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## A Digital Mathematical Performance to Support the Physical Embodiment of Balanced Equations

Stephen Chen, Steven Khan, Alireza Moghaddam, Immaculate Namukasa, and Darren Stanley

One of the topics of exploration at the UWO Symposium on Digital Mathematical Performance was on how the combination of physical performances and digital technologies could improve mathematics teaching.  Our working group developed, performed, and recorded a skit that attempts to help students learn the concept of balanced/imbalanced equations with respect to the process of solving equations.

The following eight video clips in .wmv (original recording and smaller downloads) and .mp4 formats represent the key highlights of our presentation at the UWO Symposium.

Video 1 -- The Skit

In the skit, one of the actors is carrying a heavy load of single items and grouped items.  The actor wants to balance this load -- i.e. carry the same amount in each arm.  Upon reaching his destination, the actor needs to be unloaded in such a way that he remains in balance at all times.  A result of this unloading process is that a set of single items eventually balances a set (of one) grouped items -- and thus isolating and solving for the unknown.  In the process of unloading, several common errors are enacted that cause the actor to become imbalanced.  The actor over reacts to these situations and creates a physically embodied understanding for the concept of balanced/imbalanced equations.

There are several goals to this skit.  First, there is the goal of teaching the topics of balanced/imbalanced equations and the solving of equations.  To help students learn the concept of imbalanced equations, a physical movement/full-body gesture is created.  It is hoped that teachers and students will be inspired by our performance and re-enact the skit.  It is hypothesized that the physical embodiment of the mathematical concept and the opportunity to role-play will both help the students to learn the mathematical material better.

Video 2 -- The Mathematics of the Skit

The actor in the skit is carrying one bag and three cups in one hand and two bags and one cup in the other.  Thus, the equation is x + 3 = 2x + 1.  To isolate x quickly, a student might take the 3 away from the left hand side -- which leads to an imbalance.  Upon learning that they have to take away from both sides, a common error is to take away a unit from one side and a group from the other.  These two common errors are part of the skit.  Another error that was not acted is to remove all of the units (e.g. 3 from the left and 1 from the right).

It is hoped that the students will enjoy the skit.  And, just as important, it is hoped that by performing the skit, it will help them learn the associated mathematics more easily.  Similar to the expected classroom use of this digital mathematical performance, the skit and the role playing are done first and then followed by the mathematical lesson.

Video 3 -- The Digital Mathematical Performance

In this video, we discuss the purpose of the skit.

Video 4 -- The Lesson Goals

In this video, we discuss the objectives of the skit in terms of a classroom lesson plan.

Video 5 -- A Relevant On-line Resource

To follow up with the skit, we suggest the following on-line resource which we believe is highly relevant.  This on-line resource allows equations to be balanced and solved in an animated visualization.

Video 6 -- Why use Digitial Mathematical Performances?

In this video, we discuss the potential benefits to using a digital mathematical performance to teach the topic of balanced/imbalanced equations.

Video 7 -- David Pimm discusses Physical Rituals as a part of Learning

In this video, David Pimm provides an example of how a gesture used to help students learn a mathematical concept then become almost a required physical ritual that was a part of the mathematical concept.

Video 8 -- John Mighton discusses how to handle Negative Values

One of the limitations of the skit (and of the selected model/metaphor) is that it doesn't handle negative values.  In this video, John Mighton describes how he has taught the topic in such a way as to avoid this limitation. 