Through the years: individualizing instruction in mathematics

Through the years: individualizing instruction in mathematicsAuthor(s): E. GLENADINE GIBBSource: The Arithmetic Teacher, Vol. 17, No. 5 (MAY 1970), pp. 396-402Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186223 .

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Through the years: individualizing instruction in mathematics E. GLENADINE GIBB The University of Texas, Austin, Texas

Glenadine Gibb is professor of mathematics education at the University of Texas. She contributes richly to mathematics education through her activities of speaking, teaching, and writing. She is well known to the readership of The Arithmetic Teacher as a past editor of the journal.

/jL nniversaries provide a time for cele- bration, a time for reflection, and a time for speculation. The Golden Jubilee Year (1970) of the National Council of Teach- ers of Mathematics is no exception as one identifies issues of long standing that con- front teachers of mathematics. Among these issues is that of individualizing instruction. Let us assume that individualized instruction provides ways to teach a group of students so that each pupil can take what is for him the "next step" in his development of mathematical understandings and competencies at the time when he is ready to move ahead. Individualizing instruction requires developing ways to permit the student to progress at his own rate according to his own style of learning and ways to motivate him to think creatively in formulating his mathematical concepts and knowledge of mathematics.

In the field of elementary education, selected headlines relevant to the concern for individualized instruction through the years capture one's eye. Among these are the ungraded school, promotion plans, the Burk plan, the Winnetka plan, the Dalton plan, the contract method, ability grouping, departmentalization, the nongraded school, the Program for Learning in Accordance with Needs (PLAN), Individually Pre- scribed Instruction (IPI), Comprehensive School Mathematics Program (CSMP),

continuous progress, and computer assisted instruction (CAI). Also, there have been local efforts not as widely publicized whereby schools, individual teachers within schools, and school systems have attempted to resolve the problem of individualizing instruction in operational terms for children in their schools and classes. Can last- ing breakthroughs be made in an effort to resolve this ever-persistent problem? What will lie in the mysteries of the future for individualizing instruction in mathematics?

A time for reflection

During the colonial and early American period, schools were essentially ungraded and most instruction was tutorial in design. This type of organization enabled each child to progress at his own rate through those few texts that were available. By 1870, however, with pressures to educate more children, nearly all elementary schools in the United States were changed from ungraded to graded systems. This movement was conceived and established in the faith that all men are created equal. Grad- uates of normal schools, although lacking an understanding of child development and individual differences, were confident of what was to be done and what was to be learned at each level in the graded school.

Throughout the years, areas of "cer- tainty" in mathematics have been main-

396 The Arithmetic Teacher/May 1970



tained. Guided by textbooks and by teacher-education programs in the colleges and universities, teachers have attached knowledges, skills, attitudes, and abilities to each grade level. These certainties have been accompanied by frustrations. What does one do with children who have al- ready obtained knowledge and skill before they were supposed to do so? What does one do with children who have passed through a lock-step graded system and lock-step mathematics program without ac- quiring the certain required skills or abilities they were supposed to have? How can children be adjusted to fit the curriculum?

By 1875, numerous means of promotion were used for adjusting the grade place- ment of children in order to accommodate learning differences. If children had not achieved the "standards" of a grade level, they were not permitted to move ahead to the higher grade. If they were too far advanced, they were permitted to skip a

grade. There were semiannual promotions, quarterly promotions, subject promotions, special promotions, as well as nonpromo- tions. And, by increasing the amount of instruction for slow-learning pupils by means of employing extra teachers, out-of- school tutoring, and summer vacation classes, it was possible to get more children through the required curriculum. Other adjustments included differentiating the time required to complete the elementary school.

The current individualized instruction movement began in 1888 in Pueblo, Colo- rado, when Preston Search not only advo- cated but practiced a program of individual instruction within the graded school system [6]. When the first educational tests were given in Detroit schools in 1910, Stuart A. Courtis noted that the data secured made clear both the inefficiency of mass methods and the need for adjusting work to meet individual needs. He stated:

The only conclusion to be drawn from these results seems to be that improvement in arithmetic must be brought about through some de-

vice that will reach each individual and enable him to progress at his own rate [3:113].

