Sunday, February 26, 2017

Math in Action 17

A highlight of February in these parts is Math in Action. Our local, 1 day math fest. Having been at the U for 20 years now, part of it is just great reunion, with our former students coming back to present and knock 'em dead. The last two years have felt stepped up, though, with a keynote from Christopher Danielson in 2016 and Tracy Zager, the math teacher I want to be, this year.

After taking a year off presenting last year, first ever, this year I was back at it to talk Math and Art with Heather Minnebo, the art teacher at a local charter that does arts integration. I've consulted with her, she's helped me a ton and we get to work together sometimes, too. (Like mobiles or shadow sculptures.) The focus this session was a terrific freedom quilt project Heather did with first graders. Links and resources here.

Next up for me was Malke Rosenfeld's Math in Your Feet session. Though I've been in several sessions with her before, I always learn something new about body scale mathematics. She ran a tight 1 hour session using Math in Your Feet as an intro to what she means by body scale math. One of my takeaways this time was how she made it clear how the math and dance vocabulary was a tool for problem solving. I often think about vocabulary in terms of precision, so the tool idea is something I have to think about more. Read the book! Join the FaceBook group!

On to Tracy's keynote. She was sharing about three concrete ways to work towards relational understanding. (From one of her top 5 articles, and one of mine, too.)

  1. Make room for relational thinking.
  2. Overgeneralzations are attempted connections.
  3. Multiple models and representations are your friends. 
Illustrated by awesome teacher stories and student thinking. She wrote her book from years of time with teachers and students looking for real mathematics doing, and it shows.  Read the book! Join the FaceBook group!

Plus, just one of the best people you could meet. She gave her keynote twice, and then led a follow up session. One of the hot tips from that was the amazing story of Clarence Stephens and the Pottsdam Miracle. 

 The only other session I got to was a trio of teachers, Jeff Schiller, Aaron Eling and Jean Baker, who have implemented all kinds of new ideas, collaboration routines, assessment and activities, inspired by Mathematical Mindsets. I was inspired by their willingness to change and by the dramatic affective change in their students. We had two student teachers there last semester, and it was a great opportunity for them as well.

Only downside of the day was all the cool folks I didn't get to hang with, including Zach Cresswell, Kevin Lawrence, Rusty Anderson, Kristin Frang, Tara Maynard... So much good happening here in west Michigan. Check out some of the other sessions and resources from the Storify

See you next year?

Friday, January 27, 2017


New game! But a story first.

The idea came to me just before class, and the preservice teachers in my geometry & data for elementary course were willing to try and playtest. (Thank you!)

The class before we had defined and catalogued all the pentominoes. (Shapes made of 5 squares that only meet adjacent squares by sharing a full edge. In general, polynomioes.) I introduce them by asking about dominoes, and how do- is for two here. There's only one domino; that's when I impose the edge matching rule. Then triominoes, of which there are two. That's where we introduce the rule that if you can turn them to match, they are the same. On the board I drew
We skip right over 4, and I ask them to find all the pentominoes. We skip tetrominoes for several reasons. The objectives for this lesson are SMP 3 (construct and critique arguments) and running a mathematical discussion, in addition to the math content. We've been talking about persevering in problem-solving, too, so I'm trying to get them to be explicit about how they're trying to solve problems. Finding all the tetrominoes is sometimes a strategy that comes up for our big question: how do we know we have them all? I also want them to make the connection to tetris.

They work in groups (as usual) and occasionally I just ask the tables to say how many they've got. The first round was between 7 and 15. Second round between 10 and 13. Third round between 11 and 14. Time to put them on the board. The argument that usually comes up here is whether two pentominoes are the same if they are flips of each other. This day was a particularly lively discussion. Unusually, most of the class decided that the flips were different, with one main hold out. At one point, the chief counsel for flips are different asks "are we thinking of these as two-dimensional or three-dimensional?" "Ooh, good question!" I say. People argue both ways, and the square tiles we're using are the main argument for three. Then the holdout says "but a flip is just a turn in three dimensions!" We sort that out with lots of hand-waving and reference to snap-cubes, even though we don't have those out this day. (Point for Papert and the importance of physical experience.) Finally, they decide. Flips are different. They iron out to 18 and think they have all of them, despite the lack of a convincing argument that they do. And the frustrating refusal of the teacher to settle it by proclamation.

Next day, we're going to use the pentominoes for area and perimeter. The HW was there choice of questions about puzzles or making rectangles. One student found a 6x15 rectangle, which settled a question. I ask them for the area and perimeter of the pentominoes, and quickly someone says it's always 5 and 12. Conjecture! Rapidly disproved conjecture! Then I give some combo challenges: 3 pentominoes for a perimeter of 30 or more, 4 for 20 or under, 8 for exactly 26, 8 for exactly 36. The first is easy for most, but everyone gets stuck on one of the other three.  (So hard to get at the thinking here, though.) After a reflection, finally I ask if they're willing to try a new game. Here's the rules we finally decide:

Materials: Two teams and a set of pentominoes.
Players will add pentominoes to a figure and get points = to how much the perimeter increased. 
First team picks a pentomino and plays it. Instead of 12 points (unfair) they get one point for starters.
Second team picks a pentomino and adds it to the figure following polyomino rules. (Shared square edges.) 
Alternate until all pieces are played.

Sample game:

Wow, team one was on fire at the end! It was pretty fun, and surprisingly strategic.  Students invented more and more efficient ways to find perimeter, moving from one by one counting, to side counting, to eventually getting to a  covered this many, added this many strategy. They were surprised you could score 0, and astonished when someone shared they scored negative points. The interesting question of whether trapped empty spaces count towards perimeter came up.

In the long run, I think the game gets repetitive, but it has given students a lot of experience with perimeter by then. If students wanted to play more, I'd challenge them to make a game board with obstacles. You could play this with the Blockus pentominoes, if you have a set, but making the pentominoes is a really good activity, too.

We're not sure about the name. Pentris was suggested. Reduce the Perimeter. Perimeduce. For now the placeholder is: Pentiremeter. But we're open to suggestions

PS: finally made a GeoGebra pentomino set that I like.

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.