Mike Butler


B.A., The Evergreen State College, 1997


Concept Externalization


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Shari Auth's book cover, Introduction to Auth Method

Publication Type


Latest Publication Title

"Concept Externalization as a Tool for Teaching Mathematics," a chapter in Imagination in Educational Theory and Practice: A Many-Sided Vision, ed. Thomas William Nielsen, Robert Fitzgerald and Mark Fettes, Cambridge Scholars, 2010.

Publication Excerpt

In this chapter I present a deceptively simple approach to engaging creative aspects within students to reach new ends. By empowering these learners to reassess their conception of complex and sometimes intimidating subjects, I hope to suggest a new route by which this information may be accessed; for convenience sake, I call it concept externalization (CE). The crux of this technique involves developing external representations, visualizations if you will, of what are usually presented as abstract concepts and processes. Sometimes these are mere physical examples of the processes, like combining one ball with two to demonstrate addition. Other times, more creative generalizations come into play such as when showing the evolution of geometric knowledge from simple two-dimensional shapes into multi-dimensionality and ultimately analytical geometry with its connection to algebra through the use of coordinate planes. The application of this technique involves tapping into the imagination with skills such as generalization and judgment, among others, to create disarming, visceral forms using shape, color, texture and spatial relationships to help convey meaning. As one example of this method, I will use a three-dimensional model of the conceptual progression of mathematics I built during my graduate studies. This model should be seen as my personal view of the subject area while students are encouraged to develop their own models, along with their teachers, that are directly relevant to their studies. I will also touch on some of the implications for educators and their curriculum posed by concept externalization. By using creative thinking and imaginative modeling to tackle fields of study long restricted to standardized, linear learning methods, friction may be encountered and power relations shaken as old boundaries are challenged. Additionally, the role CE plays within a given curriculum needs to be carefully planned out since it acts only as a bridge between the student and traditional tools such as mathematics notation and not as a substitution. The technique presented also has ramifications within the curriculum for learning disabled students, given the proven benefits of appealing to multiple modalities for these learners. In conclusion, concept externalization adds one more important tool to the kit of educators and students alike. When applied in a supportive, constructive environment, this technique has the potential to unlock talent and open doors for students of all types and in many different fields of study.

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