DOCUMENT RESUME

ED 197 526 EC 131 728

AUTHOR Bartkovich, Kevin G.: George, William C.TITLE Teaching the Gifted and Talented in the Mathematics

Cla ssrcom.INSTITUTION Nation;t1 Education Association, Washington, D.C.REPORT NO IS9N-0-9106-0738-7PUB DATE. BONOTE 49p.: A part of the Teaching the Gifted and Talented

in the Content Area Series, Fred B. Tuttle, Jr.,Series Editor. For related documents, see EC 131721-727.

AVAILABLE FROM NEA Order Department, The Academic Building, NewHaven, CT 06516 ($3.50, Stock No. 0738-7-00): Also apart of "Educat±ng Gifted and Talented Students"(Edupak) comprised of filmstrips, cassettes, books($147.50, Stock No. 070-1-00).

EDRS PRICEDESCRIPTORS

MF01 Plus Postage. PC Not Available from EDRS.*Acceleration: Educational Methods; ElementarySecondary Education: Enrichment: *Gifted;*Mal:hematical Enrichment: *Mathematics Instruction:*Talent: *Talent Identification: Teaching Methods

ABSTRACTAs part of the "Educating Gifted and Talented

Students" series, the booklet offers strategies for teachingmathematics to gifted and talented students. Methods for identifyingmathematical ability are reviewed and a two stage method ofidentification is highlighted which involves screening students in analready chosen grade through two different measures of mathematic-alaptitude. General approaches are considered, including enrichment,busy work, general academic enrichment and cultural enrichment,specific academic enrichment, and educational acceleration.. Detailedare procedures for implementing teaching strategies--fast pacedmathematics class and individualized classroom approach--foundsuccessful in the Study of Mathematically Precocious Youth. Alsoaddressed are program evaluation, teacher selection, textbookselection, and other accelerative options. (SBH)

***********************************************************************Reproductions supplied by EDRS are the best that can be made

from the original document.***********************************************************************

It°:Nri

U S DEPARTMENT OF HEALTH.EDUCATION & WELFARE

or%NATIONAL INSTITUTE OF

EDUCATION

.6...... THIS DOCUMENT HAS BEEN REPRO-

IN DUCED EXACTLY AS RECEIVED FROMTHE PERSON OR ORGANIZATION ORIGIN-

''--IATiNG IT POINTS OF VIEW OR OPINIONSSTAT ED,00 NOT NECESSARILY REPRE-SENT ;OFF ICIAL NATIONAL INSTITUTE OFEDUCATION POSITION OR POLICY

1.11

Teaching theGifted acid Talented

in theMathematics

Classroom

byKevin G. BartkovichWilliam C. George

Consultant: Julian C. Stanley

"PERMISSION TO REPRODUCE THISMATERIAL IN MICROFICHE ONLYHAS BEEN GRANTED BY

TO THE EDUCATIONAL RESOURCESINFinMATION CENTER (ERIC)."

r

NEA's Teaching the Gifted and Talentedin the Content Areas series

Editor: Frederick B. Tuttle, Jr.

National Education AssociationWashington, D.C. 2

Copyright © 1980National Education Association of the United StatesStock No. 0738-7-00

The Series EditorNEA's Teaching thc. Gifted and Talented in the Content Areasseries editor is Frederick B. Tuttle, Jr. Dr. Tuttle is AssociateProfessor at the State University of New York, College at Brock-port ; Director of the Program for Gifted and Talented in theFoxborough Schools System, Massachusetts ; and Consultant Bos-ton College/Boston City Schools District 3 Gifted and Talented.Among his publications are Composition: A Media Approach,Technical and Scientific Writing (with Sarak. H. Collins), andWhat Research Says to the Teacher: Gifted and Talented Stu-dents, published by NEA.Library of Congress Cataloging in Publication DataBartkovich, Kevin G.

Teaching the gifted and talented in the mathematicsclassroom.

Bibliography: p.1. MathematicsStudy and teaching.

2. Gifted childrenEducation--Mathematics.I. George, William C., joint author. IL Title.QA11.B319 510'.7'1 80-5166ISBN 0-8106-0738-7

Contents

Introduction 5

Identification 7Approaches 14Teaching Strategies 18Implementation 31Evaluation 35Teacher Selection 39Textbook Selection 41Other Accelerative Options 42Conclusion 44Appendixes 45Selected References 47

The AuthorsKevin G. Bartkovich and William C. George are with the Studyof Mathematically Precocious Youth (SMPY) at The Johns Hop-kins University, Baltimore, Maryland. Mr. Bartkovich is an In-structor in the program and concurrently a graduate student inelectrical engineering at Johns Hopkins.Mr. George is Associate Director of SMPY and Director, Officeof Talented Identification and Development (OTID). He is alsoa Consultant for Illinois' Gifted Children Program, Springfield;Chicago Area Service Center for Gifted and Chicago PublicSchools; Minnesota Talented Youth Mathematics Project's Ad-visory Committee ; and a member of the Student Science SeminarSteering Committee for the Maryland Academy of Science.The ConsultantJulian C. Stanley is Professor of Psychology and Director ofMathematically Precocious Youth (SMPY), The Johns HopkinsUniversity, Baltimore, Maryland. Dr. Stanley and Mr. Georgehave co-edited (with Cecilia H. Solana) The Gifted and the Crea-tive: A Fifty Year Perspective and Educational Programs andIntellectual Prodigies, and (with Sanford J. Cohn) Educatingthe Gifted: Acceleration and Enrichment.

INTRODUCTION

Educators have focused a great deal of effort on devel-oping programs for children labeled "mentally handicapped."However, they have often overlooked that gifted youths alsoneed special academic programs. In particular, mathemati-cally gifted youths should be given special opportunitiesthat enable them to use their mathematical capabilities inthe fullest possible way.

The facilitation of mathematically gifted youths cantake many different forms, ranging from largely ineffectiveto extremely successful. The program ideas presented herehave been used with success by various state and local agen-cies as well as by private organizations such as the Studyof Mathematically Precocious Youth (SMPY) at The JohnsHopkins University. These ideas should be interpreted notas rigid guidelines, but as flexible methods of facilitation.Since each school and each student is unique, the modelsmust be shaped to fit the individual situation.

Giftedness in mathematics is unique in comparison togiftedness in other subject areas. Until the student reachesthe level of advanced college courses, the acquisition ofmathematics follows a clearly defined sequence of courses.Thus the mathematically gifted individual who is learningat a rapid rate can follow an unambiguous path of courses.This aspect also facilitates the evaluation of the student'sprogress. Since the majority of existing curricula follow thesame basic sequence, the student has a benchmark againstwhich a comparison may be made. In addition, the sequen-tial nature of mathematics means that the material in agiven mathematics course builds upon and utilizes the con-cepts learned in the preceding courses of the sequence. Anindividual gifted in mathematics can learn mathematicsrapidly because concepts are continually reinforced in thismanner. Mathematics also differs from other subject areasin that it is not dependent on the chronological age and theassociated life experiences of the student. 'Thus even veryyoung students can excel.

Mathematically gifted students who are in a regularmathematics class waste countless hours. They spend muchof their time going over topics already learned when theycould be learning new concepts. However, the problem in-

5 C

volves more than just wasted time. These students experi-ence boredom that eventually may lead to resentment andalienation from the learning process. This alienation canmanifest itself in the form of poor study habits, laziness,classroom behav for problems, daydreaming during class,and possibly arrogance. When "turned off" students arcfinally challenged, often they are unable to succeed. Thusthese individuals become mathematically gifted collegedropouts. Clearly, these youths must be identified and givenalternatives that challenge their abilities. It is importantthat each individual be able to develop his or her potentialto the fullest.

