Composition, Decomposition, Relationships, Representation, Context

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Composition

The first and most fundamental concepts of mathematics are the idea of counting and the more developed idea of using a system to represent numbers. Dehaene (1997) presents the intriguing notion that several species of birds and mammals can distinguish between a single item and quantities of items. Further, there is some evidence that these birds and mammals might be able to distinguish between two objects and three objects or even between three objects and four objects. Dehaene's hypothesis, which is thoroughly developed and documented, is that certain structures within the brain seem to be specifically devoted to these primitive mathematical abilities. If this is true, then what exactly do these brain structures do? What is involved at the most fundamental level in this thing we refer to as counting?

To begin our exploration of this idea, consider two (imaginary) individuals, one who has the ability to count and one who does not. As these two individuals look out on the scenery of their lives, what do they see, and how does what they see differ? On one level we can say that what they see is exactly identical. They are both receiving the same sensory signals. What they do with those signals that they receive from their senses can differ, however. The non-counting individual probably perceives that there are multiple leaves on a tree, for instance, but beyond this perception he is unable to proceed. The counting individual is able to identify a single leaf as a unit that is repeated, he is able to momentarily focus attention on each leaf once and only once, he is able to apply a counting word to each leaf in a sequence, he is able to use the exact right sequential word for each successive leaf, and after he has focused momentarily on each and every leaf he is able to report a final "count" to tell us how many leaves there are. Furthermore, he understands that he could repeat the process using a different order of leaves but the same sequence of word-labels, and if he has been careful he will come to the same ending word each time he does this. All of this can take place without any physical evidence that he is doing anything at all; that is, it can all take place in his mind. For both individuals the scene is a single scene, but the counting individual can perform this complex analysis on the scene to provide us with some abstract information about what the scene contains. It is unlikely that he would ever do this, however, unless there were some pressing need for that information. Mathematics, while abstract, rarely makes its presence known except in response to a particular goal a person wants to achieve.

One of the most interesting parts of this counting process is the very beginning. How does one begin to count? In order for our imaginary counting person to even begin to use his ability he must first decide on an object to count. He must take what he sees as a single scene and see within it some repeated objects that he considers to be units to be counted. That is, he must mentally compose what he will consider to be units. In the example given, the countable objects were, of course, leaves. But did he have any other choices? What else could he have counted? Trees? Blades of grass? Clouds? Rocks? Any of these are possibilities. It is all a matter of choices made by the counting individual. How about clusters of leaves? What if the counter determines that each branch has, invariably, six leaves? Then, when he counts branches he can determine the total number of leaves without having to count each one. The idea of counting things in groups and then counting the number of groups is an extremely powerful idea in the development of a counting system (or a number system). In the case of counting clusters, the counter has simply composed a different unit. He has considered a branch rather than a single leaf to be his countable unit. The concept of composing various counting units is a theme that we will return to again and again.

Composition occurs in geometry as well. A key element of spatial reasoning is the ability to see a shape and mentally manipulate it. But what constitutes a shape? Is it the overall outline of the shape? Is that overall outline composed of parts? Do the parts have characteristics that can be analyzed and/or found in common? Do the parts have names? Does the overall shape have a name? Most important, can the parts and the whole be seen in more than one way? The ability to compose and manipulate units of shape is one example of the importance of composition in a geometric context.

Decomposition

Once we have composed a countable unit we can go in an entirely different direction. Aside from simply counting the number of objects of that unit, we could also decide to partition, or decompose, one of those units. There is an entire world of mathematics found in decomposing units. A unit can be a geometric shape, and decomposing it into parts can begin a traditional study of fractions. A unit can be a group of objects, and a consideration of a portion of the group involves a different use of fractions. If our unit is readily divided into ten equal pieces, then we can introduce the idea of decimal numbers. If we decompose a unit into two different parts and then we study the relative sizes of those parts (Are the parts equal in size? Is one of the parts twice as large as the other?), then we introduce a study of ratio. If we can readily decompose our unit into one hundred equal parts, then we can work with percents. All kinds of measurement activities and understandings rely on decomposing as well. Rarely can we accomplish the precision that we need using whole numbers of measurement units exclusively. Precision in measurement, then, relies on the idea of decomposing units into meaningful parts.