A very legitimate pride

A very legitimate prideAuthor(s): BARBARA B. DUNNING and MEREDITH D. GALLSource: The Arithmetic Teacher, Vol. 18, No. 5 (MAY 1971), pp. 339-345Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186397 .

Accessed: 12/06/2014 14:44

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

.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 [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 195.34.79.79 on Thu, 12 Jun 2014 14:44:44 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nctm

http://www.jstor.org/stable/41186397?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Forum on teacher preparation

Edited by Francis J. Mueller

A very legitimate pride BARBARA B. DUNNING and MEREDITH D. GALL

Barbara Dunning and Meredith Gall are staff members of the Far West Laboratory for Educational Research and Development, where they developed the teacher-training program described here. Barbara Dunning, who currently works in the Laboratory's Dissemination Program, has taught high school and trained student teachers. Meredith Gall, a psychologist, directs a research and development team that produces skill-training programs for teachers.

lna lecture in the 1920s, Alfred North Whiteheadsaid:

In ... the reform of mathematics instruction, the present generation of teachers may take a very legitimate pride in its achievements. It has shown immense energy in reform, and has ac- complished more than would have been thought possible in so short a time.

Today's mathematics teachers continue to be leaders in reform. Yet even though curriculum content has been revolutionized for post-Sputnik generations, the nation's schools and colleges have not yet experi-

The Far West Laboratory for Educational Research and Development is a public non- profit organization supported in part by the U.S. Office of Education, Department of Health, Education and Welfare.

enced a comparable revolution in teaching techniques.1

At the Far West Laboratory for Educa- tional Research and Development, the Teacher Education Program, directed by Walter R. Borg and Ned A. Flanders, has concentrated its efforts on creating, field testing, and evaluating a series of inservice and preservice Minicourses that enable

1. "Repeatedly, deficiencies in the teacher's education to teach mathematics have been highlighted as a barrier in implementing improved programs in mathematics. Obviously, efforts to effectively individ- ualize instruction are dependent upon the teacher, who likely has never experienced professional 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" (E. Glenadine Gibb, "Through the Years: Individualizing Instruction in Mathematics," Arithmetic Teacher 17 [May 1970] :401).

May 1971 339



teachers to learn essential skills during regular school hours, without leaving their own buildings.

The newest of these self-contained, self- evaluative Minicourses is "Individualizing Instruction in Mathematics." It helps teachers learn to work effectively one-to-one with their students and to deal successfully with their students' mathematics problems. Use of this Minicourse multiplies teacher effectiveness and adds new classroom skills to the teacher's repertoire.

Teachers today are steadily moving away from their "traditional" roles as lecturers, organizers of content, and testers. Now -

using "modern" mathematics and armed with new knowledge about how youngsters learn - the best teachers are becoming fa- cilitators, tutors, and stimulators of small- group or individualized activities. And now that rigorous field testing has validated the Laboratory's newest Minicourse for national installation,2 any elementary teacher can easily master the skills necessary for tutoring and individualizing in mathematics.

The Laboratory's development team3 be- gan the research and development cycle for this Minicourse in 1968. A review of previous research focused on the work of Bern- stein (1959),Tilton (1947), Ross (1963), and Roberts (1968). This research highlighted the very real benefits that students derive from individualized tutoring. The developers of Individually Prescribed In- struction and of Project PLAN also supported the Laboratory's assumptions.

If there is one point on which all writers about mathematics tutoring agree, it is the importance of diagnosis. Brueckner and Bond (1955), for example, state that, "the continuous application of diagnostic methods to ferret out the difficulties pupils may be having with arithmetic is vital." Five categories of diagnostic questions evolved from the literature search:

2. The completed "product" (Individualizing In- struction in Mathematics) is sold and rented by Macmillan Educational Services, 8701 Wilshire Blvd., Beverly Hills, Calif. 90211.

3. Meredith D. Gall, Barbara B. Dunning, John Galassi.

1 . General diagnostic questions For example, "How did you get your

answer?" or "What part don't you understand?"

2. Diagnostic questions about number concepts

For example, "What is the value of 4 in the numeral 46?" or "What is another name for 12?"

3. Diagnostic questions about reading difficulties

A student's inability to read a verbal problem obviously interferes with his ability to solve it. When a student has difficulty with a verbal reasoning problem, the teacher asks the student to read it aloud.

