Category: geometry

Total 29 Posts

Asilomar #5: Michael Serra

Session Title

Games And Puzzles That Develop Sequential Reasoning

Better Title





A structure not dissimilar to Megan Taylor’s yesterday, where Serra debuted games and puzzles and gave us time to tease them out.

I sat with two former colleagues in the back – all of us now at different schools. One teacher enthused over Sudoku puzzles. They challenge kids. Kids like them. It gets them comfortable with numbers. The other enjoys Serra’s games and puzzles, like Lunar Lockout. Both cite improved student disposition toward math and improved deductive reasoning.

I disagreed with them. In general, I find it dangerous to put too much distance between “fun time” and “math time” preferring, instead, to have that cake and eat it too, creating as many challenges as I can that are both fun and mathematically rigorous. (Which Sudoko, to put it plainly, isn’t.) My task is harder, I think, and I know I fail at it more, but I’m more satisfied on balance.

It was a good conversation. Feel free to interrupt us.

Serra’s best offering for my money was Racetrack Math:

It’s like this:

  1. Draw a racetrack on graph paper, however crude.
  2. You and your opponent start anywhere on the starting line.
  3. You travel along vectors. You may increase or decrease either the x-value, the y-value, or both, but only by one unit per turn.
  4. First person to the finish line wins.
  5. (P.S. No crashing.)

This gets very interesting very quickly. You start out with tiny vectors which lengthen by one unit every turn. If you fail to notice the side of the track off in the distance, though, and fail to slow down in time, you crash. (Which I did in the example above.)

I hereby toss all of my battleship exercises in the recycling bin. This is a much more straightforward introduction to positive/negative coordinates since each new turn is relative to the last turn rather than relative to this strange coordinate axis thing.

Plus, your students can create racetracks of their own, of infinite complexity, within seconds. Serra cited some kids who created a pit lane, which you had to enter on your second lap, and oil slicks, on which you could not adjust your vector at all. I’m impressed.


PowerPoint. Which is tough when you’re asking people to solve a puzzle. If someone suggests an alternative route to the one you have programmed into your slide, you have to dodge their answer a bit.


Blank puzzles and games to draw on. Again, paper is not dead. How do you do this digitally? Load each picture one at a time into Skitch and pass a stylus back and forth? Moderation, please.


  • “There is no research that demonstrates these games improve outcomes in other mathematical procedures like two-column proofs,” Serra admitted reluctantly. “It has to be there. I know it is.

The Feltron Project

[BTW: the post-mortem.]

At the start of winter semester, maybe a month ago, I told them they’d have homework every night, even weekends.

I called it The Feltron Project. I showed ’em mine and asked them to identify the mathematical forms. I told them we were going to take their lives and make math out of them.

Track Your Life In Four Ways

I told them they had to track four variables this semester. I shared with them my ownAnyone crazy enough to try this with me: it’s essential you play along with your students.:

  • where I’ve been [cities per day]
  • text messages sent / received [quantity per person per day]
  • movies I’ve watched [title per medium (dvd, theater, ipod) per day]
  • coffee drinks i’ve purchased [accessory per drink per location per day]

The Feltron Notebook

While they thought on it, we made Feltron notebooks: graph paper, folded, cut into quarters, and bound with repurposed file folders the last teacher left behind.

I showed them how I designed my own Feltron notebook (Coudal’s Field Notes, natch) to maximize page use.

How Do We Grade Your Life?

We discussed grading. What would an A look like? An F? A C? I steered the conversation towards three criteria:

  • the interesting-ness of the variables chosen
  • their consistent tracking
  • their clear & pretty design

We discussed interesting and un-interesting variables. Some students are rocking this thing all semester long, counting calories, tracking everyone they text over a semester, tallying every ounce of everything they drink.

Other students are skating, tracking the number of days they’re late to school, tracking the number of times they sneeze, etc.

