Two pieces from this year I wish I would have seen in graduate school

I grew up loving math.

Probably from the time I was in 5th grade through my second year of graduate school I would wake up every day thinking about math. My interests varied over the years, obviously. In high school it was the joy of learning all sort of new math from Mr. Waterman as well as the fun of math contests. In college I began to get interested in mathematical physics (even convincing the MIT Physics Dept to start an undergraduate General Relativity class), but also really enjoyed learning amazing subjects ranging from number theory to combinatorics. My most memorable classroom experience in college was definitely learning abstract algebra from Mike Artin.

Towards the end of my third year in graduate school, though, I completely lost interest in math. It didn’t happen gradually, either – I just woke up one day and wasn’t interested in math anymore. I’ve never known why.

Fortunately teaching my kids over the last several years has brought back the love that I used to have for math. As a result I’ve been paying much more attention to math-relatd items both in the news and online. This year I’ve found two incredible descriptions of what it is like to do math research that I wish I’d seen in graduate school. Going through them both made me wonder if my life would be different had I come across similar descriptions 20 years ago.

The first was this Numberphile interview with Ken Ribet:

The description of Ribet’s journey to prove that the Weil-Taniyama-Shimura conjecture implied Fermat’s Last Theorem was riveting. It is such an amazing story of the ups and downs of mathematical research and totally different from what I imagined that research was like when I was in graduate school. His interaction with Harvard’s Barry Mazur was especially moving.

The second item from 2015 that I wish I would have seen in graduate school was Cédric Villani’s Birth of a Theorem. A truly incredible description of the journey from initial idea to final proof of a Fields Medal winning theorem. The collaborative effort with his partner (that essentially covered all parts of the globe!) is incredible. All of the ups and downs in the research process were fascinating to see. Even though I know better, I tend to think of Fields Medal winners as being math automatons, so I was especially grateful to hear Villani discuss the dead ends, frustrations, and failures. A the end of the book, how he handled the initial rejection of the paper was also tremendously instructive. Again, the whole process was just so different than what I imagined math research to be like when I was in graduate school.

As an aside, Villani seems like a pretty cool guy. I ran across his name for the first time in this old post from Patrick Honner:

Patrick Honner’s post about meeting Villani>

If you are a Fields Medalist and take time to do public lectures and talk to math teachers, I’m definitely buying your book.

Even though it is way too late for me to turn the math research clock back, I’m really happy to have come across these two descriptions of math research this year. I’m also grateful to have had my kids rekindle my old love of math. I hope that I’m able to instill in them some of the new ideas about learning and doing math that I’ve just picked up in these pieces from Ribet and Villani.

The volume of a dodecahedron

I asked the boys what they would like to do for our Family Math project this morning and they said that they wanted to find the volume of a dodecahedron. This seemed like a pretty good follow up to yesterday’s project of finding the volume of an icosahedron – unfortunately this one is a little bit more difficult.

However, despite the difficulty, we had a great exploration and even had the nice surprise of finding an expression for the length of the red struts. To finish on a successful note, I had them find the volume of a cube by chopping it up into pyramids.

Here’s how the boys got started with the project – one difficulty that we encounter is finding the height of the pentagon pyramid, but not for lack of trying:

We took a break from filming for a bit to see if we could make any sense out of this pentagonal pyramid. We even tried to find the height by switching over to a triple-sized pyramid like we used yesterday. Eventually the boys found an approximation to the height that seemed to be pretty close. Even with that close approximation we have two problems:

(1) We don’t know how to find the area of a pentagon, and

(2) We don’t know the lengths of the red struts.

Our pyramid from yesterday helps with problem (2), and my younger son has a nice geometric idea that helps even more. That idea really illustrates how the Zometool set helps kids build geometric intuition.

Next up I let the boys implement my younger son’s idea from the last video. Now we can see a right triangle that will let us find the length of the red struts. Fun!

With our new idea in hand, we move to the whiteboard to do a little calculating. We find another fun surprise – the length of the triangle leg we are trying to find is a length that we already knew!

The next step was going to Wolfram Alpha to have it crunch the numbers for us. We’d already found the relationship between the long and medium sides in yesterday’s blog – knowing that relationship is a big help here!

Here’s the last step – finding the volume of a cube by chopping into pyramids. A much easier problem, but still an opportunity for some good geometric discussions:

So, a nice project despite not quite finding the volume we originally set out to find. It was a nice surprise to be able to find the length of the red struts, and it was really neat to hear my younger son’s idea about finding the height of an equilateral triangle. Fun morning.