By Hortensia SotoJohnson, Professor, School of Mathematical Sciences, University of Northern Colorado
Those of us who teach mathematics know that students struggle writing the symbolism of mathematics even through they can articulate some of the concepts behind the symbolism. Those of us who interact with children know that they struggle articulating their thoughts even though they can convey their thoughts through gesture. For example, children point to indicate what they want and touch items or use their fingers as they learn to count. It is through such bodily action that children learn to recognize three objects as the quantity three without simultaneously touching and counting one, two, three. Athletes and musicians also apply bodily actions to master their sport or instrument respectively. For example, how many times have you have seen a basketball player shoot an imaginary ball into an imaginary hoop? Consider how a piano teacher places a student’s hand on top of the teacher’s hand as the teacher plays the piano. These are just a few ways in which we use our body to learn, so why not use it purposefully to promote the learning of mathematics?
In layman’s terms the philosophy of embodied cognition argues that learning is a result of interactions with our environment [1]. There are broad interpretations of this philosophy, but like others I interpret it to mean that we learn through bodily movements. Various mathematics education researchers adopt embodied cognition as a theoretical lens because it allows them to use gesture as a source of evidence as students learn linear algebra, differential equations, complex variables, etc. [2,3,4]. I too document research participants’ gestures as they tackle tasks related to my research on the teaching and learning of complex analysis, but embodied cognition also informs my teaching. In this blog, I describe how I highlight students’ gestures to help them articulate their thinking and I illustrate embodied activities designed to elucidate mathematical concepts via bodily movement.
As a first illustration, consider how I expose preservice elementary teachers to Euclidean transformations. I commence by asking students to define a translation, reflection, rotation, and dilation. Similar to children, students tend to gesture when they are unable to articulate their thoughts. For example, after asking students to define a rotation, Sammy (pseudonym) raised his hand and the following dialogue occurred. Although Sammy was unable to provide a definition, he turned his right hand back and forth as though turning a doorknob.
Sammy: Well it’s a, it’s kind of like, well you know, well I don’t know.
Me: Sammy I am going to repeat what you said and regesture your gestures.
I repeated Sammy’s words and emulated his gestures, which quickly prompted him to say, “Oh, it’s like a turn, you are just turning the object.” This exemplifies neuroscientists’ belief that if a person is “not engaged in an intentional action, or watching another person engage in an intentional action” then there is no expectation for neurons in the premotor cortex brain area to activate [5, p. 13]. These neurons are believed to be responsible for helping us interpret other’s actions as well as our own actions. In Sammy’s case, attention to his gestures facilitated creating working definitions based on the gesture characteristics. For example, the students commented that Sammy’s wrist could be perceived as the center of rotation. We have similar conversations when students alternate between turning their palm faceup and facedown as they convey their reflection definition.
After creating working definitions, the students collaboratively complete worksheets using manipulatives (see Table 1 for sample tasks). The purpose of these worksheets is for students to determine the image of a figure under a given transformation, to work backwards, and to make discoveries about the properties of Euclidean transformations. After completing the worksheets, the students present their work to the class. After this, we proceed with some embodied activities, which reinforce classroom work.
Table 1. Sample Worksheet Questions 
Suppose the point (x,y) was translated in the direction of (5,2) to obtain the image (4,8). What is the preimage point (x,y)?

In the following figure determine the line of reflection and explain your reasoning.

Consider the figure below.
1. Reflect triangle ABC about line m and label it as triangle A’B’C’. 2. Reflect triangle A’B’C’ about line n and label it as triangle A”B”C”. 3. Construct the circle with center O and radius \(\overline{OA}\). Do the same for radius \(\overline{OB}\) and \(\overline{OC}\). 4. Describe all the points that pass through each circle and explain why this happens. Use mathematical transformation ideas for your explanation. 
As part of the embodied activities, students act out many of the worksheet tasks on a giant grid where the students are the points and rope serves as segments. Figure 1 depicts students as image points after translating in the direction of (1,2). As a result of this activity the students realized that not moving at the “same rate” causes the rope to become loose. With some probing, they connect the notion of the “same rate,” to an equal slope, and to the worksheet discovery that under a given translation, the segments connecting a preimage point and its corresponding image point are parallel and congruent. It is during this activity that the students also use language alluding to rigid motions of the plane. That is, they realize that under a transformation every point on the preimage transforms simultaneously rather than one point at a time as they performed it on the worksheet.
