Pre-service and in-service education of elementary school teachers in arithmetic

Pre-service and in-service education of elementary school teachers in arithmeticAuthor(s): WILBUR H. DUTTON and AUGUSTINE P. CHENEYSource: The Arithmetic Teacher, Vol. 11, No. 3 (March 1964), pp. 192-198Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187016 .

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Pre-service and in-service education of elementary school teachers in arithmetic

WILBUR H. DUTTON University of California, Los Angeles AUGUSTINE P. CHENEY Littleton Public Schools, Littleton, Colorado

Professor Dutton is associate director of teacher training at Los Angeles. Miss Cheney is an elementary school teacher in Littleton^ Colorado.

J? or several decades leading educators and research specialists have emphasized meaning and understanding as vital aspects in the teaching of arithmetic. While these theories of learning were expressed clearly by Brownell [l],* McConnell [4], Sauble [6], and Wheeler [7] as early as 1940, it has taken almost twenty years for any appreciable change to be made in methods of teaching. During the past five years forces have been at work which have increased the emphasis upon arithmetic and mathematics to the place where a revolution is now in progress. Classroom practices have not caught up with the recommendations made in 1941 by the National Council of Teachers of Mathe- matics. Now, new mathematical systems, emphasis upon the structure of arithmetic, learning by discovery, provision for individual differences, and use of a variety of teaching aids have become the leading topics of discussion among educators, mathematicians, and psychologists. The time for clarifying and unifying these various elements into a practical plan for the teaching of arithmetic has arrived.

The Problem The purpose of this article is twofold:

(1) to report the writers' efforts to con-

* The numbers in brackets refer to References at the end of this article.

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struct instruments for the measurement of elementary school teachers' understanding of basic arithmetical concepts, and (2) to explore the results of three studies in which these tests were used to obtain possible clues for planning in-service education of elementary school teachers.

Construction of tests for measurement of teachers' understanding of arithmetic Two tests were used in the studies re-

ported in this article: (1) a test of arithmetical concepts taught at the sixth-grade level which was prepared by Dutton, and (2) an instrument designed by Cheney to measure teachers7 understanding of arithmetical concepts taught in the total elementary school program.

Arithmetic comprehension test for Grade 6 Dutton's University of California Arith-

metic Test for sixth grade contained 46 test items with four or five multiple-choice answers to word problems, diagrams, and drawings designed to elicit the studenťs understanding of arithmetical concepts.

Test specifications. An analysis of sixth- grade arithmetic textbooks was made to determine the main arithmetic concepts taught at this grade level. This list was then presented for additions and correc-

The Arithmetic Teacher



tions to a select group of sixth-grade supervising teachers in elementary schools near the university. Topics included place value, whole numbers, fractions, decimals, Roman numerals, percentage, and measurement.

Preparation of test items. Over 100 test items (at least two for each concept being measured) were constructed. A group of graduate students working in the field of arithmetic and a group of supervising teachers studied these items to see if each item tested comprehension rather than computation and to locate ambiguities. After revision, the items were tested with 200 sixth-grade children for problems in reading, ambiguity, difficulty, and test administration. A second tryout of the test was made with 500 children enrolled in 14 schools in the Los Angeles area. The biserial coefficient was used to obtain an index of difficulty and discrimination for each item.

The reliability of the test was determined by using the Kuder-Richardson formula. The coefficient of reliability was .89 with a probable error of measurement of approximately 2 raw score points.

The validity of the test was based upon a content analysis of sixth-grade arithmetic textbooks, an item analysis by experienced teachers, and the correlation of test scores with pupils' classroom per- formances.

Arithmetic comprehension test for teachers

The second test, prepared by Cheney, was constructed according to the procedures just described for Test 1. Careful analysis was made of the arithmetical concepts elementary school teachers are expected to understand and teach in Grades 1 through 6.

Test Items. Cheney used 46 test items from the Button test and constructed an additional 58 items. The total of 106 items was studied by a "jury" of experienced teachers, Cheney, and two university professors working in the field of elemen-

ti arch 1964

tary school arithmetic. The instrument was then administered to 200 prospective and regular elementary school teachers. As a result of this tryout, the test was re- duced to 52 items - 30 from the Dutton test and 22 prepared by Cheney.

The reliability was tested with the Kuder-Richardson formula. The coefficient of reliability was .88 with a probable error of measurement of approximately 2 raw score points.

Validity for the test was based upon a content analysis of 63 elementary arithmetic textbooks, an item analysis by experienced teachers, and the judgments of a "jury" of experts in the teaching of arithmetic.

Subjects. Cheney [2] tested 120 elementary school teachers in eight different schools in two large counties in the Los Angeles area. Of this number, 45 were primary-grade teachers (K-3), and 57 were intermediate-grade teachers (4-6); 18 did not designate the grade level at which they taught.

