#5 Number—Where Does It Come From?

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POSSIBLE SOURCES OF NUMBER

When we want to teach students about physical properties such as weight, color, shape, etc., we have real objects with the properties that we can show students. But where is number located. How can we show students number?

Can the student find number in objects? Can we show the student a number? No. The research is clear. Students don’t see number in objects until they have the concept and can read it into the objects. If we show young children 7 objects, they see what is in their perceptual field but their perception does not construct the concept of number as a perceptual field. They see size for number; they can perceptually recognize shapes, color, movement, size, boundaries, touching and separation, but they cannot see the concept of number in a visual scene no matter how carefully the objects may be arranged. Number is not a visual property inherent in objects. What students see are physical objects, not number as an abstract concept.

Nor is number innate. It does not come from brain growth. The brain wires itself as a result of activity. The organization that results eventually in a brain with complex organization is not due to an innate blueprint that contains number. The explanation of why 4 + 3 = 7 will never be reduced to the brain.

Nor is the concept of number contained in words. Number cannot be transmitted verbally either by explanations or reading about it. Words are representational elements but they do not contain meaning. That are attached to meaning but they do not contain meaning. The meaning of number must be organized in thought in order to ‘see’ its meaning in words or in a set of objects. Students must have some concept they can use if they are to understand the words the teacher uses.

Some writers have even suggested that number is a pre-existent idea that we are all born with or that students discover. But again the research shows clearly that there is innate concept and no ideal form of number that pre-exists for students to discover. Even if number did somehow exist in some ideal form floating apart from human thought as a form of truth, how would students’ minds make contact with this idealized realm of existence?

Perhaps number is contained in the math symbols, the numerals we teach students to say. But recognizing a symbol and saying a sound carries nothing more with it than a learned response. It has little meaning until it is invested with something far richer to give it the meaning of number. The word can be socially transmitted but meaning cannot. Words can only trigger pre-existing concepts in the child.

Perhaps students find number in the repetition of the counting experience. Perhaps students learn a pattern from the physical activity of counting objects. But the pattern doesn’t exist for the student until they have the ability to recognize patterns in repetitions such as following a set of seriated counters. An experience remains an experience on whatever plane of activity it occurs. A sensory motor activity, by being repeated, doesn’t produce thought about the activity simply by its repetition. No matter how many times the child repeats a math fact, the meaning of the fact does not increase due to the repetition. Experience remains experience until it is interpreted and given meaning.

None of these factors can account for the development of mathematical thought but all these factors do affect the development of number. If they are not the source but only auxiliary processes for development, what is the source. These factors ‘feed’ students’ mathematical activities but none of them can show or demonstrate number to students who do not yet have the concept. Students must somehow construct the concept of number before they are able to see number in the activities, in objects, in words, in patterned iterations.


THE SOURCE OF NUMBER—REFLECTIVE ABSTRACTION

The research clearly shows that the origin of number lies within an abstraction of the organization of activity directed to objects. It is a construction in the sense that it is not discovered as a pre-existent entity and it is not an invention since it derives from something pre-existing which students’ find in their own activity.


A NECESSARY FIRST STEP—ORGANIZING THINGS

The research shows how students more or less unconsciously use patterned behaviors to organize objects. The ‘do’ something to objects such as gathering toys together into separate groups or they may successively put toys into a line one behind the other. At early stages of development the ability perform organized actions on things operates without conscious intent and awareness. The actions remain subconscious as sensory motor behaviors. But once children begin repeating a behavior such as placing an object next to another object, they can repeat this little organized activity to produce something larger than isolated elements. They can produce a line of cars. The toys are no longer randomly arranged. The child has created an organization, a larger whole, that may appear as a line or as a group.


THE OTHER PART—THE ABSTRACTION

The crucial part of the process occurs after they have exercised a scheme to produce some kind of organization. That process consists of a reflective abstraction in which children begin to think about how they produced what they see. They have performed some patterned activity such as placing each object next to previous objects. They see they have produced an organized whole, perhaps a line of objects. By abstracting the method of placing objects next to each other, they can produce the organized whole once again in another situation and time. The little schemes of finding an object and putting it next to a previous object allows them to reproduce the line. They can use this behavior to construct an object they can observe as a mathematical objects, a line or serial placement of toys. This mathematical series has nothing to do with the physical properties of the objects, and it has everything to do with the organization the child put into the toys.


THE MATHEMATICAL OBJECT

Nobody taught the child to create this little organized whole out of toys (although some insightful parents may actively encourage this kind of organizing behavior by their child) nor did he see a picture in a book or read directions which he could copy. He cannot copy since he has no conceptual bucket by which to record, remember, and transport the form shown in a picture or in words. When viewing a completed organization of toys someone else has formed, the child cannot see the method of composition by which the form was produced. He even finds it difficult to think simultaneously about each of the objects as individual elements at the same time he thinks about the overall whole he makes. If he focuses on the appearance of the whole, he cannot hold in his mind the individual elements. If he focuses on individual elements, he loses track of the overall whole being produced.

None-the-less, by grouping together schemes of placing toys, he has developed the ability to form something larger than each of his separated centrations on individual objects previously allowed him to do. This whole he produced is a mathematical object. Most importantly, since he used objects, he has something to observe that he has produced. He can reproduce this ‘something’ by performing the little series of repeated, special activities on the objects, the placing. He can do it again to produce the effect.


