Month: April 2011

Total 14 Posts

Hypothetical NCTM Roster

I can’t make NCTM work this year but I checked the speaker listing (really, really quickly) and these are the people I’d check out if I were going. (Limited warranty, your mileage may vary, etc.)

  • Jerry Becker. Developing Computational Skills While Solving Problems and Avoiding Drill.
  • Akihiko Takakashi. Ideas for Supporting Students in Becoming Independent Problem Solvers.
  • Allan Bellman. Classroom-Level Assessment That Determines and Meets Individual Students’ Needs.
  • Michael Serra. Non-typical Investigations in Geometry for 2011.
  • Keith Devlin. Video Games for Mathematics: They Will Soon Get Better.
  • Bowen Kerins. Bringing Algebra and Geometry Together through Linear Algebra.
  • Bowen Kerins. Mathematics of Game Shows.
  • Glenda Lappan. Using Geometry as a Springboard to Mathematics.
  • Karim Logue. Real-World Lessons the Mathalicious Way.

[WCYDWT] Bean Counting

The Goods

Download the full archive [11.3 MB], including:

  • Dan + Chris – Question
  • Dan + Chris – Answer
  • Dan + Chris + Annie – Question
  • Dan + Chris + Annie – Answer

See, I had to do something about this problem:

This is one of those problems that wakes mathophobes up at night in a panicked, cold sweat. It’s so universally hated it’s a pop cultural cliché, a symbol of everything the layperson dislikes about math but can’t quite verbalize.

As math teachers, we step into the ring with this problem every year. In California you’re guaranteed at least one such problem on your summative student exam. But it’s a boutique problem. How much time can you really offer it in class? How will you treat it? There’s a fairly straightforward explicit formula for its solution:

Do you teach your students the formula and hope their memory serves? What kind of conceptual underpinning can you offer them without spending two weeks on it? In what ways can we improve this problem?

First, Fix The Visual

Since this problem represents itself as a real, no-fooling application of math to the world outside the math classroom, we owe it to our students to ask, “is this a good visual representation of that world?”

A: No. It’s clip art. We could upgrade the clip art to stock photography but both representations are decorative where we’d prefer something descriptive, a visual that is, itself, a useful site for analysis rather than mere drapery.

Here is that visual:

Bean Counting – Problem #1 from Dan Meyer on Vimeo.

This makeover was challenging. The task (whether painting a house or filling a cup of beans) is its own unit of measure. Think hours per house or minutes per cup. If the students manage to determine seconds per bean, you still have a math problem, but the task has changed significantly. So I sped up the tape to preserve that task. Also, the house-painting problem assumes identical houses and a constant rate of house-painting, which is the kind of unreality that exists only to serve a math problem. Here, though, we have used multimedia to inoculate that pseudocontext. Here, the glasses are identical every time and the actors are listening to a song in their earbuds that does keep them working at a constant rate, even when they’re working together.

Second, Fix The Motivation

Again, I hear you: who cares about two clowns filling a cup with beans. Try it, though. Play the video. Ask your students what questions they have. Ask them how many minutes they think it’ll take Dan and Chris together. Ask them to put down a number they know is too large and a number they know is too small. Write five of their guesses on the board.

Then move on to whatever else you had planned for the hour. Let us know how that goes.

Third, Teach

Lecture your way through the problem. Or, better, give your students a moment to reveal to you the tools they bring to the table. I’m only insisting here that when you make the brave (and, to the eyes of many students, ludicrous) claim that math has any application to the world outside the math classroom that you represent that world well and you get the motivation right. Your decisions past that will have a lot to do with your students, their developmental readiness, and your relationship to them.

But even if you lecture, don’t offer the formula. Build, instead, off their existing understanding of speed. This problem is nothing more than a strange unit for speed: percent per second. From there, what seemed complicated and dizzying becomes a straightforward application of rates: find Chris and Dan’s combined rate; divide into 100.

Fourth, Play The Answer

This is dramatic catharsis. I have no idea how to design a study to test for the effect of watching an answer rather than hearing it from the teacher’s mouth or reading it in an answer key but, anecdotally, it’s enormous.

Bean Counting – Answer #1 from Dan Meyer on Vimeo.

Fifth, Offer The Extension

Bean Counting – Problem #2 from Dan Meyer on Vimeo.

Your students’ reaction to this extensions is a meter stick for the effectiveness of your approach in step three. Will they take it in stride? Will they fall apart? If they have an inflexible understanding of the problem they may take this approach:

Which is understandable, but incorrect. (This is the explicit formula for three people.)

In both problems, though, the formula obscures the fact that nothing more complicated than speed runs beneath this problem. I’d rather my students developed that understanding than a full set of what reader Bowen Kerins calls “single-use tools”:

High school math is filled with specific tools for one purpose only. Use this box to solve a word problem about people painting houses, but this other box for this other problem. Use FOIL to expand a binomial multiplied by another binomial, but don’t try it on a trinomial! It makes no sense, and contributes to students’ feelings that mathematics is a giant toolbox you either know or don’t know how to use. [via e-mail, with permission]

Jump back into that video I linked earlier. It’s a useful depiction of a locker room-full of students who understand math to be a giant toolbox you either know or don’t know how to use. Confronted with two numbers, they multiply, they add, they average. They’re just striking the two numbers against each other, looking for sparks, looking for a number they can live with. It’s impatient problem solving.

Then the mathemagician enters the scene, reveals the “simple formula,” and computes it.

