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Individualizing instruction in elementary school mathematics for prospective teachersAuthor(s): WILBUR H. DUTTONSource: The Arithmetic Teacher, Vol. 13, No. 3 (MARCH 1966), pp. 227-231Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187201 .

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Individualizing instruction in elementary school mathematics for prospective teachers

WILBUR H. DUTTON University of California, Los Angeles, California Dr. Dutton is professor of elementary education at the University of California.

JL he importance of teachers' understanding of basic arithmetical and mathematical concepts has been studied by numerous research workers [2, 3, 4, 6].* The understanding teachers have of basic mathematical concepts is closely asso- ciated with the ability to present these concepts to children [10 ]. Considerable research has been done to show the amount of understanding of basic mathematical concepts possessed by prospective elementary school teachers. Most of these studies indicate pronounced inadequacies in teacher understanding, while a few [2, 3, 8] show that some aspects of arithmetic and mathematics are understood quite well. Few studies deal with methods and procedures for overcoming teachers' lack of understanding of these basic concepts.

The problem This study dealt with the individual-

izing of instruction in elementary school mathematics for prospective teachers who were enrolled in an upper division education course. This study was based upon three hypotheses: 1 Students enrolled in a university course

dealing with the teaching of elementary school mathematics would master the new mathematical concepts emphasized in modern programs when instruction was individualized and adjusted to their needs. The assumption was based upon the findings of previous studies by

* Numbers in brackets refer to the Bibliography at the end of this article.

March 1966

the writer [2] indicating the inadequacy of teaching procedures based upon lec- ture discussion, and group assignments.

2 Since concepts involved in understanding division of fractions seemed to be dependent upon the recall of subordinate knowledge (especially multiplication of fractions), the use of programmed materials for careful review and reteaching of subordinate knowledge would enable students to master basic concepts involved in division of fractions. This assumption was sup- ported by the writer's experience in earlier studies and by the research of Gagne [5].

3 There would be significant gains (.01 percent level of significance) in mean scores on a test measuring understanding of basic mathematical concepts given at the beginning and at the close of the instructional period.

Procedures and subjects Two sections (iV = 80) of an upper-

division methods class given at the Uni- versity of California, Los Angeles, were used in this study. Students had taken one or more years of high school algebra and geometry. All had completed a one- semester, lower-division mathematics course (3 units) dealing with the funda- mentals of arithmetic which was taught in the mathematics department. There were 4 male and 76 female students. Most students would enroll in supervised teaching during spring or fall semester, 1965.

227



Students were assigned to one elementary school near the university for three hours of participation, including the teaching of selected lessons, each week.

An Arithmetic Concept Test containing 63 multiple choice, 2 demonstration-type, and 13 true-false test items was given to the students before they began work in the course and after they had completed five weeks of intensive study on arithmetic concepts. The arithmetic comprehension test used had a reliability coeffi- cient of .89 and covered basic arithmetical and mathematical concepts that elementary school teachers must know to teach sixth grade. The test measured student understanding and did not deal with computational processes. Emphasis was placed upon meaningful understanding and application of mathematical concepts.

Tests were scored, and an answer sheet containing concepts missed was returned to each student. For example, one test item asked, "One is how many times as large as .001?" The answer sheet would show that a student having difficulty with this test item needed work on understanding of decimal place value. There were 60 mathematical concept areas (rather than specific answers) which were given to each student for additional study. The following suggestions were given: 1 Locate problems missed. Use your text

to study the process or concept. 2 Ask questions when the Problem Area is

presented in class. 3 Try teaching a new concept or difficult

step to a colleague. 4 Talk to another student about an area

where you are having difficulty. Seek active help until you understand.

5 Ask for assistance from the instructor during lab periods, or secure an appoint- ment.

6 Discover how the process or concept is taught in the school where you are participating.

Students were encouraged to overcome

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their difficulties in understanding basic mathematical concepts and were told they would be tested on these concepts at the close of the semester.

