{"id":1157,"date":"2016-02-08T08:00:09","date_gmt":"2016-02-08T13:00:09","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=1157"},"modified":"2016-02-05T08:33:54","modified_gmt":"2016-02-05T13:33:54","slug":"learning-mathematics-through-embodied-activities","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2016\/02\/08\/learning-mathematics-through-embodied-activities\/","title":{"rendered":"Learning Mathematics through Embodied Activities"},"content":{"rendered":"

By Hortensia Soto-Johnson<\/a>, Professor, School of Mathematical Sciences, University of Northern Colorado<\/em><\/p>\n

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<\/em> objects as the quantity three<\/em> without simultaneously touching and counting one<\/em>, two<\/em>, three<\/em>. 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\u2019s hand on top of the teacher\u2019s 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?<\/p>\n

In layman\u2019s terms the philosophy of embodied cognition<\/em> 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\u2019 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\u2019 gestures to help them articulate their thinking and I illustrate embodied activities<\/em> designed to elucidate mathematical concepts via bodily movement.<\/p>\n

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.<\/p>\n

Sammy: Well it\u2019s a, it\u2019s kind of like, well you know, well I don\u2019t know.<\/em><\/p>\n

Me: Sammy I am going to repeat what you said and re-gesture your gestures.<\/em><\/p>\n

I repeated Sammy\u2019s words and emulated his gestures, which quickly prompted him to say, \u201cOh, it\u2019s like a turn, you are just turning the object<\/em>.\u201d This exemplifies neuroscientists\u2019 belief that if a person is \u201cnot engaged in an intentional action, or watching another person engage in an intentional action\u201d 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\u2019s actions as well as our own actions. In Sammy\u2019s case, attention to his gestures facilitated creating working definitions based on the gesture characteristics. For example, the students commented that Sammy\u2019s wrist could be perceived as the center of rotation. We have similar conversations when students alternate between turning their palm face-up and face-down as they convey their reflection definition.<\/p>\n

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.<\/p>\n\n\n\n\n\n\n
Table 1.<\/em> Sample Worksheet Questions<\/td>\n<\/tr>\n
Suppose the point (x,y<\/em>) was translated in the direction of (-5,2) to obtain the image (-4,8). What is the preimage point (x,y)?<\/p>\n

 <\/td>\n<\/tr>\n

In the following figure determine the line of reflection and explain your reasoning.<\/p>\n

 <\/p>\n

\"Tensia1\"<\/a><\/td>\n<\/tr>\n

Consider the figure below.<\/p>\n

1.\u00a0\u00a0\u00a0\u00a0 Reflect triangle ABC<\/em> about line m<\/em> and label it as triangle A’B’C’.<\/p>\n

2.\u00a0\u00a0\u00a0\u00a0 Reflect triangle A’B’C’ about line n<\/em> and label it as triangle A”B”C”.<\/p>\n

3.\u00a0\u00a0\u00a0\u00a0 Construct the circle with center O<\/em> and radius \\(\\overline{OA}\\). Do the same for radius \\(\\overline{OB}\\) and \\(\\overline{OC}\\).<\/p>\n

4.\u00a0\u00a0\u00a0\u00a0 Describe all the points that pass through each circle and explain why this happens. Use mathematical transformation ideas for your explanation.<\/p>\n

\"Tensia2\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

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 \u201csame rate\u201d causes the rope to become loose. With some probing, they connect the notion of the \u201csame rate,\u201d 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.<\/p>\n

 <\/p>\n

\"Tensia3\"<\/a><\/p>\n

Figure 1.<\/em> Embodied Translations<\/p>\n

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 \u201cOh that\u2019s what perpendicular bisector means.\u201d 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, \u201cSo a rotation means you are traveling in a circular fashion.\u201d<\/p>\n

 <\/p>\n

\"Tensia4\"<\/a><\/em><\/p>\n

Figure 2.<\/em> Embodied Reflections<\/p>\n

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 L\u00e9n\u00e1rt spheres (Figure 3).<\/p>\n

\"Tensia5\"<\/a><\/p>\n

Figure 3<\/em>. Stereographic Projection<\/p>\n

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 team-member remarked, \u201cSo it will map to a line and there is a break at the North Pole.\u201d This dynamic engagement did not go unnoticed by the other students and I asked Neil to re-gesture his discovery. Furthermore, when we started the unit on inversions the students quickly recalled Neil\u2019s 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.<\/p>\n

\"Tensia6\"<\/a><\/p>\n

Figure 4.<\/em> Using eyes for stereographic projection<\/p>\n

 <\/p>\n

\"Tensia7\"<\/a>\"Tensia8\"<\/a>\"Tensia9\"<\/a><\/p>\n\n\n\n\n
Figure 5a<\/em><\/td>\nFigure 5b<\/em><\/td>\nFigure 5c<\/em><\/td>\n<\/tr>\n
Figure 5.<\/em> Stereographic projection of a great circle<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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\u2019 gestures instructors can hypothesize about students\u2019 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 \u201caha\u201d moments \u2013 these aha moments are the best part of teaching.<\/p>\n

References:<\/strong><\/p>\n

[1] Anderson, M. L. (2003). Embodied cognition: A field guide. Artificial Intelligence<\/em>, 149, 91-130.<\/p>\n

[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<\/em>, 21(2), 287-323.<\/p>\n

[3] Rasmussen, C., Stephan, M., & Allen, K. (2004). Classroom mathematical practices and gesturing. Journal of Mathematical Behavior, 23<\/em>, 301-323.<\/p>\n

[4] Tabaghi S. G. & Sinclair, N. (2013). Using dynamic geometry to explore eigenvectors: The emergence of dynamic-synthetic-geometric thinking. Technology, Knowledge and Learning, (18)<\/em>, 149-164.<\/p>\n

[5] Gallagher, S. (2014). Phenomenology in embodied cognition. In L. Shapiro (Editor), The Routledge Handbook of Embodied Cognition<\/em> (9-18). London: Routledge Taylor and Francis Group.<\/p>\n

[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, \u00a0\u00a0 engineering, and mathematics. Proceedings of the National Academy of Sciences, 11<\/em>(23), 8410-8415.<\/p>\n

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By Hortensia Soto-Johnson, 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 … Continue reading →<\/span><\/a><\/p>\n

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