He assumed that no progress is made if one does not master each phase as meas- ured in some way. Practice tests, designed to enable students to do their own correct- ing, were constructed. Thus each individual was permitted to progress at his own rate. These tests made it possible to adjust drill work to the needs of individual students. The teacher received papers only from those students who could not find their own mistakes. Furthermore, the self-scoring devices, daily individual records, and graphs that were used, served to motivate children to be in charge of their own development. By using these materials, it was claimed that one competent teacher could ade- quately take care of the needs of fifty children and yet completely individualize her instruction.

In 1912 Frederick Burk (San Francisco Normal School) initiated so-called self- instructional bulletins in an effort to enable every student to progress as rapidly as his individual ability permitted. The arithmetic curriculum was divided into short-step "goals," each goal representing one specific principle to be mastered. Carefully graded explanations of new steps were written in simple language so that a child could pro- ceed individually with as little or as much guidance as he needed from his teachers. Again, self-corrective tests were used to reveal to the student his weaknesses or strengths on any unit of work. Special supplementary exercises were available for drill on each specific difficulty. When all tests were passed, the student received a promotion slip. The amount of time needed to complete a grade of work varied with individuals since each student progressed at his own rate, some taking longer to complete certain goals than others.

One of Burk's faculty members, Carleton Washburne, moved on to accept the posi- tion as superintendent of the Winnetka (Illinois) schools. Although the Burk plan had worked in a laboratory school setting, could it be effective in a larger school sys-

Exce Hence in Mathematics Education - For All 397



ton plan under the direction of Helen Park- hurst (Massachusetts). The Dalton plan involved freedom, cooperation and in- teraction of groups, and learning to budget class time. The curriculum was individually paced with emphasis on personalized contracts and self-corrective practice materials. The work for each grade in each of the academic subjects, beginning with the fourth grade, was laid out in a series of related jobs or contracts. Each job was to be done within a school month of twenty days. The contract was completed across all subject areas before a student progressed to any subject area in the next job.

In England, Jessie Mackinder was individualizing work for those children entering school. Using concrete materials, the work- ing of a new process was shown to groups of children. Children were then left to work many examples until, having grasped the underlying principle, they discarded the apparatus in their own time. Records of each child's progress in each subject, in- cluding test results, were kept by the teacher from grade to grade.

During these years the one-teacher, eight-graded rural schools were not for- gotten. Brown (Connecticut) and Hoffman (Illinois) realized that the objective of individualized instruction was to teach the study of arithmetic and not to hear recita- tions. These leaders believed that the lack of specially-constructed textbooks for individualization was not a stumbling block. Textbooks could meet the conditions if reasonable care was taken in the selection of those books to be used. In an effort not to overburden the teacher with a mass of records, each pupil kept his own assignment book and progress sheets in loose- leaf covers. If a grade of В was made, the teacher gave supplementary assignments. At the end of each day, each student gave to his teacher the written work completed. Keeping record sheets and assignment books was found to be a powerful motive for good work. Also, the competitive ele- ment entered into motivating a student as one member of a group advanced a little

398 The Arithmetic Teacher/ May 1970

tern? Success of the Winnetka plan was attributed to "whole-hearted, clear-headed, and cooperative efforts" of carefully selected teachers. Teachers spent time teaching, helping individuals or groups, encour- aging and supervising. They no longer sat at their desks but were among their students as they worked. No child ever failed nor did he skip a grade. The student began in September and worked at this individual pace until school stopped in June. He then resumed his work the following September. Work was done on a piecework basis, not a time basis. The school program was divided into ( 1 ) knowledges and skills that everyone needed to master and (2) art and shop courses that provided opportunity to develop individual interests and abilities in group efforts.

In developing knowledge and skill, teachers discontinued recitation methods in favor of a system whereby each student prepared his unit of work with an answer sheet. After a group of units was completed, the student used practice tests to test himself. If practice tests were 100 percent right, the student asked for the real test. If he did not attain 100 percent on the real test, he returned to practicing until he felt ready to try the real test again. Upon mastery of a goal, the student worked toward the next goal.