A mathematically gifted person should not be thoughtof as a computational wizard able to perform sizable numeri-cal calculations mentally. Rather, the mathematically giftedstudent is one who exhibits extremely high mathematicalaptitude. To qualify for entrance into a special program, thegifted youth should be in the upper one percent or one-halfof one percent in mathematical reasoning in his or her agegroup. (21, 26) These gifted youths should not be confusedwith students in an "academically talented" or "honors"mathematics class. Typically, such a class consists of stu-dents in the top 10 to 20 percent in mathematical aptitudein that school. A class organized on, this standard will notmeet the needs of a mathematically giftea youth. The pupilgifted in mathematics is capable of learning mathematicsmuch more thoroughly and at a significantly faster ratethan the above-average student.

Some people expect the mathematically gifted studentto be male, socially awkward, wear thick glasses, and havea calculator strapped to his belt. However, none of thecommon stereotypes is valid, as research has clearly shown.(17, 20, 28) Mathematically gifted youths'are as diverseas any randomly selected group of people, except in the areaof mathematical talent. Such a youth may be introverted,extroverted, athletic, clumsy, diligent, lazy, male, or female.It is important to remember that these people are individ-uals first. As a result, a procedure that is effective for onestudent may not suit another.

Mathematical giftedness is a specific aptitude ; there-fore, such individuals may or may not be gifted in other

academic areas. (24) It 13 quite common to find, however,that students who excel in areas of pure and applied mathe-matics are often quite talented (and interested) in thesciences that use mathematics as a tool, such as physics,chemistry, and the engineering sciences.

Since mathematical giftedness is a specific aptitude, itshould not be equated with a high intelligence quotient (IQ)as measured by a general abilities test such as the StanfordBinet or Wechsler. An individual's IQ is composed of manyaptitudes and factors of which mathematical aptitude is onlyone. A person with a high IQ may have high mathematicalaptitude and relatively low verbal aptitude ; another personwith the same high score may have low mathematical apti-tude and extremely high verbal aptitude. The IQ is primarilya measure of verbal ability. The mathematical componentsof such a test tend to build on computational and recall skills,rather than on the reasoning component of mathematics.In addition, there are generally not enough items in themathematics area to come up with a valid and definitivemathematics score. Put differently, the IQ score tends tomask an individual's strengths and weaknesses, making itan undesirable tool for identifying the m athematicallygifted. (4, 16)

IDENTIFICATIONA variety of methods in addition fo the IQ score have

been used over the last quarter of a century to identify themathematically gifted. (31) These methods include diag-nostic testing, teacher nominations, standardized testing,nonverbal reasoning measures, peer nomination, and a com-bination of the aforementioned measures. Many of thesemethods have been found to be inadequate and/or subjectivein nature. The SMPY has found that test measures that arenationally administered and pose sufficient challenge in thearea of mathematical reasoning ability are one of the moresuccessful alternatives. To date, it has screeneci over 10,000youths via this method. (10, 11, 7, 1, 6) While specific apti-tude tests are not a perfect indicator, when properly usedthese tests are often much better indicators than, for ex-ample, teacher nominations or grades. Pegnato and Birchdemonstrated this point in a study that found that neither

7

teacher nominations nor the honor roll were good indica-tors of high IQ students. (14) For other studies verifyingthis point see Terman (29) and Stanley (25). The problemwith identification on the basis of grades or teacher evalua-tions is that these mea,sur( s are based on achievement,motivation, and attitude. The mathematically gifted studentmay well be inattentive, a behavior problem, or a lowachiever 1 ffiecause of boredom. Teachers do not think highlyof this type of student, and the student's grades reflect this.Renzulli has shown that classroom environment and teacherstyle will affect a student's performance negatively if thestudent's learning style different from that of classmates.(30) In contrast, identification criteria based on appropri-ately difficult standardized measures will catch these alien-ated studentsthe ones in greatest need of special educa-tional program:,.

The first decision to be made in the identification proc-ess is degree of selectivity. What constitutes the criterionfor entrance into a special program? There is a real dangerof yielding to the common tendency of being underselec-tivethat is, choosing students who are not sufficientlyqualified. If some students are given a difficult test or placedin a class that is much too difficult for them, it could behighly damaging to their academic outlook. As,a result, it isbetter to err in the direction of overselectivity. Put differ-ently, it is wiser to limit the program to a smaller number ofstudents who demonstrate a high probability of success,than to choose a larger number of students who show onlymarginal probabilities of success and put them in a situa-tion where they may indeed fail. This consideration is espe-cially crucial in a new program. Since there are so manydetails to handle, it is better to concentrate, at least initially,on a small, highly select group. In many cases there may bemany more mathematically gifted youths than the programcan accommodate. It is best to concentrate on the reallyexceptional ones instead of being diverted by slower stu-dents and underachievers.

The criterion of limiting selection to students whodemonstrate mathematical reasoning ability in the upperone percent or upper one-half of one percent has workedextremely well for a number of state and local educational

agencies. A small population base may necessitate less selec-tivity; it is unwise, however, to dip below the upper fivepercent for entrance into special programs in mathematics..If there is time and teachers are available, several levels ofstudents may be selected. Expectations and curriculumdesign wouk vary from group to group. For example, inChicago public schools, fast-paced and enriched mathe-matics progtains were designed for the top students identi-fied in an initial mathematics talent search. These classesmet for one two-and-a-half-hour sessicin each week. Studentsat the next level were placed homogeneou sly within learningcenters across qchools, but math was taught on a daily basis.The curriculum, was taught at a level one year ahead of thenormal content schedule. All students participating in theprogram were offered career education opportunities in thefield of mathematics.

After determining the standard for selectivity, person-nel must decide in which school grade to identify the mathe-matically gifted. Since mathematical giftedness is relatedto reasoning ability rather than to computational skill, spe-cial programs should generally start with subjects moreconceptual in nature. These subjects (for example, algebraI r:ad II, geometry) do require a certain mastery of basiccomputational skills. Therefore, the identification processshould take place after the school grades in which arithmeticis taught but before the earliest grade in which algebra istaught. In most school systems, this means either at the endof the sixth grade or during the seventh and eighth grades.Before students reach the seventh grade the computationalskills of even those who are mathematically gifted areusually not developed sufficiently to permit the learning offirst-year algebra.

Of course, there are exceptions. However, if an ele-mentary school student is deemed ready to learn algebra,her or his computational skills should be far superior tothose of the other students in the same grade. These skillsshould at least match the skills of the "average" algebrastudent. Diagnostic testing via a standardized computa-tional skills test, such as the Sequential Tests of EducationalProgress (STEP) , can be administered in order to deter-mine the skill leVel of a particular student.

9

Two-Stage Met hodThe two-stage method hos been one of the more suc-

cessful methods of discovering, mathematical talent. Theprocedure invoives screening students in the already chosengrade through two different measures of mathematical apti-tu le. Using this approach, in six talent searches coveringthe states of Delaware, Mnrvland, Pennsylvania, Virginia,and West Virginia, as well as the District of Columbia, over10,000 youths have been identified as mathematically tal-ented. Although there are other models of identification, thismethod has been used successfully by a number of schoolsystems to be cited later.

The first stage of the identification process consists ofestablishing a pool of talent from which the mathematicallygifted can be chcsen. The purpose of this step is to reducethe number of students to be screened further. Clearly, onlythe ablest students need to be examined further in order todiscover those who are truly mathematically gifted. Theivitial screening is accomplished by examining student per-formance on nationally administered in-grade measures ofmathematical aptitude or mathematical reasoning ability.To qualify, these screening measures nu st contain a sep-arate mathematical reasoning or aptitude section that canbe reported to the students. Preferably, these measuresshould provide in-grade national norms. Some commonlyused screening measures are listed in Appendix A.

After choosing the instrument to use in the initialscreening, one must decide what minimum percentile rank(on national norms) will qualify a student for the secondstage of the identification process. This criterion will varydepending on the size of the population in a given locale andthe general ability level of the students. The cutoff mayrange from the upper one or two percent to the upper fivepercent. In general, students below the 95th percentileshould not be eligible for the second stage. Most studentsnot scoring in the top five pe .c..ant would find the secondscreening a depressing and possibly traumatic experience.