4. Diagnostic questions about word defini- tions

Even when a student can read a verbal problem, he may not understand the meaning of all the words. During one field test, a teacher worked with a student who could not solve a problem asking for the number of stamps found on a page of a stamp album. A diagnostic question revealed that the student was familiar with record albums but couldn't visualize a stamp album with pages.

5. Diagnostic questions about the selection of a number operation for solving a verbal problem

For example, "What number operation do you use to solve this problem?" These types of questions enable a teacher to identify common student difficulties.

After the teacher diagnoses the nature of the student's difficulty, he is ready to move into the second phase of the tutoring sequence, demonstration. At this point, tutoring involves demonstrating to the student the concepts and procedures needed in the solution of the problem. The following six demonstration techniques are particularly useful with students who have difficulty with number operations and verbal problems.

340 The Arithmetic Teacher



1. Estimation This technique can be used prior to

developing an understanding of the ex- act procedures. For example, a student must solve this problem: 42 - 29 = 1 I. If he figures that 42 is almost 40 and 29 is almost 30, he can esti- mate that his answer will be about 10.

2. Expanded notation Expanded notation is helpful in ex-

plaining such concepts as place value and regrouping.

3. Number line Through the use of the number line

a teacher can supplement his verbal explanation of new mathematics concepts with pictorial demonstrations.

4. Manipulative materials Although seldom used beyond the

primary grades, manipulative materials such as Cuisenaire rods, pieboards, and place-value charts are highly recom- mended for use with all students who have a weak understanding of number concepts.

5. Picture or diagram This method is excellent for demon-

strating verbal problems. Words and numbers become more comprehensible to a student when depicted in a picture.

6. Number sentences The number sentence is a useful

technique because it helps the student see the relationship between the quanti- ties given in a verbal problem and because it develops his understanding of mathematics as a quantitative language. The step that logically follows the dem-

onstration phase of the tutoring sequence is evaluation. At this point the teacher evaluates the student's learning by assigning an example for him to solve. This example should be similar to one that originally gave the student difficulty. If the student is unable to solve it, the teacher must recycle the tutoring sequence through further diagnostic questioning and demonstration. But if the student can correctly solve the evaluation example, the teacher

should conclude the tutoring session by assigning additional examples for practice. However, the teacher should assign examples only after the student demonstrates his understanding by solving an evaluation example correctly. Otherwise, practice simply strengthens incorrect habits (Glen- non and Callahan 1968, p. 81; Brownell andChazal 1935).

The basic Minicourse tutoring sequence, then, consists of four easy sequential stages: diagnosis, demonstration, evaluation, and practice.

Two other skills have proved effective in tutoring: verbal praise, particularly specific verbal praise, a commendation tied closely to the student's performance (for example, "Good, you know how to regroup!"); and prompting questions, that is, questions asking the student to do or tell something (for example, "Will you please draw a picture of the problem?"), rather than the teacher doing or telling.

Although the main objective of this Mini- course is to increase teacher skills in mathematics tutoring, the course also aims to increase the amount of time spent in individualized mathematics instruction. There- fore, the Teacher Handbook includes a final lesson that shows how the teacher can reorganize the classroom to provide increased time for tutoring. Suggested techniques include providing partial scoring keys for students' seatwork, tutoring at a worktable rather than by walking about the room, and using student tutors for routine tutoring tasks.

The Minicourse, as developed and field tested, met virtually all the objectives listed in table 1 . The teacher learns these specific skills during the regular school day by practicing them, on released time, with his own students. "Individualizing Instruc- tion in Mathematics" - like all other Mini- courses produced to date - follows a care- fully planned instructional sequence.

First, the teacher reads a handbook lesson describing the skills to be practiced. Next he views a short instructional film that illustrates each skill, with appropriate cues

May 1971 341



and verbal reinforcement. Then he views a brief model film that lets him check his understanding and discrimination of what he has seen demonstrated.

Table 1 Minicourse 5 objectives and skills

Instructional Sequence 1

Objective To improve teacher skill in re- warding pupils' correct responses and encouraging their active par- ticipation in the tutoring process

Skills covered Using verbal praise to reward correct responses. Asking prompting questions to increase pupils' active involvement in the tutoring process.


Objective To increase teacher skill in diagnos- ing pupils' deficiencies in understanding of mathematical concepts and computational procedures.

