Last year I began the year posting about the genius award I hand out from time to time. I stole the idea from John Benson, a high school math teacher of mine in Evanston, IL who handed out a sculpture of Rodin’s “The Thinker” to sit on your desk when you made an unusually insightful observation in class.
After just 3 weeks of school, I have handed out this award twice. Once, to a geometry student who during a sometimes/always/never activity around basic geometry concepts (2 points determine a unique line, a plane intersects a plane at a line, 3 points determine a unique plane, etc), called me over with the following comment:
“Ms. Krasnow, if I ever decide to have a career as a table-maker, I will only make 3-legged tables.
“Why,” I asked (quite excited because I knew where this was going)
“Because 3 points determine a plane, so it would be far less likely that it would ever wobble as badly as these 4 legged tables do.”
Full disclosure…my favorite part of this exchange was the idea that my class might set him on a career path as a table-maker…
I stopped the class, had him share his excitement over 3 legged tables, and allowed him and the class to relish in his moment of genius. Yes..that’s a bib with a sketch of Leonardo da Vinci which I bought in Italy. Students sign it in sharpie and have the option of keeping it on their desk or wearing it for the rest of the period. And yes, these are 8th graders so yes, they all choose to wear it (and photograph themselves wearing it at the end of the period).
His insight was exciting, but one which someone comes up with every year.
The second genius award of the year was something which came out of left field, in a very exciting way. I also teach 8th grade Pre-Algebra which is a course for students who took and failed Pre-Algebra in 7th grade. As part of my Math For America fellowship, I am only teaching 40% this year at my middle school and spending 60% of my time working at UC, teaching an undergrad education class on math methodology, working with Alan Schoenfeld, coaching my math department as we transition to using the Gates Foundation formative assessment lessons and working towards the common core standards. I spend my mornings learning about ways to determine and document how teaching problem solving and sense making leads to student learning. So, naturally, I am trying harder to make every day of my pre-algebra class involve open-ended problem solving, focused on student thinking. While I love and am quite comfortable teaching in this way, it’s a challenge to always hold myself to these guidelines in a class filled with so many struggling learners whose skills are far below grade level. Multi-day problem solving lessons, such as the Gates Foundation formative assessment lessons, lend themselves to designing class around student thinking, but day-to-day topics are often harder to craft in this way.
Last week, my pre-algebra class was looking at partial products in multiplication and how to use generic rectangles (instead of an algorithm) to solve multiplication problems in ways which supported mental math. I had modeled breaking factors up into expanded form and creating generic rectangles. BUT…this was not open ended and although it got away from an algorithm, it didn’t really allow for student thinking to guide the class discussion.
So, day, 2, I tried create a more open-ended approach to this work. I traced a rectangle on graph paper with simple instructions, “How many squares are inside this rectangle? Use any method you want to find the answer.”
Although it’s hard to read this student’s work, she divided the 26×25 rectangle into 10+10+6 by 10+10+5 which left her with simple calculations to find the product. Another student solved this by doing 20+6 by 20+5, while a third did it with lattice multiplication and a fourth by the ‘old fashioned’ algorithm (meaning the one I was taught). NOW..we had a real discussion on our hands as we took the time to discuss and compare each method.
But this student got the genius award for the ease with which she solved this problem using what I showed the class, but applying it in a far more elegant fashion. And this is a student who got a similar problem wrong on a pre-assessment the previous week when her only approach was the traditional algorithm. Designing everyday tasks to elicit student thinking…a work in progress!
Pi Crust 
You really have become a very good writer.
Interesting extension to the last problem: “How many squares of any size are there in this rectangle?”
That was my first reaction, too. There are a LOT more squares in there than the student originally calculated! That is a really interesting problem to consider. I love this blog. Another observation from a pedant: Three NON-CO-LINEAR points determine a plane. Three CO-LINEAR points are part of an infinite number of planes. I also loved the three-legged chair observation. Fantastic, especially from such a young student.
David…that’s a nice one for them to ponder in the coming weeks! Thanks
AND . . . replying to myself about the requirement that three points only define a plane if they’re non-co-linear, you can use that to enter into a discussion about dimensionality (one-dimensional line, two-dimensional plane, three-dimensional volume, zero-dimensional point) and even beginning concepts of linear algebra — basis vectors for spaces (Why is it that the three points must be non-co-linear? You must have the second independent dimension. Similarly for three line segments determining a volume — they must be independent line segments or everything collapses into a lower dimensionality because of the dependence.). Middle school kids should be able to grasp these concepts, particularly if you demonstrate them with things like string and metre sticks (or yard sticks if you’re living in the only industrialised country on earth that still uses the Imperial measurement system).
@Not Dannevirke High School: all great points. Yes, maybe my post was confusing, but the question of 3 points defining a plane, that was part of a sometime/always/never conversation and as you said, the answer is sometimes. we always have a lot of fun playing around with the difference between true/false questions versus sometimes/always/never where false means a single counterexample (as in 3 colinear points don’t determine a unique plane), but sometimes could mean a single counter-example or many. if there were time in september, i’d show the movie, Flatland as it gets at the idea of dimensionality. Do you know that film? I always save it for the end of the year, but it actually fits quite nicely in this conversation.
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