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<ul><li><p>Individualization in a Mathematics Coursefor Prospective Elementary</p><p>School TeachersA. Matthew Bazik</p><p>Associate Professor of MathematicsElmhurst College</p><p>Elmhurst, Illinois 60126</p><p>The recognition of individual differences and individualized instruc-tion has been the concern of educators for centuries. Today the use of in-dividualization and individualized instruction are widely advocated, es-pecially in the area of school mathematics.</p><p>If programs of individualization are to be effectively implemented inthe elementary school, it is indeed reasonable that such procedures bepracticed in the instruction of students preparing to teach at that level.This and the fact that prospective teachers enter the elementary curricu-lum with backgrounds ranging from one year of questionable-qualityhigh school mathematics to four years of high-level high school mathe-matics have been the underlying rationale for instituting a mathematicscourse based on individualized instruction described here. This coursewas patterned after the Committee on the Undergraduate Program inMathematics (CUPM), Level I recommendation "(A) a two-course se-quence devoted to the structure of the real number system and its subsys-tem."1 Because the program consisted of a sequence of only two two-hour courses of mathematical content, the above CUPM recommenda-tion could only partially be implemented.</p><p>In an attempt to define what individualized instruction would meanfor this particular course, it was deemed desirable to study the literatureon such procedures for the purpose of establishing an instructional for-mat which would be most beneficial to the audience for which it was in-tended. At one end of the spectrum, Suppes2 advocates the use of thecomputer in the role of a tutor for adapting an education curriculum toindividual learners. While DeVault and Krierwall3 view mathematics as ahighly individual matter, they also recognize that one-to-one tutoringposes economic difficulties and has an inherent lack of interactionamong peers. Among others, DeHann and Doll4 recognize the viabilityof the group concept for individualization and that such programs mustprovide a variety of approaches.</p><p>1. Committee on the Undergraduate Program in Mathematics, Course Guide/or the Training of Teachers of Elemen-tary School Mathematics (Berkeley, Calif.: The Committee, Mathematical Association of America, 1968), p.l.</p><p>2. Patrick Suppes, "The Use of Computers in Education," Scientific American,CCXV (September, 1966), 207.3. M. V. DeVault and T. Krierwall, "Differentiation of Mathematics Instruction," Mathematics Education, Sixty-</p><p>Ninth Yearbook of the National Society for the Study ofEducation, Part I (Chicago: University of Chicago Press, 1970),408.</p><p>4. R. F. DeHann and R. C. Doll, "Individualization and Human Potential," Individualizing Instruction, Yearbook ofthe Association for Supervision and Curriculum Development (Washington, D.C.: The Association, 1964), p. 18.</p><p>241</p></li><li><p>242 School Science and Mathematics</p><p>After careful consideration of the various conceptions of individual-ized instruction described in the literature and from consideration ofsuch practices in the local community where elementary education ma-jors frequently observe individualized programs, the concept of individu-alized instruction for this course was formulated. Here it refers to a planof instruction in which students are permitted to work at their own paceon tasks prescribed in written form by the instructor.A central concept in the plan of individualized instruction is referred</p><p>to above by the words "tasks prescribed in written form by the instruc-tor." Another phrase which can be substituted for these written tasks is"behavioral objectives." Again the literature was appealed to for a pre-cise definition of this term. Tyier5 defines objectives as a device used toclarify the kind of behavior which the course should help to developamong students. Bloom6 specifically states that objectives classify howthe students are to act, think, or feel as the result of participating in someunit of instruction. Consistent with these views, the term "behavioral ob-jectives" for this instructional format referred to statements intended toreliably communicate to the learner what he should be able to do when hehad successfully completed the learning exercise.Once clear definitions of individualized instruction and behavioral ob-</p><p>jectives had been established, the Spring Term, 1970-71, and the 1971Summer Session were spent in preliminary testing of sets of behavioralobjectives which had been written as supplemental to a traditional lec-ture-discussion classroom setting. In addition, supplementary material inthe form of programmed exercises or additional exposition to augmentthe text were provided with some of the behavioral objectives. The stu-dents reaction to the behavioral objectives and supplementary materialswas generally favorable.On the basis of the experience with the development of behavioral