Dutton measured 55 students enrolled in two sections of a lower division mathematics course for prospective elementary school teachers and 79 students enrolled in two sections of an upper division arithmetic methods course given at the University of California, Los Angeles.

Findings. Results of these studies will be reported under the following headings : (1) analysis of experienced elementary school teachers' understanding of basic arithmetical concepts, (2) results of lower- division students' comprehension of arithmetical concepts, and (3) upper-division students' understanding of arithmetical concepts.

Experienced elementary school teachers9 understanding of arithmetical processes Tests were scored one point for each

correct answer. The total possible score was 52 points. The mean score for 120 elementary school teachers was 34.57, and standard deviation was 8.602.

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The 45 primary-grade teachers (K-3) in the group had a mean score of 29.91. The 57 intermediate-grade teachers scored a mean of 38.74. A ¿-test computed for the difference between these means was significant well beyond the .01 level. Failure to use many of the advanced arithmetical concepts, work with primary-grade concepts, and differences in preparation ac- counted for the low scores of primary teachers. Since 18 teachers did not in- dicate a grade-level preference on the questionnaire part of the test, only 92 teachers' scores were used in this part of the study.

An item analysis was made of the incorrect responses for the 120 teachers who took the Cheney comprehension test. (See Table 1.)

Table 1 should be read as follows: In the first main division, place value, 50 percent of the teachers did not understand place value or the uses of zero as measured by this test. A summary of the main difficulties for each major section of the test follows. 1 In work with whole numbers, 46 percent

did not recognize the value of the "one" carried as one hundred, 54 percent did not understand the value of a partial

Table 1 Arithmetical concepts measured by Cheney comprehension test and percent of incorrect responses for each item by 120 elementary school teachers

Concept Percent Concept Percent measured incorrect measured incorrect

Place value Decimals Recognition of 5 Size of decimals .33 in thousands' place .08 Recognizing 5 hundredths .20 What 267 stands for Reading ten and one-tenth . 39 in 4,267, 917, 811 .12 A number less than 2.02 . 20 Base of our number .50 Reading .04000 .48 Uses of zero . 50 Placing decimal in add. . 67 Rounding off to nearest Placing decimal hundred-thousand .27 in multiplication .50

Moving decimal in division . 48 Whole Numbers Comparing decimal fractions .48

Number carried (hundred) add. .46 Number carried add . 15 Roman numerals Number borrowed subt. . 07 Meaning of XL VI .31 Partial product in tens' place . 54 Meaning * of CM .33 Partial product 40 X 50 .32 * Place value of quotient figure .50 P t Remainder in division .53 P

е1ло- t

i_ A x í»iaao on $93 is what i_ A percent x of í»iaao $100? .22 on

Fractions 2 is what Percent of 7? -40

Recognition of a group i . 24 Figure showing } of i .10 Measurement Adding unlike fractions . 12 Meaning of gross .32 Subtracting unlike fractions . 10 Meaning of ream . 50 Multiplying unlike fractions .33 Length of a centimeter .42 Value of iXS .15 Subtracting feet and inches .44 Multiplying 2 X Ц !07 Formula D=r+ .22 Multiplying i XÍ .48 Perimeter of rectangle . 23 Dividing f by i .31 Application of area . 14 Dividing 2f by i .20 Meaning of square measure .27 Picturing division Meaning of volume .33 of fractions . 50 Volume of rectangle . 48 Decreasing the numerator . 33 Decimal equivalent . 32 Time

Subtracting hours, minutes, Liquids and seconds .42

Use of liquid measures .17 Use of time zones .47

194 The Arithmetic Teacher



product or how it was obtained, 50 percent did not understand place value in quotient figures, and 53 percent thought the remainder in division must always be placed in the quotient as a fraction.

2 Fractions were difficult for many teachers. There were 33 percent who did not understand multiplying unlike fractions and 48 percent who had difficulty multiplying f X|. Division of fractions caused difficulty for 21 percent of the teachers. When asked to draw a picture showing 3 divided by l£, 50 percent could not do the work.

3 Reading a large decimal was difficult for 48 percent of the teachers. Other difficulties centered around placing the decimal in adding (67 percent), moving the decimal in multiplication (50 percent), and moving the decimal in division or comparing decimal fractions (48 percent).

4 Approximately one-third of the teachers tested did not know the meaning of the Roman numerals XLVI or CM.

5 Work with percent is limited in sixth grade, so only the simple aspects were presented. Forty percent of the teachers did not know the answer to "2 is what percent of 7?"

6 The major difficulties in measurement centered around meaning of gross (32 percent), ream (50 percent), length of centimeter (42 percent), regrouping in subtraction with feet and inches (44 percent), and volume of a rectangle (48 percent).

7 In the subtraction of hours, minutes, and seconds, 42 percent did not understand the regrouping concept. Forty- seven percent could not use the time zones in telling time.