We conferenced, each student and I, and I suggested changes, both to add value to their final project and to make the assignment easier for themFor instance, 100 kids decided to track “TV Watched.” “What does that mean?” I’d ask. “Uh.” they’d reply. “So make it min/channel/day or min/show/day, whichever you prefer.”.


This thing runs on bi-weekly checkpoints [pdf] where I move around the class and verify that everyone’s keeping up.

One Indication This Assignment Wasn’t Stupidly-Conceived

Not one student has taken exception to the workload. Several students, without my prompting, have integrated a notebook update into their daily classroom routine.

The Moment I Fell In Love With The Thing

One freshman decided to track the cigarettes she smoked each day. Not because she wanted to scandalize me or her classmates. She just “always kinda wondered.”

One Month Later

I surveyed 99 students last week: “how much time do you spend updating your Feltron notebook each day?”

The average response was 5.5 minutes with a maximum of 31 minutes and a minimum of 0 minutesNo idea what the minimum’s about..

Next Steps

  • I ordered a hard copy of Nicholas Felton’s annual report (to which my assignment pays seeerious homage). We’ll pass pages around and develop a written narrative of his year.
  • Then I’ll fabricate entire data sets. eg. some girl’s caffeine intake over the course of a semester. We’ll run through several infodesigns and discuss which ones tell the most effective, truthfulAll better? story. We’ll use other data sets (eg. hours spent studying) to introduce some superficial correlation.
  • Uh. That’s all I have.

The Big Questions

  • Do we make the graphs in Excel or work out the math by hand? One option gets ’em dirty with the math. One is more useful to their post-grad experience.
  • What do I do when a student comes to class a month into the project and claims her dog ate her Feltron notebook? The question, as of first period today, ain’t hypothetical.

The Regret

I should’ve collaborated with someone here. I don’t know another teacher, period, who’s out there sweating the connection between language and math like I am here which makes The Feltron Project something of a blind jump off the high dive when it ain’t altogether obvious that the pool is filled with water, thumbtacks, or nothing.

Snowflake Math

[BTW: Mimi Yang’s remix is highly recommended.]

I’m about to give you what I’m convinced are good blueprints even though the house I built off of them today was pretty raggedy.

Here, three days before winter break, I wanted an activity that injected math into something mindless. I thought about snowflakes, you know, how you fold some paper, cut it here and there, and open it up only to discover you’ve recreated The Storming of the Bastille.

So here’s (what I’m convinced is) an awesome exercise in spatial intelligence for you and your students: predict what the snowflake will look like before you open it up.

I’m tempted to leave it there and let you decide how this oughtta shake out, encouraging you to please get back to me and let me know. Because what I did today didn’t have the same loose-limbed energy my best stuff usually does. This was second-rate but maybe we can spin something better out of it – you and me:

  • I passed out a sheet of standard letter paper and some scissors to each student.
  • I had them square the paper and fold it into fourths – now a smaller square.
  • I put up a series of slides. Each one asked them to make one cut.
  • They made the cuts and I said, before you open up the snowflake, sketch what you think the snowflake will look like.
  • They sketched it.
  • I walked around, observing, sometimes making comments.
  • They opened it up and checked themselves.

Then, without passing out more paper, we went backwardsWorking backwards from a solution to the problem, incidentally, is the most reliable way to carry your kids a few rungs up Bloom’s Taxonomy..

  • I gave them the result and asked them what cuts had been made to get it.
  • I called up five volunteers to the board to show their solutions, most of which differed only slightly from each other, a fact which offered up some good conversations starting with words like “compare” and “contrast.”

Then I passed out this worksheet, which asked for eight visualizations, the second half doubling in complexity by adding one fold to the snowflake.

Typing all that here at the end of the day, it’s kinda obvious to me that this was too much even for my Geometry sophomoresNine of whom apparently read this thing so, hey, team, no disrespect.. The spatial learners had a blast but I didn’t manage to transcend that division and pull the other intelligences over the wall like my better stuff tends to. This thing lacked a certain scaffolding. In other words, buyer beware.