ob-</p><p>jectives, self-evaluation exercises, and supplementary problems duringthe time period described above, similar materials were written for use ina mathematics course offered in the 1971-72 Fall Term for the prepara-tion of prospective elementary teachers. The materials included: state-ments of behavioral objectives for each of five units of mathematicalcontent along with suggested problem assignments in the required text-book adopted for the course, suggested readings from books maintainedby the instructor in a Textbook Reference Library, self-evaluation exer-cises which were keyed to specific behavioral objectives, and answers toself-evaluation exercises.At the first class meeting of the term, two mathematics tests were ad-</p><p>ministered along with the Dutton Attitude Scale and a questionnaire to5. R. W. Tyier, Constructing Achievement Tests (Columbus: Ohio State University, 1934), p. 18.6. B. S. Bloom, ed.. Taxonomy ofEducational Objectives, Handbook!: Cognitive Domain (New York: David McKay</p><p>Company, Inc., 1956), p. 12.</p></li><li><p>Individualization in Mathematics 243</p><p>ascertain the high school mathematics backgrounds of the students. Onthe basis of these initial measures, the members of the class were dividedinto four groups, three of which met twice weekly during the regularlyscheduled class period. Group membership was determined through ashort conference between the student and the instructor. One group wasformed to consist of students with weak backgrounds, usually havingtaken two or fewer years of high school mathematics with average or be-low-average grades and with little confidence in their mathematical abil-ity. Members of the second group had generally done average work intwo, and in a few cases three, years of high school mathematics and ap-peared to possess a moderate level of confidence in their ability to domathematics. Members of the third group usually had backgrounds ofthree years of high school mathematics and a moderate to strong level ofconfidence in their mathematical ability. Students attitudes towardmathematics were reflected by some as they chose to begin in a grouplower than their record of high school mathematics would indicate. For asmall number of students with exceptionally strong high school recordsin mathematics, it was decided that they would fulfill the course require-ments working on an independent-study basis.Classroom instruction for the group was loosely structured and varied</p><p>daily according to the expressed needs of the students. Often there weremore than three groups working in the classroom as students with similardifficulties left the original groups to seek additional help from the in-structor. Still others who progressed more rapidly joined a higher groupor worked for a time on a level different from those of the existinggroups.</p><p>Class time was utilized in the following activities: providing each ofthese groups with an introduction to and an explanation of the objectivesto be attained before it embarked on a new topic, further explanationand group discussion of topics introduced earlier, working with specificproblems within individual groups, and for quizzes taken as evaluationpoints were reached. Board work by the students was continually used todiagnose and correct student difficulties and misconceptions. Studentswith average or weak backgrounds or confidence in their ability to domathematics were discouraged from trying to work independently. Be-fore embarking on this plan of instruction, it was recognized that work-ing at ones own pace could be academically disasterous if it were inter-preted to mean studying independently or alone, except in the case of stu-dents with exceptional high school mathematics backgrounds. An effortwas made to continually adhere to this principle throughout the course.The exceptional students who worked independently met with the in-structor weekly for a short time (usually 10 to 15 minutes) for the pur-pose of resolving any problems encountered during the previous week be-</p></li><li><p>244 School Science and Mathematics</p><p>fore taking quizzes.Before taking quizzes students were encouraged to answer the appro-</p><p>priate self-evaluation exercises as a sample of the level of difficulty whichthey would encounter on quizzes and tests and as a diagnostic tool foreach quiz. Although the purpose of the material provided for each unitwas to ensure a high level of performance on the quizzes, students werenot held back from pursuing a later unit simply because they were unableto attain a prescribed proficiency level over material which was particu-lary difficult for them. Often when students pursued a subsequent unit,they found that they could then more easily understand the material ofthe preceding unit. It was for this reason that the objectives did not speci-fy a minimum level of achievement before students were allowed to beginwork on new material. Alternate forms of each quiz were available forstudents who had taken quizzes prematurely or who wished to determinewhether work on a new unit had indeed provided them with a better un-derstanding of