Year: 2011

Total 140 Posts

Upgrading QR Codes

Riley Lark outlines the infinite layers of abstraction between a QR code and a trivial task like classroom voting and offers an upgrade:

If I wanted students to vote as they entered or exited the classroom, for example, I’d consider a piece of paper or the whiteboard with a few markers laid out. The process for voting is “pick up this marker and scrape it across the board in a check mark shape.”

CMC-N 2011 Reax

I didn’t have sound at my morning presentation at CMC-North last weekend. That was on me, and it wasn’t an enormous deal anyway. I told the people there I was sure we’d get through it together. I was right, but I had no how right I was until we were starting into my explanation of mathematical storytelling. I was showing shots from the first acts of Star Wars, Jaws, and Raiders of the Lost Ark and the absence of music in each of those scenes was pretty conspicuous. But then we heard the music. It started in the back with a few people humming out The Imperial March from Star Wars and then it rolled over the rest of the group. I added the “pew pew pew!” of the ships volleying shots back and forth.

I love this conference and I love these people. They’re smart, optimistic, and funny. The lineup is unusually strong. The venue is world class. You should stop by sometime. Here are a few things I saw and did and the digital goodies I grabbed for you while I was there.

Bix Beeman on the Math of Surveying

[pdf handout]

The official word on the Asilomar grounds is that it’s 107 acres. But how do they know? It’s a weird shape. How do you capture the area of weird shapes and, in this case, the area of weird shapes that are larger than the paper on your desk? Beeman took us up through different solution strategies, all the way from counting squares on grid paper to the matrices and determinants of the Surveyor’s Area Formula [pdf]. It would have been great if he had given us much time to play with the different methods and discuss the tradeoff between complexity and accuracy. As it was, we rushed through (eg.) Pick’s Theorem so that in the last ten minutes we could use TI calculators to calculate the area formula across all the coordinates bounding the Asilomar grounds.

All snark aside, I do not get TI technology. I’ve used an iPhone for the last four years and the C++ programming language for the last ten weeks and, suddenly, I cannot find the appeal of a device that displays a total of 128 characters at a time in black and white. I mean, try to read a single line of the spec sheet without pitying the poor copy editor who had to write it. “2.5 times the processor speed of the TI-83 Plus.” Well! “Lists store up to 999 elements.” That many! “24KB of available RAM memory.” It just feels mean, at a point.

Beeman hinted at various times that your students will become docile and engaged as they’re copying coordinates and instructions into their calculators.

I don’t actually doubt this is true. Given the choice between a demanding task and grinding, students regularly choose grinding. There’s a lot less risk. But if you give a student this page of instructions [pdf], what work is left for her to do? What are the odds she’ll be able to conceptualize and solve her own problem later?

BTW. Here’s a fundamental difference in approach. Beeman told us early on that the official survey of Asilomar was 107 acres. Then he said, “Let’s see how close we can get to that.” Me, I would have asked the class to guess how many acres comprised Asilomar. Then, after we applied our best mathematical analysis to the task, we would have checked our work against the official survey.

Dan Meyer Doing His Dan Meyer Thing

[session website]

I’m finding the contours of this talk a little too familiar lately. I’ll be giving it in February at GSDMC, then in April at NCTM, at which point (fingers crossed) I’ll film it, post it to the archive, and move on.

Harold Jacobs’ Mathematical Snapshots of 2011


Someone said this was the fortieth edition of Harold Jacobs’ annual “Mathematical Snapshots” talk at Asilomar. It was my first. Basically, Jacobs has a sharp eye for mathematical moments in the news or in life. He summarizes them annually in his session and then passes them out at the end on a CD-ROM, which I have uploaded for you here. Jacobs said this was his first snapshots talk using a digital projector (rather than projecting transparencies) and, I have to say, he was a total natural with the medium.

