Article Summary: Seeing the graph vs. being the graph
This exploratory study navigates ideas of gesture in graphing from Gerofsky's reflection of its natural presence in their teaching practice and in their students communications to its formation into initial exploration with people they knew by having them video themselves gesturing to describe graphs and its development into the more formalized research with grade 8 and grade 11 students of varying abilities using gesture to describe a varying set of 5 graphs. Predominately there were three types of responses that emerged from the students gestures that were broken down into 3 categories:
"Category 1 students were precise and followed rules carefully, but often depended on memorization and algorithmic thinking rather than engaging fully with math concepts. ... Category 2 students' visceral, experiential approach to the graphs and multiple metaphors and verbal/kinesthetic/visual representations allowed them multiple potential entry points for sense-making and the creation of more robust mathematical conceptual objects. ... Category 3 students were in urgent need of help in learning to see graphs as whole objects and in bringing attention to those features of graphs considered mathematically salient." (Gerofsky, 2011, p. 253)
After reviewing and reflecting with the participants it was revealed which students were considered top students, average students, and struggling students and this strongly aligned with which of the categories students fell of the three described. This leads to further opportunities for exploration into the impact of intervention and the ability for observation of students gesturing in graphs as a a way to determine students patterns of noticing and engagement with secondary mathematics.
I had initially read this article in August during the MACAS conference after attending Gerofsky's keynote Opening up New Liminal Spaces for Learning, Research and Creativity with Mathematics and the Arts. I looked back to my original notes from my first read through of the article after reading it again this week to see what things caught me my first read through and what caught me this time, which ended up being very similar.
Stop 1: An observation within the exploratory study about gestures in response to given graphs, "Although the gestures produced were not spontaneous gestures accompanying speech and were produced deliberately at the researcher's prompting, subjects were largely unconscious of features of the gestures they made." (Gerofsky 2011, pp.248-249).
Throughout this article I found it fun to stop and engage in the physical gestures myself, and even when doing so with full intentionality I still was not consciously considering many factors that surprised myself. The origin being a significant point that I was not consciously considering or identifying in my choice of movements, upon reflection I believe I have identified the center of my chest to be where my gestures for this correlated the origin to be, though I don't think that is consistent when I have gestured in discussions of graphs in front of classes in the past.
Stop 2: Figure 1: The five graphs used in the second pilot study (in schools).
I took a minute to try to gesture the graphs used in the study myself. I found that I used a mixture of speed, sounds to communicate features of the graph, I consider the straight parts of the graph to be faster so my motions were faster when describing them. Sounds were often indicators of turning points as if the line was bumping into something to change its direction and turn. Reading through the categories I had features of both category 1 and category 2 in my gestures of the graphs.
The gesturing of graphs as an alternative way of expressing them connected with the video of the week and its discussions on perspective and your ability to change perspective being tied with your understanding. The language used in Rodger Antonsen 2015 TED talk video Math is the hidden secret to understanding the world, discussing thinking of equal signs like metaphors felt like a light turned on, the connection felt so applicable, having just been finishing up a unit on fractions with my grade 8 classes so many conversations reviewing and discussing equivalent fractions I wonder how those could have gone with more integration into different perspectives and using language like metaphors in our conversations around equality.
Teaching grade 8 and 9 where graphs are more formally entering the students math class worlds I have been wondering about ways to possibly gamify the gesturing to make it feel like something they may be more willing to engage with in a more full bodied way, almost like a graphing charades. Wondering what this would look like in practicality and if it would push for them to participate in the bigger motion exaggerated way or if they may start to find a uniformity and fall more to the category 1 grouping from the article.
Question: In the gestures you make to communicate, how consciously are you considering the details that make up your gestures? When you read through this and saw the graphs did you also immediately start trying to gesture through them yourself as well?