There should be room for flexibility in establishing thecutoff point. For instance, one may want to incorporateteacher nominations into the first- screening. This can bedone by allowing teachers to nominate for the second screen-

1 i 10

ing those students whose first screening measure scores arejust below the cutoff point. Such nominations should not behaphazard, since the second stage screening may damagethe egos of students who do not really excel in mathematics.

After the first stage is completed and a highly able roolof ,a.lent has been identified, it is necessary to perform asecond screening of this group. The second screening isintended to separate the mathematically gifted studentsfrom those who are merely vi:ry 5.1-ile in mathematics. Thein-grade measure of math aptitude used in the first screen-ing did not adequately demonstrate individual differences.Since all scored in the upper few percent of their age group,the :al-grade measure did not possess a sufficiently highceiling to reveal differences in ability.

In the secor.d screening, the test must be difficult enoughto spread these students i-. the upper few percent out overthe whole range of possi1:1 scores. Measures that are idealfor this purpose are those that, are designed for students inhigher grade levels. Since the identification process prefer-ably occurs in a junior high grade, a suitable second screen-ing measure would be one that is normally administered toand gives norms for a rational sample of high school juniorsand seniors. Some examples are listed in Appendix B. Notethat all measures listed in Appendix B have a separate sec-tion dealing with mathematical aptitude.

It is a mistake to believe that a group of students whoscore within a few percentile points of each other on anin-grade measure of mathematical aptitude are quitehomogeneous with respect to this ability. That may be truein a lower range of the scale, such as 64th through 67th per-centile. However, a group of students in the upper few per-cent are actually extremely diverse. This is because there isno top to the measurement scale and no limit to h .nv high anindividual can be placed on the scale. Thus, two studentswho score in the upper two percent of mathematical reason-ing ability can differ tremendously in their actual abilitylevels.

Thus a measure with a sufficiently high ceiling willspread the top few percent out over the whole range ofscores. For example, consider the 1973 talent search con-ducted by the SMPY. The group being screened were sev-

11

enth and eighth grade students. In the intital screening ascore in the top two percent on national norms for an in-grade standardized measure of mathematical reasoningability qualified students for the second screening. A groupof 953 students were eligible for the second screening andwere administered the Scholastic Aptitude Test (SAT)under standard conditions. On the mathematics CM) sec-tion, the scores of this seemingly uniform group of studentswere spread out over the whole range of ability levels foreleventh and twelfth grade students who take the SAT-M,from the lowest possible score of 200 up to a near perfectscore of 800. In other words, a very able student who scoredat the 99th percentile on the initial in-grade screening meas-ure earned a score of 360 on SAT-M, whereas a mathe-matically gifted student who had the same rank on thein-grade measure scored 710 on SAT-M. It is for studentsin the latter category that we need to develop special pro-grams in mathematics.

After obtaining the .second screening scores, one mustdetermine what will be the cutoff for eligibility to a specialprogram. Although such a cutoff may seem arbitrary andunfair, limited resources often dictate this approach.

There is some flexibility in the choice of a cutoff scoreat this point, although one must be careful not to be under-selective. Especially when a program is just getting under-way, it is better to select only a few students who are sureto succeed. Several school districts that have used the SATand have developed successful programs for junior nighstudents require participants to have a score of at least 500on SAT-M. This score corresponds to the 57th percentile onnorms for college-bound seniors and corresponds to aboutthe 80th percentile on norms derived for all eleventh andtwelfth grade students. In its six talent Searches, the SMPYhas found that of the 10,000 seventh and eighth graderswho scored in the upper three percent on the initial screen-ing, about one-quarter scored at least 500 on SAT-M. Thisimplies that the students who score 500 or more are at leastin the approximately top one percent of their age group inmathematical reasoning ability. We educators should beconcentrating our efforts toward this group of mathemati-cally gifted youths.

-4. 3 12

Table 1 shows how some school systems have used thistwo-stage me-lc.1 fo/. the identification process. Please notethe different meashres used at the second screening level (s).

TABLE 1.SCREENING MEASURES USED AT THE SECOND LEVEL OF THE TWO-STAGEIDENTIFICATION PROCESS BY VARIOUS STATE AND LOCAL SCHOOL DISTRICTS

Grade at time Screening measureof testing usedProgram

State of Delaware(Rural, suburban, andurban)

State of Illinois(Rural, suburban, andurban)MarylandBaltimore County(Suburban)Charles County(Rural)Howard County( Suburban)

MinnesotaMinneapolisSt. Paul(Urban)Duluth(Urban)

NebraskaEducational ServiceUnit No. 11(Six rural counties)Omaha(Urban)OklahomaBartlesville(Rural)PennsylvaniaMontgomery County(Suburban)WisconsinEau Claire region(Suburban)

7

7

Scholastic Aptitude TestMathematics (SAT-M)

School and College Ability TestQuantitative (SCAT-Q)

7 School and College Ability TestQuantitative (SCAT-Q)

7, 8 Preliminary Scholastic AptitudeTest (PSAT)

7 Orleans-Hanna AlgebraPrognosis Test

7, 8 School and College Ability TestQuantitative (SCAT-(1)

7-9 School and College Ability TestQuantitative (SCAT-Q)

Pre-10, 11, and12 Summer

7

7

1971-72 practice test inmathematics of the CollegeBoard Scholastic Aptitude TestSchool and College Ability TestQuantitative (SCAT-Q)

Orleans-Hanna AlgebraPrognosis Test

8, 9 Scholastic Aptitude TestMathematics (SAT-M)

7, 8 Scholastic Aptitude TestMathematics (SAT-M)

At this point, the process of identifying mathematicallygifted youths has been completed. It is important to realizethat at any stage a student should have the option not toproceed further. A special program should only include stu-dents who are eager and highly motivated.

Before proceeding to a discussion of teaching strate-gies, there are some terms that need to be clarified. Con-cepts such as acceleration, enrichment, and creativity must

13 X4

be defined in order to avoid the misunderstanding that theseterms often generate.

APPROACHESAccelerated education is based on the concept of indi-

vidual differences and connotes the idea of a person movingthrough one or more subject areas more quickly than in thestandard program designed by the various school sys-tems. (8) This alternative is suited for the upper few per-cent of students in a particular age group or grade. Thepurpose of acceleration is to meet the intellectual needs ofgifted studentsneeds that are not met through the stand-ard age-grade lockstep approach to education. The rate andneed for acceleration varies from individual to individual.The rate of learning should be commensurate to the stu-dent's ability and motivation.

Enrichment of education implies that sL 'aject matter ispresented in greater depth for the gifted student than forother students. Material is added to the standard course,but the grade or course placement of the gifted youth isneither advanced nor retarded.

Creativity is concerned with thinking in unique andinteresting ways. For example, a creative thinker couldprove a geometry theorem in an original way. Despite theclassical definition of creativity, it seems that unique, inter-esting methods of learning in themselves do not generallystress the content or curriculum.Enrichment

Whenever professionals discuss teaching strategies formathematically gifted students, enrichment of curriculumrather than educational acceleration is the most popularalternative. The four most common forms of educationalenrichment might be termed busy work, general academicenrichment, cultural enrichment, and specific academicenrichment. (20)Busy Work

Some teachers use busy work to keep the brightest stu-dents occupied while the rest of the class proceeds in thenormal manner. Busy work consists of having bright stu-

1 ,514

dents do more work relating to a specific subject than theaverage students. It is an easy method to implement, sinceit requires virtually no extra effort from the teacher. Thisextra work is not designed to help students understand newconcepts, only to give additional practice in performingwork that has been learned previously. For instance, whilemost students in a mathematics class may be assigned everyother problem in the textbook, a gifted individual might beasked to do every problem. Because mathematically giftedstudents are often forced to do extra work in a subject inwhich they are superior as compared to classmates, theycan become bored and frustrated and lose interest in thesubject. Actually mathematically gifted students need to dofewer problems than average students, since they can learnmathematical concepts far faster and with less practice. Inthe previous example, it would be better for the mathe-matically talented youngster to do every fourth problem,rather than every problem.

