Japanese

An Investigation on the Theme of "Number/Quantity" ABSTRACT

---

I hold that teachers cannot manage the instruction of NUMBER without teaching what the number is. In the present system of mathematics instruction, teaching <what> is changed for teaching <how>. That is, teaching NUMBER-use is frauduously substituted for teaching NUMBER. This way of instruction results in students' inability in studying/understanding/using NUMBER. By stages, using NUMBER without understanding what NUMBER is becomes more and more difficult.

The learning leaded by <how> is extentive and is destinated to blow out. We can innerly maintain mathematics only if it is conceived generatively.

Furthermore, such instructions as substitute <how> for <what> make students misunderstand the way of learning --- for them, to learn mathematics is to learn some kind of <know-how>'s --- and make teachers design those instructions that fail the points of mathematical subjects.

Therefore, it is requisite to recover <what> in the teaching of mathematics. For example, we would introduce the system of natural numbers as "sequence", the fractional numbers --- "integral ratios of quantity-elements", the "positive and negative numbers" --- "the symmetrization of coeffitients/operators in a QUANTITY-system Q corresponding to the sysmmetrization of Q. In particular, as the instruction of the semantics of NUMBER, we would, to some extent, characterize the NUMBER-use as the QUANTITY-system.

From the standpoint described above, the author tries to investigate the theme under the title "number/quantity".

NUMBER is a self-consistent system. But whenever it is in our daily use it is a subsystem of a system that we should call "quantity". In particular, to make NUMBER a subject in mathematics class is to make QUANTITY a subject.

NUMBER/QUANTITY occures as a reading. That is, all that exist are our daily lives --- "language games"(Wittgenstein) --- and NUMBER/QUANTITY is figured out there. In this sense, the first meaning of "number/quantity" is NUMBER/QUANTITY-form.

Anything figured is imaginary. But "to be imaginary" does not imply "no to be realized" --- we can set the subject "realizations of NUMBER/QUANTITY". The fiction of NUMBER/QUANTITY is endorsed by two kinds of reality. That is, any language games from which NUMBER/QUANTITY is figured out and any realizations of NUMBER/QUANTITY.

To define the NUMBER-form is to determine an algebaraic structure of a set. Systems that are determined in this way are NUMBER-systems.

The realization of NUMBER is a practical and itemized subject. In fact, the realization of NUMBER starts in the system of natural numbers. The essence of is "sequence", and it is formalized by the Peano's Axioms. Henceforth the realizations of NUMBER are done in the form of "extending a NUMBER-system" --- a well-known story.

The meaning of NUMBER can be formalized to a form to which the NUMBER-form is embedded as a factor. This form is the QUANTITY-form or the POSITION-form. Each POSITION-form contains a QUANTITY-form as a factor, and each QUANTITY-form can be embedded in a POSITION-form. In fact, the meaning of NUMBER is "operators/coefficients(scalars) in a QUANTITY-system". Whenever numbers are used, they are embedded in some QUANTITY/POSITION-systems. The QUANTITY/POSITION-form introduced here is understood as a "universal object" in the category of QUANTITY/POSITION-systems.

As for the NUMBER, QUANTITY and POSITION-forms, chief subjects are as follows:
1. NUMBER/QUANTITY-systems can be displayed in the form of "line". The status of the NUMBER/QUANTITY-line is "picture of NUMBER/QUANTITY-system".
   
   
2. The subjects "measurement", "alteration of units-systems", "proportional relation" belong to the theory of the category of NUMBER/QUANTITY-systems. They can be formulated by using such concepts as "morphism", "functor", "natural transformation".
   
   
3. There exist ways to duduce a new QUANTITY-system from any QUANTITY-systems. The proportional functions between two QUANTITY-systems Q1,Q2 form a QUANTITY-system Hom(Q1,Q2). From Q1,Q2 we can also deduce their "product" Q1Q2.
   
   
4. "Computation" is a sequence of expression-transformations under a syntactic rule. This rule of transformation is of two kinds. That is, one that depends to the form of realization of NUMBER/QUANTITY (e.g., "digital number") and one that does not. We can call the former "formula" and the latter "law". Laws are implicatons of NUMBER/QUANTITY-form.
   
   
5. The form of description of a QUANTITY-element is <unit x number>, and to measure a QUANTITY-element is to make this type of description. Making such descriptions is a practice --- that is, out of determinism. Treatments of quantities start in practices of measurement. And here occure various practical problems, such as choosing units, determining a units-system, realizing a units-system-community, realizing NUMBER through making descriptons of quantities, measuring, treating of "fragments".

In fact, above-mentioned contents are exactly those contents that are explicitly or implicitly subjected in mathematics class under the title "number/quantity". Furthermore, almost all subjects of school mathematics already exist in elementary school mathematics. We must explore for each mathematical subject such presentation that enable students to understand the subject.

Of cource, the problem of the exploration of instructional presentations is classical. But, today, it can be newly enlightened. That is, those miss-leading instructions in the present system can be now returned to "poverty of technology-for-presentation". And our problem is the reconstruction of the mathematics teaching by introducing new-age instruction/learning media --- in fact, "multimedia".