CONSTANT ADJUSTMENT MEANS THERE IS ACTIVE REGULATION OF ACTIVITY

But when he attempts to use his schemes to re-produce the larger organized whole, he must constantly attend to them to adjust the actions as he applies the schemes according to the situation and objects in order to make the patterned whole appear again. He cannot blindly, randomly repeat something. He must locate a toy, grasp it, place it, adjust its position so it looks right. He must give at least some thought to making sure each action is successful in forming the organized whole he intends on re-producing. This natural attempt to fit and adjust his activities to make them successful is the work of the function of the regulators that are inherent in the dynamics of all living activity. Humans aren’t passive storage containers for knowledge. They are the seat of self-regulated activities.

Regulations are the little adjustments of the activity as it compensates for small disturbances or problems it encounters and must overcome in carrying out the project of arranging the toys. The regulations compensate for differences among the toy, their size, whether they belong, the area needed to extend the arrangement, and so on. Regulation is the essential property of intelligence. An unusually large toy might be rejected because it doesn’t seem to fit to the notions of what toys can used. To compensate for its size that makes it difficult to use, the child simple rejects it. To successfully use the toy, the child might have made it smaller in some way so it could then fit in. Or the child might anticipate what kinds of toys will work and select them before starting. This anticipation is the mark of a more advanced form of activity.


MATHEMATICAL FORM PRODUCED BY THE CHILD

The essential mathematical idea is to see that a mathematical form was put into the objects by arranging them. The child has some simple coordinations of activity, some little schemes, that he can use to produce an arrangement. As soon as he produces some kind of organized form with the objects, he can attempt to reconstruct the activities he used to produce the form. This attempt to re-use his coordinated activities and his beginnings of thinking about these activities as he must adjust and adapt them is the process of reflective abstraction.


AGAIN—REFLECTIVE ABSTRACTION AS THE SOURCE OF CONSTRUCTION

Reflective abstraction is the process of abstracting the form or method of activities that is used to produce a repeatable, organized whole. The child must first use an existing scheme, pattern of activities, or method to organize objects. The abstraction comes from attempting to think back on how the actions were performed. It attempts to reflect on the plane of thought the unconscious form of the schemes used in the behavioral activity. It moves activity a tiny step higher in making more conscious. But reflection is also the process of re-organizing all the individual schemes into a thought that can replace them. Thought is simply activity occurring on a different plane, the plane of consciousness, as mental activity begins to disconnect itself from physical activity. Just as the physical activity employed an organizer, a scheme, for initiating the activity, thought also must organize activity on the plane of thought. Thought reflects and reconstructs what occurred at the behavioral level thereby making the child more aware of the operations used to produce the mathematical form of the line of toys.

Reflective abstraction—this fundamental process of drawing out the implicit coordinations of our activities to reconstruct and reproduce them on a higher and more explicit level of mental activity—is the essential constructive mechanism that is the source of the development of new mathematical thought at every level from the new born infant up to the most advanced mathematician. The child does not see number in a set of objects shown to him but he can organize the objects and abstract his methods of organizing the objects to develop the idea of number. Mathematics doesn’t come from verbal explanations or drill and practice. Counting is a rote activity imposed on the child by the sequence of numeral names but it is a behavioral patterns that is learned and does not allow a reflective abstraction by which to pull out the concept of number.


TWO FORMS ABSTRACTED AND COORDINATED INTO NUMBER

To construct the idea of number, the child must use his existing methods of organizing objects to do two things. He must mark off objects to turn them into a set of counters. He must distinguish among them by putting them into some kind of order. He can then use the learned sequence of numeral names to follow the order of the counters and to stop when all counters of the set are enumerated. The simultaneous putting into a set and ordering the counters are the twin operational schemes that form the concept of number, not all at once, but by gradual step-by-step constructions from activities and reflective abstraction.

For example, imagine a child playing with toy car and a toy auto parking garage. At an earlier point in development, the child can only apply a scheme of assimilation—like putting the car into the mouth, turning it over, pushing it, throwing it—to each object but without relating objects and schemes one to another. At a later point, when schemes are organized into larger wholes and can incorporate several steps, the child plays with the toy cars and auto ramp as a whole object. He may more or less think about which group of cars he is placing into the auto ramp and which groups he is putting outside or in a different part of the ramp. Or he may more or less think about lining the cars up as they enter the parking ramp. He may move the first car and create a gap which he then fills with the second car creating another gap which he fills. He may try to think about how he moves the cars forward and the gap move backward. These patterned activities are the sources of grouping into sets and ordering in series, the two properties of number.

Thus to form the concept of number, the child must coordinate set making of objects simultaneously with ordering the objects to produce the cardinal and ordinal properties of number. The child doesn’t receive number from outside sources, from counting, from words, from teachers, from books, nor does he discover number as pre-existing in objects, from being show a set of objects that are supposed to demonstrate a number, from ‘seeing’ patterns in things, or from experiences practicing procedures given to him such as counting. for there to be development, the child must do the organizing work and must do the reflection. Each child constructs the concept of number from a process of putting organization into objects and then reflecting on the methods he used Thereby he abstracts and reconstructs the behavioral operations on the plane of mental operations.


MATH EDUCATION

Student activity-based mathematics doesn’t mean students invent or discover. It means there is a disciplined adherence and careful attention to requiring students to organize objects, to abstract their methods, and to represent their methods. Certainly the instructional process can employ objects, discussions, coaching, direct telling, practice, and all the rest but all at the appropriate point in the cycle and all in the service of promoting the process of organizing objects and reflective abstraction. When students are required to organize objects and then to pull out and represent the form of the operations they used, these student constructions and representations will reflect how they actually understand mathematics concepts and that allows authentic instructional assistance because the teacher can follow the development of students’ constructions. Students’ organization of activity and their products of reflection depends on how the student understands, not what they have copied and learned. And that process of organization, reflective abstraction, and represesntation is the essence of math education and indeed, all of mathematics itself.