Another example might help amplify the dilemma facedby mathematically gifted youths. Mark, who had an IQ of187 in kindergarten, was a 12-year-old eighth grader takingAlgebra I. Hoping to challenge him, Mark's teacher askedhim to work every problem in the book while the rest of theclass was assigned every other problem. This mathemati-cally gifted student already knew Algebra I quite well andneeded very little practice in the material. He naturally re-sented the burdensome and boring task he was given.

Ironically, this situation motivated Mark to attain veryhigh standards. Bored in Algebra I and encouraged by amathematics specialist, in the spring semester of eighthgrade Mark took a computer science course at The JohnsHopkins University and received an A. From then on hetook the remainder of his mathematics coursework at thecollege level. The summer after eighth grade, Mark took .acourse in college algebra and trigonometry and received aB. In the following two years, he completed the collegemathematics courses through calculus and linear algebraand took two chemistry courses, receiving all As. WhenMark was only 15 years 2 months he entered JHU as a full-time student, with his freshman year and 30 percent of hissophomore year completed. Today he is a superb graduate

15

student at Massachusetts Institute of Technology.In a way, Mark's eighth grade algebra teacher did this

brilliant young man a favor by causing him to find anotherway of challenging himself. However, without the specialopportunities for educational acceleration in mathematics,Mark would have been forced to sit through a number ofroutine full-year mathematics courses far below his abilitylevel.

General Academic Enrichment and Cultural EnrichmentGeneral academic enrichment does not provide the

specific type of advanced stimulation that a gifted studentneeds. Instead, it often consists of offering such subjectmatter as a special high-level social studies course for allhigh IQ youths, or perhaps something essentially nonaca-demic like chess or creative training (process-oriented). Forthe most part, these offerings are not related to specific sub-ject matter and generally do not meet the particular need-.or interests of mathematically gifted youths.

Cultural enrichment involves providing special culturalexperiences beyond those offered in the usual school cur-riculum. Classes in music pprechttion, the performing arts,and foreign languages may be offered. There is nothingwrong with this kind of program ; however, these classescertainly do not meet the specific needs of a mathematicallygifted youngster. Indeed, such experiences are often bene-ficial for all students regardless of ability level.

Specific Academic EnrichmentFinally, there is the form of educational enrichment

termed specific academic enrichment. This approach wouldprovide, for example, a special mathematics curriculumfrom kindergarten through seventh grade designed forthose students whose abiilty to reason mathematically putsthem in the upper 10 percent of their age group. But sup-pose that such a student goes through this program andthen takes a regular Algebra I course in eighth grade. Obvi-ously, the student will be bored and frustrated with thecomparatively slow pace. Even if the special curriculum iscontinued through twelfth grade, there are still problems.If the enriched program is truly designed for the mathe-

1 7 16

matically gifted, participants will have learned mathematicsthat nonparticipants will not learn until they attend college.Unless ..-ame provision is made for participants to earn highschool and/or college credit for what they have learned,they will end up repeating coursework they have alreadycovered. Thus the boredom and frustration are merely post-poned. The resulting lack of stimulation and decelerationof potential may be worse than if no enrichment had takenplace. Dr. Julian C. Staniey, Director of SMPY, states, "Themore relevant and excellent the enrichment, the more it callsfor acceleration of subject-matter or grade placement later.Otherwise, it just puts off the boredom awhile and virtuallyguarantees that eventually it will be more severe." (19)

Educational AccelerationIt follows then that relevant academic enrichment in

mathematics is a good approach only if it is pursued withan eye to long-range planning and if the student receivescredit for what is learned. When these conditions are met,then what is taking place may be called educational accel-eration.

The best support for the implementation of educationalacceleration is found in the many examples of mathemati-cally gifted students who have accelerated their educationswith good results. (28) As previously stated, mathematicsis ideally suited for acceleration because the rate of learn-ing is more dependent on intellectual ability than on chrono-logical age and life experiences. Nevertheless, there mustbe strong student motivation. Educational acceleration mustbe the student's choice ; it cannot be forced by parents,teachers, or administrators. If the student is not willing toaccelerate, he or she will not want to do the amount of worknecessary for success. But for those students who are eagerto move ahead, educational acceleration has proven to be anexcellent alternative.

For example, as of spring 1979, 250 of the studentsidentified by SIVirY (in a five-state region) had taken 589college courses prior to entering college full-time. Theirmean grade -post average was 3.52 out of 4.0 (A = 4,B = 3, et -.). The average age of this group at thetime thatthe courst_ was taken was 15 years. Although SMPY began

17

in 1971 by facilitating se renth and eighth graders, alreadymore than 220 young men and women identified by SMPYhave entered college one to s;x years early.

Many highly talented professionals have acceleratedtheir educations in some way. Norbert Weiner, the origina-tor of cybernetics, received his doctorate at the age of 18.Charles Louis Fefferman became a full professor of mathe-matics at the University of Chicago when he was only 22years old. Two years before he had received his Ph.D. Hebecame the first winner of the National Academy of Sci-ence's $150,000 Waterman award when he was barely27. (12) The list of successful early graduates is long andfar outweighs the case of William James Sidis, who failedsociety's norms of accepatable behavior after having grad-uated from Harvard at the age of 16. (13)

Those opposed to educational acceleration are oftenconcerned that the social-emotional growth and creativedevelopment of these students are hindered. This concerngoes to the extreme of ignoring the other crucial factor,namely intellectual development. There are numerous casesof accelerated students who have made extremely goodsocial adjustments and have not been creatively disadvan-taged by acceleration. (27, 33) Furthermore, no studiesshow empirically that acceleration is bad, while manystudies demonstrate that it is good. (2)

This is not to say that acceleration is a perfect solutionto the problem of the mathematically gifted. In nearly allcases, there is no perfect solution. One must determinewhether if this direction is not taken, a serious problem mayresult. The danger is that when a student's learning ratedoes not keep pace with his or her ability, there is a greatrisk of diminishing or even extinguishing a student's aca-demic motivation in areas where he or she is quite talented.Again, the decision must be made for each individual stu-dent; an educational program that is right for one individ-ual may not meet another's needs.

TEACHING STRATEGIESFrom the previous discussion, it is clear that any teach-

ing strategy geared to meet the needs of the mathematicallygifted student must be based on altering the rate at which

18

students progress through the typical mathematics sequence.This rate must not only exceed that of an average mathe-matics class, it must also be faster than the rate of a typicalhonors class consisting of students in the upper 20 percentin mathematical reasoning ability. The focus must be on theupper few percent. This select group is able to learn mathe-matics at such a great speed that a class designed for theupper 20 percent will only bore and frustrate the mathe-matically gifted. An excellent example of the tremendousspeed at which mathematically gifted youths can learn pre-calculus mathematics is demonstrated by a special classconducted from June 1973 to August 1974. (9) During 14months of meeting for one two-hour class per week, 16ninth graders from the Baltimore area completed AlgebraII, Algebra III (college algebra) , plane geometry, trigonom-etry, and analytic geometry. Thus by the beginning of tenthgrade these students were prepared to take a rigorouscourse in calculus.

