Tuesday, December 6, 2016

Why Math?

I'm teaching a preservice teacher math for high school course this semester. You wouldn't know, since I've been so bad at blogging this semester. This is the best group of writers collectively I've ever had, I think.

There's an odd issue, though. They're already leaving the profession! Here's some last blog posts:

Then Dan Meyer had this amazing group keynote at CMC-some direction. We need math teachers to teach good reasoning so that people will not spread fake news. And a lot of other good reasons. And Bowman Dickson wrote his teaching philosophy, which motivates me just reading it.

I think about why teaching a fair amount, but don't know that I think about why math teaching. I'm so far in, there's no getting out. But what about our students? One thing I'm hearing more and more is how many people are telling young people to not go into teaching. But if they are persevering in pursuing teaching, why should they teach math?

Math is power for their students. If they are successful in math, their choices for future careers expand. If they learn the mathematical practices, they will be more successful in any career. But beyond that, it will support them in living a better life, making better choices and being more informed.

The very first course I taught (30+ years ago!), and I use taught loosely because I was not a good teacher, I was impressed by how after a good lesson, students could do something that they could not do beforehand. They had literally expanded their capabilities. What a privilege to teach a subject like that.

Math is beautiful. It's not often taught that way, but the sheer power of the ideas that underly what is taught is bewildering. The complexities of the infinitely small and large, the realms of pure thought can be traversed, and the ineffable mysteries of what is possible. WOW. Eugenia Cheng describes math as the logical study of logical things. How does that humble beginning become star-spanning cosmologies and quantum field theories? Jamie Radcliffe described math as a language in which you can only write poetry. There is some bad poetry, but the best has a power and grace that is preserved through the centuries.

I teach math because it is worth knowing and I want to share it. Because I want more people with whom to play!

Friday, September 9, 2016

Book Celebration

To celebrate the release of the newest great and greatest new children's math book... by which I mean Which One Doesn't Belong? by the #MTBoS' own Christopher Danielson, of course... I thought I'd recap some of my favorite math picture books. This list also has MTBoS support, as I solicited suggestions from the MTBoS for a colleague.

The request was for a parent with a mathematically curious child (really, could be anyone then, am I right?) of 4 or 5 years.


  • Moebius Noodles, Maria Droujkova's great book about big math ideas to explore. There were articles about calculus in kindergarten when it first came out.
  • Great new book: Which One Doesn¹t Belong. OK, I'll say more. I love this book because it's clever and pretty, but also because it can teach you how to read mathematically rich literature.
  • Math Curse, Lane and Scieska: just the best math book ever written. Nearly anything can be a problem, you know.
  • Anno's Mysterious Multiplying Jar, or anything by Mitsumasa Anno. Just charming books, and lovely besides.
  • Spaghetti and Meatballs For All, Marilyn Burns: my favorite of the explicitly mathematical genre. Tang and Murphy have their place but Burns is the queen of the genre. (Greedy Triangle, Smarty Pants, $1 Word...) 
  • Princess of the genre, Elinor Pinczes: One Hundred Hungry Ants, A Remainder of One, ...
  • Infinity and Me by Kate Hosford
  • Tessallation!  by Emily Grosvenor
  • Grandfather Tang’s Story, by Ann Tompert
  • The Dot and the Line, by Norton Juster


What to do:

Possibly for older, but like Madeline L'Engle, I think people underestimate kids:

  • The Phantom Tollbooth, by Norman Juster
  • The Man Who Counted, by Malba Tahan
  • Flatland, by Edwin Abbott 
  • The Number Devil, by Hans Magnus Enzensberger
  • The Adventures of Penrose the Mathematical Cat, by Theoni Pappas
  • The Cat in Numberland, by Ivar Ekeland and John O'Brien
  • A Wrinkle in Time, Madeline L'Engle (First I heard of a tesseract.) There's an audiobook where L'Engle reads it herself. Highly recommended.
And please add your own suggestions!


