the nctm film-text series "mathematics for elementary school teachers"

The NCTM film-text series "Mathematics for Elementary School Teachers"Author(s): JOSEPH MORAYSource: The Arithmetic Teacher, Vol. 14, No. 4 (APRIL 1967), pp. 296-299Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187301 .

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The NCTM film -text series "Mathematics for Elementary School Teachers"

JOSEPH MORAY San Francisco State College, San Francisco, California

Joseph Moray was the film teacher for the NCTM film-text series "Mathematics for Elementary School Teachers" He is a professor at San Francisco State College.

JVLy worst subject has always been mathematics, but I was just getting to be confident about teaching second-grade arithmetic when along comes this 'new math' and I'm all shook up again."

"I signed up for that course given at the college math department. After two meetings a lot of us dropped out. The stuff was over our heads, and we couldn't see how it related to our teaching."

These comments were fairly typical a few years ago, and we hear a few similar re- marks today. Changes in teaching, however, have not kept pace with changes in the curriculum. It must be acknowledged that a great deal of effort has been made to prepare teachers to cope with the recently developed mathematics programs and the new text materials. School districts and colleges have provided workshops and courses in new mathematics for teachers, but they have sometimes been handicapped by a lack of fully qualified instructors and consultants, as well as by the sheer magnitude of the task.

In appraising the progress of the revolution in elementary school mathematics, John L. Marks concluded that "the vast majority of our elementary teachers are unprepared to tackle any of our newer programs," and he suggested that "an in- service program, of such magnitude that

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it is difficult to imagine, must be devised immediately."1

In reflecting on the challenge of the Cambridge Conference Report on Goals for School Mathematics, Marshall H. Stone stated that "the real obstacle lies not in our knowledge of mathematics or our knowledge of pedagogy, but in the out- moded and inadequate preparation we are still giving to our future mathematics teachers."2

Much has already been done to provide content background materials for teachers. Fred Weaver listed forty-two books and several film series which have been issued recently to help with pre-service and in- service programs.3 But the training prob- lem has been compounded by the speed with which experimental programs were put into textbook form. Teachers tended to be intimidated by the authoritarian tone of many of the new texts, overwhelmed by the new notation and upgraded content,

1 "The Uneven Progress of the Revolution in Ele- mentary School Mathematics," The Arithmetic Teacher, X (December 1963), 476.

2 Review and critique of Goals for School Mathe- matics, The Mathematics Teacher, LVIII (April 1965), 360.

3 "The Mathematics Education of Elementary School Teachers: Pre-Service and In-Service," The Arith- metic Teacher, XII (January 1965), 71-76.

The Arithmetic Teacher



and confused by the seeming lack of agree- ment in the "precise" language featured in the different text series.

Teachers needed help, and in 1962 the National Council of Teachers of Mathe- matics under the leadership of Frank B. Allen decided to do something special about it. A pilot set of in-service training films was produced in the summer of 1963; and, on the basis of the promise of that project, a National Science Foun- dation grant was secured, resulting in the production of a series of ten color films on the system of whole numbers, en- titled "Mathematics for Elementary School Teachers" (MET). As part of the project, a coordinated text was designed to be used either along with the films or inde- pendently. Harry D. Ruderman served as project director, Julius Hlavatý as chair- man of the advisory panel, and Joseph Moray as film teacher. Advisory panel members, writers, and consultants are listed in the introduction to the text. The films were produced by Davidson Films, San

April 1967

Francisco; and they are available for sale or rental from United World Films, 221 Park Avenue South, New York, New York 10003. The texts are available from the Washington office of the National Council of Teachers of Mathematics.

The first film of the series, Beginning Number Concepts, deals with one-to-one correspondence, equivalent and nonequiv- alent sets, number, order, and counting. The presentation is simple, and "on lo- cation" classroom and pastoral scenes are included. The next film, Development of Our Decimal Numeration System, gives some historical background, citing several early notation systems, and presents a clear analysis of our decimal place-value system. This film features models of arti- facts and an animated sequence.

The next four films treat the addition, multiplication, subtraction, and the division of whole numbers. Each of these films uses a variety of approaches for developing meaning, and classroom scenes are used to illustrate some of these ap-

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proaches. The vocabulary was carefully chosen to reveal the meaning behind what- ever terms or expressions are used in the various new textbooks. For example, the number 3+2 is called a "sum" and the numbers 3 and 2 are called "addends" of the sum 3 + 2. The teacher asks first-grade children to "rename," "find the

standard name for," "find the simple name for," or "compute" the sum (rather than "find the sum of 3 + 2," - a possibly con- fusing request, since 3 + 2 is a sum).