A number of techniques pioneered by programs suchas SMPY have proven to be extremely successful withmathematically gifted youths. Prior to presenting some ofthese techniques, it is important to note several strategiesthat should be avoided. Above all, it is clear that somethingmust be done for the mathematically gifted. Ignoring thespecial needs of these youths often results in boredom, frus-tration, and possible behavior problems. Without sufficientchallenge these students may demonstrate poor study habitsand little ambition and initiative. These students eventually"turn off" to academics and never realize their full poten-tial. (32)

As mentioned previously, busy work is not a solution.Self-pacing or independent study is another technique thatis not very successful with this group. There must beteacher-pacing, accompanied by a program of evaluationand feedback. Any student who is able to pace himself orherself through an independent study course without feed-back clearly has no need even to attend school.

The Fast-Paced Mathematics ClassThe accelerated mathematics class is one strategy that

SMPY has found very successful with gifted mathematics

19;1 a

students. The accelerated mathematics class is specially de-signed for those in the top few percent in mathematicalreasoning ability and speeds up the rate of learning to apace that is consistent with students' abilities. In a regularmathematics class, much of the class time is spent goingover and over a topic. The pace usually is geared to the mid-dle or lower ability range so most everyone in the class cankeep up. The average student often takes a long time tolearn a single topic, then forgets it, and must relearn it, andcannot build upon that topic when working further alongthe sequence. A youngster gifted in mathematics does notneed repetition. He or she can learn a topic quickly and canprogress further in the sequence of mathematics becauseone topic builds on another.

One type of accelerated mathematics program is thefast-paced, enriched mathematics class in which all studentsmove at the same rapid pace. (5) In order to organize sucha class, the gifted students must be homogeneous in termsof ability and level of knowledge in mathematics.

Instructors who teach a fast-paced, enriched mathe-matics class should gear their teaching to the top one-thirdor one-quarter of the class. When this group of studentsgrasps a concept, the instructor moves on to the next topic.It is essential that this pace be maintained. The instructormust move ahead before every student understands a con-cept. These highly talented students generally can learnthose topics not mastered in class by doing the homework.Homework, as a result, becomes a learning tool as well as aconcept reinforcer. It is important not to slow down thepace of instruction for the sake of a few stuc';A.hts. Studentswho truly belong in the class will be able to keep up with thepace so long as they are willing to spend extra study timeoutside the classroom.

The style of instruction should not be strictly lecture.Student participation should be maximized. Rather thanpresent fully explained topics to the class, the instructorshould try to involve the students in the development of aconcept. For instance, in the course of a geometric proof,the instructor can continually solicit suggestions as to howto proceed. Or the instructor can ask the class to deduce thedefinition of a previously undefined geometric term such as

20

a triangle. Frequently a student will solve a problem orprove a theorem in a way that is different from the tech-nique the instructor would have chosen. In addition, stu-dents will constantly be jumping ahead to the next topic.The instructor must be flexible enough to allow this creativ-ity and should not try to make students conform to her orhis way of thinking. The goal of the instructor is to encour-age the students to be actively involved in the instructionalprocess. A skillful teacher, however, will not allow this kindof spontaneous participation to sidetrack the class in irrele-vant directions.

The interaction of the gifted students is further facil-itated by periodic blackboard work. By having the studentswork a particular problem at the board, the instructor ob-tains immediate feedback on concept mastery. This approachalso facilitates individual help from the teacher and fromclassmates. One of the chief benefits of a fast-paced, en-riched mathematics class is the chance to have these brilliantstudents interact with their peers in doing board work andin contributing to the overall lesson.

Homework is an essential aspect of any fast-pacedmathematics program. The material is covered in class soquickly that it must be reinforced by extensive homeworkassignments. The amount of homework should not be sogreat that it places a burden on the student or becomesrepetitive or busy work. Nevertl-sless, homework can helpa gifted youth make the most of his or her abilities. Studentsparticipating in such a program cannot do all of the home-work at the last minute. They must develop good studyhabits in order to space out the homework over the intervalbetween classes. The discipline of pacing themselves throughthe homework assignment also leads to better recall of thematerial.

The instructor should spend very little classtime dis-cussing homeworkperhaps 10 or 15 minutes for a two-hour class. Even then the teacher should wait until aftergrading the homework, unless some students have experi-enced serious conceptual problems while attempting to dothe assignment and are unable to understand further ma-terials. While grading the homework, the instructor shouldbe on the lookout for conceptual errors. Classtime should be

21

spent discussing only those concepts that caused problemsfor a significant number of the class. It is also benefic7a1 todiscuss a few challenging problems that stumped many ofthe students. If, for instance, only one student solved a par-ticular problem, ask that student to explain it to the rest ofthe class. Problems or concepts that are missed by smallnumbers of students can be handled by making appropriatenotes on the homework or through extra h.t1r, outside ofclass. Therefore, the cycle of a homework atrAgnment is tomake the assignment during one class, collect the work atthe next class, and go over parts of it at the beginning of thefollowing class (two classes after it was assigned).

The pace of the class is so rapid that it is sufficient tomeet only once a week for two to two-and-a-half hours. Ap-propriate pacing will keep students busy between classesas they learn the concepts they missed or reinforce the con-cepts the rest of the class might have missed.

It is important to realize that the fast-paced, enrichedmathematics class will function properly only if the groupof participating students is quite homogeneous. Frequently,in a group of gifted mathematics students, some studentsknow more mathematics than the others, despite being inthe same grade. In addition, individual learning rates maydiffer significantly. In such a situation, one does not want touse the fast-paced approach with the whole group. Somestudents might be bowed, and some would be forced to dropout. In short, these gifted students would be no better offthan they were in the regular mathematics class. If the poolof students is small, it is impractical to break them up intosmaller, homogeneously grouped classes that function inde-pendently. Such a situation calls for an individualized class-room approach.

Individualized Classroom ApproachSMPY has formulated a technique that accomplishes

the goal of individualized instruction : Diagnostic Testingfollowed by Prescriptive Instruction (DT PI) . TheDT PI model has been replicated by several school dis-tricts. Its use as a classroom strategy can best be explainedby first detailing its application to a situation in which thereis only one mathematically gifted youth.

The Single Gifted StudentSuppose that it has been established that a certain

youth has an extremely high ability to reason mathematic-ally. Furthermore, assume that this youth has just com-pleted seventh grade and has had no formal course entitledAlgebra I. It is highly probable that this student alreadyknows a significant. amount of Algebra I. By administeringa diagnostic test to assess the student's knowledge of first-year algebra, the student's mentor (a special kind of tutor)is able to discover just how much the student actually doesknow. The mentor then designs a program of prescriptiveinstruction to help the student learn rapidly only those con-cepts the student does not know. When the mathematicscurriculum is streamlined to teach only what the studentdoes not already know, the rate of learning increases dra-matically. Note, however, that the mentor must use many ofthe same instructional techniques used by the teacher of afast-paced, enriched mathematics class.

It is especially helpful if there are two parallel formsof each diagnostic test used to assess knowledge of AlgebraI, II, III, plane geometry, trigonometry, analytic geometry,and calculus. This permits pre- and post-evaluation ofknowledge learned. Another useful tool, if available, is ananalysis chart that breaks down the items on each test intotopics. Such charts are especially helpful in decipheringwhat a student does and does not know. One example of adiagnostic test series that meets all these requirements isthe Cooperative Mathematics Test series developed by theEducational Testing Service.

Following is an example of the DT .--> PI procedure asused by a hypothetical mentor who is facilitating one hypo-thetical mathematically gifted seventh grader who has hadno formal Algebra I course. This outline is adapted from ashort article that appeared in ITYB (Intellectually TalentedYouth Bulletin). (22)

1. Administer to the student a diagnostic test of first-yearalgebra knowledge. Observe carefully the instructionsthat come with the test, giving special attention to thetime limit. Provide plenty of scratch paper and extrapencils. Do not answer any questions about the content

23

"4

of the items. Let the student know when half the testingtime is over and when five minutes are left. In additionto notifying the student in advance of the time limit andnumber of items, the following points of test-takingstrategy should be explained.a. Since the final score on the test i; the number of cor-

rect answers, the student s1,9uld mork an answer forevery item, even if it is a gue.;..

b. The student should work as qui and -.fficiently aspossible, not spending too much ti- ..)n any one item.

c. The student should place a quesi.:a;i mark on theanswer sheet next to those items of which he or sheis unsure. If time permits, the student can return tothe marked items for further study. The studentshould make all marks on the answer sheet or scratchpaper, not in the actual test booklet.

