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<ul><li><p>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 .Accessed: 17/06/2014 07:09</p><p>Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp</p><p> .JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.</p><p> .</p><p>National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.</p><p>http://www.jstor.org </p><p>This content downloaded from 195.34.79.79 on Tue, 17 Jun 2014 07:09:03 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/action/showPublisher?publisherCode=nctmhttp://www.jstor.org/stable/41187201?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>Individualizing instruction in elementary school mathematics for prospective teachers </p><p>WILBUR H. DUTTON University of California, Los Angeles, California Dr. Dutton is professor of elementary education at the University of California. </p><p>JL he importance of teachers' understand- ing of basic arithmetical and mathemat- ical concepts has been studied by nu- merous 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 mathe- matical 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. </p><p>The problem This study dealt with the individual- </p><p>izing of instruction in elementary school mathematics for prospective teachers who were enrolled in an upper division educa- tion course. This study was based upon three hypotheses: 1 Students enrolled in a university course </p><p>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 </p><p>* Numbers in brackets refer to the Bibliography at the end of this article. </p><p>March 1966 </p><p>the writer [2] indicating the inadequacy of teaching procedures based upon lec- ture discussion, and group assignments. </p><p>2 Since concepts involved in understand- ing division of fractions seemed to be dependent upon the recall of subordi- nate knowledge (especially multiplica- tion of fractions), the use of pro- grammed materials for careful review and reteaching of subordinate knowl- edge 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]. </p><p>3 There would be significant gains (.01 percent level of significance) in mean scores on a test measuring understand- ing of basic mathematical concepts given at the beginning and at the close of the instructional period. </p><p>Procedures and subjects Two sections (iV = 80) of an upper- </p><p>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 teach- ing during spring or fall semester, 1965. </p><p>227 </p><p>This content downloaded from 195.34.79.79 on Tue, 17 Jun 2014 07:09:03 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>Students were assigned to one elementary school near the university for three hours of participation, including the teaching of selected lessons, each week. </p><p>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 arith- metic concepts. The arithmetic compre- hension test used had a reliability coeffi- cient of .89 and covered basic arithmetical and mathematical concepts that ele- mentary 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 understand- ing and application of mathematical con- cepts. </p><p>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 understand- ing of decimal place value. There were 60 mathematical concept areas (rather than specific answers) which were given to each student for additional study. The follow- ing suggestions were given: 1 Locate problems missed. Use your text </p><p>to study the process or concept. 2 Ask questions when the Problem Area is </p><p>presented in class. 3 Try teaching a new concept or difficult </p><p>step to a colleague. 4 Talk to another student about an area </p><p>where you are having difficulty. Seek active help until you understand. </p><p>5 Ask for assistance from the instructor during lab periods, or secure an appoint- ment. </p><p>6 Discover how the process or concept is taught in the school where you are participating. </p><p>Students were encouraged to overcome </p><p>228 </p><p>their difficulties in understanding basic mathematical concepts and were told they would be tested on these concepts at the close of the semester. </p><p>Instruction centered around the follow- ing procedures: brief class lectures to pro- vide depth of background for major mathematical concepts studied; selected film strips with sound recordings for dem- onstration lessons; visitation in an ele- mentary school near the university and the use of closed circuit television for dem- onstration 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 multipli- cation 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, pro- grammed materials, and instructional aids were kept. The individualizing of instruc- tion 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. </p><p>Findings Results of this study will be reported </p><p>under the following headings: (1) mastery of mathematical concepts, (2) pro- grammed instruction and concept under- standing, and (3) significance of gains in understanding of basic mathematical con- cepts between pre- and posttests. </p><p>In the table the total scores on pre- and posttests have been tabulated on a fre- quency distribution. On the pretest cor- rect responses ranged from 37 percent to 96 percent, with a median of 77.00. After </p><p>The Arithmetic Teacher </p><p>This content downloaded from 195.34.79.79 on Tue, 17 Jun 2014 07:09:03 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>Distribution of student scores on two mathematical concepts tests </p><p>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 </p><p>tf=80 Mi = 77.00 </p><p>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 re- sponses or above on the posttest. </p><p>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 par- ticipating 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 an- other base; rationalizing the meaning of f X6; place value in division with whole numbers, and rationalizing the meaning of </p><p>Programmed instruction on fractions </p><p>All students in this study were required to read a series of three chapters of pro- grammed instruction on division of frac- tions. 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- </p><p>March 1966 </p><p>terial. Incorrect responses required re- reading and additional study. </p><p>After completing this program, students were able to answer those test items re- quiring an understanding of basic con- cepts involved in multiplication and divi- sion 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 +h There was sufficient evidence to indicate that stu- dents 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. </p><p>Statistical treatment of pretests and posttests Two matched samples were used in this </p><p>study. Each studens score on the pre- test 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: </p><p>* N </p><p>/23094.16 , ~ </p><p>'D=V ~~ ( , } </p><p>"I>='/288.68-180.63 'D= 10.39 </p><p>229 </p><p>This content downloaded from 195.34.79.79 on Tue, 17 Jun 2014 07:09:03 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>Then *MD was computed by the follow- ing method: </p><p>MD-vw=r MD-2JL </p><p>V80-1 </p><p>8.89 MD = 1.17 </p><p>The -test for matched groups was </p><p>MD MD 13.44 </p><p>t == - 1.17 </p><p>=11.49 </p><p>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. </p><p>Conclusions </p><p>Prospective elementary school teachers in this study made marked progress in the mastery of mathematical concepts empha- sized in modern programs when instruc- tion was individualized and adjusted to their needs. Students showed understand- ing of the basic mathematical concepts in 52 out of 60 concept areas. Specific diffi- culties encountered by students in five concept areas were identified, and in- adequacies in instruction and learning noted. </p><p>The use of programmed instructional materials for the difficult concepts in- volved in understanding multiplication and division of fractions proved helpful to about two-thirds of the students. The importance of the recall and use of sub- ordinate knowledge and skills as impor- tant factors in readiness for and under- </p><p>230 </p><p>standing of new learnings was substan- tiated. </p><p>The direct-difference method was used to determine whether the difference be- tween the means on pre- and posttests was significant. The conclusion was that the instructional program, as measured by the test, was effective, and the differ- ence between the means was significant at the .01 level. </p><p>Implications On the basis of the writer's experiences </p><p>with this study, several implications for college teaching of prospective elementary school teachers seem worth noting: 1 The learning of basic mathematical </p><p>concepts requires intensive study, a variety of instructional experiences, subordinate knowledge and skills, and opportunity for individual exploration and development. </p><p>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 re- quired work to secure continuous stu- dent involvement in the learning pro- cess. 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 in- dividual is given freedom for explora- tion and individual study. </p><p>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 </p><p>The Arithmetic Teacher </p><p>This content downloaded from 195.34.79.79 on Tue, 17 Jun 2014 07:09:03 AMAll use subject to JSTOR Terms and Conditions</p><p>http://www.jstor.org/page/info/about/policies/terms.jsp</p></li><li><p>video tapes of demonstration lessons must be explored as means of preparing elementary school...</p></li></ul>