PME Special Issue: Bodily Activity and Imagination in Mathematics Learning
Based on my own learning experience and teaching experience, I agree to the authors’ opinion that students can more easily get engaged with concrete materials that they manipulate with their hands and various activities in which they can move their bodies, hands, feet, and etc.(including different parts of their bodies) around. The advantage of the use of concrete materials and devices can facilitate the embodiment of math through the sensory perceptions, such as touch, movement, vision, kinesthesia. This article let me recall my own learning experience. I still remember clearly how my classmates and I learned the concepts of a circle, an ellipse, and a hyperbola in the high school. At that time, computer was not so prevalent as it is today in the classroom due to its expensiveness, but my math teacher just used very simple materials to demonstrate the formations of a circle, an ellipse, and a hyperbola and helped us conceptualize the understanding of these concepts quickly and explicitly. For example, what is the definition/concept of an ellipse? Every student was given a string, two thumbtacks, and a white card board. First, I fixed the two ending points of the string with the thumbtacks on the white card board and kept the distance between the two fixed points less than the length of the string. The string can be seen to be comprised a set of points. Then I picked up the string at any point to form an angle with my pen and moved my pen while keeping the string which has already formed an angle tight. The trace obtained with the pen moving is an ellipse. In my view, this is a very good hand-based activity with simple materials, but it embodies the concept of an ellipse efficiently. I still remember I deduced independently the definition of an ellipse based on this bodily activity quickly without referring to the one in the textbook which seemed more abstract and built up a good conceptual understanding of the concept. More importantly, the bodily activities usually can impress students with a longer term memory. I inherited this kind of activities from my math teacher and have kept them in my teaching over the past ten years even after the integration of technology permeated into each subject.
The second point I’m interested in is that the authors discussed the notion of a humans-with-media system. In the authors’ view, all technological means, including calculators, graphing calculators, computers, printers, videos, etc, are interconnected. Computer is not an isolated unit. So, students’ math learning can be a product made by collectives of humans-with-media. All of the technological means can create links between body activity and math representation. For example, the simulation function offered by computers and graphing calculators can help students visualize the concepts, representations of math. Vision is a part of body sense and action with the eyes moving. Through the simulation, students can visualize the math first, and then convey the visual information to brain cells through eyes moving, and then bring about thinking, prediction, decision making, verbal discussions and arguments through language and gestures. This is a learning process involving a series of bodily activities and actions. Also, I recognize that there is a major overlap between perception and imagination. To some extent, they are reciprocal to each other. Let me go back to the example above, simulation can visualize the imagination and then imagination can prompt further needs of different types of perception.
The third point I’m interested in is that teachers’ belief about the use of bodily experience in the math classroom, just as the authors stated: “We caution that widespread use of bodily experience in classrooms will depend on teachers being able to articulate how such activity is mathematical activity that is legitimate for the mathematics classroom.” This is the exact same thing as teachers’ belief about using technology in teaching. If a teacher is able to justify the intentions/purposes of the use of bodily activities and how to use them in his teaching, he can embed them in the math teaching purposefully, appropriately rather than randomly. In my view, only will these purposefully, appropriately designed bodily activities benefit students’ math learning. Meanwhile, the appropriate use of bodily activities should be an organic component which can not be isolated from the whole instruction and provide students with the insights and feelings that are hard or difficult to fully sense in other ways.