As reported by Washburne [17], the Winnetka plan saved time, especially for more able children and those who would normally be repeaters, and allowed for a broader and deeper education. Individual promotions appeared to decrease retardation and "overage-ness," and to increase efficiency in tool subjects without placing undue burden on the teacher. Also, there was no evidence that it cost more to individualize instruction using the Winnetka plan. Difficulties were noted in securing suitable textbook materials, in the proper training of teachers, and because of its new- ness, in establishing the program.

Other efforts to break the lock-step graded system were also being made during these years. Among these was the Dal-



to seek the best methods of helping the individual child to realize his full capacities.

Following World War II, the focus was again on subject matter in the elementary school curriculum. Emphasis was placed on making mathematics more meaningful to children and on the developing of system- atic instruction. Also, developing out of the context of military instruction were ideas of programmed instruction, programmed learning, automated instruction, programmed materials, and teaching machines. The idea of programming and systems development became popularized as a process of determining empirically a sequence of interactions or operations to assure a de- pendable performance at an established standard. Programmed instruction became another means of providing for individualized instruction in the 1950s.

Among those who accepted the chal- lenge of guidance in the improvement of mathematics programs for elementary schools in the 1950s and 1960s were those persons identified with such inno- vative mathematics programs as: the Stan- ford Project, the University of Illinois Arithmetic Project, the Madison Project, the School Mathematics Study Group, the Greater Cleveland Project, the Entebbe Project, and the Nuffield Project. Their efforts have been paralleled by other groups seeking to develop basic textbook series that would reflect and extend the leadership provided by these projects. Enrichment activities, selected problems, variations in computational techniques, suggestions in teaching guides, supplementary materials, programmed materials, and mathematics laboratories have been and are being used to provide at least minimal resources that can assist the teacher in providing for individual differences in mathematical ability and interest.

Improved mathematics programs with new intent on content have been accompanied by explorations in varying the organizational design of the school and its curriculum. These include the nongraded school, increased provision for independent

Excellence in Mathematics Education - For All 399

ahead of his companions. And so individualizing instruction reached its peak in the latter 1920s.

With the exception of the Winnetka plan, various plans for individualizing instruction seemed to pass out of existence in the 1930s. This change was due pri- marily to a shift of emphasis in the elementary school. Whereas major concern in the preceding years had been on mastery of subject matter, the child now became the focus of attention. Led by Dewey and others who were concerned with an education closer to child life than that pre- sented in the then current subject-centered curriculum, the attention of leaders turned away from subjects. Units, activities, in- tegration (of knowledge), experience cur- ricula, the problem method, and other non- subject-centered curricular plans became the topics that held the attention of educators. Although the Progressives, as most of the innovators of that time were called, emphasized the importance of attention on the individual child, little in the way of specific instructional materials designed to pro- mote individualization of instruction was produced. It was believed that units, problems, or activities would permit each child to work at his own level. While the problem-unit-activity movement received much publicity, the need for instructional materials and trained teachers plus some inherent weakness prevented the movement from being put into actual practice.

In brief, inadequate education of teachers, lack of prepared texts and tests for individualized work, and less concern for subject matter might be said to be factors hampering the continued efforts of the 1920s to individualize instruction in subject areas. Yet, compromise plans did spring up as a replacement for this early work. These included ability grouping, dif- ferentiated assignments, enrichment, identi- fication of minimum essentials, and group projects in which each child participated according to his own level of readiness. These replacements have continued to be used through the years as teachers continue



study, team teaching, and adaptations of departmentalization. In addition, advancement in the field of technology has made available many new teaching aids such as computers and educational television. New designs in school buildings have made "innovation" a more attainable goal. How- ever, hasty and superficial implementation of the organizational plans simply to suggest "innovation" does not provide for the needed change in instructional procedures nor for the education or reeducation of fac- ulties to function effectively in such or- ganizations. Nongraded schools do not nec- essarily provide nongraded instruction. Team teaching or specialist teachers in mathematics do not automatically assure advancement from the "patent medicine" age of education whereby one single rem- edy is prescribed, by topic and page, as the next mathematical experience for all children. Mass methods still prevail. Teach- ers still stand before one group of children explaining the page in the book, often without understanding themselves the specific learning objective for which it was designed. Completing the textbook, page by page, regardless of how it is done, is still used as the indication that children have indeed learned. Despite such prac- tices, lip service continues to flow freely proclaiming: "Education is a personal, individual process." "The individual, not the group, learns." "The purpose of education is to develop the individual."

