Does this math course exist?

I’ve spent the last few days thinking about how students can learn about math that is normally outside of the school (both k-12 and college) curriculum.

The topic has been on my mind for a while, actually – pretty much since seeing this Ed Frenkel interview several years ago:

Frenkel’s talk has inspired several of my blog posts.

I wrote this one after seeing a project that Dan Anderson did with his students:

A list Ed Frenkel will love

Then, after seeing Lior Pachter write about how some unsolved problems in math fit nicely into the Common Core:

Lior Pachter’s “Unsolved Problems with the Common Core

I sort of combined Pachter’s idea and my thoughts about Frenkel’s interview into several different posts in the last couple of years:

Sharing math from Mathematicians with the Common Core

10 pretty easy to implement math activities for kids

A partial response to Sam Shah

This week I ran across two new ideas that got me thinking about sharing math, (and not just with kids). The first (I saw thinks to a SheckyR comment on a recent post) is this interview with Keith Devlin:

Keith Devlin’s interview: On learning and what it means to be human

This quote right at the beginning (around 3:40 into the interview) really struck me:

“If the last experience with mathematics is what you learned – certainly up to the middle level of high school – and to a large extent to the end of high school . . . you’ve basically never seen mathematics.”

Then I saw this tweet from TJ Hitchman:

I think the Hitchman and Devlin ideas are connected – if all you are seeing as a student is the math that is part of the normal school math programs (which, at least where I live, seem to be driven by what’s on the state tests) it would be pretty hard for anyone at all to get excited about math.

So, how do we, as Frenkel asks, get students to “realize that mathematics is this incredible archipelago of knowledge?”

A new idea crossed my mind this morning – and it isn’t that well thought out, but . . . .

One of the most influential-after-college classes that I took in college was a year-long physics course called “Junior Lab.” The idea in Junior Lab is that over the course of each semester you’ll do 6 (I think) famous experiments in physics (out of maybe 20 total choices). The website for the course is here:

Junior Lab’s website

After you do the experiments you present the results to your instructor as if you were the one doing the original experiment. As I wrote half-jokingly to my old lab partner, this is the most scary room on campus!

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You, of course, learn about the experiments, but there are so many lessons beyond that. The class teaches you about the breadth of physics, about experiments not working the way they are supposed to (!!), about presenting and defending results, and about writing papers.

It seems like the Junior lab format would be a great format for showing students math that isn’t typically part of a k-12 or college curriculum. It is a few steps beyond what Dan Anderson did with his “My Favorite” project, but, I think, would give students a totally different perspective on math.

It would be about as far away from a “learn this fact / take this test” type of math class as you can get. The students would have a wonderful opportunity to learn about many different areas of math and math research, and, as I mentioned above, the lessons from this class would reach far beyond the math.

In any case, I was wondering if there is a course like this anywhere. I hope there is because I’d like to think through the idea a little more carefully.

A challenge for professional mathematicians

[March 24th, 2016 update – I’m going to link some articles at the end of the blog as I see them. There are two from today. I’m really happy that people are writing about this!]

I saw this article on gravity waves via a Steven Strogatz tweet this morning:

Seeing the article reminded me of the interview that Numberphile did with Ed Frenkel a while back – in particular, the part from roughly 5:00 to 7:00 when Frenkel discuses the need for mathematicians to do better at sharing their ideas with the public:

Frenkel’s point is that even though the ideas in fields such as biology and physics are just as complicated as the ideas in math, these other areas of science are much better at communicating with the public than mathematics is.

