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A message to teachers of elementary mathematicsAuthor(s): JULIUS H. HLAVATYSource: The Arithmetic Teacher, Vol. 15, No. 5 (MAY 1968), pp. 397-399Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185793 .Accessed: 23/06/2014 05:46

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A message to teachers of elementary mathematics

JULIUS H. HLAVAT New Rochelle, New York

Dr. Hlavat is president of the National Council of Teachers of Mathematics. His term of office began in April 1968, and he will serve as president for two years. Dr. Hlavat has a dedicated interest in mathematics education for all educational levels - kindergarten through university. We are pleased to bring our readers his message for teachers.

-High school teachers of mathematics - and I include myself among them - have long been accustomed to criticisms from colleagues at the college level about the teaching of mathematics in the high schools. Occasionally we have reacted to these criticisms by criticizing our colleagues in the elementary schools. Yet years of teaching have convinced me that we all, by and large, do the best we can at all levels. Indeed, I have a profound admira- tion for the elementary school teachers who, facing very broad responsibilities, achieve great success in all areas, including mathematics.

The proposed and actual reforms in mathematical education in the past ten years have introduced what I consider a fallacy in our thinking. This is the notion that at every level the teacher of mathe- matics must be a mathematician. I do not think that is true.

The teacher of mathematics in the pri- mary and intermediate grades, and, in fact, in the high schools, should have a panoramic view of mathematics. Of course, the teacher needs to know completely whatever is on the agenda for the day or the year, and he or she must also know in broad outline what preceded and what will follow. Yet it is not really necessary that he or she be a mathematician.

This is not to say that we should ever be satisfied with only a minimum knowl-

Fabian Bachrach Photo

edge and understanding of mathematics (or, for that matter, of any of the disci- plines that make up education). On the contrary, we should all continue our study, by means of courses, lectures, reading, in- dividual investigations of special topics, etc.

There are at least three good reasons for such continued study. In the first place, it deepens and broadens our understanding of the subject matter. This enables us to enrich our offerings to the students and en- ables us also better to appraise pupil re- sponses which may be a little off the beaten path. Secondly, if we go through a learning experience of our own, periodically or con- tinuously, we can more readily appreciate and sympathize with the learning difficul- ties our pupils may have with what we are teaching them. Thirdly, our own lives are more meaningful if, through continued learning, we remain intellectually alive.

May 1968 397

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What are some of the principal ideas of which the teacher of elementary mathe- matics should have a deep understanding? Perhaps the overriding one is the develop- ment of the number system.

The teacher of elementary mathematics needs to have a clear vision of how the number system will expand and develop as the student goes through the grades and of how the "laws" or "properties" (commuta- tive, associative, etc.) persist and are re- tained even though the concept of number and its interpretations change. The notion of "operation" also changes in its interpre- tations, and this the teacher should know.

Incidentally, the concept of "operation" is one of the most difficult concepts to de- velop in the grades. By long usage, we tend to confuse the mathematical meaning of "operation" with the everyday language meaning of the word. In sophisticated mathematical language, an operation (in any particular number system we happen to be using) is an association or mapping of ordered pairs of the number system with or onto elements of the system. In ordinary language, we think of "operation" as what we do when we multiply two 2-digit num- bers, or when we divide a 3-digit number by a 2-digit number, for example. A better name for such activities is "algorithm."

The first and simplest concept of number arises from equivalent sets, and it is used to answer the question "How many?" The notion of addition is abstracted from the union of disjoint sets, and it also answers the question "How many?" At this stage, and only at this stage, it makes sense to talk about multiplication as repeated addi- tion.

When the notion of number is extended to the problem of investigating the world of size and shape (geometry), we are seek- ing answers to the question "How large?" We are entering the world of fractions, or rational numbers. Here addition can be interpreted as joining two lengths together and then asking about the resulting length, "How long?" One interpretation of mul- tiplication at this stage is not "repeated ad-

dition" but finding the area of a rectangle whose dimensions are, let us say, one-half unit and one-third unit.

Negative numbers enter when we use the concept of whole numbers (or rational numbers) in combination with the idea of direction. If you walk three steps to the right and then four steps to the left, where are you? Here we have a simple interpretation of (+3) + (~4) = ("1). And, when we wish to retain the distributive property, we define multiplication in the familiar rule: "A negative number multiplied by a negative number gives a positive number."

Let us look at one way of showing this. Now we know that 5X3= 15. But we can rename 5 as "8 - 3," and 3 as "5 - 2." If we multiply

(8 - 3) X (5 - 2), then by using the distributive law we get or 8 X (5

- 2) - 3 X (5 - 2),

8 X 5 + (8) X (-2) + (-3) X (5) + ("3) X ("2) or

40-16 - 15 + ?

The first three terms give us 9. What must we make ("3) X ("2) to be in order that 5 X 3 be equal to 15? Clearly, we shall have to say

(-3) x (-2) = +6! That is, if we wish to retain the distributive property, we must say

(-3) x (-2) = +6. Now, the distributive property is worth re- taining. Let us remember that it is the dis- tributive property which justifies the usual algorithm for multiplication of, let us say, two 2-digit numbers.

After this come the real numbers (in- cluding the mysterious irrationals) and the complex numbers. Beyond those, there are even "hypercomplex" numbers. It is important to leave doors open for further assimilation and understanding as both "number" and "operation" change mean- ings and interpretations.

What is it, however, that does remain through all the stages of the expansion and development of the number system?

398 The Arithmetic Teacher

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It is precisely the properties, which do not seem so exciting at the beginning of the story with the whole numbers: the com- mutative property, the associative property, etc. As the number system expands, we wish to hold on to these properties as long as we can. It does happen that after a cer- tain stage we are finally reconciled to sac- rificing some of them. For example, when we get to matrices (a form of hypercom- plex numbers), we give up the commuta- tive principle in multiplication.

Beyond the development of the number system, the teacher of elementary mathe- matics should have some idea of the fasci- nating worlds of algebra and geometry that lie ahead for the students.

An introduction to the exciting world of algebra is provided by the "open sen- tences" (equations and inequalities) that are now commonly dealt with in the grades. For example, in the following sentences what replacements will make the sentences true? Or false?

5 - 3 = 12. 2x - 1 > 15.

There are fascinating avenues in geome- try: the names of the various shapes we see

around us in nature; their sizes, and how we measure them; tracings, and when shapes are congruent; using mirrors to look at shapes (reflections, symmetries); etc. A number of recent issues of The Arithme- tic Teacher (and issues to come) will be a great aid in developing these ideas.

Over and beyond matters of content in mathematics that we should be aware of, at the elementary and high school levels, methods of teaching are of utmost im- portance. Of course we all know that. It