Tuesday, June 21, 2016

Book Club - Summer 16

In my math capstone class, the students can pick their own book from a list. Then we have a day for book chat. These are my notes. Links on student names go to their reviews on their own blog.

Journey through Genius, by William Dunham: Nick. Read most of this... explores a handful of the most important theorems and proofs from math history. If you're reading this, the book won't be that bad. If you're trying the proofs, it can be very difficult, and I wouldn't recommend it. Dunham claims that Archimedes is the greatest of the Greek mathematicians, the crown story. Personal stories of the mathematicians, too; for example, Cardano's tortured life. Not a lot of fun reading, but some really good explanation of why they use the methodology.

Love and Math, Edward Frenkel: Rebecca, Kourtney, Erin. "It's hard work being a teacher..." Great about the hard work being a mathematician, and the difficulties in being Jewish in an anti-semitic world. He had to literally scale the walls to get his math education. He was also moving in discussing how important collaboration is, and how important making mistakes is. The math is really high level - we could see some familiar things from abstract algebra, but there were good analogies for a lot of the ideas. "Where does love come in?" Love of math. Started in physics, but was convinced to go deeper, mentor by mentor.

The Math Book, Clifford Pickover: Anthony. Each page is like a wikipedia article about an idea, but nothing in depth. References are given so if you want to go deeper you can. "The integers came from God, and all else are hand made."  - Kronecker. It covers big ideas, inventions and famous mathematicians. Lots of fun ideas, like the birthday paradox or the infinite monkey theorem. The Johnson Theorem, riddles like the barber paradox. History of zero... I'd recommend this for teachers for the history.

Joy of X, Steven Strogatz: Jordan Drake, Nick. Easy read, a narrative. He explained negatives, but noted how we as a society avoid them (building floors, bank statements, temperatures). Strategies for finding your soulmate. What made it so easy to read was taking complicated ideas, like sine and cosine, but then gives real life examples and good visual images to support it. I went through math doing it because I could, but this gets at why these things work or are true. He gives a lot of practical applications. He gives the reasons for "why are we using this?"  Really fun to read with a lot of 'aha!' moments.

e: The Story of a Number, Eli Maor: Marty. I thought the whole book would be leading up to e, but we see it already in the third chapter. It starts with Napier, goes through logarithms, explores finance, and then calculus (Newton and Leibniz), ... It was at times interesting and boring. The most beautiful formula which makes a connection with imaginary numbers. Lots of appendices of intense proofs.

The Number Mysteries, Marcus du Sautoy:  Heather, Brianna
5 different math mysteries, got into the history of the ideas,
  • primes; the building blocks of all numbers.
  • geometry; nature is as efficient as possible.
  • tricks for games; confusing but interesting, Monopoly and more.
  • codes; Everything is a code, languages, DNA, ISBN, modular/clock arithmetic.
  • prediction; patterns are detectable, which make things predictable. Seasons lead to the calendar, etc.
"Coolest thing?" How I could relate math to all these things I had never noticed. There was just so much information.

Mathematician’s Lament, Paul Lockhart: Hannah.  K-12 math needs to be scrapped. Math is an art, it's about playing and imagination. Instead, teachers give facts and formulas for memorizing. Takes away the creativity and engagement of solving. Teachers try to relate it to life when it doesn't. It can be fun because it doesn't relate to your life. A good problem is anyone you don't know how to solve. You want students to struggle and be frustrated. Geometry is the most destructive because it destroys proofs. Instead of being charming, it's a boring list. Write a paragraph of a proof, tell the story of your thinking. Why can't 1 + 1 = 0? Even/odd + even/odd, sum of odd numbers, ... so many ways to reveal the nature of math. "As a future teacher did you find yourself agreeing or disagreeing?" The get rid of the curriculum and let every student figure out what they're working on - I disagree. But the emphasis on memorization needs to go. "Math can be fun when not related? That's really clever. Counter to the message we get."

The Calculus of Friendship, Steven Strogatz: David. Less of a math book, and ever increasing life lessons. The teacher retired and became a famous white water rafter, which is connected with limits and infinity. Irrationality, chaos theory, etc. The monk and the mountain. Will a monk who walks up the mountain and down in irregular patterns ever be at the same point at the same time. Inspiring about going into being a teacher and the effect you can have. Readable even if you don't know calculus.