  • Cindy Whitehead saw that I missed the Sir Cumference books, by Cindy Neuschwander, and suggested the Go Figure books, by Johnny Ball.

Tuesday, August 16, 2016

Quilt Show

Lots of pictures of quilts!

I see this as just #mathart appreciation, but also can be some inspiration for lessons. In fact, #MTMSchat this month (August 17th, Wed, 9 pm ET) is on this math and quilt article:  Quilt Block Symmetries by Matt Roscoe and Joe Zephyrs.

Every year for our local Coast Guard Festival, the local quilting guild puts on a show. Elizabeth, quilting friend and queen, usually gives us the insider's tour. I didn't take a picture of every beautiful thing, but did get most of the math that caught my eye. First up, the fractal quilt about which I have that whole blogpost and GeoGebra.

 Lots of interesting design choices from Elizabeth here in addition to the neat pattern. Some of the squares even have lissajous quilting.

Of course, hexagon tessellation, but there's also a permutation aspect to this one. With the center the same color, how many fabrics do you need to make 81 different hexes? This would also be nice to do some edge identification for a toroidal quilt!

Apologies: forgot to get separate shots of the labels for these, two.

On the left (Whirligig by Marcia Knorr), I love all the different quilting designs in the same quilt. I was wondering if they were organized, or what classes you might use to sort them. Kind of a giant which one doesn't belong.

On the right, I love the creation of near circles overlapping, with the decahexagons. Also some nice positive/negative space. Really this is made with just square tiles (2 sizes) and 2 kinds of triangles!

The design here is really special. The small scale rhombus tessellation underlies using color to get the effect of rhombs of different scale overlapping periodically. What are the scale factors linking these different sizes?

I don't know if any of you have this problem, but I am an obsessive counter. This has a large number of balloons in groups of 3 and 9. Each balloon is distinct, and each group of nine is arranged differently! But I disagree with Mrs. Johnson's count. (There were no balloons on the back.)

There are designs that people learn from a teacher or that are bought and sold. These are classic op art, but the deformations are interesting to look at, as well as these three variations.

Another by Elizabeth. Amazing to me how the overlapping of two similar patterns creates a third repeating shape of a white T. This one is for her daughter Honore.

Another nice use of negative space to emphasize the pattern, and interesting choices about the size of the circle relative to squares.

Here are my quilt show compatriots, Debbie, Karen (mí esposa), and Filiz. The discussion about the quilts, and the different levels of expertise is a lot of what makes the show so fun. Debbie is an experienced quilter as well.

Pretty neat symmetry variations here, also feeling like a #WODB.

Made me think I have not thought about the tiling possibilities of right trapezoids enough.

These trapezoids are arranged to almost make a hexagonal spiral. The irregularity is the part of the charm of the quilt to me, and I like the comparison with the concentric circle quilting.

Another case of two simple patterns over-lapping to make a much more complex design.

 This reminded me of quarter cross, and I like the effect of dividing up the squares into squares or triangles and alternating them. I can't tell if there's a pattern to the color choices, but I think there might be.

At the Tessellation Nation session at TMC16 (best coverage: Joe Schwartz)  I got interested in these nonperiodic tessellations with rotational symmetry, so I liked this immediately, and THEN I noticed the SPIRAL. Immediate mathquilt crush. I also like the "making it your own" aspect.

Sometimes quilters exchange work, with restrictions imposed. In this exchange, the original square had to be moved off center by the later additions. The idea of riffing on someone else's design/math is something I'd like to bring into my classes.

Attendees were fascinated by the 3-D and labyrinth aspect of this quilt.

 The description of making this is what sold me here. A square is cut along the diagonal. A quarter inch border is sewn together with a 1 inch strip of fabric to make one bar of the X, making a half inch diagonal, and then the process is repeated for the other bar, resulting in a square of the same dimensions!

Elizabeth herself, with a quilt she liked. She likes spiky designs, and thought the use of 3/4 circles was interesting and placed interestingly.

Two neat tessellations, both playing with positive and negative space.

 How many equivalence classes of squares in this quilt?

