Re-education in mathematics for elementary school personnel: in-service programs—one way to solve the problem

Re-education in mathematics for elementary school personnel: in-service programs—one wayto solve the problemAuthor(s): JOSEPH A. IZZO and RUTH KELLEY IZZOSource: The Arithmetic Teacher, Vol. 11, No. 6 (October 1964), pp. 413-417Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185004 .

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Re-education in mathematics for elementary school personnel: in-service programs- one way to solve the problem

JOSEPH A. IZZO and RUTH KELLEY IZZO University of Vermont, Burlington, Vermont Professor Izzo is a member of the department of mathematics at the University of Vermont. Mrs. Izzo is associate director of summer institutes and staff member of follow-up programs.

A he recent revolution in school mathematics - the term associated with develop- ments which have produced the so-called modern programs, in mathematics - has presented a challenge to elementary school teachers. The challenge is to acquire sufficient background in mathematics to en- able them to provide effective instruction in the mathematics program presently suggested for the elementary school.

NSF-sponsored summer institute in mathematics

Recognizing the need for a program which would help the elementary school teacher meet the challenge of re-education in mathematics, the mathematics department of the University of Vermont designed a two-part program. First, the mathematics department, with the cooperation of the University of Vermont's education department, conducted during the summer of 1963 a National Science Foundation-supported six-week institute in mathematics for elementary school personnel. Second, the participants of the summer institute returned to their local districts and conducted "courses" in background mathematics for the teachers in their own districts.

The participants in the summer institute were outstanding elementary school

teachers, supervisors, and administrators with mathematical interests and aptitudes. Of the thirty-one participants, eleven were women and twenty were men. Every grade level of the elementary school (Grades 1 through 6) was represented. The grade level and the number of participants who teach at that level (represented by the number in parentheses) were as follows: first (2), second (1), third (1), fourth (1), fifth (6), and sixth (9). There were six elementary school principals, three teaching principals (included in the grade levels above), and five elementary supervisors or coordinators. With one exception, these people were accepted to the institute with the understanding that each would return to his respective school district and conduct an in-service program.

The participants for the most part were from the northeastern part of the United States and came from ten different states: Connecticut, Maine, Massachu- setts, New Hampshire, New Jersey, New York, Pennsylvania, Rhode Island, and Vermont. There was one participant from California.

The purposes of the summer institute were to give the participants: (1) instruction in the subject matter of mathematics which the Committee on the Under- graduate Program in Mathematics

October 1964 413



(CUPM) had recommended as essential to teaching elementary school mathematics, (2) preparation for conducting an in- service program in mathematics and (3) a knowledge of effective approaches and materials which could be used in the teaching of mathematics at the elementary school level.

The thirty-one participants spent an intensive and challenging summer ac- quiring knowledge of the basic concepts of elementary mathematics - concepts which give meaning to the new programs in mathematics suggested for the elementary school. The topics studied included: the real number system and its subsystems; sets, logic, and systems of numeration; properties of numbers, relations, and operations; an introduction to algebra, and intuitive foundations of geometry.

While emphasis during the summer institute was on the fundamental ideas of mathematics, much attention was given to a consideration of ways of teaching, approaches, and materials which could be used effectively in teaching mathematics to elementary school children. The courses of the institute were planned in such a manner that the participants also gained knowledge of how to conduct an in-service program. In addition, help was given to them in the form of outlines, suggested textbooks and materials, and suggested organizational plans.

The in-service program At the conclusion of the summer in-

stitute each participant returned to his local school situation for the second part of the program which was to conduct an in-service "course" in background mathematics for the teachers in his own district.

Each local situation was unique as one would expect. The organization of and policies governing the different in-service programs varied because each program was determined by the local school ad- ministration in cooperation with the participant. In some districts much of the organization essential for conducting an

in-service program had been arranged prior to the opening of school in Septem- ber, 1963. The number, dates, time, length, and place of meetings had been determined. Policies had been established concerning, credit or non-credit for salary or certification purposes, compulsory or voluntary attendance for the local teachers, released time for the teachers and/or instructor (i.e., the summer participant), and financial support for the in- service program, including textbooks for teachers, materials, films, supplies, and, in some cases, financial compensation for the instructor. In other localities the organization and policies of the in-service program were determined after the start of school.

