NCSM 2010 — Day Two

Sessions Reviewed:

  • A Lesson Study Project: Connecting Theory and Practice Through the Development of an Exemplar Video for Algebra 1 Teachers and Students. Anne Papakonstantinou, et al.
  • Intriguing Lessons About How Math Is Taught and Assessed in High Performing Asian Countries. Steven Leinwand.
  • Problem Solving and Technology implementation in an Inclusion Classroom. Annie Fetter.
  • Western Caucus. California, Hawaii, Oregon, Washington.

Better Teaching Through Video

Houston Independent School District is huge. It won a huge grant and has a huge video production department. It created an impressive DVD of impressive practices in Algebra I and the teachers who participated in that reflection program improved impressively.

I have nothing but good things to say about the presenters, their content, and their presentation of their content. But if you believe as I do that classroom transparency is our mandate as teachers, that when you share what you do it will inevitably change what I do (even if I’m only learning from your counter-example), your questions should be the same as mine:

  • How can we scale this?
  • How can we distribute the work of editing and mastering these videos?
  • What is the least cost we can get away with for a hardware / software package for a classroom?
  • Specifically, what is the sweet spot where quality meets affordability?
  • How do we communicate this transparency mandate to parents?
  • Specifically, how do we streamline the legal release process?

Those questions weren’t addressed. Which was fine. It wasn’t my session. Nonetheless.

Problematic Problem Solving

The presenter, Annie Fetter, has been creating Problems of the Week for the Math Forum since 1993. She’s reviewed thousands of student responses in that time and she now models problem-solving techniques for other teachers in their classrooms. There was one mortifying moment in the session and a whole lot of great ones.

First, she dismissed the status quo model for teaching problem solving:

  • Underline the key parts.
  • Circle the numbers.
  • Match words to operations.

Those commands are meaningless if the student doesn’t understand the problem. (ie. the word “more” can indicate addition or subtraction, depending on the context.) So Fetter asks students instead to read a problem and complete the statements a) “I notice …” and b) “I wonder ….”

She workshopped the process with this problem:

Greta has a vegetable garden. She sells her extra produce at the local Farmer’s Market. One Saturday she sold $200 worth of vegetables — peppers, squash, tomatoes, and corn.

  • Greta received the same amount of money for the peppers as she did for the squash.
  • The tomatoes brought in twice as much as the peppers and squash together.
  • The money she made from corn was $8 more than she made from the other three kinds of vegetables combined.

You know where this went for me.

Honest to god I tried to stay cool. I asked my table buddies, “Is this problem contrived? Is there a more natural way to teach systems of equations?” The guy across from me just glared back like I had said something derogatory about his mother.

When it came time to share our noticings and wonderings, I said, “I’m wondering how Greta knows how much she made in total but she doesn’t know how much she made on peppers, squash, tomatoes, or corn. Like, what went wrong with her bookkeeping there.”

Fetter said in all good humor, “Well, sure. I mean, this is a math problem.”

That sort of response triggers my instinct for self-preservation. I start looking for a desk to duck beneath. If I pitched that response to my students, they’d already have their knives out, sharpened, and thrown.

Is there a situation where it makes sense to use systems? Did mathematicians develop systems because they makes life easier, more fun, or more meaningful? Or are they just arbitrary symbology we use to limit access to universities?

I’m glad I stayed. Her strategies for drawing students into conversation, for appealing to and strengthening their intuition, for encouraging patient problem solving were excellent. A few highlights:

  • “Students think math isn’t about them. They think math is about learning the ways to do what the dead white guys figured out how to do a long time ago. Some kids master those ways but can’t solve problems. Others can’t master those ways but have amazing problem solving skills.”
  • “Don’t listen for things, listen to students.” If you’re looking for a specific answer, your students will equate “problem solving” with “reading the teacher’s mind.”
  • A particularly sweet grace note on noticing / wondering. Give the student a mathematically rich situation but don’t yet give them a question. If you give your class a specific line of inquiry and even one student chases it all the way to the end, the moment has passed for the novice problem solvers. What moment? The moment where “we get students out of the gate who can’t usually get out of the gate.” ¶ So we take that moment away. “Find all the math here.” Then the classroom superhero will expand in every direction not just along the one.
  • “Guess and check gets a bad rap because guessing doesn’t seem like math.”
  • “Do math in pen. Learn from your mistakes, don’t erase them.”

All of which is great. But all of that requires rich mathematics. You can’t throw a piece of lumpy charcoal (see Greta’s garden above) on a student’s desk and ask her, “What do you notice? What do you wonder?”