Mathematical Mindsets, Jo Boaler: Michelle, Tabatha Lathrop.  Growth mindset = you can get better at how well you learn things, fixed mindset = you can learn things, but you can't change your intelligence in an area. This really affected me. The brain research is interesting; when you're making mistakes is when you're learning.  Feedback makes a big difference. Then she connects with math mindset. Most effective teaching is when learners explore the question, and then get  explanations of how and why. Using what the kids came up with is helpful, with engagement to start. Students will say they don't like math because it's too much answer time than learning time. The faster they can do math the better, kids think, when the reality is almost the opposite. Kids don't ask 'when are we going to use this?' in other subjects. I am literally a different person because I read this book.  (Others connected to Carol Dweck's Mindset in response.)  It's interesting as an adult learner, trying to think about where you're fixed or growth.

The Magic of Math, Arthur Benjamin: Andrew Meeuwsen. Topics align well with the course, but not a lot of the fun history. Lots of worse than dad jokes: mathematician dad jokes. Lots of tricks for doing specific problems. Many connections to his mathemagic show. FOIL, squares, magic of 9, magi of counting, magic of proofs... a lot of good math, but a lot of filler, too. The book has a steep slope, from arithmetic to calculus. My favorite was about infinity. It covers a lot of the subjects from undergrad mathematics.

Jerry missed the discussion, but has a review of The Calculus Gallery.

Sunday, June 19, 2016

World Tessellation Day 2016 Gallery

I got to hand draw a couple tessellation attempts for a service project Playing with the patterns later made two that I liked.

There's a classic semiregular tiling pattern with squares and equilateral triangles. I wondered if you could make one that had three triangles at each vertex but in different combinations.

Turns out maybe not. Everything I tried wound up with a spot with 6 green triangles. But I did find a new to me combo. I like that it has one 60, one 90 and one 120 degree angle at each vertex. (On MathToyBox)

The other one was built around a kite with a 90 degree vertex. I had to make that one afterward in GeoGebra. Not a lot of flexibility in design, but I liked the square & rhombus gaps.

Some of what I saw around Twitter and Facebook was just so delightful, I wanted to archive it.

Another great Cristóbal Vila video, Ars Qubica, via Daniel Ruiz Aguilera.

The founder of this here holiday sharing pics from a tessellation get together. With excellent toys.

I guess math doesn't suck!

Some excellent tiles. That's the Cairo Tessellation at right!

Now some awesome environments...

And nowsome action shots! Love the ones from Simon Gregg's class especially!

Happy World Tessellation Day! See you next year.

No better way to end than Jennifer Silverman and Steve Vai shredding!

Friday, June 17, 2016

World Tessellation Day One

Emily Grosvenor came up with the idea of a World Tessellation Day in connection with her charming children's book, Tessalation! June 17th is M. C. Escher's birthday (1898) and there could be no more fitting day.

Tessellations are definitely my favorite topic in mathematics. The intersection of history, art, geometry (shape and transformation), algebra, and even analysis... what could be better. Some of the greatest surprises in math have come from tilings (quasicrystals, pentagon 15) and some of the greatest mathart. I've seen them engage students of all ages.

For my post, I've been thinking about so many things, but that coalesced into a 'My Favorites' post:

My Favorite Tessellations

HM: pattern blocks.

From a recent class, Hannah made this neat dodecagon and octagon tiling. They remind me a lot of these from Simon Gregg and Daaniel Ruiz Aguilera.

10. Non-Euclidean Tilings

Hyperbolic, especially. Here's a beauty from John Baez's Google+ page.

9. Pythagorean Tiling

A tessellation that demonstrates the most famousest of theorems? That's saying a lot, that is.

8. Archimedean (Semi-Regular) Tilings

So what combinations are possible? Is this all of them? Could the semi-regular tilings be the first of these kind of problems?

And then you add the delicious topological feature of dual tessellation relationships...  The gif on the right is from thinking about a Sam Shah prompt on this idea. (On GeoGebraTube)

7. Pentagon 15

How deep are tessellations? They still surprise us. Every quadrilateral tessellates.

A monohedral tiling is a tiling where all the tiles are congruent. An isohedral tiling is a monohedral tiling where for any two tiles there is a symmetry of the tiling that maps one tile to the other. There are exactly three types of convex hexagon monohedral tilings. (Here's a good NRICH problem with one.) Every convex quadrilateral has a monohedral tiling. And we knew all 14 convex pentagon monohedral tilings. Several by one of my favorite mathematicians, Marjorie Rice. (Her website.)  And then they found the 15th. (GeoGebraTube

6. Pinwheel Tiling

Straight from the mind of John Conway.
5. Spiraling Polygrins

I went from fond of these to berserk when Christopher Danielson started making them. (On GeoGebraTube or TMWYK store.)

4. Rep-Tiling

When a tile can be composed to make a larger similar image of itself. Then it makes a tessellation by either deflating each tile into smaller images. Or inflating by composing larger and larger similar arrangements.