2. When the testing time expires, the mentor collects allof the diagnostic materials (test booklet, answer sheet,scratch paper). In order to protect the security of themeasure, the scratch paper is destroyed. Carefully scorethe answer sheet in another room away from the stu-dent. Determine the percentile rank of the score (ifavailable) from a table of national norms for Algebra Iknowledge.a. If the percentile rank is below 50, then the student

clearly knows very little of Algebra I. If this is thecase, it is best to start at the beginning of Algebra I:?rid efficiently cover all of first-year algebra at arapid pace (see Figure 1).

b. If the percentile rank is at least 50, proceed to thenext steps.

3. Give the studer t, a list of the missed items, the test book-let, and ask the student to try those items again onscratch paper. Do not let the student know how she or heanswered the items on the answer sheet or, for that mat-ter, what are the correct responses. Have the student at-tempt these items again without imposing a time limit.This time, the student should leave blank those itemsshe or he has no idea of how to solve.

24

DIAGNOSTIC

TESTING

11211101111

DIAGNOSTIC

TESTING

FIGURE 1: FLOW CHART FOR DT PI MODEL

INSTRUCTION

INSTRUCTION COVERING

ESSENTIALLY THE

WHOLE COURSE

IF PERCENTILE

RANK IS:

PROGRAM Or' PRESCRIPTIVE

INSTRUCTION COVERING

ONLY UNLEARNED TOPICS

w

NOOPIIIIM

PRESCRIPTIVE

INSTRUCTION

OPTIONAL

NEXT

LEVEL

I I r

STANDARDIZED

EVALUATION

06

REST OF

PROCESS SAME

AS IN FIGURE 2

4. After the student has finished all of these items, thementor should again collect the diagnostic materials andmark the answers. The mentor should make sure thatthe items now answered correctly are actually under-stood and were not mere guesses. This can be done byquerying the student about the method of solution. Theitems that are answered correctly at this time were prob-ably missed under standard conditions due to careless-ness or lack of time. These are hardly valid reasons forteaching the student those concepts. The same certaintyshould be sought for those items answered correctlyunder standard conditions but which had a questionmark next to them.

5. The items which the student has missed with unlimitedtime should be examined carefully by the mentor. Byreferring to the student's answers each time (i.e., werethey the same or different both times?) and by queryingthe student, the mentor should be able to deduce whichmethod of solution was used. Sometimes it will be clearthat the student had no idea how to solve an item. Ifavailable, an item analysis chart should be utilized inthe following way :a. Label the sheet with the student's name and score

under standard conditions.b. Circle the number of all items missed under standard

conditions.c. Check off those items that were answered correctly

the sezond time.d. The grouping of the remaining items gives a clear

picture of the concepts the student does not know.

6. At this point, the mentor knows enough about the pointsof ignorance in the student's knowledge of Algebra I tobe able to devise a program of prescriptive instruction.Using a good Algebra I text, the mentor should guide thestudent through those concepts that need to be covered.It is imperative that the student not work through theentire book, but cover only those points not already wellunderstood.

26

7. The program of instruction must be mentor-paced andnot student-paced. More will be said about this later.

8. The goal of the mentor is to pace the student throughthe unlearned material as rapidly as is appropriate whilepreparing that individual to achieve an almost perfectscore on a different evaluative measure of Algebra I (forexample, an alternate form of the diagnostic test).

9. If the student obtains a very high score, say above the90th or 95th percentile on national norms, then it maynot be necessary for the student to take another AlgebraI test. Nevertheless, the missed items should be reviewedusing the above procedure. If it is clear that the studenthas no holes in his or her knowledge of Algebra I, thementor may proceed directly to the next content area,probably Algebra II.

10. When the student is ready to have his or her knowledgeof Algebra I evaluated, the mentor may want to read-minister the original diagnostic measure. This can beused as a partial indicator of the student's progress,especially if the first score was very low and a few weekshave passed since the first evaluation.

11. When Algebra I has been mastered, the same diagnostictechnique is applied to the next subject, preferablyAlgebra II. The mentor may decide to skip the diagnos-tic testing in Algebra II and commence instruction im-mediately. This is a valid alternative if the studentscored very poorly, say below the 50th percentile on na-tional norms, on the initial Algebra I test. The judgmentof the mentor at this point is crucial. If it seems likelythat a great deal of Algebra I has been learned, diagnos-tic testing at this level is a must. Certainly the mentorshould use diagnostic testing in Algebra II if the studentscored well (for instance, a percentile rank of 85) on theinitial Algebra I test administered prior to any instruc-tion.The DT PI procedure described above can be modi-

fied for use with a good Algebra I text that has a reviewtest at the end of each chapter. In this scheme, the mentorasks the student to complete a sampling of the Chapter 1

27

test (for example, the odd-numbered items) without refer-ring back to Chapter 1. The mentor analyzes the student'sanswers in the same fashion as with the standardized test.Those items answered incorrectly illustrate which sectionsof Chapter 1 need to be learned. When the student has mas-tered those concei: ts, the mentor assigns the even-numbereditems in the Chr, ;ter 1 test. The student should achieve anearly perfect score on these items. Once the student hasmastered Chapter 1, the mentor proceeds in the same waywith Chapter 2. When the student has progressed throughthe Algebra I material in this fashion, the mentor may thengive the student a standardized Algebra I test. Notice thatthis modification is consistent with the model of teachingonly what the stt- 'ent does not already know. The modifica-tion is especially useful when a student scores very poorlyon the initial Algebra I test.

It is important to note that the DT PI modeldescribed earlier focused on a seventh grader who had nothad a formal Algebra I course. If, however, the studentbeing considered is in ninth grade and has had Algebra I,the mentor would start the diagnostic testing with AlgebraII. The mentor always begins one level higher than the lastmathematics course the student has taken formally in school.

When using the DT > PI model, the rate of progressionthrough mathematics is rapid enough that the mentor needsto meet with the student only once a week for two to threehours. The composition of this class is similar to that of thefast-paced, enriched mathematics class. As before, the home-work is to be extensive without being repetitive, and it mustbe done thoroughly and well. Needless repetition can createunnecessary frustration for the student. For example, con-sider the case of John, an eighth grader who scored 40 outof 40 items on a diagnostic measure of Algebra I. Whyshould this pupil be forced to sit through 180 50-minuteAlgebra I classes? As an eighth grader, John wanted to bein a ninth grade Algebra I class, but the Algebra I teacherrefused. John was sure he knew Algebra I, so he took anAlgebra I diagnostic test. True to his expectations, heachieved a perfect score. John also scored in the high 30son 40-item tests of Algebra II and III. After hearing that,the Algebra I teacher relented and told John he could take

28

Algebra I. John decided instead to take a college calculuscourse, and he did very well in it.

Teaching Small' Groups of Gifted StudentsThe DT ---> PI model can be used by an instructor who

is teaching heterogeneous groups of mathematically giftedstudents. Initially the instructor goes through the diagnostictesting procedure with each student. The results lead to theformation of small, homogeneous groups. Some students,especially those who are much more talented than the others,may have to be instructed individually. The teacher workswith each group separately, pacing the group through thematerial using the techniques described previously. Severalgroups can be functioning at once, but the teacher must becareful not to be sidetracked by any one student or anyone group.

The instructor works with a group for a short time,pacing students through the material or explai- ing a con-cept. Then the instructor's attention shifts to ant ,ther group.While one group is working with the instructor, the othergroups must be active. For instance, the instructor can ex-plain a topic to group A, assign a few problems to checktheir understanding, and then go work with group B whilegroup A is solving the problems. The teacher later returnsto group A to check the problems and pace them throughanother section.

