Mathematics and the Body
“This research [...] takes up the idea that abstract mathematical concepts are necessarily grounded in our physical, embodied experiences of the world - and that, in fact, the historical origins of these abstract concepts always emerge from empirical, sensory observations (Gerofsky, 2011, p. 246). Throughout our experience for this week’s material I was thinking about the origin of what we are teaching and the why. Teacher’s are frequently asked by students, why do we have to learn this? Math & Measurement in the Garden: Garden, Spaces & Technology reminded me of just that. By teaching students this information (the why) and allowing them to ‘play’ with their own relationship to the concepts, students can develop a deeper understanding of what they are learning through their own sensory observations of the world around them.
Antonsen (2015) argues that to truly understand we need to be able to adapt our perspectives. Gerofsky (2011) gives the perfect example of students who are able to demonstrate their understanding from various perspectives and those who are able to only show their understanding on paper. “I realized that my own schema for graphing placed the ‘origin’ (the (0,0) point) of the Cartesian graph at my navel. [...] It was clear to see that some of my students shared my internalized notions of the placement of the x-axis and my propensity for large, physical gestures of graphs, while others placed the x-axis higher and used smaller gestures within the classroom context” (p. 248). This example from her research shows how students who internalized their understanding of graphs by using large gestures, illustrating that they are becoming part of the graph, placed themselves within a different perspective. On the contrary, the students who used the smaller gestures were mimicking what this graph looked like on paper. These students had yet to internalize their understanding of the concept by thinking about it from a different perspective. They were, rather, demonstrating their understanding of the curves in the exact way they had been shown in class.
After immersing myself in this week’s course content, I am left wondering:
To what extent should we explicitly make these connections for students? To what extent should we allow them to explore the content to make their own conclusions?
How can we help all of our students see mathematical concepts from various perspectives with the time constraints and vast amount of content that needs to be covered throughout the school year?
References
Gerofsky, S. (2011). Seeing the graph vs. being the graph. pp. 245-256. https://drive.google.com/file/d/1XoLRw6hk6jyfKZCpg4FQC04uWfr-C1h7/view?usp=sharing
This week I've been thinking a lot about the connection between the physical experiences of math and abstract mathematical concepts, and enjoyed reading your post about students benefitting from gestures and kinesthetic movements. I, too, get students asking about the relevance of what they are learning and I always direct the conversation toward the fact that they are learning broad skills and concepts and the specifics are just the way that we get there and practice those more meaningful skills. I've been inspired this week to encourage students to interact with some of the math concepts we are covering in a more meaningful way.
ReplyDeleteI think allowing students to explore vs. providing them explicit connections is a delicate balance. Critical thinking is a skill and building an environment where students have the time to explore connections on their own takes a lot of trust-building. In my experience that trust tends to not happen until later on in the year when I know students won't take advantage of the freedom and extra time they're given and that they will (for the most part) remain on topic. This ties into your second question that there's a limited amount of time to tackle these things. I think interdisciplinary projects are the answer, allowing students to experience and apply math in real-world situations or scenarios as explored in other subjects.
One of the specialists in our school focuses on technology with the students and we've asked her to supplement our math by using Minecraft to have the students explore area, perimeter, and volume. Not math with your body, per se, but a way that you could take advantage of another subject to have students experience math in some other way. I can think of a lot of connections to music, too.
I really liked your second question (How can we help all of our students see mathematical concepts from various perspectives with the time constraints and vast amount of content that needs to be covered throughout the school year?) because it is something I find myself struggling with every year. Allowing our students the time to play and discover concepts vs. providing them with these connections is such a balance, one I feel I will never truly win with. One of my biggest hurdles has been finding balance with this while also working to ensure everyone in my class is getting what they need. I am not currently in the classroom as I am on maternity leave but last year alone I had students working at the kindergarten level in the same class as kids working well above grade level. Sometimes it feels so impossible to provide those rich learning environments with such a vast level of understanding.
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