Figure 1. Embodied Translations
Figure 2 illustrates the students determining their image point under the given line of reflection. The combination of the rope and large right triangle helped them make meaning of the fact that a line of reflection is the perpendicular bisector of the segment connecting a preimage point and its image point. While working on the worksheet some students generally forget about the perpendicular aspect or the bisector aspect of the definition, but somehow using the rope facilitated attending to both facets of the definition. This could be because they are able to simply fold the rope over the line of reflection. It is not uncommon to hear comments such as “Oh that’s what perpendicular bisector means.” Similar comments are made with the rotation task. As the students rotate about the center of rotation (another student) while holding the rope they remark, “So a rotation means you are traveling in a circular fashion.”
Figure 2. Embodied Reflections
I now highlight an example where students unconsciously engage in bodily movement. As part of a second semester geometry course for prospective secondary teachers, the students performed stereographic projections using Lénárt spheres (Figure 3).
Figure 3. Stereographic Projection
One group of students was not satisfied with their image sketches because the string should go through the sphere. Thus, they relied on their eyes (another example of embodied cognition) to determine the image of the circles (Figure 4). The group progressed quite rapidly through the tasks until they arrived at the great circle that passes through the North Pole. At this point one of the students, Neil (pseudonym) got up and pointed both of his arms up to denote the North Pole (Figure 5a). While engaged in bodily motion he mentioned that one half of the great circle would be projected down (Figure 5b) and the other half would get projected in the opposite direction (Figure 5c). During this action both he and his teammember remarked, “So it will map to a line and there is a break at the North Pole.” This dynamic engagement did not go unnoticed by the other students and I asked Neil to regesture his discovery. Furthermore, when we started the unit on inversions the students quickly recalled Neil’s bodily action as they hypothesized about the image of a circle that passes through the center (O) of the circle of inversion. They knew the circle would break at point O and map to a line.
Figure 4. Using eyes for stereographic projection
Figure 5a  Figure 5b  Figure 5c 
Figure 5. Stereographic projection of a great circle 
Currently, there is much buzz about active learning of mathematics [6] but the definition of active learning is vague and sometimes it is difficult to determine if students are truly engaged in learning. By paying attention to students’ gestures instructors can hypothesize about students’ mathematical reasoning and ask probing questions that help students convey their mathematical reasoning. It is also an effective technique for assessing mathematical misconceptions, but this can only occur if instructors are attuned to gesture. Furthermore, tasks such as embodied activities help bring to life the mathematics where students are actively learning and have “aha” moments – these aha moments are the best part of teaching.
References:
[1] Anderson, M. L. (2003). Embodied cognition: A field guide. Artificial Intelligence, 149, 91130.
[2] Nemirovky, R., Rasmussen, C., Sweeney, G., & Wawro, M. (2012). When the classroom floor becomes the complex plane: Addition and multiplication as ways of bodily navigation. Journal for the Learning Sciences, 21(2), 287323.
[3] Rasmussen, C., Stephan, M., & Allen, K. (2004). Classroom mathematical practices and gesturing. Journal of Mathematical Behavior, 23, 301323.
[4] Tabaghi S. G. & Sinclair, N. (2013). Using dynamic geometry to explore eigenvectors: The emergence of dynamicsyntheticgeometric thinking. Technology, Knowledge and Learning, (18), 149164.
[5] Gallagher, S. (2014). Phenomenology in embodied cognition. In L. Shapiro (Editor), The Routledge Handbook of Embodied Cognition (918). London: Routledge Taylor and Francis Group.
[6] Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., and Wenderoth, M. P. (2015) Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences, 11(23), 84108415.
Your post is very impressive for me. Actually, I liked the examples which you have given to explain that how students can learn through a touch. The example of a basketball player is a great example because we all have seen this kind of activity that a player shoot the imaginary ball into the imaginary hoop. Yes, it will be easy for the students to learn through embodied activities.