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.