material which had earlier been especially difficult forthem.Ten quizzes, each covering previously designated objectives, were</p><p>given as students felt able to do the self-evaluation exercises which exem-plified the mathematical content of the objectives. When all objectivesfor a given unit, along with the appropriate quizzes, had been completed,students began work on the following unit. Some of the independentstudy students had completed one and two units, respectively, of the sub-sequent course in the two-course sequence of mathematics for prospec-tive elementary teachers. During the first year of this program, all stu-dents were required to take the midterm and final examinations togetherat dates determined at the beginning of the course. ,To illustrate the form and nature of the objectives with which each stu-</p><p>dent was provided, a small sample from various units of the course aregiven in the following list:</p><p>1. You should be able to specify a set verbally, or in writing by listing (tabulating) itselements or by use of set-builder notation.</p><p>2. You should be able to define the "product" of two whole numbers, and illustratethe application of the definition for two given factors.</p><p>3. You should be able to count in any place-value numeration system with wholenumber base of two or greater.</p><p>Keyed to these objectives were the following self-evaluation exercises:1. Complete the following table on set notation:</p><p>Verbal Tabulation Set-builderdescription ofelements notationThe set ofletters in ___________ ___________</p><p>word "floor"{1.2.3.....99 }</p><p>{x|x is an odd number}</p></li><li><p>Individualization in Mathematics 245</p><p>2. a. In2x3 = 6,2 and 3 are called______and6 is called______b. The product of whole numbers ______ = n({a, b, c, d, c}) and == n({ a, b}) is defined as______x______ = n){___________}).</p><p>3. a. Write the first twenty numerals in the binary system.b. What follows 99 in base twelve?______c. What precedes 100 in base six? ______</p><p>At the end of the material for each unit, a complete set of answers foreach self-evaluation exercise was given.</p><p>In the subsequent two years, the program has continued in much thesame way that it was instituted. The lists of behavioral objectives havebeen continually refined, and students are permitted to take midterm andfinal examinations whenever they are prepared to do so, rather than at adate predetermined by the instructor. Use of the Textbook Reference Li-brary, however, was discontinued after the first year of this program be-cause most students who attempted to use this material were usually con-fused by even slight differences in notation and definitions from those ofthe text adopted for the course.At the end of each of the past three academic years, the students were</p><p>asked to anonymously complete a questionnaire on the instructional set-ting and the value of being informed of behavioral objectives. In re-sponse to the question on whether or not being informed of behavioralobjectives for each unit helped foster a better understanding of the ma-terial, over 90 percent of the students answered affirmatively.As for the usefulness of the self-evaluation exercises which accompa-</p><p>nied each set of behavioral objectives, all members of the classes usingthe material during the past three years found them helpful. The finalitem of the questionnaire was a free-response question about the ap-proach used in the course. A sample of actual statements written in re-sponse to this inquiry is given in the following list:</p><p>"The self-pacing is good especially in a college situation because it eliminates somepressure. It makes mathematics more enjoyable."</p><p>"I find I have more incentive to learn in a more structured class type situation. It wasvery easy for me to fall behind.""Math is a hard subject area which causes many people to have mental blocks. An in-</p><p>dividualized program gets rid of any mental block because there is no pressure to learnthe material as fast as anyone else is learning it."</p><p>"I liked the approach. I know I would be bored greatly if I had to go at the same rateas other class members because of my background in math. *</p><p>REFERENCES</p><p>1. BLOOM, B. S., ED. Taxonomy of Educational Objectives, Handbook I: Cognitive Do-main. New York: David McKay Company, Inc., 1956.</p><p>2. Committee on the Undergraduate Program in Mathematics. Course Guides for theTraining of Teachers of Elementary School Mathematics. Berkeley, Calif.: The Com-mitee. Mathematical Association of America. 1968.</p></li><li><p>246 School Science and Mathematics</p><p>3. DEHANN, R. F. AND DOLL, R. C. "Individualization and Human Potential,"Individualizing Instruction. Yearbook of the Association for Supervision and Curricu-lum Development. Washington, D.C.: The Association, 1964.</p><p>4. DEVAULT, M. V. AND KRIERWALL, T. "Differentiation of Mathematics Instruction,"Mathematics Education. Sixty-Ninth Yearbook of the National Society for the Studyof Education, Part I. Chicago: University of Chicago Press, 1970.</p><p>5. FLANAGAN, J....</p></li></ul>