8 While there were many aspects of arithmetic not understood by experienced elementary school teachers, there were specific areas of strength in each area measured by the test.

Cheney concluded that experienced ele-

March 1964

mentary school teachers had not been taught arithmetic by meaningful methods, and were making the adjustment to newer methods of teaching slowly. These teachers need assistance in their specific areas of weakness, a point that will be pursued in the next two sections of this research report.

Lower-division students* comprehension of arithmetical concepts

Only the results of this study will be given, since a detailed account may be found in The Arithmetic Teacher of February, 1961 [3]. The study was the basis for the follow-up investigation of upper-division students reported in the next section.

An Arithmetic Comprehension Test which covers basic concepts pupils are expected to know by the close of sixth grade was administered to 55 lower-division students at the University of California before and after they had completed a re- quired mathematics course for prospective elementary school teachers.

Changes made in students' understanding of arithmetical concepts during one semester were determined through the use of two matched samples and two distribu- tions of scores for the two sections measured. The means for these tests were: Mi = 33.94 and M2 = 38.78. The t was significant at the 1 percent level.

Students in this study still clung to traditional methods when they were asked to explain partial products in multiplication, placement of quotient figures in division, and placement of the decimal point in answers to problems involving decimal fractions. Lack of understanding of denominate numerals seemed closely asso- ciated with the lack of emphasis placed upon these processes in our industrialized society. While students made much progress in this lower-division mathematics course, a systematic approach to eradication of student misunderstandings of arithmetical concepts was recommended.

195



Upper-division students9 understanding of arithmetical concepts The arithmetic comprehension test

prepared by Dutton and discussed earlier in this paper was given to 79 students enrolled in two different sections of an upper- division methods course on the teaching of arithmetic at the University of Cali-

fornia, Los Angeles. The study was limited to women students, since there were few men enrolled in these sections. Students were planning to begin supervised teaching either during the next spring semester or in the fall. Students were tested at the beginning and at the close of the fall semester.

Table 2 Incorrect responses of 79 prospective elementary school teachers on arithmetic comprehension test

Test item Description of J test item ^Г°Пд WrOng of J 1st test 2nd test

1 Number of hundreds in 160 5 0 2 Locating thousands' place 2 0 3 Rounding off 6 0 4 Carrying in addition 7 4 5 Subtracting with regrouping 8 3 6 Understanding large numbers 13 2 7 Liquid measures 18 ] 1 8 Using partial products in multiplication 40 8 9 What does a remainder mean? (division) 40 6

10 Multiplication terms 20 4 1 1 Subtracting with decimal fraction 4 0 12 Place value (10,000) 15 2 13 Using time zones 27 23 14 Moving the decimal in division 20 7 15 Placement of quotient figures 42 11 16 Locating hundreds' place 15 4 17 Number less than 2.02 18 13 18 Number greater than 2.04 3 0 19 Base of our number system 40 3 20 Reading decimal fractions 6 2 21 iX8(show) 16 1 22 3-Mi (diagram) 45 12 23 One- third used in rectangles 20 12 24 Part of rectangle (decimal fraction) 6 0 25 Perimeter 15 7 26 Uses of zero 45 26 27 Proportion of a number 10 6 28 Reducing a fraction 0 0 29 2Í-J-Í (diagram) 40 9 30 Pounds in a ton 14 2 31 Use of "gross" 30 28 32 Reducing fractions 14 2 33 Square measure 18 9 34 Size of decimals (.001) 10 2 35 Meaning of (.10J) 4 0 36 Volume 27 18 37 Addition with decimals 38 25 38 Multiplication with decimals 38 5 39 Use of "ream" 40 27 40 2ХЦ 0 0 41 f-f-i 14 2 42 Use of percent (100 percent) 0 0 43 Applying decimal fractions 12 2 44 ÌXb 8 2 45 Regrouping with denominate numbers 34 20 46 Area in measurement 16 4

!96 The Arithmetic Teacher



Upper-division university students, upon entering a curriculum and methods course dealing with arithmetic, made the same general mistakes on the arithmetic comprehension test as the lower-division students reported on in the last section. The major difficulties encountered by these students were understanding the meaning of multiplication, showing what a remainder means in division, rationaliz- ing the placement of quotient figures, understanding the meaning of base ten, drawing figures to show division of fractions, understanding the meaning of zero, knowing terms such as ream, gross, and ton, understanding the placement of the decimal point in decimal fractions, and regrouping in work with denominate numbers.

The instructor returned students7 test papers and helped each student identify the areas and concepts he did not understand. Teaching procedures centered around building a background for the understanding of crucial arithmetical concepts taught at each grade level. Stu- dents made lesson plans, taught selected lessons in arithmetic in elementary schools near the university, and participated in an elementary school classroom three hours each week for one semester. At the close of the course the sections were re- tested to see if they had ovecome the difficulties shown on the pretest.