Citation: Gerofsky, S. (2011). Seeing the graph vs. being the graph: Gesture, engagement and awareness in school mathematics. In G. Stam, & M. Ishino (Eds.), Integrating gestures (pp. 245-256). John Benjamins Publishing Company. https://doi.org/10.1075/gs.4.22ger
“Category 2 students' visceral, experiential approach to the graphs and multiple metaphors and verbal/kinesthetic/visual representations allowed them multiple potential entry points for sense-making and the creation of more robust mathematical conceptual objects”
ReplyDeleteI resonated with this quote because I have witnessed this in my classroom. Although, I must admit that not all my students will use their whole body to describe graphs. I have some that like to ‘hear’ graphs on Desmos (and would use sounds to demonstrate salient features of graphs), many rely on verbal representations/metaphors, and some with small-scale gestures. As for myself, if I had to do this experiment, I would probably have put the origin where I can see it and use verbal metaphors and small-scale movements to describe graph features. I think it is because I am an observer; I look carefully at things trying to understand them and find patterns. It makes me wonder, do personal characteristics or learning preferences make us choose some metaphors over others?
Also, what I like about this statement is that there is an emphasis on the use of verbal, kinesthetic, and visual representations. What I take from it is that we do not have to make big gestures to show we understand graphs, but there are many different perspectives we can use to demonstrate understanding. This shows that it is important to use all of them in our classroom to make sure we reach as many students as possible.
Finally, this quote makes me wonder if using different perspectives could become a type of assessment in my classroom. Afterall, Antonsen (2016) mentions that when one truly learned a concept, they will be able to use different perspectives to represent it. Could I make my students use different perspectives to show their understanding of a concept?
Antonsen, R. (2016, December 13). Math is the hidden secret to understanding the world. [TED talk]. Youtube. https://youtu.be/ZQElzjCsl9o
I really enjoyed your anecdote throughout the summary of the article, that allowed me to better understand it, without having read it. So, thank you for that!
ReplyDeleteI also reflected about how conscious I am about the gestures that I make, specifically when teaching, and turns out I am not very conscious at all, because the most I can say about it is: I know I do it, but I don't know what I do or why.
I had a conversation with a colleague this week who teaches grade 8 math and is currently introducing linear equations and graphing. She is a seasoned teaching who keeps up with new ways of teaching and works really hard to make every topic fun, engaging, and easy to understand, BUT she finds that it is really hard to make this topic entertaining! She has reached out to math consultants and the beautiful internet but is just struggling. I do think that adding in a kinesthetic piece to it, or adding sounds, movement, or art could add the a bit of spice to this!
Thoughts?
Lovely! Thank you so much, Colleen, for actually trying out gesturing these graphs as you re-read this piece. I love your insights into the different speeds for gesturing straight and curved portions of the graphs, and your thought that the graph makes turns where it might be bumping into something! (When we were working with high school students, some characterized those turns as 'boing, boing' -- places where the graph bounced off something, a similar concept that we used with teaching about the absolute values of functions that 'bounced off' the x-axis). It is surprising too that, even when we are deliberate and intentional about our mathematical gesturing, there is so much we are unconscious of!
ReplyDeleteNoemi, very interesting to consider people's cultural and individual characters in the ways that we all interact with movement, observation and graphs (and other mathematical phenomena). I agree with you that it's not necessary for everyone to do things the same way...for example, not everyone might feel ok with making big gestures in class, for lots of reasons. It's a good challenge to think about how to integrate everyone's ways of observing and noticing as part of a whole class' sense of what a graph might mean or how it works.
Taylor, I'm interested in what your colleague might take up if you share some of these embodied math ideas with them. If you do that, please keep us posted on how it goes for them!
Just a note that you are currently in a reading group of 4 people (which I'll reshuffle this week...) Thanks to Noemi and later Taylor for responding to Jean-Jacques -- Colleen N., are you reading Jean-Jacques' writing? And Jean-Jacques, are you reading and responding to your other group members? I will soon be re-shuffling the groups, and I will make sure that the four of you get into 3-person groups for the next part of the course!
ReplyDelete