Steve Leinwand

Steve Leinwand is a young up-and-comer, a researcher at AIR who spits hot fire on the track, and deserves his own section heading. If you get a chance to hear him speak on anything, take it. I can put the dy/dan stamp on three guarantees:

  • He will call you “gang.”
  • You take offense at least once in his talk — either by his tone or by a rhetorical overreach.
  • He will be the most well-researched speaker you see that day.

In particular, he has an encyclopedic knowledge of the content standards and assessment questions of countries in the East and the West. He noted right away the irony of totalitarian eastern countries assessing their students more constructively than their democratic counterparts in the West.

For instance, a sample first-grade assessment in China:

What is the approximate thickness of 1,200 sheets of paper? What is the approximate number of classes that may be formed by 1,200 students? What is the approximate length of 1,200 footsteps?

And one from the third-grade:

Estimate the number of characters contained in one whole page of a newspaper.

The most interesting aspects of his presentation, then, compared assessments of similar standards between the different countries. The only people who would dispute the inferiority of ours to theirs, Steve suggested, were “tired, old, white men who still told time with analog watches.” (One got the sense here that Steve was doing his damnedest not to name names.)

Decide for yourself.


  • “The National Math Panel report was a complete and total disaster.” So there you go.
  • “I do this [present at conferences] to entertain myself and I’m surprised whenever anyone comes.” Naturally the hall is packed.
  • “The Common Core math standards are what we’ve been praying for for a generation.” He anticipates improvements in the next draft.
  • “By 2014, we’ll have a national test in the 4th, 8th, and 11th grade. It will be on a computer and feature both constructed response and multiple choice. Students will have a four-week window to take it, two weeks after which they’ll release 75% of the questions and data to all stakeholders.” By “constructed response” he referred to a problem where a student saw two egg rolls on a screen and had to drag a knife around with the mouse, clicking to create cut lines to show how to feed three people equally.

He noted the inequities in models of teacher development between the East and West that made adopting the eastern model of student instruction difficult. In Korea, for instance, the ratio of teacher salary after fifteen years to GDP per capita is 2.5.

The Chinese, meanwhile, apply a model to teacher training that looks suspiciously like medical residency.

Final Steve-ism: “We do the same thing [at these conferences] we don’t want our kids to do: we sit, we get, and we forget.”

Western Caucus

Math supervisors from California, Washington, Oregon, and Hawaii gathered together in a room to update each other on old and new business. We stood one-at-a-time and introduced ourselves.

Aw screwit, I thought, and went there: “I’m Dan Meyer. I’m a high school math teacher and I blog.”

“You what?”

Caucuses strike me as a relic of an age when you couldn’t e-mail or otherwise connect instantaneously with your long-distance colleagues. I’m not saying they aren’t useful or even fun but it will be interesting to watch these stodgy national organizations try to persuade young members of that usefulness.

Gratuitous iPad Review

  • Typing is getting easier.
  • Auto correct is still confounding.
  • It’s an attention-getter. Leinwand accused me of buying it just to make friends.
  • I love the sound of typing, or rather the lack of sound. There’s always some guy in a session clattering away noisily and unselfconsciously on his laptop. The iPad “keyboard,” meanwhile, is whisper quiet.
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. I like Leinwand and Leinwand likes Common Core math, so there’s that. I realize that’s pretty weak argumentation but it’s like that with so many things in my life where I just draft off the ideas of smarter people in fields I know nothing about.

    Of course, I know something about this field so there’s really no excuse not to dig in myself. Jason Dyer’s done some of that yeoman work and, while he’s seems to have resisted a categorical thumbs up or down, his overall tone has been positive.

  2. I give a thumbs up on the math, but a long spiel why is going to have to wait.

    The short version is the crew working on this has put great pains to both compare with existing international standards and justify their decisions with research. While I quibble with parts, at least the standards are backed by *some* sort of reasoning.

  3. Two things.

    First: I agree with Jason. I like that the Common Core are coherent. They are really thought out from beginning to end. Like Jason, I don’t think they are perfect, but they are way better than many states’ standards. That said, my favorite thing about the Common Core is that they exist. I’d like to see states start supporting the Common Core as the core of their standards. This would help to get states in synch and help to reduce the mile-wide inch-deep thing we got going on.

    Second: My favorite quote from the post was “‘Don’t listen for things, listen to students.’ If you’re looking for a specific answer, your students will equate “problem solving” with ‘reading the teacher’s mind.'” Bravo. Your job as a teacher is NOT to pave the road and make sure the student is on it. Your job is to define the destination and make sure the students are taking good paths to get there.