Instruction centered around the following procedures: brief class lectures to provide depth of background for major mathematical concepts studied; selected film strips with sound recordings for demonstration lessons; visitation in an elementary school near the university and the use of closed circuit television for demonstration lessons; specific assignments in new textbooks adopted for state use - Greater Cleveland Mathematics Program, Grades K-3, and Silver Burdett Arith- metic Series, Grades 4-6; assigned study of a set of three chapters of programmed instruction covering a review of multiplication of fractions and the teaching of division with fractions, including division of a fraction by a fraction, e.g., (§ -s-D ; the preparation of selected lesson plans, and opportunities to work with the instructor or his teaching assistant in a curriculum laboratory where new textbooks, programmed materials, and instructional aids were kept. The individualizing of instruction centered around each student's identifying his areas of weakness and re- ceiving directed guidance in overcoming these difficulties. Six students with pretest scores of 90-97 percent correct responses were given the opportunity to do enrich- ment work and to study research reports on elementary school mathematics.

Findings Results of this study will be reported

under the following headings: (1) mastery of mathematical concepts, (2) programmed instruction and concept understanding, and (3) significance of gains in understanding of basic mathematical concepts between pre- and posttests.

In the table the total scores on pre- and posttests have been tabulated on a fre- quency distribution. On the pretest correct responses ranged from 37 percent to 96 percent, with a median of 77.00. After

The Arithmetic Teacher



Distribution of student scores on two mathematical concepts tests

98-100 6 94-97 2 20 90-93 3 26 86-89 5 17 82-85 17 6 78-81 13 1 74-77 8 0 70-73 6 3 66-69 6 1 62-65 4 - 58-61 4 N = 80 54-57 6 Af2 = 91.84 50-53 4 46-49 1 42-45 0 38-41 1

tf=80 Mi = 77.00

five weeks of instruction the test was re- peated, and scores ranged from 66 percent to 100 percent correct responses with a median of 91.84. There were 69 students out of 80 who had 86 percent correct responses or above on the posttest.

There were 52 concept areas which were mastered by all students. Eight specific concepts caused difficulty for slightly more than one-third of the students participating in this study. The concepts (drawn from five concept areas) which caused difficulty were using the language of sets to show union or intersection; changing numerals from base ten to another base; rationalizing the meaning of f X6; place value in division with whole numbers, and rationalizing the meaning of

Programmed instruction on fractions

All students in this study were required to read a series of three chapters of programmed instruction on division of fractions. A scrambled-book plan was used which required the student to read and then select a response to questions from three or more multiple-choice statements. Correct responses were rewarded by allow- ing students to advance through the ma-

March 1966

terial. Incorrect responses required re- reading and additional study.

After completing this program, students were able to answer those test items re- quiring an understanding of basic concepts involved in multiplication and division of fractions. However, even though students achieved mastery of the concepts called for in the program, about one-third of the students encountered difficulty with the rationalization of two problems which asked for the application of newly learned concepts: (1) showing the meaning of f X6, and (2) demonstrating on a number line the meaning of Ц+h There was sufficient evidence to indicate that students were not thorough in their study of the programmed materials and that more branching and additional learnings on these two concepts should be built into the program. The importance of the recall of subordinate knowledge and readiness for new learnings was especially crucial for these two concepts. Additional atten- tion must be given to testing for and secur- ing mastery of subordinate knowledge as an integral aspect of readiness for the new learnings.

Statistical treatment of pretests and posttests Two matched samples were used in this

study. Each studenťs score on the pretest was compared with his score on the posttest. The means for the tests were Mi = 74.11 and M 2 = 90.80. To determine whether the difference between the means was significant, the direct-difference method was used [9]. °D was obtained by computing the standard error of the mean difference {aMD). Letting D stand for difference, the SD formula was:

* N

/23094.16 , ~

'D=V ~~ÍÕ ( , }

"I>='/288.68-180.63 'D= 10.39

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Then *MD was computed by the following method:

■MD-vw=r •MD-2JL

V80-1

8.89 °MD = 1.17

The ¿-test for matched groups was

MD °MD 13.44

t == - 1.17

¿=11.49

With matched groups the number of de- grees of freedom for evaluating the t is N- 1, where N is the number of pairs of subjects. With 79 d/, this t is significant at the 1 percent level; the conclusion is that the instructional program, as measured by the test, was effective.

Conclusions

Prospective elementary school teachers in this study made marked progress in the mastery of mathematical concepts emphasized in modern programs when instruction was individualized and adjusted to their needs. Students showed understanding of the basic mathematical concepts in 52 out of 60 concept areas. Specific difficulties encountered by students in five concept areas were identified, and inadequacies in instruction and learning noted.

