I will often bring these rich tasks to our class for students and I have found that students not only enjoy them, but that these tasks have also been great for developing and reinforcing the "habits of a mathematician." I usually title these tasks "POWs" (for Problem of the Week). Yesterday, as part of the progression in figuring out the area of the "infinite star" (aka Koch Snowflake) I gave students this problem hoping that we could emphasize the importance of 'taking things apart and putting them back together':
I was pretty surprised by the difficulties students had with this problem. In fact, I had even included a more difficult one in anticipating students blowing right through this one. About 50% of the class was completely stumped by the original problem (which I appropriately named the 'Crooked House'…in case you haven't noticed, I appreciate a cheesy title). I asked them at the end of class to reflect on and discuss strategies that were helpful and even asked them why this problem was so difficult. They said things you might expect:

"we didn't have a formula"
"I needed the distance formula"
"I didn't know where to start"
"rusty geometry"

A little piece of my soul died. We have spent all year de-emphasizing formulas and talking about habits that give you a place to start. Then, this happened:

student: "It was like a POW, but not."
me: "What do you mean?"
student: "Well, it seemed like a POW-type problem but it wasn't one."
me: "So, just because it didn't say 'POW' at the top of the page you didn't approach it in the same way?"
student: "Yeah…kinda."

It was clear that students see math compartmentalized into separate worlds. I haven't helped them see that these habits ARE mathematics. They aren't just useful for puzzle-like problems, these habits are at the foundation of the creation and authoring of ALL mathematics. With my last two units I have started to include 1 or 2 of our habits as the central focus. Today's prompt looked like this:
It was a great reminder of the importance of redefining mathematics for students, helping them see connections, and making sure the message we send is clear and consistent in ALL that we do.


02/20/2012 6:14pm

Constance Kamii, a Piagetian scholar, writes about common pedagogical strategies to teach (she has a fairly broad conception of this) algorithms is harmful for children's learning. Rather than measure learning by the number of correct answers produced quickly (i.e. by achievement tests), she & Piaget measured learning by evaluate a child's ability to reason logically and numerically, and their confidence in being able to figure things out. And she states, "Algorithms are harmful for two reasons: (a) They encourage children to give up their own thinking, and (b) they "unteach" place value, thereby preventing children from developing number sense (Kamii, 2000, p. 83).

In your experience last week, it is quite evident that algorithm-based teaching has untaught your 11th graders about area, and that they clearly have given up on believing they can utilize their own thinking.

This is a problem I can watch a 5th grade class of students take on with ease! (that is not a surprise to you)

The challenge -- how can you reteach 16-17 year olds to re-engage their own thinking?

02/21/2012 5:42pm

I accept that challenge!! BUT, I wonder what we can do to reduce that type of algorithmic teaching earlier on. As we have talked about, the way in which we grade, give feedback, etc must support this goal, no?!?

02/22/2012 12:34am

I am currently at a loss. I believe in my soul that a true paradigm shift, actually TWO paradigm shifts, will have to occur--and probably simultaneously. Both of these on the order of Kuhn's Scientific Revolution.
The first will be an actual shift in the enactment of modern (constructivist) learning theory into teaching theory. Today's teaching theory remains entrenched in behaviorism.
The second will be a social-political shift that replaces hierarchical-centralized-depersonalized structure with heterarchical-decentralized-personal. [I must attribute this second though to Seymour Papert (and colleagues, cf. Turkle & Papert, 1990 at http://bit.ly/xRyjuD)

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