I was reminded of Frenkel’s point again this morning when I learned that earlier this month Maryna S. Viazovska solved the 8-dimensional sphere packing problem. Viazovska’s paper on is here:

The sphere packing problem in dimension 8

Maybe I’m a little biased – especially right now because I’ve been spending this week playing around with 4-dimensional shapes with my kids . . .


but I think that the sphere packing problem (i) is something that can be explained to the public (it certainly seems less complicated than gravity waves) and (ii) is something that the public would find to be interesting. There’s not been much of any coverage of Viazovska’s result, though. Here’s what I found doing a simple Google news search:

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So, it sure seems this new result is something that would be great to share with the general public. There are, of course, many different directions an article could go – just off the top of my head:

(A) Jordan Ellenberg does a great job explaining the sphere packing problem and the connection to things like the Leech lattice and Hamming codes in How not to be Wrong,

(B) John Cook and Keith Devlin both have recent blog post with connections to higher dimensional spheres / cubes:

The empty middle: why no one is average by John Cook

Theorem: You are Exceptional by Keith Devlin

(C) Two years ago, Steven Strogatz shared this wonderful paper on N-dimensional spheres:

(D) The 2-dimensional problem of circle packing is something anyone can understand and is pretty fun to play with – here’s an old project I did with the boys using disc golf discs, for example:


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Sphere packing (well . . . circle packing)

Also, a version of the circle packing problem was in Jim Propp’s most recent blog post about mathematical thinking:

Believe it, then don’t: Toward a Pedagogy of Discomfort

So – come on professional mathematicians!! – here’s a great opportunity to promote a neat result and bring some really cool math to the public’s attention. Don’t let the physics crowd have all the fun!

A few articles that I’ve seen:

On Gil Kalai’s blog:

A Breakthrough by Maryna Viazovska lead to the long awaited solutions for the densest packing problem in dimensions 8 and 24

Kalai’s blog post also led to a question on Quora:

Why is the solution in dimension 8 such a breakthrough?

Proof in math

There are been three recent pieces about mathematical proof that have caught my attention in the last couple of months:

(1) Most recently Keith Devlin’s piece here:

(2) Evelyn Lamb’s coverage of Leslie Lamport’s talk at the Heidelberg Laureate Forum (including, importantly, a link to a paper in which Lamport takes a critical look at a proof in Michael Spivak’s Calculus textbook)

A Computer Scientist Tells Mathematicians How To Write Proofs

(3) Numberphile’s “All Triangles are Equilatleral” video featuring Carlo Séquin whose Art and Math collection is always a pleasure to look through:

These three pieces have kept me thinking about proof in math for a while now. I was sick over the weekend and spent a little time browsing through the Museum of Math’s public lectures and found this nice one from Steven Strogatz where, by happy coincidence, the topic of proof in mathematics also comes up:

In this lecture Strogatz discuses a proof of the fact that the area of a circle is equal to \pi r^2. He also wrote an article about the same proof in the New York Times:

Steven Strogatz discusses the proof that the area of a circle is \pi r^2

The combination of the recent writing on proofs in mathematics and watching Strogatz’s lecture gave me an idea for a fun way to talk about proofs in mathematics with my kids. Not in any formal way, but (1) just to show them some easy “proofs,” and (2) to show them that it is ok to question something even if it is supposedly a mathematical proof.

I’ve used this format for a talk previously after learning about Jordan Ellenberg’s concept of “algebraic intimidation” (and you’ll see in our first video from today how much that talk still bothers my younger son!)

Jordan Ellenberg’s “Algebraic Intimidation” and the series 1 + 2 + 3 + 4 + . . . = -1/12

The two topics for today were (1) the proof that Strogatz used to show that the area of a circle is \pi r^2 and (2) the proof that \pi = 4. The proofs are actually quite similar and it turns out that one of them ends up with an incorrect result (though I won’t say which one!).

At the end of each proof I asked my kids what they thought. Funny enough, my younger son does not believe either of them because he is uncomfortable with the use of infinity in the proofs. My older son believes the first one but not the second one.

Here are those two talks – the first is about the area of a circle:

and the second is about the value of \pi:

So, a fun morning with the boys talking through a few “proofs.” I really like the lessons in both Devlin’s piece and in Strogatz’s lecture about using proofs to tell stories and to illuminate. Lamb’s piece about Lamport reminds us that details are important, though, and the two pretty similar proofs I went through with the boys this morning serve as a reminder that the details can actually be pretty subtle (but fun, of course).

