shocked because it doesn't seem very sophisticated to count on your fingers. I can think of a few friends who would never lower themselves to do such a thing. That doesn't bother me. Let's just get the job done.

Does anyone have any other applications of using your fingers to count, or experiences in doing this?

Let's stop making excuses for Aunt Sally

Let's stop making excuses for dear Aunt Sally. You know who I mean: Please Excuse My Dear Aunt Sally. Many students can still remember the mnemonic PEMDAS for the rules of order of operations, but PEMDAS omits or confuses some situations. Most prominently, even my current college algebra textbook does not make it clear that in the M and D of PEMDAS multiplication and division are of equal priority, so a division is performed before a multiplication if the division appears to the left of the multiplication.

I like to invite my students to use all their calculators on expressions like 300 - 200/50*10 to see which calculators obey the rules of operations and which do not. I am always surprised when students first turn to the calculator on their cell phone to answer this.

PEMDAS doesn't address fraction bars, nor that with multiple grouping symbols you start with the innermost grouping symbols first. And I suppose it's too much to expect that PEMDAS address that in -3² the negation precedes the exponentiation; by the way, Excel disagrees with this.

Speaking of exponentiation, what do you do with x ^{y z} ? Exponentiation is not associative. Johnny Lott in A Problem Solving Approach to Mathematics says exponentiation is done in order from right to left (page 276), although Excel disagrees with this one too.

So there's more to the rules of order of operation than PEMDAS. But if you are getting a little tired of Aunt Sally, why not switch to Please Email My Dad A Shark, by the authors of xkcd.com

Anyone have any good ideas on teaching order of operations?

GFU2TT2BG3ZC

The color-challenged math student

If you are part of the 92% majority (that was not supposed to be a political comment), then you can see a number inside that circle.  But I can't.  The common description for my condition is being colorblind, although I prefer the more politically correct "color-challenged".  I am not truly colorblind.  I see some colors, just not as many as you see, and I get especially confused between red and green, shades of red and green, and colors containing either red or green (for example blue versus purple).

I will stop my car for the top (or left) traffic light, but I don't really think it is red. I would wear a red tie with a green-striped shirt if no one stops me, so I try not to own shirts with green.  Or to wear ties.  I'm never quite sure if the meat is cooked enough. You won't see me choosing colors of house paint or women's makeup shades, and I won't be disabling colored wires for the police bomb squad.  So generally I have learned to live with this minor inconvenience, and the accompanying jokes, which has not impeded my career choice.

But the world of math is not as black and white as it used to be.  A textbook author would never graph two curves in the same color.  But if y = .5e^x  is in red and y = x² is in green, I am going to be confused as to which curve is greater in 1 < x < 3.  I also get confused by multi-color pie charts where some of those colors start to blend together.  And if you really want to lose me, just show me one of those colored maps with ten or so different colors representing ten different levels of the amount of rainfall.

So if you are a teacher with control over such things, please be aware that some small percent of your students could be color-challenged and they may not even know it.  The best thing you can do is provide labels in addition to different colors.  Another possibility is to provide different kinds of patterns (see Excel's FORMAT, FILL, PATTERN STYLES).  Lastly, choose among colors we are more likely to recognize, and especially avoid red AND green together.

Have you had any difficulties in math with a color-challenged math student?

Beyond the scope of the course

Hi.  Occasionally a student will ask me a question, or will make a mistake, that leads to a topic that is beyond the scope of the course.  Recently this occurred in college algebra where students confused algebraic functions and algebraic numbers.  This entire subject is beyond the scope of this course - we don't prove a function is not algebraic, but we merely state it without proof;  and we really don't care for this course whether a number is algebraic or not.

My philosophy is that I always try to answer a student's question, or try to correct a mistake, even though  this gets us outside the scope of the course.  Students will not be tested on whether a particular function is algebraic or not.  But I always have this nagging feeling that I'm providing more information than anyone really cares about.

