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Archive for June, 2011

Well-Formed Math Problems

Sebastian Deterding, in his keynote on gamification:

I would argue that these are not only principles for good games, but for any human activity to be well-formed: we enjoy situations with clear, structured, unconflicting goals, clear, limited action spaces with choice, clear and fair rules, scaffolded challenges and complexity matched to our abilities, and clear, actionable short- and long-term feedback.

Here are two very similar #anyqs entries from Dan and Nancy — two teachers I worked with in Grand Forks, ND. I asked my readers to decide which was the better first act and I disagreed with most of them.

Dan

Dan's features a spigot leaking into a bucket. It's just leaking. Drip drip drip drip drip.

Nancy

Nancy's features a faucet leaking into a measuring cup. It's just leaking. Drip drip drip drip drip. But Nancy also includes a timer on her iPhone. And at the end of 39 seconds she draws the measuring cup close to the camera so you can see how many ounces have leaked out so far.

Why Dan Has Told The Better Mathematical Story

The first act of a good story introduces a conflict. It does very little to solve it. Think of the shark in Jaws munching on the lady swimmer. At that point, we have no idea what tools, resources, and information will be brought to bear on the task of killing the shark. We only know we want it dead.

The first act of a good story asks very little of the viewer's intellect. It appeals, instead, to the gut. The viewer of Nancy's first act would ideally think, "My word. How much water is that faucet going to waste?" Instead, because Nancy has already foregrounded the tools, resources, and information that belong in the second act of the story (just several minutes later in the lesson!) the viewer thinks, "Oh. This is a math problem, isn't it?"

We need to curb our natural tendency as math teachers to burn up interesting problems on an altar to our math gods. In this case, all that means is you wait until after your students have formulated a question that interests them before offering them tools, resources, and information to solve it.

BTW: Picky? Absolutely. But where's the fun in this job if not in negotiating the details. For whatever it's worth, if you called me out for featuring timers prominently in the first acts of my own stories (as Bowen Kerins did recently) you'd be right on. The timers came from a position of insecurity that no one's going to wonder "how long?" if I don't explicitly call out time in the first act. That's done now.

Brief Remarks Encapsulating Spring Quarter

It's hard to know how much disclosure is worthwhile here. For my own sake, I'm going to post a reminder to myself that this was the quarter I thought I was juggling everything like a champ only to have basically everyone in my life, in the same week, point out that I was only going through the motions of a world-class juggler. All the people, tasks, and things I thought I was juggling with such verve and style were lying on the ground around me.

There were basically endless ways to invest a few dozen hours this spring. That included interesting classes and projects at Stanford. It included the week I spent in Singapore learning from and working with ten of the world's best math curriculum designers. It included speaking, workshops, and webinars. It included Graphing Stories, a project that seemed too fun not to pursue even though editing 160 stories cost me a pile of time during finals week.

I half-assed my way through much of it, convinced the entire time that I was owning all of it. In my first-year review, my advisers were rightly concerned about me, and about Stanford's investment in me. I'm putting in a hectic pace this summer (see below) after which I need to sit down and take a machete to my calendar and day planner.

One outcome of the first year of grad school is that I became a faster, better writer. Blogging for years on an if-I-feel-like-it basis didn't do much for my proficiency and speed at assembling an argument. Reading great writing daily (see below) and being asked to write a few thousand words about it every few weeks has done a lot of good for me. (I need to get faster at reading, though.)

The other outcome of this last year is that I've gone a long way to shed what Labaree (see below) calls the "normative view" of education. I'm less concerned with how I think things should be, with proving out my own pet theories, and more interested in accurately describing how they are. At the same time, other professors will insist that your pet theories are the reason why you were invited to doctoral study at Stanford. This is a tension I don't expect to ever resolve. It's a feature of grad school, not a bug.

The Sum Of My Research Interests

We submit a paper at the end of year two — a fun-sized research project, basically — that qualifies us for doctoral candidacy. The final project of the first year was a proposal for that study. My exact research question for that paper is this:

What teacher moves during a task's launch lead to its productive implementation by the students?

Elaborating further, I taught a class a few weeks ago at my old high school. I popped in to say "hi" and wound up leading two activities for my old department head. In both cases, I had to launch the tasks. I set a scene and questioned the students about it to the point that I thought we were ready to work within it. With one problem, the task transitioned smoothly from launch to productive work. In the other, the task made a rocky transition. I find that moment of transition suspenseful, highly motivating, and worth some study.

