Saturday, June 17, 2017

World Tessellation Day 2017

Happy Tiling!

Our 2nd annual World Tessellation Day, celebrating the birth of Maurits Cornelus Escher on June 17, 1898.

Pat Bellew has a good quotation in today's On This Day in Math:
By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I have made, I ended up in the domain of mathematics, Although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists. - M. C. Escher, Quoted in To Infinity and Beyond, E Maor (Princeton 1991)
The best post I saw in advance was Evelyn Lamb's Look Down. (Which also sounds like it could make a good horror movie.) Also note the founder of this here holiday's post, Emily Grosvenor's post on making the holiday. Also also note that Eric Broug's  TED-Ed on Islamic Tessellations came out in time for today.

Best way to see what is happening is probably the twitter hashtag #worldtessellationday. Lots of groovy there.

I had to make something, so here's a GeoGebra applet. Instead of an Escher-style, I made a square you can fill and then fill the plane in three different ways. The matching conditions make for some cool patterns to me, like infinity tiles. If you want an Escher style applet, check my collection of tessellation sketches.

My most recent tessellation work was with Heather Minnebo's art students. I helped some work out the mathematics, but got interested in her art directions.

I was having to eavesdrop while taking to other students, and we'll talk more about. But what I got was her talking about size of the tile compared to the final paper (encouraged to be quite large, like a meter by 1.5 m.) She talked about the designs within the tile, that were going to have to fill the whole paper. It struck me that the same things she was looking for are what I would want to emphasize the mathematical structure. Plus some nice visual estimation. Afterwards, she was highlighting craftsmanship as a growth area for some students, and I was wondering what that looks like in the math classroom.  Heather says "craftsmanship for my kids (excluding some of my quirky friends who are owned by their ideas) comes down to ownership and then honing skills." She notes that at least half-ish of them were able to mesh conceptual understanding with some solid technical skills and creativity.

Heather notes: I have more questions than observations.  I've taught this lesson now half a dozen times, how long before I really get a handle on how to best teach it? Every time I get an idea for how to improve the instruction and pacing, but it's a long ways from solid.  (A solid chunk of this weakness/uncertainty is the knowledge that I don't have the full breadth of understanding of the mathematics behind it.)  How do I get them to keep the spontaneity of their creativity and experimentation, but add in the understanding that this game has rules to follow. For  instance, when I say midpoint I mean get a ruler and find the midpoint.   
The enthusiasm and interest is palpable from the introduction – their minds are blown by the Islamic architecture and Escher's work – to their own initial trial and error.  How do I encourage that same level of interest and enthusiasm all the way through to the end?  I have two groups of kids who are able to do this from beginning to end: the ones who get it, own it, and have the discipline and mastery of skills to carry out their vision, and then there are my creative geniuses who are owned by their ideas and they are consumed with fulfilling the vision in their heads and I don't believe they see the issues of craftsmanship or skill we see.  The bulk of the kids are in the middle – their enthusiasm and effort peters out (in varying degrees) as the repetition of shapes try their need for instant gratification, brains.  This harsh judgment includes myself as I too often fall into this  low attention span category. 
Lastly and most complex, how do I convey the challenge to see each individual shape as a separate defined image/area, yet also view and plan them work together as a whole?  Unbelievably, More than a few kids ignore the pattern they've established by tracing their shape, and they add color and additional patterns over all – sometimes obliterating the tessellating pattern they created.  This blows my mind.  I struggle against being frustrated as I think they absolutely must be missing something to choose to do that.
Those are just a few challenges and questions I have… I'm sure there are many more this was just off the top of my head.

Wow. These are some of the biggest teaching questions that there are! Any suggestions or comment from readers?

I always love how bold elementary students are (compared with college) and willing to try when college students often need to know if it will work first.

Monday, May 22, 2017

Math Teachers at Play 108

I requested to host 108 because I have an absurd attachment to this number.

  • 108 beads on a Tibetan Buddhist Mala.
  • 108 pressure points in the human body.
  • 108 suitors courting Odysseus' wife Penelope.
  • An official Major League Baseball baseball has 108 stitches. In 2016 the Chicago Cubs won the World Series for the first time in 108 years, ending the longest championship drought in North American professional sports. The Cubs' win came in the 10th inning with 8 runs. From what perspective would you see the maximum fraction of stitches?
  • Police departments in India are warning iPhone users to ignore this Siri prank:  users say "108" to Siri, then close their eyes. When a user says "108" to Siri, it automatically dials local emergency services. (The prank ask users to close their eyes because there is a five-second window where a user can press cancel, in case a call was placed accidentally.)
  • Wikipedia notes: "The well known bas-relief carving at the famous Angkor Wat temple in Cambodia relates the Hindu story of a serpent being pulled back and forth by 108 gods and asuras (demons), 54 gods pulling one way, and 54 asuras pulling the other, to churn the ocean of milk in order to produce the elixir of immortality." Also, in karate, "The ultimate Gōjū-ryū kata, Suparinpei, literally translates to 108. Suparinpei is the Chinese pronunciation of the number 108, while gojūshi of Gojūshiho is the Japanese pronunciation of the number 54. The other Gōjū-ryū kata, Sanseru (meaning "36") and Seipai ("18") are factors of the number 108."