In the 1960s, the renaissance of individualized instruction brought to the fore- front: (1) the need to define objectives and to state them in behavioral or performance terms; (2) the need to develop both premeasurement and postmeasurement and assessment devices for monitoring progress in the attainment of each objective; and (3) the need for procedures for planning each individual's mathematical program in terms of the learning objectives of mathematics programs. Projects as Comprehen- sive School Mathematics Program (CSMP), a Program for Learning in Accordance with Needs (PLAN), Individually Prescribed

Instruction (IPI), computer-assisted instruction (CAI), programmed learning and local school projects have addressed themselves to revitalizing earlier attempts to individualize instruction in mathematics and to identifying the objectives of this type of instruction. The instructional model of IPI, for example, makes use of place- ment tests to locate the individual child in the continuum or hierarchy of prescribed learning objectives in terms of ob- servable student behavior. Attempts have been made' to design tests for assessing student competency and to determine what instruction is needed to help the student achieve what is for him the "next" learning goal. Instruction, often in the form of lessons for individual study, is provided. Posttests are given to indicate to the child and his teacher whether or not he has sat- isfactorily attained the goal identified. Indi- vidual records of progress are kept. If the student has been successful in his work, he moves ahead. If the student has en- countered difficulty in attaining the learning objective, he has the opportunity for further work and consultation with his teacher.

A time for speculation If education in mathematics is to be

truly individualized, then what type of instructional materials make for most progress in attaining the goal of individualized instruction for all children? The answer to that question lies in the future. It would seem, however, that survival or nonsurvival is dependent upon the success or failure to resolve other problems that have con- fronted teachers of mathematics throughout the years. Among such problems are:

1 . Can agreement be reached as to what mathematical understandings, skills, and competencies are required in mathematics? If so, can these goals be stated in terms of learning behaviors in such a way as to avoid substituting memorization for in- depth learning?

2. How can one provide the stimulation

400 The Arithmetic Teacher /May 1970



of thought and cognitive growth in mathematics of each student by asking the "right" questions at the "right" time?

3. Can research be designed so as to contribute knowledge concerning how children develop mathematical concepts and skills? Certainly, more knowledge is needed about ways to identify learning styles, aptitudes, and interests of individual children, and about how to identify components of an individual learning style.

4. What instructional strategies can and should be employed in order to predict with confidence that children will develop the abilities to think independently, to make choices, to plan, and to evaluate? Further- more, can instructional strategies be designed to match with individual learning styles and individual learning potential? Can the curriculum be adapted to the needs of students, instead of students being adapted to fit the curriculum?

5. Is it possible to design suitable materials and at the same time not make the "package" so expensive that their use is prohibitive? Through the years the lack of appropriate materials has been noted as a handicap in moving toward individualized instruction programs.

6. Can the instructional programs of our schools provide guidance for the ad- ministrative organization and structure of our school buildings rather than be re- stricted because of them?

7. What changes must be made in teacher-education programs to truly pre- pare prospective and experienced teachers to assume their responsibilities in the ca- reer they have chosen? Can plans be im- plemented so that the aptitudes, interests, capabilities, and learning styles of adults can be used to maximize each individual's potential as a teacher of children? Re- peatedly, deficiencies in the teacher's education to teach mathematics have been highlighted as a barrier in implementing improved programs in mathematics. Ob- viously, efforts to effectively individualize instruction are dependent upon the teacher,

who likely has never experienced profes- sional preparation in the individualization of instruction. Just as materials, planning, organization, and opportunity are needed in the schools, these same needs exist in teacher-education programs. Each teacher must be prepared to accept his responsi- bility both as an individual and as a member of a group.

