Triangle problem

The site Five Triangles has lots of good mathematical problems for middle schoolers. I really liked a recent one:

I wanted to see what configurations with this property looked like, so I made this brief animation:

Braid groups for nine year olds

Homeschooling my daughter in math has been a very interesting experience so far. As a mathematician, I try to re-imagine the topics in our curriculum with the goal of introducing deeper concepts as soon as I can.

Recently thanks to Allen Knutson, on Google+, I found out about a great video series by Ester Dalvit on braids and knots. My daughter has made it though about half of it, which already covers quite a lot of mathematical concepts (especially from a nine-year old's perspective). So I am trying to use it as a springboard to do some algebraic concepts that most K-12 students never see, like noncommutative groups.

So far I've just made a worksheet on braids along with a reference sheet, which went over pretty well today. I'm not sure how much farther I should push it now.

MOOCs - the hype and the pleasant reality

MOOCs - massive online open courses - are all the rage right now, and many people are wondering what their eventual impact will be.  For example, an opinon piece in the New York Times quotes someone as saying that students might be told to "take the following online courses over the summer or over a certain period, and then, when you’re done, you will come to campus and that’s when our job will begin.”

I don't think that's going to happen to a great extent.  Before explaining why, I'd like to make it clear that I am a big fan of MOOCs.  I've signed up for many courses on Coursera, EdX, and Udacity over the last two years, and learned a lot.  I use some of them to supplement my homeschooling, and I plan on incorporating some of their material into courses I teach.  I wish there had been MOOCs when I was in middle school and high school - I'm certain I would have eaten them up.

But the current hype can be pretty unrealistic.  Its interesting to compare the expectations of MOOCs to a very similar phenomenon of 100 years ago, the correspondence course.  Here's the beginning of an article in the journal Science (volume 24 from the year 1906):

There was tremendous interest in the general population in correspondence courses.  Here are some enrollment numbers for one group of correspondence schools (I'm sure there were many others in addition):

Consider that the population of the United States was about a quarter of what it is now, so this is comparable to enrolling a total of almost 4 million people today.  Its not hard to imagine that people thought this would radically change the nature of higher education.

There are many things that make colleges and universities attractive and useful and important to students that are not easily captured by a correspondence course or MOOC.  To be fair, the social and collaborative aspects of MOOCs make them far superior to a correspondence course.  But even for the limited goal of learning a well-defined list of concepts and skills, I think the shared commitment of physically showing up to class is psychologically crucial for most people and it will continue to be so. 

More math worksheets for the adventurous

These are still a work in progress, but here is another collection of worksheets I made for my daughter as part of her homeschooling in mathematics.

These include the ones I posted about previously.

I am planning on polishing up the complete collection once I stop making them for the year, sometime in June.  One thing that is missing is acknowledgements - I have taken many ideas from many sources, such as Fawn Nguyen, Dan Meyer, Kate Nowak, Alfred Posamentier, and many others including state math tests and standards from around the country.

Top mathematics videos

These are all worth watching.  The first two are especially good for kids:

Vi Hart has many, many awesome math videos.  Here's just one to get you started.  OK, I can't resist linking to this one too.  Very hard to pick a favorite.

Donald Duck in Mathmagicland.  This is from 1959, so parts of it are a bit dated but it had a significant influence on me when I saw it sixth grade.  I know many other friends and family who remember it fondly as well - a real classic.

A video from 1988 from Caltech, very well done: http://archive.org/details/theorem_of_pythagoras.

Möbius Transformations.  Short but very well done.

Niles Johnson's Hopf Fibration video.  Hopefully Niles will continue making more videos like this.

Not Knot part I and part II.  This is a window onto some really advanced mathematics, but I think its worth showing to just about everyone.  Perhaps more than any other video I know, it visually conveys one of the many amazing structures of modern mathematics.  Its a good antidote to anyone who thinks that math=arithmetic.

Dimensions is a 9-part series of short movies that have excellent animations.  Of course some episodes are better than others.

There are some other series which are more educationally oriented (and generally less visually stunning) but worth checking out: the Mathalicious project, the TED-Ed math section,  and the Numberphile youtube series.

Finally, two movies which are good are "Between the Folds" (on origami and mathematics), and the BBC Horizons documentary "The Proof" which is about Wiles' solution to Fermat's Last Theorem.  These are both about an hour, so they require a longer attention span than the ones listed above.

If you have any favorites which aren't listed here, let me know.

The Singular Value Decomposition and Congressional Voting

I am teaching a class about the SVD (Singular Value Decomposition) of a matrix this week.  I was inspired by a nice article of Carla Martin and Mason Porter, "The extraordinary SVD", to compute the SVD of the voting record of the 112th Congress (House of Representatives) to show to my class.

If you're interested in how this is done, here is the Sage code I used as a Sage worksheet. 

Here is the projection onto the first two singular vectors:

Third grade math worksheets

These are some (about 76 I think) worksheets that I have used in homeschooling my daughter this year:

http://www.d.umn.edu/~mhampton/ThirdGradeWorksheets_1_to_76.pdf

She was 8 and 9 during this time.  Some of them were too hard, and she required quite a bit of assistance.  Also, since she is learning German, there are a handful of questions that use German number names. 

They would require significant editing to be appropriate to a broad audience, but I am proud of some of them.  Its hard to find good workbooks that include more than the bare minimum material that kids in the US are over-tested on.

Motion sensitive LED belt

Recently I built a motion sensitive LED belt for a dance performance by my wife. In case it is of interest to others, this post will briefly describe how I made it.

I started with the LED belt kit by Adafruit:
http://www.adafruit.com/products/332,

for which they supply an excellent tutorial:
learn.adafruit.com/digital-led-belt 

To this I added a 3-axis accelerometer, the ADXL335 from Adafruit:
http://www.adafruit.com/products/163

Adding the accelerometer is very simple.  The 3V in and ground are soldered with short wires to the corresponding pins on the Atmega32u4 breakout board, and wires from the X,Y, and Z outputs can be soldered to the F4, F5, and F6 pins (see http://github.com/adafruit/Atmega32u4-Breakout-Board for more details or http://www.adafruit.com/datasheets/af_at32u4bb_pinout.pdf for a good diagram).

After a lot of tweaking, here is the code we actually used for the performance:
http://www.d.umn.edu/~mhampton/LED_belt_accelerometer.pde

Here is a short video (with a special guest star - our dog - in the background):

Sorry about the graininess - hopefully I can get a somewhat better version soon.  I'm also hoping for good closeup pictures of the soldered connections.