A lucky save courtesy of Khan Academy

We are in the process of moving right now and I discovered last night that I accidentally moved the math books up to the other house. Whoops!

Because of that goof up by me, this morning I had my older son play around with spheres, cylinders, and cones on Khan Academy. Turned out to be a lucky break as those exercises revealed a little gap in his understanding that I hadn’t noticed previously.

The exercises themselves aren’t necessarily super special or anything, but they were different enough from the exercises in the Art of Problem Solving book to reveal this little gap. The problems he was working on were from this section:

Khan Academy problems on volume

The slight difference between this problems and the ones in our Introduction to Geometry book was that (most of) the problems asked you to round the answer to the nearest integer. One of the answers was something like My son first rounded 100 / 3 and then multiplied by on this one.

It was interesting to see the difficulty he had seeing as a number rather than as a symbol. I’m happy that these problems from Khan Academy helped me discover and address (hopefully!) this issue.

Here’s the last problem we did this morning.

So, a fun and instructive morning for both of us. Lucky to have a positive outcome arise from leaving the math books in the wrong house!

Kate Owens on concrete vs. abstract

Saw a fantastic sequence of tweets from Kate Owens yesterday:

+part of what I aim to teach is the jump into abstract thinking & into abstraction of theory, apart from any particular application.

— Dr. Kate Owens (@katemath) May 19, 2015

I think about (and struggle with) similar ideas constantly and try to make sure that I show the boys some abstract ideas from math regularly.

One of the pieces that got me thinking about this subject more carefully is Numberphile’s interview with Ed Frenkel from last year:

I love Frenkel’s “painting the fence” analogy, and I super duper love his simple idea: “So how do we make people realize that mathematics is this incredible archipelago of knowledge?”

Since seeing this interview I’ve been trying to pay much more attention to the math that is being shared both in popular culture and online, and I try to mold some of these abstract mathematical ideas into fun projects for the boys. It doesn’t always work, but for the most part I think they have enjoyed these projects which are far from what they see in their school math books.

One of the first tries at sharing some abstract math was using what Jordan Ellenberg calls “algebraic intimidation” to talk about some infinite sums, including the sum made famous by Numberphile’s video last year:

1 + 2 + 3 + . . . . = -1/12

Jordan Ellenberg’s Algebraic Intimidation

Also last fall I happened to find a link to the public lectures that mathematicians have delivered at the Museum of Math in New York. Terry Tao’s lecture – “The Cosmic Distance Ladder” – inspired three projects with the boys:

Three projects from Terry Tao’s MoMath lecture

Another MoMath Lecture – this one from Bryna Kra – turned into a fun project with snap cubes and angry birds!

Using Bryna Kra’s MoMath lecture with my kids

Along the same lines as the MoMath lectures, Jacob Lurie’s public lecture after winning the Breakthrough Prize is a beautiful way to share some profound mathematical ideas with kids:

Using Jacob Lurie’s Breakthrough Prize lecture with kids

Recently I’ve seen two amazing pieces of math that University of Colorado professor Richard Green has shared on Google+. The remarkable problem in the tweet below even attracted the attention of Tim Gowers – so click to Green’s post to see that comment!

Some remarks by me on @Mathistopheles's very interesting blog post about the number 82000: https://t.co/gRlE6aUjF7

— Richard Green (@edfromo) May 17, 2015

Another great piece of math to share with kids from Richard Green

Of course it isn’t just math professors sharing great ideas. Fawn Nguyen is a constant source of inspiration:

A 3D Geometry proof with few words courtesy of Fawn Nguyen

and I cannot wait to give the project she shares in this blog post a try:

Fawn Nguyen share’s John Conway’s rope game

Tina Cardone shared a neat geometry problem last fall that turned into a fun 3D printing project for us:

A cool geometry problem shared by Tina Cardone

(our blog has probably 20 more 3d printing projects inspired by Laura Taalman, Steven Strogatz, Patrick Honner, Evelyn Lamb, and others!).

So, I was glad to see those tweets from Kate Owens yesterday. They made me think finding more ways to share some abstract math ideas that kids don’t usually get to see. I was also happy to see that other people are thinking about how to share more abstract ideas with kids and students, too. I really do think that people learning math not only benefit from seeing these abstract ideas, but that they also really want to see them.

Shortly after I saw the Ed Frenkel interview last year Dan Anderson shared a wonderful project that he did with his students – Just click through to Dan’s blog to see the list of topics the kids chose to see why I really believe that kids want to see more of these ideas:

A list Ed Frenkel will love

Looking forward to the next fun idea I find online 🙂