Jacobs’ snapshots fell into one of two categories. In one case, he’d read off tidbits from the news that hinted at something mathematically interesting. Like this cornball who insists he’s proven that pi is rational and is, in fact, 3.125. These were interesting but nowhere near as impressive as the snapshots which he attempted to turn into some kind of challenge for students, giving them some part to play in the interesting mathematics. Like the Italian woman who received a $44,500 parking ticket because the police officer set the date at 208 instead of 2008. Jacobs asked students to calculate the correct fine.

Excellent curation, really. I’m not sure I needed the presentation, though. Let’s work out some kind of rotation for next year and share the CD, okay?

Michael Serra’s Math Games

[pdf handout]

I had lunch with Jodie T, whom I hadn’t seen since our days learning to teach in the same cohort at UC Davis, and we worked off the lunch coma in Michael Serra’s session on math games. There were a lot of classics (Knight’s Tour, Battleship / Treasure Hunt) but Serra applied some kind of twist to each, ratcheting up the demand (and fun) of the task just past the point where I would have quit. Definitely check out the handout.

Alan Schoenfeld on Common Core Assessments

Schoenfeld advises the SMARTER Balanced Assessment consortium. He made a case for the quality of the consortium’s work and he laid down high odds for the success of its Common Core assessments. I can’t speak to his second point. The politics of math education (particularly in California) go back to when the Hatfields accused the McCoys of de-emphasizing procedural fluency. I felt he made a strong case for the quality of the assessments, though, particularly if your alternative is the California Standards Test, as it was for everyone in the room.

Here’s the case for quality. SBAC draws heavily from the talent pool at the Shell Centre in England, which includes Malcolm Swan, whose exemplary work I’ve covered here and here. These are exceptional educators and task designers, but you don’t have to take my word for it. The Shell Centre has released a pile of sample assessment tasks. Here’s one:

Compare that to the released questions from California’s Geometry CST [pdf]:

More? Here’s CST Algebra [pdf]:

Compared to a Shell Centre algebra assessment:

The new assessments are more challenging and they reveal more about a student’s thinking. (They’re critiquing arguments. On a math test.) Check out the rest and let me know your reservations.

CMC-North on Technology

Here’s a terrifying thought. It’s 2032. I’m fifty years old, still a CMC member, still attending Asilomar, but Merrill Hall is only half-full for the closing keynote and everyone attending has white hair. Nobody came up the ranks in the last twenty years, in large part because the CMC-N conference-going experience would still be totally at home in the 1990s, right down to the TI calculator sessions. (Sorry. No more jokes.)

There were nineteen tweets on the #cmcn11 hashtag this year, a third mine. Someone at CMC-S e-mailed me a long-ish note to ask permission to livetweet my session, which to the best of my knowledge never happened. They passed out CD-ROMs of session materials in the conference bag. There are better options for conference scheduling than PDFs but a PDF of the program would beat whatever this is:

I’d like to start an off-the-books sub-committee, a place to brainstorm some ideas to present to the conference planning committee, many of which we’ll implement ourselves at CMC-N 2012. If you’re a CMC member (either South or North) and want a piece of the action, email a good idea for upgrading the CMC conference-going experience (so I know you’re serious) to

Interesting Things To See At CMC-North This Weekend

Just impossibly excited about tomorrow’s Northern California math conference. Here’s my tentative dance card, in chronological order, plus a few alternates.

  1. Bix Beaman. Breathtakingly Gorgeous 107 Acres? How’d They Get That? Also: Tom Murray. Blood Count: Are You At Risk?
  2. Steven Leinwand. Converting Typical PD into Real Teacher Development Practices. Also: I’m doing the same talk in this slot I did at Palm Springs last month and will do again at NCTM in April before taking it out back and giving it the Old Yeller.
  3. Harold Jacobs. Mathematical Snapshots of 2011. Also: Brian Lim. Make Use of Structure in High School Mathematics Classes. Scott Farrand, Rick West. Polynomial Surprises. Allan Bellman. You’ve Checked for Understanding – Now What!?
  4. Michael Serra. Teaching Sequential Reasoning Through Games and Puzzles. Also: Breedeen Murray. Beyond Sudoku: Use Logic Puzzles to Develop Reasoning Skills.
  5. Christopher Mackenzie. An Appropriate Tool for Algebra is a Dynamic Spreadsheet! Also: Avery Pickford. Making Common Core Process Standards More than an Afterthought

Other places you’ll find me:

  1. Seeley’s opening night keynote.
  2. Schoenfeld’s Sunday morning keynote.
  3. I’ll be doing one of the talks at Key Curriculum Press’ Ignite event Saturday night.
  4. Lunch on Saturday you’ll find me at the picnic tables as you walk from the venue towards the water. All welcome.

