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Saturday, July 30, 2011

Summer Learning, PCMI Edition: Deeper Criteria

This is a series of posts that are reflections from the Park City Mathematics Institute Secondary School Teachers Program.

One afternoon we listened to a lecture/powerpoint by Douglas Corey of Brigham Young University about comparisons of effective teachers at home and abroad. Toward the end of his talk, he seemed to have partaken of a generous serving of the edureformer kool aid and came across as anti-teacher or at least teacher-concern-dismissive, which obviously turned many people off. However I took away some notes about his research that struck me as important.

Based on classroom-level comparisons between different countries from the TIMSS video study, researchers found
  • there is no single effective teaching method
  • all high-achieving countries teach quite differently
  • we can not judge a lesson's effectiveness by methods used, but rather
  • effective lessons have deeper criteria in common he called "instructional principles"
He asked us to predict the instructional features which must be present for students to learn with understanding. These were the guesses that my group brainstormed. If you want to play the home version of the game, take a moment to jot down what you think they are too.
  • teacher content knowledge
  • problem-solving
  • students have to be working and thinking
  • deeper explorations
  • students making connections
  • continuous assessment that informs instruction
  • deliberate metacognition is part of instruction
  • teacher believes all students can learn rigorous, conceptual mathematics
  • students need to spend time thinking about math outside of class
However researchers only found two:
  • "intellectual engagement" - the teacher has to get the kids thinking about a problem. Students have to struggle. "Struggle" means students expend effort to make sense of math, to figure something out that is not immediately apparent. It does not mean needless frustration.
  • "connection-making" - the focus of the teaching has to be on making connections. Connections don't come by accident but must be an explicit focus of planned instruction.
The struggle thing rang true for me. At some level I internalized that idea long ago. I'm still coming to terms with the connection-making point. The same concept was approached earlier by Gail Burrill with respect to the Common Core standards. She pointed out that in American classrooms, teachers can plan and ask connection-making questions and activities, but students mostly still end up doing procedures. A big question I am still grappling with is how to design and deliver instruction so that the students are doing connections. I have only vague notions about what that would even look like. I don't really know what to do with this yet beyond hang a sign on the bulletin board next to my desk at school that says "make the students do connections."


  1. I like to see people quoting studies to show what's effective, but I'm wondering where in the study it says this. I was so excited to get a good quote from the study I skimmed most of it and read through the conclusions section without reading what your presenter shared. Perhaps I missed it. If you can find out what page(s) of the larger study it's on, I'd love to have those results in my back pocket. Thanks!

  2. I'm probably unfamiliar with this, because I'm not a math teacher, but what exactly do you mean by "making connections"?

    I've always interpreted it as casting new knowledge in the context of old knowledge because the new and old concepts are related and/or fundamentally the same.

    For instance, after learning modular arithmetic, one can go back to looking at the divisibility rules and understand why they work and how one might create a novel rule (for say, 37, since 3*37*9 = 999).

    I'm wondering how much this type of connection-making can be influenced by something the teacher does (e.g. by encouraging an environment of exploration and question-asking) vs. a mentality towards learning that has become systemic, especially for older kids. (In the sense that schools/teachers/parents pressure kids to always get the correct answer, not ask "stupid" questions, etc.)

  3. Really nice post, Kate. I think the two items you list there encapsulate much of the more detailed descriptions the group came up with.

    Struggling seems to encapsulate the notion that students have to be thinking, and in particular, thinking to expand there thoughts and abilities. Reaching into the embryonic goo of their brain to find what works.

    Making connections is solidifying those embryonic notions and latching them onto each other (obviously).

    I think a lot about "making connections," because it seems to be eat the heart of what math is - the idea that there is an essential and abstract underlying structure that can be pinned to much of, if not all of, the world.

    I'm a huge fan of Douglas R. Hofstadter, and I think about analogy as the fundamental way we can make connections. It's what morphisms are. Here's a link to Hofstadter's lecture called, "Analogy as the core of cognition."

    I do this with Bongard problems to name one example, but analogy seems to run through mathematical thinking quite naturally. Explicitly sharing my analogies with the class allows them to analyze them, break down weak points, and come up with there own.

    I'll probably post about this on my own blog, "Lost In Recursion," relatively soon, but I wanted to share.

    Thanks again!

  4. @Patti, I don't think Doug was necessarily quoting the TIMSS study for all of this, but rather citing a number of studies including his own. He may have provided citations in his presentation but I don't have a link to them, I'm sorry.

    @Hao I don't know exactly what they mean, either. I think your modular arithmetic/divisibility example is apt. As far as the systemic mentality, I have to believe that changing that with my students is entirely within my control.

    @Paul, thanks for offering analogy as a way of doing connections. I'll have to think on that.

  5. I liked the modifier "intellectual" when it comes to engagement. I've had people (cough::smartboard reps::cough) try to sell me a number of times that seeing things move around or blink or some other shiny new tech toy will help me because it will increase my students' engagement.

  6. Am I the only one who has had (many times) the experience that some students will refuse to make connections because it is "too hard"? For these students there is no payoff for struggling and the penalty is - according to their lights - worth it.
    I am going to take issue with Kate - I don't think it is entirely within our control.

  7. @iodean,
    Interesting point; I wonder if there aren't ways to raise the (intrinsic) payoff for stuggling to make connections. In that case, some connections may need to be with regular life rather than solely within mathematics. Which I bet is one reason I like teaching stats to the mathematically disaffected. Not that I have the answers, but I can see the benefit of at least starting out by making myself a couple postits ("make connections!" "struggle!")...

  8. I think that all we can do is encourage the connections, and create en environment in which they are more likely to happen organically, based on their own interests. Otherwise, it won't be authentic. For example, I showed them your video, Kate, of the crosswinds landings, and the connections they made to the vectors lessons depended on their own interests, like one wants to be a pilot, another never wants to fly again...What might work, then, is using a classroom model that allows for differentiation, like flipping for example.

  9. Oh and also I like the word "struggle" much more than the one I was using, which was "suffer". Clearly I still have a few issues with being the all-powerful-immortal teacher...


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