#3 Classification—Extension and Intension

.
MEANING AND SET MEMBERSHIP

Classes are concepts; all nouns refer to concepts, or potential concepts not yet fully understood. The two properties of classes, their extension and intension, are more important than the obscure terms suggest. Extension simply refers to what objects belong in a class. Intension refers to the characteristics or properties that define membership in a class. Intension is the meaning of a class. It’s important for teachers to understand how challenging the problem of classification is for young children. The challenge comes from developing the coordination between what concepts object s belong to, their set membership, and the what properties of the objects are essential to defining set membership. A child may put cows and pigs together and call them animals but only because they look different. They may be unaware of the distinguishing features that make cows different from pigs or that define them both as animals. Snakes may not be animals to them. And they almost never classify insects as animals. Perceptual appearances don’t form a solid basis for conceptualizing things that children need as a foundation upon which to build their academic knowledge.


NOUNS REFER TO CLASSES AS CONCEPTS

Classes are concepts that apply to separate objects. Concepts like volume, straight line, time apply to a continuous pat/whole aspect of reality. Classification is important since every noun, every reference by name to a group of similar things, depends on the classification system at work. Instruction in classification begins in the primary grades with helping students form the concepts of living and non-living things. Teachers help students with the concepts of shapes, numbers and numerals, perhaps various occupations, weight, quantity, and so on. At the primary grades, few nouns are yet conceptualized at the level of being true classes. Helping students conceptualize the important nouns in academic instruction forms the basis of the descriptive knowledge of the primary grades. Primary students tend to see only isolated categories of things that go together, not inclusive classes. Their nouns tend to be unstable categories tied more closely to activities with the objects. "Cows are for giving mild and living on a farm." "We ride horses." That kind of thing. These categories aren't related logically to other categories. For example, cows may not also be animals.


EXTENSION—THE HIERARCHY IDENTIFYING CLASS MEMBERSHIP

Classification seems simple enough, just putting things together that go together, but understanding the unseen psychological system itself turns out to be the challenge. We can't just do classification if we are to teach it. We must be consciously aware of the underlying system that remains hidden to most people. As teachers, we have to understand how the hidden mechanism of classification we are unconsciously using itself works to coordinate the features of classes with their inclusion into a classification hierarchy of inclusions to produce classes. Not only that, we have to understand the earlier forms of classification; these are psychological systems we may no longer be using but which are the forms primary children are using.

Classification is a system of logic because to represent cows and animals means cows and all other animals are combined into the class of animals. The relationships of a classification hierarchy are formed by combining sub classes into more inclusive super classes. Or the reverse also is available. We can take apart animals into cows, pigs, and other animals. This combining and taking apart mathematically could be represented as the addition of classes and the subtraction of classes. It expresses itself in language as we talk about things. "All cows are animals. Some animals are cows, some are pigs but there are other kinds of animals as well. There are more animals than cows." We can tell whether this operational system is present as a specific ability in a student by giving him special problems and listening to his reasoning.


INTENSION—CHARACTERISTICS DEFINE MEANING

But classification is more than forming and combining classes or taking them apart. All this has to be done based on the characteristics or properties that the objects have such that they belong to a class. Obviously, a cow doesn't have some of the characteristics of pigs but it has all the characteristics of being an animal. So a class has two aspects, one has to do with which objects go in which classes. Do all cows belong in the class of animals? Do all animals belong in the class of pigs? When we see an object standing out their in the field, what classes can we put it in? We haven't mentioned the properties of the object that we are using to classify it. We are only referring to which objects go in which classes. The technical name for that is the extension of a class. The extension of a class is the identification of the class membership, which objects go in which classes.

Obviously, we can't put objects into classes without seeing some kind of characteristic features that let's us know it belongs in a class. We look at certain characteristic features of an object and see these are characteristics every cow has and no pigs have. We also see characteristic features of the object that every animal has including all pigs. So the characteristic features are what the object means to us. If we refer to an object as a cow, we can imagine features that we know all cows have, all mammals, all animals, all living things, etc. But seeing the object as a cow means we don't see it as a pig. The meaning of the object is also that it is an animal. The meaning of things are those properties that distinguish it from others and that identify its similarities to other objects having the same name. The technical name for the properties that form the meaning of a noun is comprehension or intension of a class. Of course, all this logic and meaning are not yet present in young children's representations of things.


THE DEVELOPMENTAL PROBLEM—COORDINATING INTENSION AND EXTENSION

Young children face a difficult problem, a developmental problem, not a problem of learning. The developmental problem means there is no direct way to solve the problem except by a series of attempts at organizingthe objects into classes, checking them out as to the properties, and then trying to go back and readjust the classification method. This developmental problem for children needs to be clear. Since representing things as concepts with names means children have to know what characteristic features an object has. They have to identify only those features that are the key and common for every member of a class. What properties make all these objects of this class similar? Then these properties have to be used to define the objects that belong to a class.

But which comes first—identifying the properties all members of a class have in common or forming the class of all members that belong? To figure out what characteristic features all cows have in common requires that the child get a collection of cows together to see what characteristic features they have in common. But to gather together a set of objects that are all cows requires knowing what characteristic features to use to identify objects that are cows. The child's problem is how to coordinate the features that seem to define similarity with the names applied to objects such that the resulting classes form a hierarchy of inclusions each with its defining properties. Thus, the researchers label this developmental problem as "the coordination of intension and extension"—the simultaneous identification of properties with the grouping into nested nouns, i.e., classes.

If we go deep into this developmental problem of how to form classes, we find we can't really show or tell students how to form classes. They can only make an attempt to group things and name them in a certain way and then test out how they tried to organize their understanding of the objects by comparing to what others say. If others are using inclusive classes but the child has only categories, the child has no means by which to understand what is said as classes. The child can only form categories. So what the teacher thinks she is saying and what she knows she means is not the same meaning the child hears in the teachers' words—the child has no other choice. The child can only use whatever system he has developed so far.

Once the student can coordinate the characteristic features of objects with a hierarchy of naming and he understands classification, then he can receive and understand the teacher's classification talk. Her words have conceptual meaning. But until then, all he hears and has available to understand with are simple categories and a variety of perceptions from experiences with the objects. The objects he calls cows may mean that the object for him has many features and is tied to his experiences with it, most of which are irrelevant to classification in order to conceptualize and thereby understand the object. The students particular experience may not have much to do with the class of cows as such as adults conceptualize them. Adults know that "we drink milk from cows" and "cows have horns and tails" but those characteristics are not relevant to conceptualizing the class of cows (except inferentially such that they are mammals and vertabrates but of course that's not what the child means). The properties, perceptions, and meanings the child attaches to cows come out of his experiences and activities with cows. So the child has to identify just which characteristic features are the proper ones to define the class of cows and which characteristics belong to other classes as well and which ones are irrelevant.


CONCLUSION

The primary teacher’s task is to help students learn about the properties of things and to develop the ability to classify things. When we ask someone to define what they mean by a term, we are asking for their classification. A definition always has both the extension and intension. "A fork is an eating utensil with tines." From this we can see the extension, the hierarchy of inclusion relations—forks are included in the class of eating utensils. We also have the specific defining property, the tines as the distinguishing property from other eating utensils such as spoons. This "is a" "has a" definition is the goal for every important noun in the primary curriculum since every noun carries with it the possibility of being conceptualized as a class with defining properties and therefore properly understood. That conceptualization coordinating the properties with class membership is an essential development of primary students as they construct an understanding of the world .