Thursday, January 12, 2017

Mathematical Autobiography

For Tracy Zager's amazing new book (book, FaceBook, forum, Twitter), she's asking for mathematical biographies. I used to ask my preservice teachers to do that, but haven't in a while. Thinking that I'm going to again for this read... so I should, too. I'm not sure what lessons there are to glean from it, but we don't get to choose our story!

My home was centered on art and literature. Father a lawyer, mother an artist, both avid readers. (When we had to clear out their house there were at least 20,000 paperbacks in the attic. Crazy.) So I always loved art, reading and writing more than math. Science I loved, though, and my parents were generous with books and museums for it. Math, I was good at, but it was boring. And more so each year. I was a competitive little jerk in elementary, though. In third grade I poked a pencil through my finger when I was peeved at missing an answer on a timed test. That was pretty much the end of the competitiveness.

Math got more and more boring as it went on into middle school, because there was so much repetition. I didn't understand why we did the same ideas every year. The details were barely different, but the same ideas over and over. And the lessons day to day involved so much repetition. I was lucky to have the kind of brain that this stuff just stuck. Although that made homework feel like hitting your head against a wall. But then we had an experimental self-paced program in 7th grade and I got to do 2 years of math in one. Only had to take assessments, so practice didn't have to be repetitive.

Bad news was in 8th grade my folks switched me to a small Catholic school. (In preparation for going to a Catholic high school; my father was in the first graduating class and my grandmother helped found it. Not optional.) The math was entirely repeat, so after a month they arranged for me to take algebra at the nearby junior high. I got the book and the assignments, and tried to catch up on my own. Without reading the text. Are you kidding? I was amazed at how long the homework was taking. I was good at guess and check, but that was so slow. The first day the teacher was doing the problems that people had put on the board. The first problem she wrote the equation, and subtracted something from both sides...

... and the heavens parted. I still remember that feeling 40 years later.

I enjoyed the math a little bit more after that. The ideas had gotten more interesting. But the homework was still terrible and classes excruciating. There was no AP at my small high school, and I got to go to the community college for calculus. Best thing about that was time with my friend Mark who was in the same boat. Class was uninspiring and I got an uninspired A-. Plagued by falling asleep in class most every day. (A problem that continued through all my schooling and still today in some meetings, church services and watching tv. It was me, not the teacher. I apologized but...)

My guidance counselor hated me for some reason, and never filed the forms for transfer credit that he was supposed to do. (More troubling was the request for scholarship info he never filled. He was the yearbook advisor and tried to convince my parents that I was failing that. Weird little monk he was.) But that was my big break. Michigan State placed me in honors calc, and I got to meet John Hocking. He was a real mathematician and shared topology with us. He convinced several of us to switch to or to add a math major. Because there was all this math we just had to know. Bill Sledd, John McCarthy and why can't I remember the name of my awesome tensor calculus prof? Awesome profs, and choosing math teaching over physics lab assistant for a job sent me off to grad school in math. (After a year doing art in Spain... story for another day.) I was going to still do cosmology or super string theory, but just come at it from the math side.

In grad school at Penn, my future advisor was our analysis prof, Nigel Higson. Awesome mathematician, barely older than us, fun and inspiring. When he got hired away by Penn State, he let me follow. When I was considering quitting to go get secondary certification, he encouraged me to finish - "you're so close, and you never know what it could lead to." Right as always, Nigel. Nigel's enthusiasm and curiosity for math are still inspiring me. But it was also then I saw the next level. His view of what was true and how things worked were beyond me. I could do Ph.D. mathematics, but I didn't have the drive and/or capacity for results that birthed fields of mathematics or got published in Annals. But to get to the point where I could see that... I'll always be grateful. Invited to dinners with Field medal winners who were also charming company? That was only going to happen at Nigel's house. Not to mention getting to hang around the effervescent Paul Baum.

My last years at Penn State were also when I got introduced to math ed, by my friend Sue Feeley, who was a math ed Ph.D. student. Putting Polya into someone's hands is a dangerous gateway book, Sue! I was trying to reform a math for elementary education class, and started to find out what I should be doing to teach. Blew my mind. Teaching went from something I liked a lot to my first love. And teacher's mathematics along with it.

Yotta, yotta, yotta, 20 years later, badaboom badabing, here I am. Loving math, math art, math games, math history and loving the teaching of it.

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." :)