#4 The Logic of Seriation

THE TWO DIFFERENCES BETWEEN CLASSIFICATION AND ORDERING

A classification of things does not have a necessary corresponding spatial arrangement that can be perceived whereas a series of ordered differences does have a spatial sequence that corresponds to the seriation. We can perceive a relation, a difference between two ordered objects, whereas classes cannot be perceived. Since we can perceive the form of an ordered series of sticks according to their differences in length, does that make teaching or understanding seriation easier?


THE SERIATION PROBLEM

Does the understanding of seriation start with the perception of a series of things from which understanding is then abstracted? Or does the understanding of seriation depend on the development of an operating logic the student uses? If there is a logic in students’ ordering activities that does not come from seeing an ordered series, then what is the difference between perception of a series and the logical seriation of the same series?

A prevalent misunderstanding of learning by lay people is that perception can be the source of knowledge; this seems particularly compelling as for example, in trying to show students an ordered series of things from which they are expected to understand seriation. One of the main explanations for the origins of knowledge is that it begins with sensory input.


RESEARCH SHOWS A LOGIC DEVELOPS

The researchers studying the growth of knowledge have used a variety of methods to investigate how seriation develops and whether it derives from activity or from perception. For example, we can show primary students an ordered series of sticks and ask them draw what they have been shown. We can use a box and watch how students use tactile seriation to explore the sticks and then draw the configuration that he intends to construct. Or we can give unordered sticks and ask students to put them in order.

The experimental research shows clearly that the development of the logic guides activity such as ordering, drawing a series, inserting sticks, and describing a series. The logic always begins with a simple, isolated activity of comparing two sticks to see one as big and the other as small. The student at this level can only organize isolated duples of big and small, or two groups of several long lines and then extremely short lines when drawing. Then as the student attempts to coordinate two separate duples another difference appears, and the student produces big, medium, small. The students still seems apparently unconcerned with the graphic character of the unconnected duples or triples they produce.

But at the next stage, the graphic character of a series emerges for the student. They attempt to make the whole series look right, and they use trial and error to give the perception of the whole series a good form. Now the set of elements that belong to the series (the extension) includes more than duples or triples, and the order (the intension or property of the sticks) begin to emerge. But the order is tied to perceptual comparisons of proximal sticks and the overall extension depends on a perceptual good form. Neither is yet logical or coordinated with the other. As a result, the student has difficulty inserting a stick or composing a method of always choosing the largest stick of the remaining ones to add to the smaller sticks already starting the series.


THE LOGIC OF ORDERING

To understand the seriation, students must be able to identify the specific ordered differences they are combining and coordinate these into an overall series or whole. The student logically must add together the ordered differences into the whole by isolating differences and adding the relations together to make the series ‘get’ larger and larger. Likewise, the student must be able to take apart or undo the differences by isolating each of the relations and undoing the difference it added to go back down the series to get smaller and smaller.


THE GROUPING TOGETHER INTO A COMPLETED SYSTEM

But there is one more step in the development of seriation. It is one thing to add together the relations of difference to compose a series and it is another operation to take apart or undo the series. In both cases, students may not ‘remember’ exactly what they started with so they are not sure if they can exactly undo things. Both the operations of putting the relations together into a series and taking apart to undo the series have to be coordinated for there to be a non-contradictory, logical understanding. Simultaneously coordinating both operations requires that the student group them together into a mental system so that each action is controlled by the mental system. The system started out as a small, local system that coordinated only the relations of big and small relations of two objects and ceased to operate as soon as attention shifted to the next objects. Then it grouped together little, medium, big. Then a more continuous but one way series. Finally it completes itself by grouping together both the addition and subtraction of relations making up a series.

This addition of combining individual relations between elements and taking them apart to undo the series operates in logic like the addition and subtraction of numbers operate. Numbers form a system which logically controls the operations. For example, in student language, "if you add 3 and then take away 3, you haven’t changed anything." The result is 0, and adding 0 means exactly the same as you started with. So the operations fit together into a system that insures the operations always work out logically, that is, so they aren’t contradictory.

Thus, the action of inserting a stick can be accomplished logically by simultaneously coordinating the relations of smaller and larger to find the position in which the stick is larger than the one preceding it and smaller than the one following it. Rather than perception, the student is using thought to organize activities, solve the problem, and explain it. This logic did not come from perception; it came from the successive organization of activities. Logic corrects perception by enriching it with relations that let the student ‘see’ the order. This seeing is not perceptual but is an mental object of thought.


CONCLUSION

Understanding seriation does not come from perception. We can’t show students a series of sticks or two different stick and expect them to abstract an understanding of differences in length and the seriation of differences. Understanding seriation depends on the development of an operating logic students use, and that logic evolves out of the students’ increasing coordination and grouping of their activities.

To the layperson it appears easy to teach since all one has to do is simply show students what they need to know about seriation. They don’t see that there is a logic in students’ ordering activities that takes a long time to develop and does not come from just seeing an ordered series and copying it in thought. There is a huge difference between perception which is a sensory motor contact with objects and the abilityh of logical thought to compose and take apart ordered differences of a series. A perception cannot be composed and taken apart. It doesn’t allow reasoning or problem solving, the contribution of logical understanding of seriation.