Each of these areas of general concern contains many more specific questions for which answers must be sought. Research in mathematics education has made little headway in these areas in the past. Will more progress be made in the future? One can speculate that educators in our schools, colleges, and universities will continue to strive for educational ideals and engage themselves in those scholarly efforts to help both children and teachers realize their full capacities. The realization of those goals lies in the future.

A Selected Bibliography

1. Ccok, Walter W., and Theodore Clymer. "Acceleration and Retardation." In In- dividualizing Instruction. Sixty-first Year- book of the National Society for the Study of Education, part 1. Chicago: Uni- versity of Chicago Press, 1962.

2. Cooley, William A., and Robert Glaser. "The Computer and Individualizing Instruc- tion." Science 166 (October 31, 1969): pp. 574-82.

3. Courtis, Stuart A. "Ability Grouping in De- troit Schools." In Adapting the Schools to Individual Differences. Twenty-fourth Yearbook of the National Society for the Study of Education, part 2. Bloomington, 111.: Public School Publishing Co., 1925.

4. Dale, Edgar. "Instructional Resources." In The Changing American School. Sixty- fifth Yearbook of the National Society for the Study of Education, part 2. Chicago: University of Chicago Press, 1966.

5. Dagne, Frank A. "Personalized Instruction." Illinois Education 56 (October 1967): 68- 70.

6. Dean, Ray B. "What Ha.s Become of the Individual Instruction Movement." School and Society 58 (September 4, 1943): 164- 67.

7. Flanagan, John C. "Functional Education for the Seventies." The Phi Delta Kappan 49 (September 1967): 27-31.

Excellence in Mathematics Education - For All 401



8. Glaser, Robert. 'The Design of Instruction." In The Changing American School. Sixty- fifth Yearbook of the National Society for the Study of Education, part 2. Chi- cago: University of Chicago Press, 1966.

9. Gray, William S. "An Illustration from the University of Chicago." In Adapting the Schools to individual Differences. Twenty- fourth Yearbook of the National Society for the Study of Education, part 2. Bloomington, 111.: Public School Publish-, ine Co., 1925.

10. Heathers, Glen. "School Organization: Non- grading, Dual Progress, and Team Teach- ing." In The Changing American School. Sixty-fifth Yearbook of the National So- ciety for the Study of Education, part 2. Chicago: University of Chicago Press, 1966.

11. Knight, Frederick B. "Arithmetic." In Psy- chology of the School Subjects. Review of Educational Research, vol. 1. Wash- ington, D.C.: American Educational Re- search Association, 1931.

12. . "Pupil Classification and Grouping." In Psychology of the School Subjects. Re- view of Educational Research, vol. 1. Washington, D.C.: American Educational Research Association, 1931.

13. Mackinder, Jessie. "Individual Work in In-

fants' School." Adapting the Schools to individual Differences. Twenty-fourth Yearbook of the National Society for the Study of Education, part 2. Bloomington, 111.: Public School Publishing Co., 1925.

14. Parkhurst, Helen. "The Dalton Laboratory Plan." In Adapting the Schools to Indi- vidual Differences. Twenty-fourth Year- book of the National Society for the Study of Education, part 2. Bloomington, 111.: Public School Publishing Co., 1925.

15. Sutherland, A. H. "Ability Grouping in Los Angeles." In Adapting the Schools to In- dividual Differences. Twenty-fourth Year- book of the National Society for the Study of Education, part 2. Bloomington, 111.: Public School Publishing Co., 1925.

16. Tyler, Fred T., and William A. Brownell. "Facts and Issues: A Concluding State- ment." In Individualizing Instruction. Sixty-first Yearbook of the National So- ciety for the Study of Education, part 1. Chicago: University of Chicago Press, 1962.

17. Washburne, Carleton. "Burk's Individual Sys- tem as Developed at Winnetka." In Adapt- ing the Schools to Individual Differences. Twenty-fourth Yearbook of the National Society for the Study of Education, part 2. Bloomington, 111.: Public School Pub- lishing Co., 1925.

31st YEARBOOK HISTORICAL TOPICS FOR THE MATHEMATICS CLASSROOM

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402 The Arithmetic Teacher/ May 1970



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Through the years: individualizing instruction in mathematics