The in-service class

There was no specific pattern for the local programs but certain similarities did exist. For the most part the in-service program was similar to a class or course with the summer participant serving as the instructor. In presenting a mathematical topic the instructor used a multiplicity of approaches. These varied approaches illu- strated for the teachers the idea that change in the ways of teaching mathematics also is necessary if there is to be significant improvement in the mathematics programs for elementary schools.

In situations where each class session was of sufficient duration (e.g., one and a half or two hours), one class design em- ployed in the in-service program was a division of class time into two parts. Dur- ing the first part the instructor presented and developed some subject matter topic. The second part was devoted to a work- type session during which time the teachers worked examples and problems which further developed the mathematical topic or which served to introduce a new topic. Some exercises led the teachers to "discover" concepts and in this way illus- trated a valuable and effective approach in the teaching of mathematics.

The materials used during the work sessions were those which had been pre-

414 The Arithmetic Teacher



pared by the instructor or were commer- cially prepared mathematics workbooks or textbooks. Many of these commercial materials had been developed for high school courses and, with some adaptations, were very effective in introducing or de- veloping basic mathematical concepts for the teachers.

For situations where the class session was for a shorter duration, the class time was devoted to presentations and discus- sions and then ' 'homework" in the form of exercises and problems was also as- signed.

The major objective of the in-service programs was to help the teachers develop a better comprehension and appreciation of the structural aspect of mathematics which would in turn help them understand the "why" in the procedures of arithmetical computation. The topics presented to the teachers were the same as some of those studied in the summer institute, but the treatment was less rigorous. In most programs the following topics were included: early systems of numeration, scientific and expanded notation, the number line, different number bases, set concepts, operations within the system of whole numbers, concepts of elementary number theory, the system of rational numbers, and selected topics from geometry using a set approach.

Keeping abreast with subject matter in any area is of concern to elementary school teachers. Of equal importance to teachers is: "How do I use this knowledge to improve my teaching?" The particular needs of the local district determined how the question was met. For example, some districts had already made partial or com- plete mathematics textbook adoptions. During their in-service work the local teachers made an analysis of the teacher's editions of these textbooks. They found that because of their recent study of mathematics they were able to make a more effective analysis and evaluation of the material.

Other districts were preparing to pur-

chase new textbooks and the local teachers, under the supervision of the in-service instructor, studied various textbooks which were on the market. For these teachers an analysis of textbooks became a part of the in-service program, but only after study- ing the basic concepts of elementary mathematics.

The use of materials, a study of methods of teaching, the viewing of recent films concerned with the "new" mathematics, and the sharing of techniques and procedures which had been used effectively in the classroom were included in many of the in-service programs. This part of the program was conducted in different ways. For some, alternate meetings were devoted to subject matter and to materials or approaches; for others, each meeting was devoted to both subject matter and approaches; and in a few cases the subject matter part of the program was completed first with a series of meetings on approaches conducted afterwards.

Reactions to the in-service program

Local teachers

The teachers in each district who participated in the in-service program reacted very favorably to the program which had a fellow colleague (the elementary school classroom teacher, supervisor, or administrator) return to the local district to share with them the knowledge which he or she had gained.

One of the authors, Dr. Ruth Kelley Izzo, followed up the summer participants and their in-service programs. She received many unsolicited comments from the local teachers. Among the comments were the following :

"Are you going to take this course? He's a good guy. You will learn a lot of mathematics." (During one of her visits, Dr. Ruth Izzo, at the request of the instructor, had joined the in-service group without introduction.)

"This class has done so much for me. I have always disliked mathematics, but

October 1964 415



now I am over that fear. Never thought my supervisor would help me that way."

"Thanks to [a principal] I have con- fidence enough now to undertake a mathematics course at ... University."

"She not only has taught me mathematics, but has helped me to see how I can improve my arithmetic classes. Dur- ing our sessions I find idea after idea com- ing to me as to how I can help the children in my class."

"I have gotten a great deal out of this. I understand better why some of the children have trouble with arithmetic."

"The sessions have been very profitable and it wasn't easy for him. You know, it isn't easy teaching one's peers. However, . . . had the backing of our Superintendent and most of us really wanted to learn."

