Math is math, except in social science

"So how are the eggs?" "Eggs are eggs." "Eggs are eggs. That is very profound. By the same token, couldn't you say fish is fish? I don't think so." 

So goes a Seinfeld dialog.  Similarly Sigmund Freud is alleged to have said, "Sometimes a cigar is just a cigar," although researchers question whether he really said it.

And by the same logic, math is math, whether it is taught in high school or college, a 2 year community college or a 4 year college, etc. Right?  I found a counter-example of this in statistics.

I recently taught an intro to statistics class in a psychology department using a statistics book for the behavioral sciences.  This book defines the sample standard deviation for descriptive purposes as S_X with an N denominator while defining the sample standard deviation for inferential purposes as s_X with an N-1 denominator.  I found a second statistics book for behavioral sciences that agrees with this.

Is there a recent textbook in the math or statistics world that defines the sample standard deviation with an N denominator?  I haven't seen it.  And not only will the student of this psychology class find this definition conflicts with the math world, but she will also find (and did find) it conflicts with the Excel world, not only for the Excel standard deviation function but for the Excel statistics Data Analysis add-in functions.

Why can't we all just get along?

After decades of suffering from her poor attitude toward math with her "Math class is tough!" quote http://www.youtube.com/watch?v=NO0cvqT1tAE which ironically was misreported by the press as "math is hard," Barbie is apparently putting her life on the line in Barbie Bungie Jumping to help students learn algebra, physics, and statistics: https://youtu.be/W5f7QoHOXq0

This appears to be an experiment relating the distance of a jump to the number of rubber bands used, and then estimating the line of best fit.

NCTM has a lesson on this: http://illuminations.nctm.org/Lesson.aspx?id=2157

I guess blondes do have more fun!

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 π 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 π 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 π, 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 π 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?

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