Week One - Mathematics and the body

Reading the introduction reminded me that the tension between concrete and abstract in mathematics has been around for centuries and is unlikely to disappear any time soon. It is readily apparent in the academic and workplace math streams with the latter often viewed by students (and some teachers) as a lesser form of mathematics. 

 

I think it is the same with manipulatives, though I do wonder if manipulatives can become a crutch if used incorrectly. Yet, it is a false dichotomy and we should embrace an “and” understanding of the two ideas rather than an “or” understanding. As suggested by Antonsen’s Ted Talk, we need both, and more, to see the whole picture. I was fascinated by all the different geometric representations he used for 4/3! In my experience, geometry is an area that some students grasp quickly while others struggle greatly. “Down with Euclid! Death to all triangles!” honestly made me laugh as I can easily envision that being the rallying cry of some students.

I enjoyed the measurement activity as a fun beginning to embodied mathematics. My final paper/project in EDCP 552 was related to embodied learning with a focus on the link between gestures in spatial reasoning. My husband and I calibrated our body measurements, then I went outside to pace the length of the retaining wall next to our house. My body-based measurement was 27 paces, 1404 inches, or 117 feet in length.

We are planning to add a second tier to the wall so my extension activity was calculating the number of blocks needed. Using my hands, each block is 16 inches. At 1404 inches the wall is 88 blocks long. If we do a second wall that is 6 blocks high we will need 528 blocks. My number was within 10% of the actual number my husband had calculated so my body-based measurements made a good estimate. 

 

The activity reminded me of the “back of the envelope” physics calculations from my university days used to check that a result was reasonable. Physicists also like making assumptions to simplify things, so I also appreciated Antonsen talking about making and playing with assumptions. 

The idea of playing with math seems to be missing in many secondary classrooms, including my own. Likely a combination of pressure to cover all of the material and a significant proportion of students not finding math fun. They would relate to the idea of math as “a complicated but ultimately meaningless game of moving symbols on paper according to algorithmic rules”, but not in the way Bourbaki mathematicians intended. Embodied activities could help students take math a little less seriously and help shift the perspective of math as symbolic manipulation.

My reading was Gerofsky’s Seeing the graph and being the graph. The chapter was an investigation of the gestures used to represent and describe graphs. Students in grades 8 and 11, as well as 2 teachers, were given graphs of functions and asked to describe the graphs using gestures, sounds, and non-technical language. Students who struggled with mathematics had significant difficulties with describing the graphs and their gestures were often inconsistent with the features of the graph. 

Students with an algorithmic understanding of mathematics had a tendency to recreate the graphs using their arms and fingers as if they were drawing on a white board in front of them. They also focused on specific details and numerical values present in the graphs they were give. Students with a deep understanding of mathematics had a tendency to use their arms and bodies to enact the features of the graphs, being the graph. I wonder if Plato and the "enlightened" mathematicians would be horrified by the top students debasing themselves in this way? Though I think it clearly shows that a physical understanding does not mean a lower level of understanding. 

It was quite an interesting read and the results made sense to me. Students with strong understanding are able to view and represent the graphs using multiple perspectives. The others were only seeing from one perspective and simply attempted to re-create what they saw. I also really appreciated the tempered suggestion that gestures and embodied learning could make a useful addition to the teacher’s toolkit, but do not nullify other approaches. In my experience, there is a lot of emphasis placed on visual and auditory learning, but kinesthetic learning is often overlooked at the secondary level, especially in courses that are considered more rigorous or academic in nature.

She writes, “an embodied gestural approach to the teaching of graphs and functions would be helpful in offering a multimodal resource for learners to draw on in their studies. That is not to say that current teaching methods using algebra, word problems, tables of values and drawn diagrams ought to be abandoned – quite the contrary. Rather, these more traditional methods ought to be supplemented by elicited large, close-up gestures, especially in the initial stages of teaching mathematical functions." (p. 254)

I wonder, would using gestures as a teaching tool in math help students with an algorithmic understanding, or could it muddy the waters since they don't "experience" math?