Obviously, there will be gaps, as when a group of stu-dents is waiting for assistance from the instructor. This iswhy it is important to choose a good text. The text must beclear and succinct enough that students can rapidly teachthemselves without constant help from the teacher. Thus ifthe teacher is unavailable at some point during the class, thestudents can still progress through the material by readingthe text. This approach also permits some individuals tomove faster than others.

The purpose of the small groups is to encourage inter-action as students ask each other questions and assist oneanother. The composition of the groups must !--r2 flexible ; thebrighter students should be able to move ahead and out ofthe group, possibly into another group.

As with previous strategies, this technique is alsoteacher-paced, not student-paced. The instructor's chiefresponsibility is to stimulate the students to move throughthe material at a pace consistent with their abilities. Theteacher must guide students through only the necessaryparts of the text and not force them to relearn conceptsalready known. The students should know at all times whatpoint is to be learned and how fast it is to be mastered.

For an accelerated mathematics class, the best sequencefor covering precalculus mathematics is Algebra I, AlgebraII, Algebra III, geometry, trigonometry, analytic geometry,and calculus. The rationale behind this sequence is that eachcourse is reinforced by the next. Mathematics that is learnedquickly must be reinforced by building upon it immediately,as Algebra III builds upon and reinforces Algebra II.

Some believe that holes develop in a student's knowl-edge when mathematics is learned rapidly. A skilled instruc-tor will not allow this to occur. The feedback generated fromhomework and testing should be scrutinized carefully tomake sure that no holes develop. An accelerated pace ofinstruction does not imply that material is omitted. It meansonly that the students' level of reasoning ability is highenough that the instructor does not need to predigest ma-terial, give lengthy explanations about concep s, or go overpoints, more than once or twice. In fact, because :::he youthsin an accelerated mathematics class are so brilliant, theinstructor can cover topics in greater depth than an averageclass is able to handle. Ironically it is more likely that gapsin knowledge will occur when a mathematically gifted stu-dent is placed in an average class. Because the studentalready knows many sections of the material, the courseseems repetitive and slow. The student will frequently be-come inattentive and miss points that he or she genuinelyneeds to learn.

These two strategies for teaching mathematically giftedyouthsthe fast-paced, enriched mathematics class and theDT -a. PI modelhave each been implemented successfullyin various secondary school systems as well as in specialprograms such as SMPY. The following examples shouldhelp instructors appreciate the effectiveness of these Iwostrategies.

11

30

IMPLEMENTATIONThe Fast-Paced, Enriched Class Model

There are a number of programs that have been basedon the homogeneously grouped fast-paced, enriched mathe-matics class model. One such class sponsored by SMPYlasted for 14 months, from June 1972 until August 1973.The class began with 19 students from the Baltimore areawho had just finished sixth grade or were of that age. Sixstudents left the class by the end of the summer. In Sep-tember three more students were added. All 16 of thesestudents quickly completed Algebra I and II. Ten of the 16students then completed Algebra III and trigonometry. Nineof the 10 continued in the class and completed analyticgeometry. Eight of these nine studied further material dur-ing the summer of 1973 and completed plane geometry.These eight students were thus prepared to begin calculusin the fall of 1973. (3, 23)

Throughout the duration of the program, this class metfor two hours each week on Saturday mornings. Therefore,the total amount of instructional time for eight students whoremained in the class for all 14 months was approximately120 hours. These eight completed the entire precalculussequence in 120 hours. Students in the regular school cur-riculum take 180 50-minute classes for each of four-and-a-half years! Even the students who left the program beforeits conclusion made astounding progress through part ofthe precalculus sequence when compared to the averagesecondary school mathematics student.

In Minnesota (Minneapolis-St. Paul and Duluth areas)fast-paced mathematics programs are currently in theirfourth year of existence. The programs are designed as atwo-year bridge of the limited precalculus course offeringsavailable in junior high school. In approximately 120 hoursof instruction spaced over a two-year period, students coverAlgebra I and II and plane geometry. When these studentscomplete junior high school, they are then permitted to con-tinue at the next appropriate level in the mathematicssequence taught in the high school. The suburban school dis-trict of Howard County (Maryland) and schools in the stateof Delaware rely on this approach. In each case instructors

31

have found it an excellent way to bridge the age-grade gapin mathematics content between elementary school andsenior high school.

In rural Charles County (Maryland), a fast-pacedmathematics program that starts in the second semester ofthe seventh grade h- s resulted in the development of a cal-culus program for the school system. Until this fast-paced,enriched mathematics model was introduced, calculus wasunheard of as a high school subject.

There is one factor common to all school districts, re-gardless of the talent population and transcending the rural,urban, and suburban classifications. Each district that hasinstituted the concept has found that students develop anew attitude towards mathematics, resulting in increasedexpectations as to what can be accomplished in a learningatmosphere that is both stimulating and challenging.

When mathematically gifted youths are placed togetherin one class they can learn calculus rapidly and at the rigor-ous level of a college course. Consider, for example, onecalculus class that lasted from September 1974 to May1975. (23) This special class, which met for two hours onSaturday mornings, supplemented the regular high schoolcalculus classes each of the 13 students was taking. Most ofthe students previously had been in a fast-paced, enrichedprecalculus mathematics sequence. This supplemental cal-culus class delved into material in greater depth than theregular high school class. The purpose was to prepare stu-dents to take the more difficult (BC) level Advanced Place-ment Program (APP) exam in calculus, administered na-tionally each May. While many high schools offer calculus,only some offer it at a level high enough so that the studentsare prepared to take the less difficult (AB) level APP cal-culus exam. Very few high schools have the resources tooffer a calculus course sufficiently advanced to prepare stu-dents to take the more strenuous BC level A PP exam. How-ever, due to the ability level of the students in this specialcalculus class, the group was able to cover thoroughly theadvanced topics tested by the BC level exam.

In order to measure student progress, six instructor-designed tests were administered. In May all 13 of the stu-dents took the BC level calculus examination. Nine students

32

earned scores of 5, three received 4s, and one received a 3(on a 1 to 5 scale, where 5 is highest) . The significance ofthese scores is that a 4 or 5 on level BC is worth two semes-ters of calculus at most selective universities, enabling par-ticipants to get a head start on their college educations. (AtJohns Hopkins University, two semesters of calculus areworth eight credits, where 15 credits is the normal course-load for one semester.) It also means that high-scoring stu-dents have the opportunity, while still in high school, totake more advanced college courses that build upon the cal-culus, thereby offering even greater stimulation and chal-lenge.

The DT PI ModelThe classroom strategy of diagnostic testing followed

by prescriptive instruction has not been used as much asthe strategy of hothogeneously grouped fast-paced mathe-matics classes. It has, however, been extremely successfulin the situations where it has been implemented. SMPY usedthis model with two groups of students that met for eightweeks during the summer of 1978. The older group con-sisted of 44 mathematically brilliant youths who had justcompleted the eighth grade. They met with skilled mentorson Tuesdays and Thursdays for three to five hours each day.A summary of the courses completed by this group is givenin Table 2.

The younger group consisted of 33 mathematicallygifted students who had just completed the seventh grade.This group met only on Wednesdays during the eight weeks,for five hours each day. A summary of the accomplishmentsof this group is also given in Table 2.