The balance between “answer getting” and “mathematical thinking”

So – two similar posts on twitter this morning just a few minutes apart:

David Coffey here:


Keith Devlin here:

I struggled with the balance between answer getting and mathematical thinking just this morning with my older son.  I had him working through an example problem in the “angle-angle similarity” section from Art of Problem Solving’s Geometry book.   This is a topic that he’d not seen before, but he seemed to be able to work through the first couple of sample problems reasonably well.  The problem we recorded gave him a little more difficulty, though – particularly when it came to identifying how two similar triangles would match up.

I’m not satisfied with how I interacted with him in this problem and that dissatisfaction has been nagging at me all morning.  I’m not sure what specific thing I should have done better, but I definitely wish I had found a better balance between the answer getting part of this problem and the math ideas.

Echoing and maybe amplifying a point in Keith Devlin’s latest article

Today (August 1, 2014) Keith Devlin published a nice article touching on both mathematical ideas and math education:

Most Math Problems Do Not Have a Unique Right Answer

If you have any interest at all in math or math education this article is well worth the 5 minutes it’ll take to read. I want to extend one of the points in the piece a little, though. Maybe it was my own poor reading of what he wrote, but I felt that one of the ideas near the middle of the article did not come through as clearly as I would have liked:

“Knowing how to solve an equation is no longer a valuable human ability; what matters now is formulating the equation to solve that problem in the first place, and then taking the result of the machine solution to the equation and making use of it.”

I think that Devlin has condensed a lot of information into the phrase “formulating the equation” so I’d like to un-condense it a bit. I worry that it is easy to think (as I did at first) Devlin’s comment means that you just write down some equations, head off to the computer, get your results, and charge full speed ahead.

The mathematical thinking that Devlin focuses on lies in understanding the results, not in simply writing down the equations. Rather than discussing this idea in the abstract, though, I want to give some fairly concrete examples from my own life and development in understanding this process.

The first example comes from my undergraduate thesis in college. In the spring of 1992 researchers (who would go on to win the Nobel prize in physics for their work) released the first detailed map of the cosmic microwave background radiation. I sat in the auditorium for this presentation and couldn’t help getting caught up in the excitement. After the talk I asked my adviser, Professor Ed Bertshinger, if I could try to study some piece of these results for my undergraduate thesis. He liked the idea and mentioned that he’d been wondering how the microwave background radiation would be distorted if the entire universe was rotating. Seemed like a fun problem – especially since my interest in physics was more math-y than hands on physics-y, so I dove in.

After 6 months or so I presented my findings to Professor Bertshinger. I was quite happy to have both an equation and some computer modelling. Though we had spent plenty of time talking over the course of the year, this was the first time that he saw the final results. Upon seeing the equation he drew a couple of diagrams on his chalkboard and eventually arrived at a picture nearly identical to the one drawn by my little computer model. It was stunning for me to see – six months of work for me and he drew the damn picture in about a minute just by waving his hands! Given enough time I could write down the equations, but he understood them. Seeing first hand the difference between those two acts was a powerful lesson for me.

The next examples are two fun billion dollar prize promotions that I’ve have come across my desk at work. The first was Pepsi’s “Play for a Billion” in 2003 and the second in Quicken’s billion dollar bracket game from this past spring.

In the Pepsi game 1,000 contestants took a guess at a six digit number from 000000 to 999999. So, 1,000 guesses at a number selected at random from one million possibilities. The chance of someone guessing correctly was 1,000 / 1,000,000, or more simply 1 in 1,000. As long as you’ve got the right security to prevent cheating, the math behind this game is not particularly hard (or even particularly interesting). This is a case where understanding the formula and solution does not take much time.

