Interestingly,
I have always had a minor bone to pick with Plato and now I feel even more
justified. When I first read Plato's Republic,
I detested the idea that in his utopia, people would be born knowing exactly
where they belonged and the occupations they would hold. For me, the joy in
life is through discovery. Logic alone hasn't held my interest as a human;
rather, I long to approach most things from a feeling perspective. In fact,
when I decide to become more logical, I come across as cold and that's not my
authentic self. This week's introduction helped me understand why Plato's
philosophy has always bothered me—his view that the physical world is inferior
and that only 'pure thought' leads to truth excludes people like me who learn
through feeling and doing. The course reading explains how this binary thinking
has made mathematics seem like 'an otherworldly realm' accessible only to an
elite few, when in reality, math comes from and connects back to our bodily
experiences.
Roger
Antonsen's TED talk resonated with me for several reasons. I embraced the idea
of perspective taking-especially the way he demonstrated 4/3 in a variety of
ways. I was particularly tickled by his four triangles. It really uses
the imagination and demonstrates deep understanding. It also reminded me that
embodiment doesn't equal understanding and how imperative it is to help
students connect ideas to how and why they are using manipulatives and so on.
This week,
we were learning about standard form and expanded form. I decided to throw in
some movement in the hopes of aiding student understanding. For standard form,
students went from sitting on the carpet to standing with their arms straight
up and overhead. For expanded form, students were asked to spread their arms
wide. The physical contrast between 'compact and contained' versus 'spread out'
matched the mathematical concept perfectly. In the spirit of using our bodies,
my class also tackled traditional measurements of length. For many students,
this was a chance to use a ruler in a low-stakes way as it was fun, and they
all enjoyed seeing what their personal body measurements calibrated to.
Collaborating with partners was part of the experience and the room was alive.
What I really noticed about this exercise was that students were genuinely excited and enthusiastic. I had zero students sitting to the side—it was 100% participation. We extended this to a practical application, using our body measurements to figure out spacing for salmon cutouts along our fence. However, we ran out of time after measuring our salmon template, so we'll continue calculating fence spacing next week. This activity challenged the Bourbaki idea that mathematics should be abstract and divorced from the physical world. When math is grounded in embodied experience, every student can access it.
I love your idea of having students physically contract and expand their bodies to explore standard and expanded forms—it’s such a brilliant way to make abstract concepts tangible. I also really resonated with your note about running out of time. I often feel that tension too: when students are fully engaged in a hands-on math activity, the time flies, and I’m left wondering—do we keep going and risk falling behind in other subjects, or pause and hope they’re just as engaged next time?
ReplyDeleteI remember a teacher once suggesting that I put exact times for each activity on my schedule, but I resisted. I worried that if we went past that time, students would start asking why we weren’t where the schedule said we should be, like gym at 10:00 instead of 10:15. It made me realize that sometimes we need to make intentional space in the day for experiences that take longer, because the learning and engagement are worth it. I’ve also heard of school systems that dedicate an entire day to math or art, and I always thought that was fascinating.
Reading your post reminded me that it is possible to spend a larger portion of our day on one rich, hands-on activity. Even if it feels risky or like we might not “cover everything,” the depth of understanding and excitement that students gain can make it more than worthwhile.
You used gesture to help make meaning of the math! The arms-up-for-standard-form and arms-wide-for-expanded-form is such a clear example of how gesturing helps students build conceptual understanding first, and then translate that understanding into language (Goldin-Meadow et al., 2009). In the classroom I so often see students know what expanded form means way before they can say it cleanly or teach it to another. The gesture gives them something to anchor the words to, so when we later say “expanded,” it gives them an extra cue to pull from. Its like the gesture is a bridge the gap between embodied understanding and verbal acquisition.
ReplyDeleteAntonsen’s triangles are delightful, but as you point out, imagination and manipulatives only do the heavy lifting if we see how and why they work, especially in math when mimicking is extra common!
I love an alive classroom! So many ah-hahs. One hundred percent participation is amazing! I agree with Vannessa that this learning sometimes takes longer and is "weighed" against what it is taking time from which can add stress to the day. Overall, this is showing how math can be as a social and include such physical activity. Challenging Bourbaki by accident, or maybe on purpose, feels like a double bonus. It seems like this brings down many of the barriers to entry and hooks them in with fun! I look forward to the salmon spacing update. What a fun tie into where we live!
Ahh, I also rail against Plato's supposedly ideal world! It all seems so predetermined and hypercontrolled. Where's the mystery, and the joy of making and discovering? Kristie, great insight too that embodiment alone does not equal understanding. There is a subtle dance among sensory experiences, discussion, explanation, symbolizing, drawing, back to sensory experiences, as we use all the resources at hand to explore and double-check our meaning-making with a new set of ideas. Thanks everyone for the great discussion! Isn't it wonderful when kids are fully engaged and enjoying math? We can find ways to make the timing work with that, I'm confident of it!
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