Usually one would see that exclamation as “Represent!” and it conjures up images of fans in team colors, a spirited school group or some other proud group member.  The individual (or individuals) is recognized as a part of the larger group, and while the whole group is not present, this individual should perform or act in such a way as to be a reasonable substitute for the group. In fact, many leaders andor companies will send someone to represent them at a meeting or celebration and, while the representative is not the leader or the company, her attendance is enough to suggest that the leader or company is present.

This is much the way that I think of mathematical representations.  They serve as a sort of substitute for “the real thing.” For example, I could represent the number four as “♣︎♣︎♣︎♣︎” or as “IV” or as “4.” I might draw a line segment to represent length or an area (array) model to represent the product of two numbers.  Sometimes a simple sketch is included to help us see the key elements in a situation. If I know that Sue is 54 in tall, which is 5 in taller than her brother Sam, I can represent their heights with sketches of stick figures (or line segments or tape diagrams), and that can assist me in finding Sam’s height.

Representation is a familiar word.  But what then is re-presentation? I first encountered this form in some writing by Ernst von Glasersfeld.  Dr. von Glasersfeld was a philosopher and professor of psychology at the University of Georgia and a colleague of Dr. Steffe (whose research is the catalyst and foundation of our work). When von Glasersfeld uses re-presentation, it describes the mental action of imagining an experience.  It differs from imagination in that it is rooted in a real experience. It differs from representation in that it occurs in the mind. Think to yourself “what does the dashboard of my car look like?” Can you picture it? Do you know where the gas indicator is? Do you know where the air conditioning [knob] is? Of course, you do!  You have seen it many times. In fact, you may have re-presented your gas indicator before when trying to decide whether or not to leave home early enough to stop for gas on your way to work.

Children can use re-presentations of past images or experiences to advance their concept of number.  They need many, many experiences counting perceptual items. That is, they need lots of practice counting things they can touch, see, or hear. It is through these experiences that they begin to construct quantities.  Then they might be able to engage in a situation where the table is set with five plates and is asked how many plates there would be if three more plates were added. She might point to each of the five plates while uttering “one, two, three, four, five” and then re-present an image or sound pattern for three to finish counting “six, seven, eight.”  The re-presentation becomes more critical as the second value increases. A re-presentation can help the child keep track of their additional counts. From here the child will eventually count abstract items (nothing perceived or re-presented), count-on and make sense of number sentences in symbolic form (such as 5+3=?). Who knew that such a tiny punctuation mark could radically shift our perspective?

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