Late breaking news from biology via Paula Beardell Krieg: 108 ways to be crazy. 108/52 = 3^3/13...

  • The numbers 48151623, and 42 appeared in LOST, both in sequence and individually. The numbers add up to 108, and the button in the hatch had to be pushed every 108 minutes.

More mathily:
  • So why can we add 100 and 8 and get a number divisible by 9? That's weird, right?
  • It can be factored as 1x2x2x3x3x3. (Next up, 27648.) This means 108 is hyperfactorial
  • 108 degree angles in a regular pentagon, which makes it Golden ratio adjacent. Also 36 and 72 degrees are the angles in the golden triangle, which Euclid directs as the isosceles triangle with base angles double the other angle. 36, 72, 108...
  • I got wondering if many other numbers are multiples of the same number with (non-ending) zeroes removed. 
  • New to me was the idea of a refactorable number: a number divisible by the count of its divisors, also called the tau numbers. 108 is the 18th tau number. Can you find all 17 prior?
I discovered many of these things when (accidentally?) hosting Carnival of Maths 108

There weren't many submissions this month, so many of these are blogposts I've noticed in the past month. I encourage you to think about submitting - either your own or others - and hosting. Denise Gaskins is the originator of this carnival, and has a submission form. Last carnival at Give Me a Sine; next carnival at Math Mama Writes. (Hi, Sue!)

Long preamble! Let's Play Math, shall we?

Just, Wow!
Fawn Nguyen has many excellent posts, sites and presentations, but still occasionally has one that makes you say just wow. Her post on doing the Euclidean Algorithm with her middle school students was like that for me. Mike Lawler did an excellent follow up on the material with his family mathematicians.

Lana Pavlova explored what it means to be good at mathematics in this mindset-rich post.

Megan Schmidt blew me away with her rigorous honesty in her Ignite presentation on We Are Powerless. Link goes to the playlist, and you will enjoy every one you listen to. Only time for two more? José Luis Vilson, My Kids Can Do That Math, Too, and Robyn Drew, The Mystery of the Circle are where I'd start. 

Mike Lawler also got his boys exploring convex pentagon tilings via Evelyn Lamb's post on Math Under Her Feet. These are amazing geometric patterns.

Nice post about teacher's Math Circles by Katrina Schwartz at KQED/Mindshift. tl;dr: Doing math helps us empathize with students and teach better.

Chase Orton shared numberless data problems, extending Brian Bushart's numberless story problem idea. 

Denise Gaskins has a quick note about how even a workbook can be a moment for play.

Rupesh Gesota does an in-depth interview with a student on different division methods


Sasha (AO) Fradkin has a Kickstarter on for the book Funville Adventures, a math-inspired fantasy. (Her post to introduce it.)

Edmund Harris has some math-art shirts for sale. I ... can't decide.

Simon Gregg shared his students' responses to MANY Which-One-Doesn't-Belong situations.

Several of us enjoyed playing with the Girih Designer that Anugrah Andisetiawan‏ shared. (Islamic/PatternBlock tessellations and patterns.)

by coronaking
Paula Beardell Krieg completed an amazing book making and money math project with 2nd graders.

Watch Out 
Kyle Pearce shared his presentation Beauty and Complexity of Elementary Mathematics.

Kristin Gray has a whole sequence of number routines at the Teaching Channel.

Somehow I had missed the amazing Infinite Series YouTube channel from PBS, with Kelsey Houston-Edwards illuminating mind-blowing topics.

3 Blue 1 Brown released a terrific introduction to calculus in 10 essential YouTube videos.

Benjamin Leis wrote a response to Peter Liljedahl's Global Math presentation, and then tried out VNPS with one of my favorite mathgames, Sprouts. (This was a new-to-me blog I've subscribed to now.)

Advanced Topics
Mike Lawler also also got exploring complex roots & juggling with 3-D printing, sparked by a John Carlos Baez post. Beautiful, etherial structures.