The use of programmed instructional materials for the difficult concepts involved in understanding multiplication and division of fractions proved helpful to about two-thirds of the students. The importance of the recall and use of subordinate knowledge and skills as impor- tant factors in readiness for and under-

230

standing of new learnings was substan- tiated.

The direct-difference method was used to determine whether the difference between the means on pre- and posttests was significant. The conclusion was that the instructional program, as measured by the test, was effective, and the difference between the means was significant at the .01 level.

Implications On the basis of the writer's experiences

with this study, several implications for college teaching of prospective elementary school teachers seem worth noting: 1 The learning of basic mathematical

concepts requires intensive study, a variety of instructional experiences, subordinate knowledge and skills, and opportunity for individual exploration and development.

2 While an individualized approach to the teaching of mathematical concepts has definite advantages, the present en- vironment of university students and the conditioning influences of past edu- cational experiences seem to necessitate considerable teacher guidance and required work to secure continuous student involvement in the learning process. There is a definite tendency for students to work on those courses where assignments are given and due-dates established. Students tend to neglect or postpone work in classes where the individual is given freedom for exploration and individual study.

3 The use of programmed instructional materials seems to provide numerous opportunities to diagnose students' subordinate knowledge and skills essen- tial for sound sequential learning and expansion of mathematical concepts. These materials seem to provide ap- propriate experiences for individualized instruction as well as supplementary learning experiences. Other communica- tions media, such as teaching machines, listening and viewing centers, and

The Arithmetic Teacher



video tapes of demonstration lessons must be explored as means of preparing elementary school teachers to understand and appreciate sound mathematical learnings.

4 This study reinforces the findings of the writer [2, 3] about the importance of identification of individual learning difficulties involved in understanding basic mathematical concepts. The time and effort required to learn these basic concepts or to overcome difficulties in- volving understanding of concepts must not be minimized or overlooked by college or university teachers. The assumption that brief, in-service classes or short, intensive courses (both pre- service and in-service) will enable elementary teachers to understand and to participate in meaningful teaching of the new mathematics must be scrupu- lously challenged.

Bibliography 1 Brownell, William A. " Arithmetic

Readiness As a Practical Classroom Concept/' Elementary School Journal, LII (September, 1951), 15-22.

2 Dutton, Wilbur H. "University Stu- dents' Comprehension of Arithmetical Concepts/' The Arithmetic Teacher, VIII (February, 1961), 60-64.

3 ^ an(j Cheney, Augustine P. "Pre-service and In-service Education of Elementary School Teachers

in Arithmetic/' The Arithmetic Teacher, XI (March, 1964), 192-198.

4 Fulkerson, Elbert. "How Well Do 158 Prospective Elementary Teachers Know Arithmetic?" The Arithmetic Teacher, VII (March, 1960), 141- 146.

5 Gagne, R. M. "Learning and Pro- ficiency in Mathematics," The Math- ematics Teacher, LVI (December, 1963), 620-626.

6 Harper, E. Harold. "Elementary Teachers' Knowledge of Basic Arith- metic Concepts and Symbols," The Arithmetic Teacher, XI (December, 1964), 543-546.

7 Melson, Ruth. "How Well Are Col- leges Preparing Teachers for Modern Mathematics?" The Arithmetic Teacher, XII (January, 1965), 51-53.

8 Nelson, L. Doyal, and Worth, Walter H. "Mathematical Compe- tence of Prospective Elementary Teachers in Canada and in the United States," The Arithmetic Teacher, VIII (April, 1961), 147-150.

9 Stoneking, Lewis W., and Welch, Ronald С "Teachers' and Students' Understandings of Arithmetic," Bul- letin of the School of Education, XXXVII. Bloomington, Ind. : Indiana University, September, 1961.

10 Underwood, B. J., et al. Elementary Statistics. New York: Appleton-Cen- tury-Crofts, Inc., 1954, p. 169.

Solving problems is a practical art, like swimming, or skiing, or playing the piano : you can learn it only by imitation and practice. If you wish to learn swimming you have to go into the water, and if you wish to become a problem solver you have to solve problems. - George Polya.

March 1966 231