Skills covered Asking general diagnostic questions (e.g., "How did you get your answer?"). Number operations: asking questions to test pupils' understanding of place value, regrouping, and other number concepts. Verbal problems: asking questions that test pupils' ability to read the problem and to decide on an appropriate number operation.


Objective To increase teacher use of techniques that help to develop pupils' understanding of mathematical concepts and computational procedures.

Skills covered Estimating an answer prior to using a computational algorithm. Number operations : depending on the situation, using expanded notation, the number line, or manipulative materials. Verbal problems: having the pupil draw a picture of the problem and having him write a number sentence to express the problem's requirements.


Objective To increase teacher skill in evaluating student progress and assigning practice examples.

Skills covered Assigning an evaluation example. Assigning practice examples.


Objective To improve teacher skill in orga- nizing the mathematics class period for individual tutoring.

Skills covered Having pupils correct their own work. Having pupils tutor each other (peer tutoring).

The following day, the teacher selects two students who need tutoring and prac- tices (about ten minutes with each student) the same skills in a microteaching situation. The teacher videotapes these microteaching sessions so that he can re- play them later and evaluate them by using checklists in his handbook. Thus, he gets immediate feedback in a nonthreaten- ing and comfortable atmosphere.

Moreover, a day or two later the teacher has a second chance to practice the same skills during a "reteach" session with dif- ferent students in front of the videotape recorder.

No supervisor is needed, but the teacher can play back his tapes with another teacher who is taking the course at the same time. One videotape recorder lets four to eight teachers in the same building take the course at one time. An administrator acts as local "coordinator" - to be sure that the videotape equipment is scheduled as needed and that teachers get an hour a day, three days a week, for their microteaching practice sessions.

The Minicourse works because teachers spend 70 percent of their time practicing classroom skills rather than just reading or hearing about them. If, during the rigorous development, evaluation, and revision stages, any Minicourse fails to accomplish its objectives satisfactorily, it is further re- vised and retested until the performance goals are reached.

The mathematics Minicourse was first tested with 47 teachers (grades 3 to 6) in three San Francisco Bay Area school dis- tricts. Their mean age was 34.6, and their mean years of teaching experience was 9.1. The effectiveness of the Minicourse was determined by videotaping two tutoring sessions for each teacher, before and after the course. Tables 2 and 3 show the key results of this analysis,4 with the most impressive gains being achieved in the amount

4. Except for evaluation, all gains shown in table 2 are statistically significant at thé .005 level or better.




Table 2

Teachers7 use of specific tutoring skills before and after the Minicourse

Percentage of teachers Pretape average Posttape average improving

Verbal praise* 7 occurrences 11 occurrences 70% Diagnostic questions t 10 occurrences 15 occurrences 79% Demonstration techniques* 18 seconds 168 seconds 65%

(tutoring on number operations) Demonstration techniques* 48 seconds 165 seconds 80%

(tutoring on verbal problems) Evaluation! 27 teachers 30 teachers (no change) Practice t 1 teacher 11 teachers 18%

* Based on about ten minutes of tutoring. t Based on about twenty minutes of tutoring.

of time spent using demonstration techniques during tutoring.

More detailed information on the field- test results, and information on a replica- tion study involving another group of teachers, may be found in a new book, The Minicourse: A Microteaching Ap- proach to Teacher Education (Borg, Kel- ley, Langer, and Gall, 1970). These find- ings seem impressive in view of the fact that nearly all the teachers who participated in the field testing had many years of experience and had previous inservice training in modern mathematics instruction. Another 150 teachers across the nation participated in operational field tests that proved the Minicourse to be fully ready for general use, without any Laboratory supervision.5

Teachers enjoy taking the Minicourse. Everything needed comes in one self-contained package - an introductory film, eight instructional and model films, a coordinator's handbook, and teacher's handbook (including self-evaluation forms and follow-up lessons). By following the daily course schedule (for twelve days, one hour per day) and the step-by-step instruc- tions, teachers acquire the skills through repeated practice and - by videotape feed-

5. Of these teachers, 70 percent reported that the Minicourse was better than any other inservice training they had received.

back - "see themselves as others see them." Yet their tapes are always "confidential" - no supervisor or colleague is allowed to view a teacher's videotape without his per- mission.