School officials School officials were enthusiastic about

the total program - the summer institute and the in-service programs. Here are some of their comments :

"[Classroom teacher] is doing a tremendous job with our teachers - first, in the understanding of mathematics and then in many other professional ways."

"One of the principals told me yester- day, 'You know, the teachers really look forward to this class.' "

"[Classroom teacher] did a terrific job with the teachers! I am strengthening my background in science and encouraged . . . to tackle the mathematics. We, teachers and supervisors, have to work together on the upgrading of programs. Teachers have different subject matter strengths and why shouldn't we capitalize on that?"

"One reads in books on theory about the cooperation of high school and elementary school teachers. We have evi- dence that it can be done. The head of our high school mathematics department and the instructor in the in-service program are working very closely on the mathematics program. He attends all of her in- service meetings."

"If I had personally designed an op- portunity for our school system, I could not have come closer to our needs. The program has helped us tremendously."

"We were more than eager to cooperate in the establishment of an in-service program because your program was a very fine and intelligent approach to the whole problem. The elementary teachers of this system have been helped immeasurably."

The participants (in-service instructors) One must realize the additional responsi-

bility which this two-part program placed upon the participants. First, attendance at the six-week institute at the University of Vermont demanded diligent and intensive study. Second, the in-service programs added more to already heavy teaching and administrative responsibilities. How did the participants react to the in- service phase of the program? Here are some of their remarks:

"Quite a task, but I've never enjoyed doing anything so much . . . gives you such a professional lift."

"I would like to share with you the tremendous feeling of satisfaction that I have as a result of the first meeting. . . ." (In correspondence sent to the authors.)

"The classes require so much preparation - hours! But, I am enjoying every minute."

Summary A tentative summary was made of

twenty-three in-service programs which were conducted during the first half of the school year. Some of the programs were not scheduled to begin until February; other programs had to be postponed or were not conducted for various reasons.

This summary indicated that more than eight hundred elementary school personnel were directly involved in the in-service programs. These eight hundred people are teaching mathematics to approximately twenty thousand elementary school children.

416 The Arithmetic Teacher



Conclusions

1 A new elementary school mathematics curriculum is emerging which necessi- tates a re-education in mathematics for most elementary school personnel.

2 The retraining program in mathematics for elementary school personnel is of such magnitude that many programs need to be planned.

3 Colleges and universities are helping to meet this problem of re-education by offering special courses. This service, however, is limited because these in- stitutions have their own problems of new curricula, increased enrollment, and limited number of staff members.

4 In-service programs in mathematics are effective in helping teachers acquire sufficient background in mathematics to work effectively with the emerging elementary school mathematics programs.

5 An excellent source of leadership and

personnel to carry out in-service re- education in mathematics is the elementary school teacher, supervisor, or administrator.

6 Universities and public school systems can work cooperatively in helping in- service teachers in their efforts toward re-education in mathematics: the universities by conducting summer institutes and the public school systems by encouraging and supporting the leadership of well-qualified teachers, supervisors, and administrators.

7 After working with the participants in the summer institute and during the follow-up program, the authors must conclude that the objectives which un- derlie the "re volution in school mathematics" will be attained. Elementary school personnel of the caliber of those who participated in this two-phase program will make certain of it!

An easier way to check long division At the present time in school, long division is checked the way shown below:

Division Check 1072 1072

61 ) 65421 §i 61 1072 ~442 6432 427 65392 ~TEl + ?9_ 122 65421 check 29

While this method provides a good way to check long division, it leaves much room for a mistake. Also, this check sometimes requires as much time as the example itself, which could prove costly in an exam or other paper.

For these reasons, I would like to propose an- other method which is superior (in most cases) to the current method, in which the division example actually checks itself!

To check by the new method (using the same example), we simply circle all numbers above the lines and add them in the columns that they are in, adding the remainder if there is any:

Division and Check 61 ) 65421

~Ï42 (327)

151

~29 65421 check

The reason that this method works is because multiplication (as used in the current method) is actually an addition anyway, just as 3X2 is the same as 2+2+2. - Thomas Sahagian, 6th grade, Carter School, Needham, Massachusetts.

October 1964 417



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Re-education in mathematics for elementary school personnel: in-service programs—one way to solve the problem