Check out the look on Junior’s face. It’s like he’s seen a magic trick. He asks, “Are you sure?” and then takes the mathemagician’s word for it. Meanwhile, we’ve upgraded the representation of the problem and not one of our students has had to take our word on the answer. They just watch it.

Featured Comment

Matt Vaudrey:

I passed out calculators for the bean counting problem, but made them give guesses (and back them up) first. Some couldn’t wait, and started crunching numbers.

The catharsis was definitely more potent in the bean counting video than the Little Big League video (“Wait, what’d he say?”). Once the time stopped at 4.5 minutes, students started with bragging.

“Ahhh! I TOLD yoouuuuu!”

Winter Quarter Wrap-Up / Spring Quarter Kick-Off

Brief Remarks Encapsulating Winter Quarter

  • Mentorship. This is new: I switched emphases from teacher education to math education. I’m retaining Pam Grossman (my current adviser in teacher education) but adding Jo Boaler (who is the math education professor at Stanford) to the Team Dan Meyer, Ph.D. roster. The education of new teachers and development of current teachers is still wildly fascinating to me, but I am asked with growing frequency to speak to and write for and work with math educators. I know enough about what I don’t know to know that I need to study up and work out some blind spots in my vision if I’m going to be effective in any of those roles.
  • Temptation. The private sector extended several invitations my way last quarter to leave Stanford – to cut a corner, basically, and go straight to work. Some of those invitations were easier to turn down than others. In every case, though, I was grateful for the opportunity to remind myself again of the reasons I committed to this difficult, frequently humbling work.
  • Music. I tend to wear out the grooves on a single record during finals week each quarter, playing the same songs over and over and over until they become useful white noise. Fall quarter it was Mumford and Sons. Winter quarter it was the soundtrack to The Social Network by Trent Reznor and Atticus Ross. Anyway.

Notes on last quarter’s classes:

  • Statistical Methods in Education. Key skill: analyze regression tables like this one for meaning. Prof. Stevens said in fall quarter he loves the moment when an author drops the tables in a paper because up until that point we’re just bobbing along with the author’s narrative. But the table tells its own stories.
  • Proseminar. One of my colleagues said it pretty well: “In any given week of proseminar, two thirds of the class simply don’t give a damn.” Which is to say the wonks don’t really care much about the pedagogy and the teachers don’t care much for policy and the social theorists have an entirely separate set of interests.
  • Casual Learning Technologies. This was a mixed bag. The field is really, really new (James Gee, the discipline’s flag-bearer, is a linguist by training who got interested in gaming all of six years ago) and has a lot of room to grow. Which is to say, I wasn’t dazzled by the literature. Remind me to post my group’s final project, though. That was fun.

Current Coursework

  • EDUC325C – Proseminar. David Labaree, Francisco Ramirez. Required. Labaree, in his initial remarks to the class: “You may have heard this course features too much reading, too much writing, that the criticism is too harsh, and our opinion of schools is too pessimistic. It’s all true.” (Labaree has written a few books of note.)
  • EDUC359F – Research in Mathematics Education. Jo Boaler. Elective.
  • EDUC424 – Introduction to Research in Curriculum and Teacher Education. Hilda Borko. Required.

Winter Quarter #GradSkool Tweets

  • Yes, this is #gradskool and, yes, Angry Birds is on the syllabus. 6 Jan
  • Today’s #gradskool throw-down: Who won in US schools and universities — Dewey or Thorndike? Great discussion. Lots of nuance. 18 Jan
  • Stats prof, reading the room: “I don’t know how to make this more lively. I really don’t know how to make this more lively.” #gradskool 23 Feb
  • Carol Dweck is speaking. I am listening. #gradskool 8 Mar
  • Dweck has no slides. She’s four-feet tall, sitting on a table, feet dangling beneath her, positively /owning/ the room. #gradskool 8 Mar
  • Five rows from Michelle Rhee. An unlikely mix of education and business grad students in the building. 11 Mar
  • Rhee: “What we did definitely made people unhappy.” She literally seems to believe that diplomacy and efficacy are mutually exclusive. 11 Mar
  • Rhee: “Is there a less controversial way to do controversial things? I don’t know the answer to that.” 11 Mar
  • Rhee: “Chris Christie? I love him. He’s a Republican and I’m a Democrat. It’s not obvious we’d get along so well.” Seriously? 11 Mar
  • Rhee: “I worry about people going into the job with longevity as one of the goals. I’m not a big believer in longevity.” 11 Mar
  • GSB student: “Did you really eat a bee?” Rhee: “I did eat a bee.” Way to pitch her a fastball, Chuck. 11 Mar
  • These moguls were the most out of place contingent at the Rhee Q&A. Good luck finding the executive washroom, fellas. 11 Mar

Michelle Rhee followed me on Twitter the next day. So look out, right?

Favorite Winter Quarter Papers

I spent a few weeks of my winter quarter trying to make sense of the PBL / anti-PBL scrum of 06/07. Those papers are below, in chronological order, with a closing paper pitched specifically at math educators.

Spring Speaking & Workshops

[PS] Assessment

It’s bad enough when you’re trying to gin up interest in math by way of pseudocontext. It’s worse when you’re trying to assess math by way of pseudocontext. If the student isn’t interested in math by now, what do you think an assessment is going to do?

If your students miss these problems, how certain are you they really misunderstood the mathematics? How certain are you they weren’t distracted by the problem design?