3. Penrose Tiling

These were my exposure to quasiperiodic tilings. There properties are many and wonderful. At one point I was stuck on my thesis and my advisor (Nigel Higson) gave me these to work on. My best ever Mathematica program generated them by projecting n-dimensional integral lattices onto an intersecting plane. For part my thesis I then made quasiperiodic integral operators out of them.

2. Islamic Tilings

Most recently, Daniel Ruiz Aguilera got me working on the Qarawiyyin Mosque Tiling. (GeoGebraTube) Endless riches with new work still being done. As a bonus, these are often interspersed with knotting, another favorite.

1. Escherized Tiling

Instead of mine, let me show some ooooold student work from a couple of preservice art teachers in one of my first courses taught at Grand Valley. I still keep these in my office.

Current: Self-tiling. Since Math Munch unveiled this great Lee Sallows self-tiling I've been curious. They deflate in only one way, but inflate in four ways - I can't figure out what that means about the structure. (GeoGebraTube)

So many types that didn't make the list. And despite the numbering, I'm just crushing on them all.

I hope one of these pave the way for you, or maybe showed you a new kind, or just reminded you of old favorites.

And happy first World Tessellation Day! Tile on, brothers and sisters.

Monday, May 30, 2016



Good for sports, bad for surgeries, okay for a test, depending on your standards.

I'm very grateful to Anne for starting #MTBoS30, and the early adopters that greased the skids for me to do it to. I've blogged more this month than I did all last year!

The posts I'm the most glad I wrote:
  • Queen of Quilts - just for the appreciation of Elizabeth, but I like that GeoGebra bit a lot as well.
  • Not Subtracting - such an opportunity, to tagalong in a conversation with Dan Meyer, Marilyn Burns and the MTBoS.
  • Commentary - for a reminder to myself to think this out. My colleague Clark Wells said this is what he was talking to me about, so there's hope for more local discussion.
  • More Tessallations - for the chance to share student work and math I love. Several students have blogged about it since, and I love that kind of resonance.
Amazingly I still haven't gotten to everything I wanted to write about. A couple of new games, some lessons from last semester, some hand drawn mathart, and the 3rd grade mathart project. (And the Hamilton song parody.)

My take away from the month is that I like blogging. Why don't I do it more?  I think because I've forgotten the reason I started, which is the reason I ask my students to write and try to get my kids to write. It is good for you on the atomic level. The conversation, the curation, the community - those are excellent side dishes. Writing makes you think.

So do I make a commitment to more? 2/month, 1/week, every Monday? I don't know - I'm not very good at these commitments. After not having given up the MTBoS30 each morning, I know I should be ready to be Dread Pirate Roberts. But I'll settle for being a member of the Goon Squad.

Sunday, May 29, 2016

All the Way

Missed another #MTBoS30 post yesterday, but it was in the service of a day chock full from 6 am to midnight, so no regrets. Time with one of the bravest people I've ever met, my son doing well on his first dan (permanent blackbelt) tae kwon do test, church, dinner with family...

With a free Sunday morning, we (as a family, even) finally got to watch the Lyndon Johnson biographical movie All the Way.

It's amazing.

I'm 51 (with considerable less grace and style than this 51 year old) and this was my birth year. I don't remember it, then, but this was the backdrop of my first memories. Kennedy and King being shot, Nixon, Vietnam and Watergate was what I knew about politics and government. It was amazing to watch this movie, with its decidedly modern viewpoint. It took decades for me to move beyond childish black and white images of these people and my black and white judgment of their actions. The filmmakers are good at filming what was actually said then in a way that makes connections to today possible.

Looking back, it's so easy to identify the right side of history. To see bigotry and name it when we are free of it (we think!). My spouse is excellent at challenging us (okay, me) to see what is inequitable now.

One of the things I enjoyed most was seeing Robert Moses portrayed as a young man, working for the Student Nonviolent Coordinating Committee. That fits seemlessly into his work as a math educator. The Algebra Project, interviews (on a Selma anniversary, NPR), or his book Radical Equation. He is one of my dearest heroes.

In All the Way, Dr. Moses is portrayed as too radical to effect change. To be so convicted to principle that he can not compromise for some gains. Dr. Moses makes direct connection between the idea of civil rights and the empowerment of mathematics education. It's so complicated, it could be easy to walk away, and understandable when people do. Education cannot solve poverty, but it's such a necessary part of any solution.

Are we not able to affect change because we need an LBJ? Someone with the conviction that can see a path to equitable education and is enough of an asshole to get it done? I think we are accountable both for holding and proclaiming the principles and doing the problem solving to get to a better place. But I am on one side of it, and often in danger of fighting the LBJs who are probably on our side.