The Quicken promotion involved contestants trying to predict the outcome of 63 basketball games. Since you don’t know the precise chance of picking the outcome of any one game, the math you need to use to understand this promotion is pretty different than the math you needed for the Pepsi promotion. A purely formulaic approach is going to have a variety of problems – not the least of which is small changes in your assumptions lead to large changes in the estimate of the odds of someone winning this promotion. Worse, it is not at all obvious what the correct assumptions should be to begin with! Nonetheless, given the amount of money on the line, you need to be confident that you have analyzed the problem correctly and that’s where the mathematical thinking comes into play. Many of the articles about this promotion assumed that each game was a 50/50 chance, and thus sadly missed an opportunity to write a really neat and math-related article. Working through the various mathematical ideas behind this promotion (even the “birthday paradox” came up!) was one of the most interesting problems that I’ve ever worked on.

I guess the last example that’s not too hard to understand comes from problems in the financial markets. Roger Lowenstein’s incredible book “When Genius Failed” talks through the problems which arose when just one hedge fund got into trouble in 1997. More recently, we are all probably way too familiar with the problems that arose in the financial markets in 2008. Not all, but many of these problems came from groups of people using mathematical models and formulas that they did not fully understand.

The financial crises, and all of the terrible consequences that resulted, probably explains better than anything else why I wanted to clarify Devlin’s point. When Devlin writes about mathematical thinking, he isn’t talking about just writing down equations. Anyone – from undergrads writing about physics, journalists writing about basketball promotions, to derivative traders playing with other people’s money can write down equations and play with computer models. If we don’t understand the equations, solutions, or limitations, we get into trouble. That trouble can range from writing a dull senior thesis to causing the collapse of the world’s financial system. When we learn a little bit about mathematical thinking, hopefully we’ll do neither!

Keith Devlin’s comments on Dan Meyer’s circle problem

Earlier in the week I wrote about a neat geometry / algebra problem Dan Meyer had posted on his blog.  That post (including the link to Dan’s blog) is here:

It had been both fun and interesting to see all of the comments that have been posted to Dan’s blog about the problem during the week.   This evening Dan posted some updates to his original post including the following commentary from Keith Devlin:

I immediately drew a simple sketch – divide the interval, fold a square from one segment, wrap a circle from the other, and then dive straight into the algebraic formulas for the areas to yield the quadratic. I was hoping that the quadratic or its solution (by the formula) would give me a clue about some neat geometric solution, but both looked a mess. No reason to assume there is a neat solution. The square has a rational area, the circle irrational, relative to the break point.

So in the end I just computed.  I got an answer but no insight.  I guess that reveals something of a mathematician’s meta cognitive arsenal.  You can compute without insight, so when you don’t have initial insight, do the computation and see if that leads to any insight.

Having personally found some of the math hiding in the problem to be pretty neat, and in particular fun to talk through with a kid learning math , I wanted to take an extra few minutes to take sort of the opposite side of Devlin’s argument and advocate for something neat relating to  math I saw in the problem.

First, a quick (50 second) review of the problem:

Next, a quick solution (again, under a minute).  This isn’t meant to be instructive, but rather just a quick review of one possible solution where something interesting (I think anyway) comes up:

Finally, about 3 minutes of advocating for what I thought was particularly interesting – how does the second solution we got from the quadratic equation enter into this problem?

So that’s why I wanted to advocate a little for this problem.  Thinking through some of the geometry that comes up from the two solutions to the quadratic equations can be an important (and fun) exercise for students.  The significance of the first solution is clear – it is right on the line after all.   The meaning of the second solution is less clear and easy to ignore.  However, extending some of the geometry videos / apps posted in the comments on Dan’s blog would show that locating the point P at this second location also produces a square and circle having the same area.  This second solution naturally leads to asking what other solutions might exist with P not on the original line segment.  It turns out there’s a really cool geometric surprise hiding in those solutions.  How fun!

All in all, a really neat problem.