This next quilt is another one interesting for the process. All the different squares are from the same fabric!

Last one: A year or so ago Elizabeth asked if it was possible to have a round robin for six people where each person worked on each quilt once, but on each exchange you received your quilt from someone new. I talked out the problem with my colleague Brian Drake, and it's now a problem in our discrete course. Hint: yes. I was doing it the same time my capstone students got absorbed in magic squares and there are great connections. Here are the round robin quilts. (My solution for six.)

Bonus material: I'm loving the quilting blog that Elizabeth recommended. Maybe start here or here if you're interested.

Saturday, July 9, 2016

Tess Elation

Special opportunity today, to meet and greet the author of a fun new math book.

Emily Grosvenor (twitter, website) is a "reporter, travel writer and essayist" who has gotten all the way to Mathland with her illustrated children's book Tessalation!. After a successful Kickstarter, the book is available for pre-order or as an e-book at Amazon (free for Kindle Unlimited) and direct order from Waldorf Books. I am between having received my electronic and my physical copy, and find the book just charming.

Emily was willing to answer a few questions, so here we go!

Q. Do you remember when you first noticed tessellations?
No. I first learned about them in 4th grade -- every part of my creativity seems to have it seeds in the 4th grade, it must be a seminal year for development. We did an activity in gifted class where we made tessellations. I made an uninspired tessellation of a seal jumping out of the ocean. But it was fun. And clearly it stuck, since I was still thinking about it almost three decades later. 

Q. Do you have a tessellation from someone else that you like especially? (Maybe a favorite Escher tessellation?)
I'm a big fan of Horseman But honestly, I think my favorite is just the simple hexagon tessellation. We just got bees at our home in McMinnville. I have great hopes I'll see one there, soon!  

Q. What makes tessellations worth thinking about and exploring for you?
I find patterns soothing to look at. Honestly, visual culture bothers me a lot. Usually there is so much going on, and I get distracted easily. But with tessellation you can take in the chaos and then let the eye, and the mind, settle on an individual part. Also, I am very compelled by the idea of seeing myself as a part of a greater whole. Not just with my family, but with my community. One of themes behind Tessalation! is that the world is not as chaotic as it seems, that there is an inherent beauty and order to it, and we can be a part of it. 

Q. What was your experience on the first World Tessellation Day?
It was a crazy day! My best friend was in town with her kids and husband and we threw a party at the McMinnville Public Library. We had a tessellation station, we screened the book on the wall, we had hexagon cookies, tiling turtles tessellated games, and coloring pages from the book. It was a BLAST! I was so tired. I probably should have been tweeting out tessellations all day, but there were a couple hundred people around the world who were posting images. In all, I was happy with the outcome. When I got a chance I checked in and retweeted, liked or posted what I could. People who like tessellations really love them. Also, it's a visual meme, which makes it easy to get behind.  
Emily created World Tessellation Day, and used it to launch the book. She has a fun post about the book launch here. The twitter stream for #WorldTessellationDay had a ton of fun participation from genuinely around the world.

Q. Why should it be an annual event?
Why should anything? It's fun. Fun to post, fun to make, fun to see all of the creativity happening around the world. I was most impressed by the posts coming in from Spain showing all of the tessellated mosaics available in plain view in public spaces. There are a lot of silly holidays. We share World Tessellation Day with National Flip Flop Day, for example. Who cares about flip flops? Well, someone does. If anyone cares about something there should be a day for it.
Didn't realize it was also flip flop day - despite Danica McKellar's tweet. Doh.

Q. What’s challenging for you when you are developing a tessellation?

I actually don't do a lot of designing tessellations. Notice I did not make the illustrations in the book, for example. But I did try to create the feeling of tessellating in the rhyme scheme and overall meter of the book. I wanted there to be a strong connection between how the text feels when read out loud and what you are looking at. Can words tessellate? I think I tried to do that. 
Q. Do you have a general process you follow?
The best part of this project has been how it has opened this entire world to me of math play, tessellation and visual culture. I launched this project thinking that tessellations are awesome but not really having any idea of the scope of talent out there or of the artists who are working in tessellation. I've been touched by people who have reached out from around the world to share in the excitement. But my favorite moments are when my 3-year-old, Griffin, finds them in plain sight. Just yesterday he got a new pair of Timberline sandals and said: Mama -- there's a tessellation on my foot! 