Favorite Spring Quarter Papers

I read the last few pages of Augier & March five times, and the last paragraph, which features one of the most satisfying turns of a phrase I've read in grad school yet, a few more than that. I'd give a finger to be able to write a tenth as well as this team. (Rumor has it that March is the poet of the two. Reportedly, he rejects a syllabus for his Stanford business courses, assigning novels, poetry, and Homeric epics instead.) ¶ Berger & Stevenson wasn't assigned but it's valuable for anyone trying to carve out a living within education, but outside the classroom. ¶ Delpit explains why some minority parents prefer lecture and drill-oriented skill practice. ¶ Doyle & Carter describes the negotiation of a task between teacher and students better than anybody. This is high drama. You're watching Ms. Dee start with a high-value academic task that her students negotiate down to nothing. ¶ Erlwanger's piece was assigned as an example of research that has a) stood up over time and b) affected policy and practice in spite of its small sample size. ¶ The piece by Jackson, et al, isn't available yet (though the author, herself, was extremely forthcoming) but it is the most forceful take on the task launch I've read yet. It comprises, like, 90% of the conceptual framework for my qualifying paper. ¶ Labaree's proseminar course could basically be described in a single line: "why reform is hard to pull off." Every time I read his stuff, I found myself thinking, "Oh so this is why Scott McLeod and Will Richardson are so angsty all the time." His second piece describes the transition from teacher to researcher in a way that had all of us classroom expatriates nodding our heads grimly. ¶ The question no one seemed to be able to answer convincingly was "What is a conceptual framework, exactly, and how does it differ from a literature review?" Lester goes a long way, though.

Augier, Mie & March, James G. (2007). The pursuit of relevance in management education. California Management Review, 49(3) (Spring), 129-146.

Berger and Stevenson. K-12 entrepreneurship: slow entry, distant exit. Retrieved June 2007.

Lisa Delpit. (1995). The silenced dialogue. In Other people’s children (pp. 21-47). New York: New Press.

Doyle, W. & Carter, K. (1984). Academic tasks in classrooms. Curriculum Inquiry, 14(2), 129-149.

Erlwanger, S. (1973). Benny's conception of rules and answers in IPI mathematics. Journal of Children's Mathematical Behavior, 1(2), 7-26

Jackson, K. (2011). Investigating how setting up cognitively demanding tasks is related to the opportunities to learn in middle-grades mathematics classrooms.

Jackson, K. (2009). The social construction of youth and mathematics: The case of a fifth-grade classroom. In D.B. Martin (Ed.), Mathematics teaching, learning, and liberation in the lives of black children (pp. 175-199). New York: Routledge.

Labaree, D. (2003). The peculiar problems of preparing educational researchers. Educational Researcher, 32(4), 13-22.

Labaree, D. (1997). Public good, private goods: The American struggle over educational goals. American Educational Research Journal, 34(1) (Spring), 39-81.

Lester, F. (2009). On the theoretical, conceptual, and philosophical foundations for research in mathematics education. ZDM, 37(6), 67-85.

Schoenfeld, A. (1988). When good teaching leads to bad results: The disasters of 'well-taught' mathematics courses. Educational Psychologist, 23(2), 145-166.

Stein, MK., Grover, B., Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.

Turner, Ralph. (2000/1960). Sponsored and contest mobility and the school system. In Arum, R. & Beattie, I (eds.). The structure of schooling (pp. 22-35). Mountain View: Mayfield.

Webb, N., Franke, M., De, T., Chan, A., Freund, D., Shein, P., Melkonian, D. (2009). 'Explain to your partner': teachers' instructional practices and students' dialogue in small groups. Cambridge Journal of Education, 39(1), 49-70.

Music For Final Exams

James Blake.

Spring Speaking & Workshops

I'll be in ten states doing ten workshops and keynotes this summer. Three of those are still open for registration. Details here.

  1. Grand Forks, ND. June 13-14. Grand Forks Education Center.
  2. Beaufort, SC. June 21. Beaufort County Summer Institute.
  3. Bowling Green, KY. June 22-23. Green River Regional Educational Cooperative.
  4. Richmond VA. June 24. MathScience Innovation Center.
  5. New York City, NY. June 27. Math for America.
  6. Grapevine, TX. July 19. Conference for the Advancement of Mathematics Teaching.
  7. Orlando, FL. July 28-29. NCTM High School Institute.
  8. Washington, DC. July 31. Siemens STEM Academy.
  9. Atlanta, GA. August 2-5. The Lovett School.
  10. Mountain View, CA. September 10. The Perplexity Session.

John Burk reflects on his online PLC and at the same time serves up a primer for anyone wondering, "What's the deal with Twitter / blogs / etc?"

The teachers I see that truly embrace the online world—those who move from simply consuming neat ideas and bookmarking cool applets to engaging in the conversation and sharing what they do—see a leap in their growth in teaching that can’t be matched by any workshop, conference, degree or professional growth plan.

Agreed. Totally. In a webinar Q&A last week, someone asked me to make a pitch for blogging. I speculated that the edublogosphere gave me a two-for-one deal: two years of growth as a teacher for every school year I taught. That isn't remotely empirical, but blogging openly, transparently, with an explicit invitation for criticism, accelerated me through a lot of my misguided early-career enthusiasms. (Worksheets, slide design, and NCLB, let's say.) A lot of commenters worked overtime to help me understand the ceiling on those devices was lower than I thought. The positive response to the first WCYDWT, on the other hand, turned me toward a productive vein which I've been chipping away at ever since.

Food Pyramid Math

Geoff Krall gets mathematical with the food pyramid redesigns:

I was all set to contest that the new (“old”) food pyramid, adopted in 2005, was garbage mathematically and visually. And the challenge was for students to come up with a better, more mathematically accurate, food pyramid. Then the United States government dropped the new MyPlate diagram in my lap.

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