Redesigned: Follow That Diagonal

Which is a better treatment of that problem with the rectangle’s diagonal? How are you defining better? Better for what purpose? Help me out here.


From Alan Schoenfeld’s 1994 Math 67 midterm:

The diagonal of the 3 x 5 rectangle below passes through the interiors of 7 of the 15 squares that comprise it. In general, consider an N x M rectangle. Through how many of the NM squares that comprise the N x M rectangle does the diagonal pass?


From Kate Nowak’s blog:

Draw a 9 by 3 rectangle on a square grid. Draw one diagonal. How many squares does the diagonal pass through? Draw some non-similar rectangles with one diagonal. How many squares does the diagonal pass through? Develop a rule to determine the number of squares a diagonal passes through for any rectangle of any size.


My own treatment, submitted for review, correction, and debate:

How many squares will the diagonal of the large rectangle cut through? [This question added because it wasn’t clear I’d ask it – dm]

I’ll follow up in the comments at some point on the decisions that went into my redesign.

2011 Dec 1. Check out David Cox’s parallel investigation of this problem, leading to an incredible Geogebra applet.

Pretending Closed Questions Are Open

I was in Avery Pickford’s session at CMC-South when he put up this image and polled his participants for questions that interested them.

They asked about the slope of the diagonal. They asked about its length. Avery then constrained their questions. “What things could we count?” he asked.

His participants responded with “the perimeter” and “the number of squares.” At that point, Avery just asked the question that interested him:

How many squares does the diagonal pass through?

His session ended on that problem but I’m extremely curious what would have happened had he presented a new image and asked his participants for new questions. I can’t be sure but I suspect they would have held out. They’d know from their last experience that Avery had a question in mind and everyone but the apple-polishers would have waited him out.

Open And Closed Questions

If you have a question you’d like your students to answer, ask it. But before you ask it, consider creating a visual – something short and sweet, one photo or one minute of video – that orients your students to the context of your question and makes that question seem like a natural one to ask. Like:

If you’d like your students to pose some questions of their own, ask them what questions they have. But questions about what? Give them something to ask questions about. Consider creating a visual – something short and sweet, one photo or one minute of video – that lends itself to different perplexing questions. Like:

What Happens On Twitter Stays On Twitter

How can you tell in advance if students will be perplexed by your closed question or if they’ll have open questions about your photo or video? You pilot it. There’s no right way to pilot curricula, only optimizations for different variables that are often in competition with one another. Like:

  1. Are you piloting with students or with some proxy for students?
  2. How easy is it for your participants to give you feedback?
  3. How many participants are giving you feedback?
  4. How helpful is that feedback to your development process?
  5. How far into the development process are you waiting to get feedback?

Here’s one optimization: show teachers your photo or video on Twitter and ask them what questions they have about it.

This means (1) you aren’t piloting with students, which is unfortunate, though no students are harmed if your idea is a dud, (2) it’s easy for your participants to give you feedback, (3) the number of people giving you feedback is proportional to your followers on Twitter, (4) that feedback is often useful – if you plan to ask a closed question, the feedback will let you know if that question is interesting; if you plan to ask for open questions, the feedback will let you know what questions to expect.

Or you might pilot your curriculum on the same day you’re teaching it, making modifications for your afternoon class based on feedback from your morning class. You can evaluate the variables for yourself on that one.

Make it work for you. Twitter, #anyqs, your classroom, your faculty lounge, whatever. Make it make you a better teacher. Just understand that when you’re using curriculum in the classroom, you’re optimizing for an entirely different set of variables than when you pilot that curriculum somewhere else.