Thanks, Emily, for the book, and holiday and interview.

Oh! I should have asked how she got connected with her talented illustrator, Maima Widya Adiputri (Tumblr, FairyFrame).

Find out much more about this book from other stops on the booktour.

Find out more about tessellations from the resources on my page.

Or start immediately making your own! One of my most recent ones to play with on GeoGebra is a funky hexagon one, with a glide reflection similar to what the Horseman has.

#TessellationNation, now #tessnat, is coming at TwitterMathCamp16. Christopher Danielson was thinking it should be based on people's questions, so hop on Twitter to chip in, or share them here.

So far:
Christopher ‏@Trianglemancsd
We proposed this session as one revolving around our questions. Maybe you could share of those here before TMC?
I would like to learn more about how to categorize tessellations.
I wonder about the relationship between "tiling" and "tessellation".
I am super curious about the tilings in mosques. Are they tessellations? Why do they so rarely appear in the math analyses of tessellations I've encountered?
#tessnat There's a start on where my mind is for #TMC16. What about you, Tessellation Nation?

Malke Rosenfeld ‏@mathinyourfeet
1. Hi #tessnat. My goals: try & try again. I would like to play with diff kinds of tiles to help me ask new questions.
2. After I play I'd like to talk abt my notices/Qs and then design a tile that is simple but creates an interesting result/design #tessnat
3. I would also like to observe someone designing/creating an anthropomorphic tiling if that ends up happening. #tessnat

Megan Schmidt ‏@Veganmathbeagle
@Trianglemancsd OH!
I want to draw the things, whatever that means. #tessnat
Ok. My needs are "be in the #tessnat morning session." :)

Friday, July 8, 2016


Quick game idea dreamed up while watching tennis. I'm excited about it, but it's untested. Maybe someone will playtest with me at TMC16?

2 or 4 players (doubles, naturally)

Materials: deck of playing cards, score sheet.

Set up: deal each player 5 cards. Randomly determine who is serving first.

Tennis scoring:

  • In a game, you score love (0), 15, 30, 40, game. But games have to be won by two points. If you get to 40-40 it is deuce (tie). One point from there is Ad (advantage). If the player with ad wins, it's game. If the other player wins it's back to deuce. 
  • It takes 6 games to win a set, but you have to win by two games. 
  • If you get to 6-6, there's a tiebreaker. The server serves once, then players take turns serving twice. First to 7 points wins the tiebreaker and the set, but, everyone say it, you have to win by two points.

(Game design aside: having to win by two points is one of the best remove first player advantage mechanics ever. The tennis tiebreaker was also a great innovation.)

A match can be one set, best of three or best of five. I recommend playing one set to start.

Playing a point: players draw up to 5 cards. The server can play a card or flip over the top card. Each card played has to be higher than the card before. Players can play two cards together as a sum. If a player can't play or chooses to pass, the other player wins the point. The same player or team serves for an entire game, then the other player or team serves the next game. Continue alternating serve throughout the match. When the deck runs out of cards, shuffle the played cards to continue.

Special cards:
Ace: 1 on a serve (flipped from the deck) 11 any other time.
Face cards: count as 10, but can only be played as part of a sum. However, King beats Queen beats Jack beats 10, so Queen+2 beats 10+2.

Variation: card pairs are multiplied rather than added.

Doubles: in a doubles match, either player on a team can play a card, or play a sum, or a sum can be made with one card from each player.

Notes: The weird scoring is the downside, but kids learning tennis scoring is not a bad thing. And it's a classic for a reason. I'm really excited by the strategy here. When to give up on a point, how to get rid of low cards, which pairs to form and play.

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!