The Laboratory has been asked, time and again, whether videotape is necessary for this type of inservice training. With American Institutes for Research, the development team conducted a study involving 35 teachers who were using Project

Table 3 Number of teachers using each demonstration technique before and after Mini- course 5

Number of teachers Precourse Postcourse

Estimation 6 21 Expanded notation 11 19 Number line 2 11 Manipulative materials 8 19 Picture or diagram 18 35 Number sentence 14 24

PLAN's individualized mathematics curriculum. This test reaffirmed the earlier results by showing that the Minicourse effectively increases the use of tutoring skills. But the Laboratory also learned that videotape and audiotape feedback are equally effective in helping teachers to master the skills. Thus, schools and col-

May 1971 343



Microteaching session with video feedback

leges6 will be able to use audiotape re- corders where videotape equipment is either unavailable or heavily scheduled - without lessening this Minicourse's effectiveness.

Minicourses have proved to be an ex- tremely valuable alternative in the professional development of inservice and preservice teachers. Moreover, they provide an excellent mode of training for parapro- fessionals, student teachers, interns, or in- experienced personnel who can tutor students under the guidance of experienced teachers.

Like all other Laboratory products,7 this new Minicourse moved through a rigorous research and development cycle before its release to the distributor for

6. The Minicourse has also been tested with student teachers. Kenneth Nelson and Douglas Rector at the State University of New York (Fredonia) have offered this Minicourse to several hundred undergraduates as part of a mathematics methods course. Since it is difficult logistically to obtain real pupils with whom to practice, some student teachers play the role of pupils while others practice tutoring them. Roles are then reversed so that everyone has the opportunity to evaluate his performance in both roles on videotape playback. As one might expect, their research data reveal that student teachers make substantially larger gains in tutoring skills than do inservice teachers.

7. For a list of Laboratory products, write Dr. Ferucio Freschet, Far West Laboratory for Educa- tional Research and Development, 1 Garden Circle, Hotel Claremont, Berkeley, Calif. 94705.

national installation. The big payoff will come when a teacher, after completing the course and noting the improvement in classroom behavior, begins to enjoy that "very legitimate pride" that caps his professional development.

Bibliography

Bernstein, A. "Library Research - a Study in Remedial Arithmetic." School Science and Mathematics 59 (1959): 185-95.

Borg, W. R., M. L. Kelley, P. Langer, and M. Gall. The Minicourse: A Microteaching Ap- proach to Teacher Education. Beverly Hills: Macmillan Educational Services, 1970.

Brownell, W. A., and C. B. Chazal. "The Effects of Premature Drill in Third-Grade Arithme- tic.'* Journal of Educational Research 29 (1935): 17-28.

Brueckner, L. J., and G. Bond. The Diagnosis and Treatment of Learning Difficulties. New York: Appleton-Century Crofts, 1955.

Glennon, V. J., and L. G. Callahan. Elementary School Mathematics: A Guide to Current Re- search. Washington, D.C.: NEA, Association for Supervision and Curriculum Development, 1968.

Roberts, G. H. "The Failure Strategies of Third-Grade Arithmetic Pupils." Arithmetic Teacher 15 (1968): 442-46.

Ross, R. "Diagnosis and Correction of Arithmetic Underachievement." Arithmetic Teacher 10




Microteaching session with audio feedback

(1963): 22-27.

Tilton, J. W. "Individualized and Meaningful Instruction in Arithmetic." Journal of Educa- tional Psychology 38 (1947): 83-88.

Note. The two pictures shown in this ar- ticle are of microteaching sessions at Hay- ward Unified School District, California, being directed by Mrs. Meedie Monegan.

Letter to the editor

Dear Editor:

Replying to the following final test question in a mathematics course - "Exactly how would you use the Twenty-seventh Yearbook of the NCTM in your classroom?" - one future elementary school teacher wrote:

"First, I would read it! "Second, apply what it says. "Third, put it on my chair and sit on it so I

could look over my class. "Fourth, put a handle on it and use it on any-

one that failed to do their outside reading or study for my tests."

"Sorry, but I just had to write something," were his final words.

In answer to a discussion question in my final examination, a very attractive future teacher answered: "I cannot discuss this topic because I do not remember what I read."

Thank you for an excellent publication. - Richard J. Donald, Assistant Professor of Ed- ucation, Bloomsburg State College, Bloomsburg, Pennsylvania.

May 1971 345



Documents

A very legitimate pride