Sam Shah did an intriguing exploration of Graham's Number. He's also looking 

I've invited the #MTBoS to follow along with my summer online calculus course. There are some cool teachers dropping knowledge on my students - awesome! 

One of the posts I'll share with those students is Simon Gregg's circle and square optimization post, with bonus generalization & striking images.

A couple of the images for this post were made with a GeoGebra factoring applet. How would you picture 108?

I'm getting to do a little (too little) in David Coffey's Math Recovery/Design Thinking course this summer. Here's his intro to SAFARI design.

Dan Meyer bids adieu to Malcolm Swan.  Dr. Swan moved the whole field forward and he'll be missed.

Wednesday, March 8, 2017

Lenten Teaching

Two things have me thinking about lent and teaching.

The first is just lent itself. An ancient word for spring, it's a fascinating spiritual practice; the idea of preparation for a holiday by engaging in disciplines. The traditional disciplines are prayer, fasting and almsgiving.

  • Prayer - most of my prayer time is in contemplative prayer. This can be thought of as emptying oneself, filling yourself with the Spirit, or relationship building with your higher power. For me it's a matter of prioritizing, because it's just as easy for me to put off prayer as it is to put off a phone call or email. (If I owe you either I apologize.)
  • Fasting - usually thought of as giving something up that you know you either particularly like or would be better off without. Lent is when my father realized he was an alcoholic, as he found himself literally unable to stop. (Later he gained sobriety through a twelve step program. for more than 30 years.)
  • Alms giving - often thought of as money for charity, more broadly it is service or caregiving for the marginalized, suffering or powerless.

For some reason, I've never thought before this year about what this personal life practice would be like in my professional life. I look forward to lent every year for it's sense of renewal. Maybe it's like New Year's resolutions, without having to pretend that you're going to be doing it forever.

What would these disciplines be like applied to my teaching life? Here's what I've got this year. (Hopefully I can do this every year as well.)

  • Prayer - on one hand, just praying for my students. I do this anyway, but have been more intentional about it this past week. Jesus knew what he was doing when he asked us to pray for our enemies. It increases compassion and empathy even there. How much more for people we already care about! But also, I'm trying to think about this in terms of relationships as well. What are the things I can do to strengthen my relationships with my students?
  • Fasting - this might be where I started. What do I do (or not do) in the classroom that I should give up? My goal is to try to interact more with individuals and groups while they're working. I tend to let them really work independently, and I don't want to start that, but it's okay, I think, to become a part of their group for a little while. Also goes with the relationship idea.
  • Alms giving - where can I be more supportive and generous to my students? All I can say so far, is that I'm on the lookout.

The second thing is a blogpost by Matt Larson, NCTM president, the Elusive Search for Balance. He gives some history about which of conceptual understanding, procedural fluency, and application have received more emphasis over the years, and advocates for balance, in alignment with the National Resource Council's 2001 recommendations:
"We want students to know how to solve problems (procedures), know why procedures work (conceptual understanding), and know when to use mathematics (problem solving and application) while building a positive mathematics identity and sense of agency."
The comments are also fascinating, with a lot of big math ed names.

My first reaction is that this is less of a pendulum swinging and more of a pendulum stuck to the procedural side of the triangle with chewing gum. When have we not emphasized procedure? I think in the research community we might swing a little, but in the teaching community emphasis on problem solving remains rare. NOT TO FAULT TEACHERS, as I have never known a community more focused on doing good for others. But we know that people tend to teach as they were taught, which does not push the pendulum.

But, of course, I do know a lot of wonderful teachers who are working in the balance that Matt is talking about. Thank you, #MTBoS. How did they get that way? We are drawn to systemic programs and sweeping curriculum changes, but that doesn't seem to change teachers.

What if it's more like discipleship? Teachers change when someone they know shares a better way with them. When their questions cause them to seek a solution and they find someone trying something that might help. It's not the person up at the front of the room with a microphone, it's the community of practice. This is something the #MTBoS gets right.

I've been thrilled with the increase in attention going to teachers like Dan, Fawn, Graham and Christopher. Their keynotes are amazing, and I've seen them light some fires in teachers' hearts. But we need to connect with those teachers and support them in this new direction. That is what's going to finally unstick the pendulum. Tell two friends and they tell two friends. Go hear Fawn together, then give a visual pattern or problem solving situation a try together, too.

To circle back to the twelve steps, the twelfth one is a doozy, and I think is what I'm trying to get at.
Having had a spiritual awakening as the result of these steps, we tried to carry this message to alcoholics, and to practice these principles in all our affairs.
Having found out what math learning can be like...

Of course, I'd love to know what you think, if you care to share.

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.