  4. Sorry I urged you to come to the caucus. Thought we would actually get to talk to one another. (Usually the purpose of introductions.)
    Great meeting you

  5. I find the info on China interesting. We were there in May–visited a school and spent the afternoon in a 5th grade class. The first thing I noted was the LARGE class size…more than 50 students in the class. That appeared to be typical. I didn’t see much in the way of supplies. No computers or any electronics in the classroom. If there were calculators, I didn’t see them. Granted, this was only one school, but from what I’ve been able to gather, schools are pretty similar, at least in cities. Also, from what I understand, in China, parents must pay to send their children to public school.

    Enjoying your blog and just watched your 03/06/10 YouTube video. Thanks!

  6. I’m glad you’re glad you stayed for the rest of my talk :-) You’re exactly right that Greta’s Garden is not the best problem choice to use in modeling our “Noticing and Wondering” strategy. All the problems mentioned in the talk were picked with the goal of helping the kids at this particular school get better at the Guess and Check strategy. Since we wanted kids practicing guess and check, we didn’t want them practicing a lot of noticing and deciphering and whatnot, or else they may have never gotten to the guess and check.

    Here’s a “scenario” (our name for problems without questions) we used in our booth in the exhibit hall as a starter for Noticing and Wondering: A regular hexagon and an equilateral triangle have the same perimeter.

    Paul, I like your addition of the destination to my idea about listening to kids, not for answers. I would add that a teacher who is really listening to their kids will also be willing to occasionally change the destination if it becomes clear that the kids are really interested in something the teacher didn’t anticipate!

  7. In the Measurement Unit that I drafted up and taught to my 9th-graders a few months back, I had included discussing with kids (and subsequently, quizzing them) on what were some reasonable estimates for various lengths of real-world objects. But, obviously, assessing them with worthwhile questions is not the problem — how do you TEACH reasonable approximations? I found, even after the discussions (and quite a few hands-on measurement activities), that the only kids consistently choosing the correct approximations were the ones who already had great number/measurement sense.

    Any thoughts??

  8. how do you TEACH reasonable approximations?

    Great question. My sense is that one calibrates reasonable approximations through trial and error, that every time you find out you’ve over- or undershot an approximation, you’ll approximate the same measurement better the next time. That’s just a guess, though.

  9. Mathsemantics, by Edward MacNeal, has a great chapter on this. He says you need to have a web of basic numbers in your head, and then estimate often, do something to commit to your estimate (say it out loud), and then confirm the true value afterward. Doing that often improves your estimation ability.

    The students have to take responsibility for doing that, of course. So a first step would be a very cool problem.

  10. Hello Dan,
    I read your blog from time to time. Thanks for making us laugh, think, and for your passion for math.
    I’m recovering from my traumatized math childhood. Now I teach two daughters at home, and they don’t have math shivers or preconceptions. They are fresh, eager to learn, and a joy to get reacquainted with math at almost forty.
    My oldest thinks that the last number is something around 40 40 40 40 40 40 40 …anyway, God knows that number for sure.

    My comment is about the famous Greta problem…that gives me shivers. My daughter after we do mental math, comes with the same nonsensical problems on her own. Her point it’s to make it difficult. She senses those problems are totally ad hoc. A different thing is when we asked her about her sister age when she is X age, or vice-versa. That she knows very well, it’s very valuable to children to know they will always be older and you can’t catch up with age. And it’s fun (for them) to know how older or younger others and themselves will be and at what years. (You were talking about a real scenario to use or how mathematicians started to investigate this area of knowledge. History and estimating ages in the future and past as well as scales for maps and drawings may have been one area to practice this.

    Thanks for your blog and your spirit. I enjoyed specially your worksheet post, as a former public school teacher and homeschooler mom now I related to it. And your humor is very appreciated.

  11. Now I’ve read the comments and they pose very good questions, it’s not about a formula of finding real scenarios for math, but more about being able to see the math in the world, which is very difficult and in the meantime there are the tests and district requirements teachers need to meet.
    One person said that you get better at teaching things with time. That’s true. Another (or you) said that at times you just work the problem straight and that’s fine.
    I’ll give you a different example with reading. There is a new and deplorable collection that intends to use difficult vocabulary words in stories that supposedly will make the children laugh. It’s so fictitious, and they still believe that’s how children gain and increase vocabulary…WRONG. You just have to talk using proper words, read the best without watering it down to them, but using age appropriate books, which are many more than many think. And if you have to face a spelling test, or a test that will ask you for definitions, look, just cram it, do it, and then one day you’ll hear the word in something you read, or a conversation, and you’ll learn it for real.