I am humbled by how generally useless academics are in society.

One of the reasons that the Math-Twitter-Blog-o-Sphere (almost as ridiculous sounding as "snick") is so encouraging to me. We are self-organizing and devoted to the education of kids independent of what the government or publishers or pundits say. Now I'd love to see the NCTM play the role of the NAACP in pursuing systemic change, too, but I'll take what I can get. And this band of teachers, working one or six classes of students at a time is getting a lot. God bless you all in your work.

We will overcome.

PS. In the Selma anniversary interview, Dr. Moses is asked what he'd like to hear President Obama say in his address. Re responds: "I'd like to hear him speak about education. We can do all we want about voting and everything else, but if we don't provide an education for every child in this country that's what they need for the 21st century then we will just be sending them to the criminal justice system. We do not have in this country an education system that is dedicated to educating every child, so I'd like to hear him speak out about that." Me, too.

Friday, May 27, 2016


Not too much time today, so I'll just post the GeoGebra thing I've been working on.

I have a Tumblr account as well as this blog, it's a fun mathematical space, and the reblogging is an interesting spin on curation. Here's a post I wrote for a 'my favorites' at Twitter Math Camp 16 to help #MTBoS folk get started on Tumblr.

One of my favorite ever mathart sources was @tilman's http://geometrydaily.tumblr.com/.  There's a site that has a bit of a similar flavor, http://www.dailyminimal.com/.

The other day, they had this:

Nice, right? Rotating ellipses... what I always try to think of is how can this be generalized? Rotating conic sections (definitely sometime will do!)... what if it didn't just rotate around a point? What if it followed a curve? One idea I really liked from the dailyminimal was having a start and stop point. So ...

el bigote
You can choose how many ellipses, make your own parametric curve or hit the random button, design your own ellipse, use the red and green to pick starting and stopping point... hopefully lots of flexibility.

On GeoGebraTube, so go play. If you make something cool, let me know!

Thursday, May 26, 2016


In the Nature of Mathematics course we were talking about China today. The main activity was the students trying to figure out the nets that make the Liu Hui solids, which I learned about from Jennifer Silverman. It's wonderful seeing the students engage in 2D/3D thinking. Today we started with the Tangram instead of Magic Squares, because the students had been frustrated with Archimedes Stomachion. They were challenged, but successful, and we got onto some neat puzzles in some groups using multiple sets and making squares of different sizes.

Here's the handout:

But what I wanted to write a note about was the idea of commentary. Mathematics in China followed a bit different path than in other ancient cultures, perhaps because there was more prevalent instruction. Lost is the origin of their ancient text, Nine Chapters on the Mathematical Arts. Instead of the advance coming from a collater, the big jump was Liu Hui writing a commentary on the text. To get the feeling of it, I asked students to solve one or more of the sample problems:

Chapter 6:12. A good runner can go 100 paces while a poor runner covers 60 paces. The poor runner has covered a distance of 100 paces before the good runner sets off in pursuit. How many paces does it take the good runner before he catches up the poor runner.

Chapter 7:1. Certain items are purchased jointly. If each person pays 8 coins, the surplus is 3 coins, and if each person gives 7 coins, the deficiency is 4 coins. Find the number of people and the total cost of the items.

7:18. There are two piles, one containing 9 gold coins and the other 11 silver coins. The two piles of coins weigh the same. One coin is taken from each pile and put into the other. It is now found that the pile of mainly gold coins weighs 13 units less than the pile of mainly silver coins. Find the weight of a silver coin and of a gold coin.

Chapter 8: 1. Top-grade ears of rice, one bundle, medium grade ears of rice, two bundles, low grade ears of rice, one bundle, makes 39 dou. Top-grade ears of rice two bundles, medium grade ears or rice three bundles, low grade ears of rice one bundle makes 34 dou. Top-grade ears if rice one bundle, medium grade ears of rice two bundles, low grade ears of rice three bundles makes 26 dou.
And then to write a commentary on it. This is new territory. So I had them share in groups, and pick someone to share with the class.

Mostly what they shared was their solution, so I asked a commentary type prompt after each. Usually I'd be hesitant to put people on the spot, but these are senior students and we've had a pretty open classroom culture so far. I asked about solving with different representations, how you would describe in general the solution method, and an extension question about the mathematics. As they got into those discussions, it occurred to me that this might be a good framework for thinking about writing in our foundations classes. As the students discussed the idea of commentary, they noted that it seemed like a good way to draw attention to the idea of generalization, and a support for student reflection.

So thanks, Liu Hui! We'll see if I can get students writing their own commentaries on the mathematical arts.