NCTM 2010 — Day One

The national conference for math supervisors ended just as the national conference for math teachers began. No breaks. Let’s jump right in.

Sessions Reviewed:

  • Making Mathematics “Real.” Donald Saari.
  • Making Math Much More Accessible to Our Students. Steven Leinwand.

Making Math Real …

In order to teach a lot of calculus quickly to a lot of college freshmen, Don Saari gave them the Kathy Sierra experience: he made them “feel like they invented calculus.”

Saari said that “when you get them to invent the ideas, they blaze through. They own the ideas.” And even if their invented ideas aren’t perfectly formal (ie. “the slice-and-add method” instead of “rectangular approximation to integrals”) it turns out to be a fairly trivial task to adjust their naming conventions once they have that firm conceptual grasp. (cf. Tom Sallee’s session from CMC-North 2007.)

… And Much More Accessible

Leinwand modulated his earlier remarks nicely from a crowd of supervisors to a crowd of teachers — many more charts, data, and citations in his session with the former; a pep rally grounded in examples of classroom practice with the latter.

The imperative was the same in both, though: “Empower the species. Give them access to math.”

I’ve attended more sessions on problem-solving this week than I can count and, if you’re looking for the most-mentioned, highest-yield technique, here it is:

Celebrate non-standard approaches.

That’s it. If a student comes up with a method for solving a problem that’s functional but outside the scope of your plan, let the plan go. Bring that student to the front of the class to explain her method and throw all kinds of enthusiasm at her.

The only prerequisite for this kind of teaching is the release of a certain kind of personal insecurity that has no business in education anyway. You won’t miss it. There are few downsides here. Like my companion said, “Do we still need to tell this to teachers?”

Leinwand put this shot over the net with some nice topspin:

We [the US] need innovation. Who’s going to innovate but the kids who look at things differently. If we continue to teach math the way we have taught it, we will continue to do right by the same thirty kids who think like nerds.

It’s even better to put students in a place where non-standard solutions aren’t just appreciated, but inevitable. Here’s one of Leinwand’s examples. Let’s say you want to chat about subtracting whole numbers.

The wrong way:

Sunil has 73 marbles. Jacinda has 63. How many more marbles does Sunil have than Jacinda.

A better way:

73 and 63. Tell me everything you can about them.

And when someone tells you (among a bunch of other fun facts you didn’t anticipate) that 73 is ten more than 63, ask your students to convince you.

Bonus Steve-ism: “Estimation, number sense, and guess-and-check are the coin of the realm.”

The Really Hard Part

Saari brought in toilet paper to describe limits and apple cores to illustrate integrals. Leinwand used a recent experience with a highway patrolman to motivate evaluating expressions.

It’s simple to encourage and embrace non-standard student responses. What’s difficult is assigning problems rich enough to allow for non-standard responses. (eg. not this.)

Leinwand didn’t address this at all. Neither did the SFSU professors in the rich problems session yesterday. Saari brought it up at the end of his session as people were starting to filter out. “I’m not saying this is easy,” Saari said. “It takes awhile to adopt this approach.”

Suffice it to say I would’ve stuck around for another sixty minutes if he had decided to elaborate on that casual disclaimer.

Not that you asked but I think one of the largest challenges facing a new teacher is to establish certain practical habits such that her entire life becomes an ongoing 24/7/365 exercise in curriculum development.

Why do it to yourself?

Because high-achieving, curious, capable students always, for whatever reason, seem to study with those teachers. Because that lifestyle will turn your classroom teaching into a daily act of creative expression, “a personal delight,” as Saari put it. Because it’s incumbent on human beings to live lives of fascination and to share that fascination with others.

Pick one.

I'm Dan and this is my blog. I'm a former high school teacher, former graduate student, and current head of teaching at Desmos. More here.


  1. re: “[a] 24/7/365 exercise in curriculum development”:

    I like that so much that I forwarded it off to the ITEEA’s IdeaGarden list, which is the professional list for technology education teachers. Great stuff, Dan. I’ve also been bouncing your NCTM reports to our middle school math teacher.

    Keep up the good work. It’s inspiring to see conference notes being made useful by turning them into public conversations.

  2. At the risk of a minor heresy, let me propose how multiple choice tests, while not very good towards most of the purposes for which they’re currently being abused and misused, can make interesting tools for provoking thought when placed in the context of Leinwand’s coins of the realm.

    As someone who tutors a lot of students for SAT/ACT prep, my coins of the realm in general are: process of elimination and “everything is a reading test”: that is, if you don’t take advantage of the answer choices and don’t read things carefully and attentively, you’re almost certainly under-performing. It’s obscene how many questions kids get wrong on various sections of these exams by not thinking analytically about why wrong answers are there and noting what exactly the question is calling for, and this is true on all sections of these tests and those like them (GRE, GMAT, LSAT, etc.)

    So teachers could do a lovely service for students by taking math problems (and some science problems from the ACT) and making them launching points (or valuable side-trips) for investigations (I assume, Dan, that you don’t teach everything with the same structure, even when you find one you really like). ;^)

    If anyone is interested, I could probably provide a few rich examples of what I’m talking about. Feel free to write me:

  3. I disagree with “The only prerequisite for this kind of teaching is the release of a certain kind of personal insecurity that has no business in education anyway. You won’t miss it.”

    I haven’t released my insecurity. I have had to learn to accept it. In getting to this point, I realized that I, in actuality, had very little control of what was really happening in my classroom. Oh, I could make it appear that i was in control of the structure and behaviors, but that was a role that both the students and I mutually agreed upon. Unspoken, but still a shared understanding. In reality, I was never in control of their learning.

    Now while I have a general idea of where I’d like my lessons to go, I never know where they’ll actually end up. Do I still get insecure about this? Yep. But I have learned to have faith in my students capabilities to do and to communicate mathematics. I have also had to develop a bit of faith in my own ability to help lead a discussion. Does it always end up where I’d envisioned? Nope. But my day is never boring. Insecurities and all.