Thinking about this dates back to when one of my sons was in middle school and says he was taught that only rational numbers can be exponents.  I then dashed off a letter to my son's teacher, reminding him that while logarithms were not part of the middle school curriculum, did he really want to teach the kids something that they would have to un-learn in a future class?  I try not to teach something that a future teacher may have to correct;  my son's teacher felt an irrational exponent was too much information for that grade level.

Which of us is right?

An algebra problem students struggle with

Hi.  I am amused when a student midway through the class will ask what grade he or she needs on the remaining material, so that the final grade will be say a B.  My amusement is because either it is a student in my algebra class asking this question, or a student in my statistics class where algebra is the prerequisite. 

I should use this example as additional ammunition to the question, "Why do we have to take algebra?  I'll never use it."

We do this in chapter 2; by chapter 8 students have forgotten

Every time I teach statistics, at least one student will ask me a question about chapter 8, whose answer is in chapter 2.

In chapter 2 we calculate the standard deviation of a sample. I invite the class to do one calculation of a sample of three numbers using each step of the formula: √ ∑(x_i - ̅x)² / (n-1). Then I show them how to do it with a single Excel function, =STDEV. At the end of chapter 2, every student can do this in Excel.

We discuss standard deviations in the subsequent chapters, such as when we do Normal distributions, but we don’t calculate any.

In chapter 8 we do hypothesis testing of two samples that are matched pairs, such as a husband’s data and a wife’s data. We take the difference between the two values in each pair, calculate the mean of the differences, and the standard deviation of the differences. And at least one student will ask how to do these standard deviations. The answer is: the same way we did them in chapter 2.

Even if you don't teach statistics, I bet you have a similar experience. Comments?

Division by zero

Hi.  One topic I cover in week one of a college algebra course is division by zero.  We are going to see this later on in the chapter on rational functions, so I think we ought to settle this issue sooner rather than later.  Most people reading this blog are well-aware that division by zero is an undefined operation in the rules of real and complex number arithmetic.  3/0 = undefined.

This is often quite a surprise to my students, who apparently never learned this anywhere in their K-12 educations.  Truthfully, I can't remember learning it in my K-12 education either.  I think my fourth grade teacher is on Facebook, and I'm going to ask her if she remembers teaching it.

I invite my students to try 3/0 on as many calculators as they own.  Sometimes the calculator will show "Error" (which is descriptive, but is not a correct answer), and sometimes it will show zero (which is worse).  Students are even more surprised when I tell them both of these calculator results are incorrect.  However, we also experiment on calculators with Order of Operations problems, so students discover that not every calculator follows those rules.

I can picture Reverend Jim on the TV show Taxi:  "3/0 = undefined?  You're blowing my mind!"  I'd love to hear from some K-8 teachers or some high school algebra teachers on whether they teach division by zero.

First post - please be gentle

Hi.  It's probably about time that I join the blogging world.  I have been teaching online college math for about four years.  I started out with pre-algebra, worked my way up the math hierarchy, and now mostly teach college algebra and statistics.

I teach for several for-profit colleges.  Sometimes "for-profit" has a poor reputation.  I am not involved in the recruiting, advising, and counseling end of education, and perhaps this poor reputation is not undeserved.  But as an analogy, I never like dealing with car salesmen, but I still drive a car.

The first question I asked during my first interview was how does an online school know its students are not cheating.  Unless there is some in-person proctoring by a certified third-party (which is possible - I have taken online courses as a student that did this), an online school can not know.  But plenty of cheating goes on in a brick and mortar school too, from paying someone to submit assignments and using cell phones or crib sheets during exams, so please don't be "holier than thou."

I have two bases of comparison of my online courses at online schools versus the content of brick and mortar schools.  The first is that my son who attends a state university took a very similar statistics course to the one I teach.  Although my course is fewer weeks than his, my course seems to cover approximately the same topics.  Second, I recently took an online liberal arts course as a student at a different state university, and I was decidedly unimpressed with how little my professor was involved in the class, how infrequently she communicated with students, and how she apparently didn't read the students' posts at all because she never commented on some of mine which clearly exceeded that scope of the course.  So I have some confidence that my online students are not getting an inferior education.

I look forward to sharing some of my online math experiences, discussing math and teaching, and learning from this vast blogging community.