Based on my own learning experience and teaching experience, I agree to the authors’ opinion that students can more easily get engaged with concrete materials that they manipulate with their hands and various activities in which they can move their bodies, hands, feet, and etc.(including different parts of their bodies) around. The advantage of the use of concrete materials and devices can facilitate the embodiment of math through the sensory perceptions, such as touch, movement, vision, kinesthesia. This article let me recall my own learning experience. I still remember clearly how my classmates and I learned the concepts of a circle, an ellipse, and a hyperbola in the high school. At that time, computer was not so prevalent as it is today in the classroom due to its expensiveness, but my math teacher just used very simple materials to demonstrate the formations of a circle, an ellipse, and a hyperbola and helped us conceptualize the understanding of these concepts quickly and explicitly. For example, what is the definition/concept of an ellipse? Every student was given a string, two thumbtacks, and a white card board. First, I fixed the two ending points of the string with the thumbtacks on the white card board and kept the distance between the two fixed points less than the length of the string. The string can be seen to be comprised a set of points. Then I picked up the string at any point to form an angle with my pen and moved my pen while keeping the string which has already formed an angle tight. The trace obtained with the pen moving is an ellipse. In my view, this is a very good hand-based activity with simple materials, but it embodies the concept of an ellipse efficiently. I still remember I deduced independently the definition of an ellipse based on this bodily activity quickly without referring to the one in the textbook which seemed more abstract and built up a good conceptual understanding of the concept. More importantly, the bodily activities usually can impress students with a longer term memory. I inherited this kind of activities from my math teacher and have kept them in my teaching over the past ten years even after the integration of technology permeated into each subject.
The second point I’m interested in is that the authors discussed the notion of a humans-with-media system. In the authors’ view, all technological means, including calculators, graphing calculators, computers, printers, videos, etc, are interconnected. Computer is not an isolated unit. So, students’ math learning can be a product made by collectives of humans-with-media. All of the technological means can create links between body activity and math representation. For example, the simulation function offered by computers and graphing calculators can help students visualize the concepts, representations of math. Vision is a part of body sense and action with the eyes moving. Through the simulation, students can visualize the math first, and then convey the visual information to brain cells through eyes moving, and then bring about thinking, prediction, decision making, verbal discussions and arguments through language and gestures. This is a learning process involving a series of bodily activities and actions. Also, I recognize that there is a major overlap between perception and imagination. To some extent, they are reciprocal to each other. Let me go back to the example above, simulation can visualize the imagination and then imagination can prompt further needs of different types of perception.
The third point I’m interested in is that teachers’ belief about the use of bodily experience in the math classroom, just as the authors stated: “We caution that widespread use of bodily experience in classrooms will depend on teachers being able to articulate how such activity is mathematical activity that is legitimate for the mathematics classroom.” This is the exact same thing as teachers’ belief about using technology in teaching. If a teacher is able to justify the intentions/purposes of the use of bodily activities and how to use them in his teaching, he can embed them in the math teaching purposefully, appropriately rather than randomly. In my view, only will these purposefully, appropriately designed bodily activities benefit students’ math learning. Meanwhile, the appropriate use of bodily activities should be an organic component which can not be isolated from the whole instruction and provide students with the insights and feelings that are hard or difficult to fully sense in other ways.
3 comments:
Thanks Julia! I enjoyed reading your blog.
Since we will be commenting on one another's blogs during class tonight, I'll hold off on giving my comments until others in the class have had a chance to write too.
Susan
What a fascinating statement-simulation can enhance the imagination.." This really struck out at me because visualizing is such an important part of the reading process-and now, to link it with math=awesome! I definitely agree and see how one's imagination can aid in conceptual development, but I haven't given it much reflective thought on my practice-thanks for opening up my eyes!
Julia --
I'm very interested in your comment that the use of bodily interaction in math instruction is very much like the use of technology, in that some teachers may resist both at first, and in the way that both need to be integrated into a balanced program of instruction. I agree that it's all too easy to either reject anything unfamiliar, or to embrace it to the exclusion of other important elements of teaching.
What I really like is the idea of a "fluid movement" between technologically-mediated experiences, fully-embodied experiences and abstract symbolization in math learning. It seems to me that this is the way we live nowadays too -- in our bodies, online and through abstract ideas and signs. In this kind of teaching, it would be the teacher's responsibility to design ways of working that let kids see the equivalences and resonances among these worlds.
Susan
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