There is also an emphasis on developing meaning for an idea before a symbol or term is used to represent that idea. Some of the new primary textbooks in- troduce new material too quickly for many children, and teachers need to develop a feeling for an effective sequence for developing mathematical concepts.

Film 7 shows how children use basic properties and an understanding of place- value notation in developing an inter- pretation of addition and subtraction computation procedures, or algorithms. The following film features an animated sequence showing how the distributive property is related to the multiplication algorithm, and Film 9 includes a sequence showing how a class develops an algorithm for division.

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The final film in this series reviews the key concepts of the whole number system and indicates that these concepts may be built upon for the effective teaching of later topics in mathematics.

While the series deals mostly with content of the early and middle grades, the films were designed for all elementary teachers. In three separate pilot studies, involving a total of 147 teachers, most of the teachers rated the content level of the films as neither too difficult nor too easy. Only one of ninety kindergarten-through- fourth-grade teachers rated the content as "very difficult," while eleven said the content was "very easy." About 78 per- cent of all the teachers rated the series as a whole either good or excellent.

A study of reactions to the text produced similar results. The text was con- sidered by teachers as being helpful in explaining and extending the mathematical ideas presented in the films.

Current use of the text and films in pre- service courses indicates a favorable re- sponse from both instructors and students. Requests have been made for showing the films to parent groups, and distribution rights for television have been cleared.

Those of us who were involved in the production of the MET materials are gratified by the wide use now being made of the films for both pre-service and in-

The Arithmetic Teacher



service programs. We are especially gratified by reactions like the one expressed by a teacher who, after viewing the whole film series, said: "I thought I didn't like mathematics but now things are clearing

up for me, and I'm inspired." We hope this will be a typical comment for teachers who "didn't like mathematics," or who have been "all shook up" by the "new math" programs.

Don't shy away from the zero exponent

There is a tendency for most elementary mathematics texts to ignore the zero exponent. When the numeral 5,621 is expressed in exponential form, it is presented as (5 x юз) + (6 X 102) + (2 x io1) + (1 x l). You will notice that all column values are expressed in some power of 10 except the unit column. Rather than as (1 x 10°), it is shown as (1 X 1). While (1 X 1) is certainly correct, it does not seem logical to drop the exponents that have been used to express the value of the other columns.

After examining a sufficient number of texts to assure myself that the practice of omitting zero exponents was rather common, I asked, "Why the omission?" There seems to be a dou- ble reason for shying away from the zero exponent.

Reason number one is probably that many people involved in writing mathematics series feel that students will not grasp the concept. Reason number two is probably that many teachers are not sure how to explain the fact that any number to the zero power is equal to one. Rather than try to explain this somewhat vague idea, there seems to be a general consensus that it should be ignored. My feeling is that both of the above reasons are very weak and most certainly will not help to promote a better understanding of mathematics for most children.

I would like to offer the following suggestions for teaching the zero exponent. Teachers must first examine their own understanding of exponents to be sure they understand the zero exponent. Secondly, there is a need for teachers not to feel guilty when something cannot be shown by a picture or in some other visual form. Mathematics, as well as all other subject areas, has certain ideas that can be observed by students in some concrete form and other concepts that cannot be seen or felt. The teacher can show that 2 2 is simply 2 X 2, or 2 two times, and 2 3 is 2 X 2 X 2, or 2 three times. When 2 ° is to be written, you simply cannot write 2 zero times. Two to the zero power is not something that can be drawn.

Teachers usually explain that any number to the zero power is equal to one "by definition."

April 1967

While this is true, it is hardly the type of explanation which gives a student much opportunity to use his powers of observation or develop his logical reasoning.

Perhaps students can be given more than one explanation of zero exponents and then choose for themselves the way that appears to make most sense to them.

The "logical approach" can be used by mak- ing use of what has already been learned. If students have learned that 2 2 equals 4 and 2 3 equals 8 and 24 equals 16, then the following series of numbers can be presented to the students:

24 = ? 2 3 = ? 22=? 21 = ? 2 ° = ?

The students will know that 24 is 16. They

will see that 2 3 is 8 and also that it is -

of the previous number. The 2 2 is 4, which

again will show that the result is - of the

previous number. At this point you can have a discussion as to the value of reasoning in certain situations. In this particular mathematical example we cannot compute the 2 °, but we

can fit it into a pattern where each answer is ̂ of the previous answer when exponents are de- creased by one each time.

24 = 16. 2 :! = 8. 2- = 4. 2i=2. 2" = 1.

Since each answer is -^ of the previous one

and since 21 is 2, our reasoning tells us that 2° should be 1.

A second explanation is the more standard computational approach often shown. By em-

[Continued on page 306]

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the nctm film-text series "mathematics for elementary school teachers"

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