A nice math-y present for kids: Grime Dice

A while back I gout a couple of sets of non-transitive dice for the kids to play with. Our project on the set of “Grime Dice” is here:

Non-Transitive Grime Dice

About a week ago my younger son found the dice sets and has been playing some sort of game with them. For our project today I asked him to explain his game.

It looks like the second of dice (the 4 white ones) are the so called Effron’s dice. I’m not sure where I got them, but the Grime dice are easy to find on line.

Our original project shows the one bit of math fun that you can have with the Grime dice. The non-transitive property that they have is really surprising.

The game my son has been playing shows what a kid might do just messing around with the dice. It is fun to hear him talk about some of the things he’s learned – around 4:00, for example, he talks about some of the dice having “an advantage” over other ones. He also remembers the strange non-transitive property.

You never know what is going to grab a kid’s attention!


Some thoughts on intro to analysis 25 years later

Saw this neat tweet from Steven Strogatz last night:

The short summary of his analysis course made me wonder how much an intro analysis course has changed in ~25 years. So, on an early trip into Boston today, I grabbed my old copy of Rudin and flipped through it. To me it looks like Strogatz’s course and Dan Strook’s back in 1991 were remarkably similar. The one topic on Strogatz’s list that wasn’t really part of Strook’s course is Fourier Series, but that was actually the next class in my undergrad analysis sequence so I’m not too surprised it wasn’t covered.

Anyway, with about 30 min to write something, here are some of my random thoughts I had about intro to analysis and intro to proof 25 years after I took that class . . . .

The reason that Strogatz’s post caught me attention is that I’ve been thinking a lot about my own math education over the last couple of years, and about that intro analysis course in particular. Some of Cathy O’Neil’s ideas have been really helpful to me – see here, for example:

How to teach someone how to prove something

She also had some great thoughts on the topic of an “intro to proof” class during Harvard’s Gender Equity in Mathematics discussion.

Two other pieces that got me thinking about ‘intro to proof” ideas are (i) Numberphile’s famous 1 + 2 + 3 + . . . = -1/12 video:

(plus the important Follow up video from Ed Frenkel )

and (ii) Jordan Ellenberg’s How not to be Wrong. In the discussion of series in that book (and I’m going from memory here, because I didn’t think to grab this book on the way in) he uses Grandi’s series as an example of defining a seemingly-divergent series to have a value. The idea that struck me (again, from memory so not quite a direct quote) went something like this:

“We can say the series 1 – 1 + 1 – 1 + . . . . diverges, or, if we want to define the series to have a value, the only value that makes sense is 1/2.”

Now, I understand that one of the main points of an introductory analysis class is to learn basic techniques of analysis and I’m not even remotely questioning that. Looking back, though, I wish I would have seen ideas similar to what Ellenberg talks about with Grandi’s series or that Numberphile presented with the -1/12 series (though probably more like how the ideas were presented by Frenkel) in that course.

That there is a way to make sense of 1 + 2 + 3 + . . . = -1/12 has to be at least as interesting as the idea that 1 – 1/2 + 1/3 – 1/4 + . . . = ln(2). In fact, after seeing the Numberphile video I went straight to the library to grab a copy of Hardy’s Divergent Series book to see what the hell was going on – I’d never seen that idea before!

Anyway, I think students would find it extremely interesting to see that there’s a bit more to series than meets the eye. Even in areas that seem as clear cut as convergent and divergent series, mathematicians have found some amazingly interesting things to say!

Now, back to the general idea of an “intro to proof” class. I know that I had to work super hard to understand proofs as an undergrad. In fact, it is a nothing short of a miracle that I got a B in that class. In terms of helping math students learn about proofs, I love the idea behind the two classes that O’Neil got going at Berkeley and Barnard. One point from the post I liked above that was completely missing from my intro analysis class is this one:

“But of course the most important thing was that I clearly stated at the beginning of each class in the first two weeks that proving things in math was a skill like any other that you get good at through practice.”

I think you can go even further than that, though, and give students a peek at the process that mathematicians going through in their work. This Numberphile interview with Ken Ribet is something I wish I would have seen when I was younger (and I know the topic isn’t analysis, but I don’t think that matters all that much):

To me, seeing the ideas about collaboration, working and revising, failing multiple times, and even not quite understanding that you’ve understood something (!) would be eye-opening for kids learning about proof. It would, I think, plant the idea that when you are struggling to understand this theorem of Abel (which, I think is the one from “Abel’s test” in Strogatz’s note):

If the series \sum a_n, \sum b_n, \sum c_n converge to A, B, C, and c_n = a_0 b_n + \ldots + a_n b_0, then C = AB.

you’d know that Abel had to do a heck of a lot of hard work to understand it himself!

Henri Picciotto’s factoring activity

[sorry for the tired write up – I’m a little sick today]

Saw this neat tweet from Henri Picciotto last night:

At the end of the blog post is this activity:

Geometry and Graphing Connection by Henri Picciotto

and the last page of this activity really caught my attention. I’d been doing a bit of work on factoring and completing the square with my older son anyway, so I thought I’d give it a try with him tonight.

One warning is that I’m pretty sick and out of gas today. I wanted to see how he’d react to using factoring ideas with geometry, but I didn’t have the energy to pursue all of the ideas. That said, I’m actually very happy with how he reacted to the ideas – especially with little help from me since I was fully in the useless zone tonight. We’ll definitely be revisiting this activity.

I started off with a quick introduction to how the blocks could be used to represent the various pieces of a quadratic polynomial. He seemed to catch on fairly quickly using the example x^2 + 3x + 2:

At the end of the last video he’d picked a 2nd example quadratic with negative coefficients. I wanted to do an example with negative coefficients, but I wanted to wait to do that example later in the project.

Here he pics the example x^2 + 5x + 6 which makes another rectangle. He’s able to see how the blocks show that this quadratic factors into (x + 2)(x + 3).

The next two examples showed what this method would look like for a perfect square, and how it could be used for completing the square.

Now we returned to a polynomial with negative coefficients. The geometric ideas here were a little harder for him to see. I need to find a better way to communicate the ideas for this part so that he sees the -2 as -3 + 1:

My only idea for helping him see the geometry in the case with negative coefficients was to use an example that was a little easier. That example was x^2 - 2x + 1. He did seem to see the geometry here, which made me happy. I was especially happy that he saw how a block was taken away twice.

So, I think this is a really great project from Henri. I’m really excited to try some more with it this weekend.