Summary: Seeing the graph vs. being the graph: Gesture, engagement and awareness in school mathematics (Gerofsky, 2011).
The article investigates how body movements and gestures can engage students’ understanding of mathematical graphs. This idea came from a daily class task: when the teacher asks students to show y =4, all students use different gestures to represent the graph.
In the research, students were selected from different grades in secondary schools and participated in the activity of using gestures to explain the given graphs to their partners. The process was video recorded to qualitatively analyze how students use their gestures and bodies to communicate the graph to others.
The results show three different ways of representing the graph:
1) Students use a small movement (fingers, hands or arms) to show the graph.
2) Students use the whole-body movement (including hands and arms).
3) Students have difficulty using gestures to accurately show the graphs.
Dr. Gerofsky argues that using larger gestures and whole-body movements can be more effective in engaging students in mathematical learning. “Being the graph” can let students make a deeper connection in learning.
STOP 1: “variations in the placement of the x- axis in relation to the gesturer’s body, and potential cognitive, cultural and semiotic interpretations of this placement,”
I like this statement that the gesture used can be influenced by one’s potential cognitive and cultural factors. This reminds me of how different cultures use fingers to represent numbers. Growing up in China, we use the “telephone gesture” to represent the number 6, while I realized many of my students from other countries used two hands to represent 6. Sometimes this discrepancy can be confusing. i.e. I can unconsciously use this gesture to refer to the number 6 in front of my students, but some of them would be confused and ask me what that means. In this research, purposeful sampling is used to select students from different grades with diversity in terms of gender, ethnicity, and math achievement and enthusiasm to make sure multiple perspectives are involved in the process.
STOP 2: “In other words, being the graph in a fully-embodied way fosters engagement and attentiveness far more than merely seeing the graph.”
This reminds me of another article written by Dr. Susan Gerofsky that I read from my previous conference course, The Aesthetics of Scale: weaving mathematical understandings (Knoll et al., 2015). It is a math workshop where three making components were provided for participants to work on, from small hand-held, tabletop, to large collaborative scales to make waves. When the participants transferred the skills that they earned from the small and medium scales to the collaborative large scale, their senses started from “seeing the graph” to “being the graph”. This idea resonates with Dr. Gerofsky argument that if students’ whole bodies are fully engaged in the activity, they can experience more attentiveness to mathematics learning (Gerofsky, 2011).
Questions:
1. Besides using the gesture, what are some other ways you can let students make and experience graphs?
2. Can the gesture activity also apply to other math concepts?
References:
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: The interdisciplinary nature of gesture (pp. 245–256). John Benjamins Publishing Company. https://doi.org/10.1075/gs.4.22ger
Knoll, E., Landry, W., Taylor, T., Carreiro, P., & Gerofsky, S. (2015). The Aesthetics of Scale: weaving mathematical understandings. Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, 533–540. http://archive.bridgesmathart.org/2015/bridges2015-533.pdf
I resonate with your point about how different cultures represent numbers in different ways. The number “7” is a particularly interesting and sometimes controversial example. Even within China, people represent it differently. After looking into this further, I learned that writing “7” as an upside down “L” is mainly common in coastal regions of southern China, where I am from. In northern China, people often use the same fingers but orient them differently. This highlights how gestures are culturally shaped and learned, rather than universal, which connects well to Gerofsky’s discussion of how meaning is constructed through embodied experiences.
ReplyDeleteDr. Gerofsky’s reading, along with your reflection, also reminded me of the idea of the unity of body and mind. Cognitively, the brain directs the body to produce gestures when representing graphs, but through physically enacting these gestures, the brain also receives feedback that can deepen understanding. In this way, thinking is not confined to the mind alone. Instead, mathematical meaning emerges through the close interaction between cognitive processes and bodily movement. This perspective helps explain why “being the graph,” rather than simply seeing it, can lead to greater engagement and awareness of mathematical relationships.
In response to your question about other ways of making or experiencing graphs, I think integrating the arts could be a powerful approach. For example, students could be asked to create an artistic representation of a graph using specific functions, allowing them to express mathematical ideas through visual and physical creativity. This kind of embodied activity could also extend to other mathematical concepts beyond graphing. As seen in the reading I explored this week, even simple finger gestures during basic calculations can have a positive impact on students’ learning. Together, these ideas suggest that gestures, whether simple or expressive, can play an important role in supporting mathematical understanding across different contexts.
Sunny, I thought I would take a crack at responding to your questions:
ReplyDelete1. Besides using the gesture, what are some other ways you can let students make and experience graphs?
Assuming 100% buy-in by students, I wonder if giving them the option to demonstrate being the graphs in various ways would actually engage them on yet another level. Could students generate even more understanding for themselves (and others through visual communication) If students were able to wiggle around while lying on the ground, or imagine traversing a map of the graph across the ground instead of being limited to just tracing shapes in the air?
What experience would give the most outcome for the least amount of resources and time? I’m thinking just gesturing in the air as it’s immediate, no set-up, no clean-up and probably counts as a first-order experience connecting to Nathan (2021).
2. Can the gesture activity also apply to other math concepts?
I wonder if pen strokes also count as gesture. When we scribble in the air or trace on each others’ backs and try to guess the image, these all count for something. As a child I used to trace many different images to develop the strength of hand, smoothness of curves, proportions of features, etc. I wonder if we aren’t emphasizing enough the importance of taking notes in class because it, too, is a form of gesture that builds understanding through the cognitive-physical feedback loop. I’m sure this is not the direction I was meant to go, but I’m sure it would apply – the chances of it not applying are too small given the overlap that gesturing is a cultural practice, and mathematics is also a cultural practice.
I think it would be really exciting to list out mathematical concepts from K-12 (and beyond), then create a bank of cultural gestures that are associated with them from across the world – it would be very interesting to see which concepts commonly have gestures, and which are lacking; which concepts have multiple representations, and which converge to a single gesture that unifies across the world. How interesting!!
Thank you for the thoughts Sunny!
Reference:
Nathan, M. J. (2021). Foundations of embodied learning: A paradigm for education (1st ed.). Routledge. https://doi.org/10.4324/9780429329098
Thanks very much, everyone! I'm particularly thinking about Oliver's responses to Sunny's questions: first, a very nice idea about having students come up with multiple ways of embodying the graphs of functions (and I immediately think of three or four more, besides gesturing in the air!); and second, the idea of drawing as the traces of gestures with your hands and a pen or chalk or pencil -- see Nathalie Sinclair and Liz de Freitas' work on mathematical diagrams as gestural, based on work by Châtelet.
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