Increasingly, during the last twenty years, the literature relating to arithmetic instruction has carried the words “meaning,” “meaningful,” and “meaningfully.” For some persons, these terms seem to be no more than words—mere items in the vocabulary of modern elementary education, adopted because, for the moment, they are fashionable. For others, these words serve as symbols of a vague protest against what they call the “traditional” arithmetic, though they have little except pious wishes to offer as a substitute. For still others, the terms are appropriate for use in connection with arithmetic experiences which arise out of felt needs on the part of children. This third usage, unlike the first two, has in its favor a certain definiteness. It implies particular conditions of learning and motivation. Children see the chance to use their arithmetical ideas and skills to further some end, and they use the ideas and skills for this purpose.
We should, however, at this point, distinguish between what I shall designate the meaning of a thing and the meaning of a thing for something else; for the sake of brevity, between meaning of and meaning for. I know little about the meaning of the atomic bomb, because I lack the knowledge of chemistry and physics which are requisite to accurate understanding, but I think I know a good deal about the meaning of the atomic bomb for other things— for peace or for the destruction of our culture, for example.
The distinction I am suggesting is no verbal quibble, no bit of theoretical hairsplitting. Failure to recognize the difference between meanings of and meanings for makes it difficult for those of us who are interested in the improvement of arithmetic instruction to agree on procedures. We use the same words but in different senses. The third usage, namely, that children have meaningful arithmetic experiences when they use arithmetic in connection with real life needs, relates to meanings for. On this account some prefer to call such arithmetic experiences “significant” rather than “meaningful.”
On the other hand, just as the meaning of the atomic bomb is to be found in the related physical sciences, so the meanings of arithmetic are to be found in mathematics. They are not to be found in the life-settings in which they are normally imbedded, except by him who already possesses them. They must be sought in the mathematical relationships of the subject itself, in its concepts, generalizations, and principles. In this sense a child has a meaningful arithmetic experience when the situation with which he deals “makes sense” mathematically. He behaves meaningfully with respect to a quantitative situation when he knows what to do arithmetically and when he knows how to do it; and he possesses arithmetical meanings when he understands arithmetic as mathematics. In arithmetic, then, meanings of may be defined as mathematical understandings, and it is in this sense that the word will be used throughout this article.
I have spoken of meanings as if they were absolute—as if, one has a meaning, or he has none. In terms of learning, however, meanings are relative, not absolute. There are degrees of meanings; degrees of what may be termed extent, exactness, depth, complexity; and growth in meanings may take place in any of these dimensions. For relatively few aspects of life, for relatively few aspects of the school’s curriculum (including arithmetic), do we seek to carry meanings to anything like their fullest development. Moreover, whatever the degree of meaning we want children to have, we cannot engender it all at once. Instead, we stop at different levels with different concepts; we aim now at this level of meaning, later at a higher level, and so on.
“Meaningful” arithmetic, in contrast to “meaningless” arithmetic, refers to instruction which is deliberately planned to teach arithmetical meanings and to make arithmetic sensible to children through its mathematical relationships. Not all possible meanings are taught, nor are all meanings taught in the same degree of completeness. Meaningful arithmetic, then, may be thought of as occupying a place well to the right on a scale of meaningfulness. On the other hand, “meaningless” arithmetic occupies a place well toward the left end of the scale but not at the 0-point; for there can hardly be a wholly meaningless arithmetic. Meaningless arithmetic is only relatively meaningless. Its content is taught with no specific intention of developing meanings, and the meanings which are learned are acquired incidentally and largely through the learner’s own efforts.
The meanings of arithmetic can be roughly grouped under a number of categories. I am suggesting four.
1. One group consists of a large list of basic concepts. Here, for example, are the meanings of whole numbers, of common fractions, of decimal fractions, of per cent, and, most persons would say, of ratio and proportion. Here belong, also, the denominate numbers, on which there is only slight disagreement concerning the particular units to be taught. Here, too, are the technical terms of arithmetic—addend, divisor, common denominator, and the like—and, again, there is some difference of opinion as to which terms are essential and which are not.
2. A second group of arithmetical meanings includes understanding of the fundamental operations. Children must know when to add, when to subtract, when to multiply, and when to divide. They must possess this knowledge, and they must also know what happens to the numbers used when a given operation is employed. If the newer textbooks afford trustworthy evidence on the point, the trend toward the teaching of the functions of the basic operations is well established. Few changes in the more recent textbooks, as compared with those of twenty years ago, are more impressive.
3. A third group of meanings is composed of the more important principles, relationships, and generalizations of arithmetic, of which the following are typical: When 0 is added to a number, the value of that number is unchanged. The product of two abstract factors remains the same regardless of which factor is used as multiplier. The numerator and denominator of a fraction may be divided by the same number without changing the value of the fraction.
4. A fourth group of meanings relates to the understanding of our decimal number system and its use in rationalizing our computational procedures and our algorisms. There appears to be a growing tendency to devote more attention to the meanings of large numbers in terms of the place values of their digits. Likewise there is a strong tendency to rationalize the simpler computational operations such as “carrying” in addition and “borrowing” in subtraction; but there is some hesitation about extending rationalizations very far into multiplication and division with whole numbers and fractions.