Table 2 shows that students did make significant improvement in their understanding of basic arithmetical concepts. However, approximately one-third of the sample still did not understand the following concepts: use of time zones for telling time while traveling; uses of a zero in our number system; meanings of gross, ton, and ream; reason for the placement of the decimal point in certain positions in work with decimal fractions, and regrouping in the subtraction of denominate numbers. Still another group of students, between 15 to 20 percent, had difficulties with placement of quotient figures, division of fractions, and volume,

March 1964

Table 3 Total scores of prospective elementary school teachers on University of California Arithmetic Comprehension Tests

Test 1 Test 2 Scores

No. of students No. of students

46 0 2 44 2 23 42 4 11 40 16 24 38 10 9 36 7 4 34 14 4 32 12 0 30 4 1 28 2 1 26 2 0 24 1 0 22 0 0 20 3 0 18 0 0 16 2 0

jV = 79 N = 79

Overall gains made in understanding arithmetical concepts are shown in Table 3. On the first test the median was 34.50, the Q3 40.28, and the Q- 32.96. The total possible score on both tests was 46. Gains in total achievement are shown on test 2 where the median was 41.70, the Q3 44.46, and the Qi 40.06. As in the lower-division courses, the progress made by upper- division students is important and significant. However, a systematic, individual- ized approach to eradication of student misunderstandings of arithmetical concepts must be provided at this level of teaching, too. Steps are being taken to provide this kind of instruction for prospective elementary school teachers.

Conclusions

While there must be considerable flex- ibility in curriculum-improvement work and in instructional practices, the need for a multiple approach to the consolida- tion of gains already made in the teaching of arithmetic and the incorporation of new theories seems imperative! At least three interrelated approaches must be pursued

197



with vigor. First, the background mathematics essential in the preparation of teachers of arithmetic must be more rigorous and must reflect modern theories of learning. As shown by Ruddell and others [б], this reform is long overdue in the majority of institutions of higher education. Second, professional educators offering courses in the teaching of arithmetic must accept a dual role; they must join hands with mathematicians to insure educationally sound instruction in pre- paratory mathematics and arithmetic courses, and provide instruction for prospective teachers which is based upon in- dividualized and measurable behavior- centered goals. Third, in-service education must be provided at the local school- district level to insure implementation of district objectives, continuous improvement of tenure staff members, and proper induction of new teachers who come from a wide variety of institutions.

References

1 Brownell, W. A. "Psychological Con- siderations in the Learning and the Teaching of Arithmetic/' The Teaching of Arithmetic, Tenth Yearbook. Wash- ington, D.C. : The National Council of Teachers of Mathematics, 1935, pp. 1-31.

2 Cheney, A. P. "An Evaluation of the Understanding Possessed by Elemen-

tary Arithmetic Teachers of Basic Arithmetic Concepts. " Unpublished Master's Thesis, University of Cali- fornia, Los Angeles, 1961.

3 Dutton, W. H. "University Students' Comprehension of Arithmetical Con- cepts," The Arithmetic Teacher, VIII (February, 1961), 60-64.

4 McConnell, T. R. "Recent Trends in Learning Theory: Their Application to the Psychology of Arithmetic," Arith- metic in General Education, Sixteenth Yearbook. Washington, D.C: The Na- tional Council of Teachers of Mathema- tics, 1941, pp. 268-89.

5 Ruddell, A. K, Dutton, W. H., and Reckzeh, J. "Background Mathematics for Elementary Teachers," Instruction in Arithmetic, Twenty-fifth Yearbook. Washington, D.C. : The National Coun- cil of Teachers of Mathematics, 1960, pp. 296-317.

6 Sauble, Irene. "Enrichment of the Arithmetic Course : Utilizing Supple- mentary Materials and Devices," Arithmetic in General Education, Six- teenth Yearbook. Washington, D.C. : National Council of Teachers of Mathe- matics, 1941, pp. 157-95.

7 Wheeler, R. H. "The New Psychology of Learning," The Teaching of Arith- metic, Tenth Yearbook. Washington, D.C. : The National Council of Teachers of Mathematics, 1935, pp. 233-50.

A request for information The School Mathematics Study Group Panel on Teacher Training Materials is conducting a survey of mathematics in-service programs now in operation throughout our nation.

The survey includes both in-service programs for classroom teachers and programs designed to train individuals to staff in-service programs in both elementary and secondary schools.

198

If you know of any such programs now in use, please send the name and address of the instructor directly responsible for the program to:

Professor John Wagner, Science and Mathematics Teaching

Center, E 37 McDonel Hall, East Lansing, Michigan.

The Arithmetic Teacher



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Pre-service and in-service education of elementary school teachers in arithmetic