In this chapter, the author presents the findings from the first two years of a multi-year study into the diverse expressions of secondary students' gestures while tasked with describing mathematical graphs.
Existing studies have shown 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 (Radford 2009, Arzarello et al. 2009, Nemirovsky & Ferrara 2009, Tall 2004). Thus, as a linguistic researcher in mathematics education, the author is interested in students’ use of gestures in communicating about mathematical graphs.
Based on the authors' informal observations, how students describe graphs through gestures in the classroom is very different. After experiencing an exploratory study with colleagues and family members as study participants, the researcher conducted the study in schools and asked the following questions:
1. Can ‘elicited gestures representing the graphs of mathematical functions’ be categorized in a meaningful way that captures the spectrum of learners’ cognitive approaches to graphs?
2. If so, can the categories of graph gestures be qualitatively correlated with student attentiveness to and engagement with the mathematical features of these graphs?
Based on video recordings of participants' behavior and interviews with them, the researcher categorized gestured graphs as follows：
1. An “arm’s-length visual model” of the graph. These gestured graphs involved small movements of a finger, hand, and arm, without a great deal of larger kinesthetic movement involving the spine.
2. “Being the graph/being in the graph”. These gestured graphs involved noticeable movement of the spine and often markedly kinesthetic, whole-body movements.
3. “Inaccurate, not aware of what counts as salient”. These students had difficulty producing gestures for the graphs, hesitating repeatedly, or rushing through the task.
This categorization was linked and compared to teachers' holistic year-long assessments of student participants, and the results were striking. Students with graph gestures in Category 1 were the average students. These students are hard-working but rely more on memorizing formulas and algorithms. Students whose graph gestures fell into Category 2 were the top students. These students were recognized by their teachers as having a deep mathematical understanding. Students whose graph gestures were coded as Category 3 were struggling and at risk of failing mathematics – except for one student.
I think this study has important implications. This study implies that students' gestures can reflect their level of math cognition to some extent. In addition, mathematics students benefit from engaging in an embodied, visceral way with mathematical objects like graphs through large gestures and kinesthetic whole-body movement. This will inspire teachers to use more gestures to teach math concepts or graphs to students.
In my personal experience, math teachers are accustomed to using multimedia such as PPTs to present graphs and charts to their students, who are usually seated in their seats taking notes. While this traditional approach isn't necessarily flawed, this study shows us another viable direction. Through gestures, students can better engage in the classroom and teachers gain more teaching methods.
Stops:
In reading informal observations of students, I was interested in the differences in the students' gestures. I wondered if the students who made smaller gestures did so because they were shy. In other words, there were other factors unrelated to math cognition that caused them to make different gestures.
While reading the results, I was struck by the categorization of gestures and the close connection to the teacher's assessment of students. Though the sample size of this study is small, it may well reflect a more generalized conclusion, that those students who prefer whole-body movements have a deeper understanding of math knowledge.
Questions:
In the context of your teaching experience, have you observed any student gestures that have made a strong impression on you? What is the math level of the students who use this gesture?
References:
Arzarello, F., Paola, D., Robutti, O. and Sabena, C. 2009. “Gestures as semiotic resources in the mathematics classroom.” Educational Studies in Mathematics 70 (2): 97–109.
Radford, L. 2009. “Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings.” Educational Studies in Mathematics 70 (2): 11–126.
Nemirovsky, R. and Ferrara, F. 2009. “Mathematical imagination and embodied cognition.” Educational Studies in Mathematics 70 (2): 159–174.
Tall. D. 2004. “Building theories: The three worlds of mathematics.” For the Learning of Mathematics 24 (1): 29–32.
Thank you for your concise summary of the readings. Your summary allows me to quickly grasp the essence of this reading. I am fascinated by the fact that the mathematical ability of students can be reflected in their gestures. However, I share the same concern as you regarding the findings that students' characters may also contribute to their willingness to express knowledge through gestures.
ReplyDeleteIn response to your question, it is common for me to check students' facial expressions to gauge their understanding of my explanations on certain mathematical concepts. When I observe most students in my class furrowing their eyebrows, I immediately recognize the need for further explanation. I then provide additional clarification and support my explanation with daily examples.
However, categorizing students' mathematical ability based solely on their facial expressions in class is challenging, as various factors can influence facial gestures. I can make educated guesses, but accurately reading their minds during the explanation of mathematical concepts is not possible.
For example, when high achievers furrow their brows at a mathematical concept while low achievers do not, it is challenging to conclude that the high achievers are actually low achievers based solely on their facial expressions, especially when those who struggle in mathematics appear unfazed.
One possible reason why low achievers may not furrow their brows is that they may have already given up on listening to the explanation.
ReplyDeleteThank you so much for the summary of the reading.
In response to your stops:
Providing students the opportunity to represent graphs individually in a private setting might alleviate social anxiety, potentially influencing results positively. In such a setting, students may feel less pressure and judgment, leading to more authentic and accurate gestures. This could enhance their confidence, allowing for a diverse range of expressions and a deeper understanding of the material. However, individual differences must be considered, as some students may not experience social anxiety in group settings. Conducting a comparative study between group and individual settings would be beneficial to draw more conclusive insights. In conclusion, a private setting has the potential to improve the accuracy and authenticity of students' graph representations, but the impact may vary based on individual preferences and comfort levels.
Considering participants' cultural backgrounds is crucial, as cultural norms influence the comfort level of using gestures. In cultures where body language is more prevalent, students may enter the study with a higher natural inclination and comfort in using gestures. This pre-existing familiarity with gesturing could positively impact their ability to represent graphs, potentially leading to more expressive and accurate outcomes.
Conversely, students from cultures that place less emphasis on gesturing may initially feel less comfortable incorporating body movements into their representations. This cultural variation in gestural norms could introduce a confounding factor, influencing the results by affecting the participants' confidence and fluency in conveying graph-related concepts through gestures.
In response to your question:
To answer your question honestly as I reflect on my teaching days, I have not noticed gestures, perhaps because I did not look for them.
Moving forward, engaging students in co-creating gestures for mathematical concepts empowers them and enhances the learning experience. This approach fosters active engagement, allowing students to interpret and represent concepts individually. By associating gestures with abstract ideas, it aids memory retention and deepens understanding. The strategy promotes cultural sensitivity, accommodating diverse perspectives in gesture development. Collaborative creation fosters a sense of community. Implementing this student-centric approach creates an inclusive and participatory learning environment, enriching the overall educational experience.
Thanks Stelios and everyone! I agree with your objections to the idea of categorizing students according to their gesture types -- other factors may well be involved, and this is not meant to be an assessment tool! But it is interesting to start noticing mathematical gestures and to work with them: not to assess individuals' understanding, but to notice some general trends that might point to different ways and levels of understanding in a large group.
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