In each of these two groups, many students already hadan understanding of various sections of the precalculussequence. Nevertheless, almost none of the Wednesdaygroup had taken any formal school courses beyond AlgebraI, if that much. Likewise, the majority of the Tuesday-Thursday group had not progressed beyond Algebra II informal coursework, if that far. Despite this, these young-sters had learned much advanced precalculus. They wereable to progress quickly through several subjects becausethey were taught only the concepts they had not already

33

TABLE 2.RESULTS OF VARIOUS PROGRAM IMPLEMENTATIONS OF THE DT PI MODEL

ProgramNumber ofstudents

Grade(s)when inprogram

Typicalamount ofcoursework

covered

Maximumamount ofprecalculuscoursework

covered

Maximumhours of

instruction

SMPY's Summer1978 WednesdayProgram

33 7-8 2 4 40

SMPY's Summer1978 Tuesday-Thursday Program 44 8-10 2 4 48

SMPY's Summer1979 Program

94 7-8 2 4 40

Nebraska's 1978Summer Programfor ESU #11

8 9-11 2 3 60*

Nebraska's 1979Summer Programfor ESU #11

16 8-12 2 3 60*

This includes homework time, since the Nebraska summer program was residential. Home-work and instructional time is included in their six hours each day.

learned. As a result, 12 out of the 33 in the Wednesday groupwere certified as having completed the precalculus sequence.Students in the Wednesday classes, meeting for a total ofapproximately 40 hours, successfully completed an averageof 1.94 years worth of precalculus mathematics. The Tues-day-Thursday group, which met for a total of 48 to 60hours, successfully completed an average of 2.20 yearsworth of precalculus mathematics. If these students had nothad this opportunity, they typically would have had to sitthrough a full year each of routinely paced precalculus sub-jects, repeating many concepts they already understood.

The DT PI mod& was also used in the mathematicssegment of the first two Summer Honors Programs spon-sored by the Educational Service Unit (ESU) #11 in ruralsouth central Nebraska (see Table 2) . The classes met sixhours a day for ten days during two-week periods in June1978 and June 1979. A total of 24 students, who ranged fromend of eighth grade to end of twelfth grade, participated inthe two classes. Under the direction of two trained mentors,the students in these two classes typically completed twoprecalculus courses in approximately 60 hours of instruc-tional time.

No matter which of the described teaching strategies isused to teach mathematically gifted students, the learningrate is so rapid that a class of extended duration needs tomeet only once a week for two to two-and-a-half hours.Depending on the situation, one model may be better suitedthan the other. For instance, the fast-paced mathematicsmodel is applicable only where there is a large enough popu-lation base to enable identification of a group of homog-eneous studentsfor example, in a metropolitan area inwhich a county-wide program is implemented. The DT > PImodel is applicable to all situations, but especially useful inlocales where the population base is so small that any class-size group of mathematically gifted students would be quiteheterogeneous. The DT --> PI model could be implementedwith a whole class on a district-wide basis, or in individualschools where in many cases less than 10 students comprisea regular mathematics class. In a small class of mostlyaverage students, it is quite feasible to use the DT > PImodel with the one or two mathematically gifted studentspresent.

Other ActivitiesIn addition to offering mathematically gifted youths a

program of instruction, it is beneficial to provide career edu-cation in the areas that make great use of mathematics.This feature may take the form of inviting speakers whohave careers in mathematics or in areas of applied mathe-matics. The speakers provide excellent role models and willsimulate student interest in fields that call for specialmathematical talents.

Math teams are particularly popular with youths whoreason well mathematically. Teams can be organized on aschool-wide, district-wide, or even county-wide basis. State-wide or even broader competitions may be organized inaddition to local contest,.

EVALUATIONIn an accelerated mathematics program, it is essential

that the teacher closely monitor progress through effectivetechniques of evaluation. There are general evaluative

35 6.

guidelines that should be followed for the two classroommodels that have been presented.

A flow chart of the evaluative process for the homogen-eous fast-paced, enriched mathematics class is shown inFigure 2. The first step of this process occurs before theactual instruction begins. The students who are enrolled inthe class should take a diagnostic test to determine whichconcepts they already know in the subject to be taught.Concepts most class members know can then be covered veryquickly or even skipped. If a few students do not understanda topic that has been mastered by the others, they can behandled individually outside of class. The initial diagnostictesting may be accomplished through standardized measures(which are particularly effective when item analysis sheetsare used, as previously explained) ; teacher-designed tests ;or textbook tests, such as chapter review tests, which fre-quently are broken down according to topic.

When actual classroom instruction begins, progressshould be continuously measured and reported to the stu-dents. The use of board work and homework as evaluativetools have been discussed previously. Periodically, the in-structor should administer teacher-designed tests andquizzes. However, frequent quizzes use up too much valuableinstructional time. Homework and tests, that indicate tothe teacher how well an individual is learning the class-room material, should be returned to the student with evalu-ative comments. Once the student is given this feedback,instruction continues. This cycle occurs over and over again.

This evaluative process will indicate when a student'falls behind the rest of the class, either because of poorhomework, a poor test score, or both. If that happens, thena conference should be arranged with the student and/orparent (s) . If it is determined that this student is below theability level of the rest of class, she or he may need to beplaced back in a regular mathematics classroom. It is bestto make such moves at the end of a subject sequence, suchas Algebra I. If, however, the struggling student is willingto work assiduously, seek extra help outside of class, andspend more time on homework, she or he may be able toremain in the class. If it is determined that the student isdoing poorly because of an unwillingness to make a suffi

36

FIGURE 2: FLOW CHART FOR A FAST-PACED, ENRICHED MATHEMATICS CLASS

DIAGNOSTICTESTING

FURTHERINSTRUCTION

FEEDBACK(TO STUDENTS)

FURTHER STUDYWITH PRESENT

PROGRAMLEVEL

STANDARDIZEDEVALUATIONOF COURSEKNOWLEDGE

This material was first published in New Voices inCounseling the Gifted, Colangelo and Zaffron, Copy-right ©1979, Kendall Hunt Publishing Company.

8

DIAGNOSTIC

TESTING

PLACEMENTINTO NEXTPROGRAM

LEVEL

cient effort on the homework, this person should be givenan ultimatum to either do the homework diligently or beplaced back in the regular instructional classroom. Mathe-matically gifted youths should view participation in anaccelerated mathematics class as a privilege and not a right,no matter how bright they may be.

When the instructor believes that a subject has beenmastered by the class, a diagnostic measure (such as theCooperative Mathematics Test series) of that particularsubject should be administered. This phase of the evaluativeprocess leads to one of two alternatives. In some cases, itmay indicate that more instruction is needed at the presentcourse level. For those students who score well (say, abovethe 90th percentile), this standardized evaluative measurebecomes the basis for giving credit for the subject mattercompleted. Those students who are awarded credit then be-gin the next level in the course sequence.

A flow chart of the evaluative process for a class fol-lowing the DT---). PI model has already been shown in Fig-ure 1. The process is similar to that of the fast-paced, en-riched mathematics class. The chief difference is that theinitial diagnostic measure determines the level of instruc-tion on an individual rather than on a group basis. Theremainder of the process is unchanged. Again, it must bestressed that the teacher should hold the students to highstandards of homework.

When grades are given for completed coursework, theyshould be seen as a reward and not as a penalty. Thesebright students would probably obtain an A if they were ina regular math class. Students who participate in an accele-rated mathematics class complete more courses, do morewokk, and cover material at a higher level than do pupilsin the regular mathemaitIcs classes. Since only those stu-dents who do put forth a strong effort are allowed to remainin the class, they should not be placed in "double-jeopardy"by receiving poorer grades than they would have receivedin a regular classroom situation. Therefore, students whocomplete a subject (i.e., score above the 90th percentile onthe evaluative measure) should receive a grade of A forthat course. In a few cases, a grade of 13 may be appropri-ate. However, no one should receive a grade lower than B.

38

Students deserving lower grades belong in a regular class-room.

TEACHER SELECTIONThe choice of instructor is a major factor in the suc-

ce8s of an accelerated mathematics class. Instructors shouldmeet several criteria:1. They should know mathematics very well (for example,

at least through calculus if teaching iprecal: ulus). It isimportant that a teacher's knowledge level extend be-yond the level at which they are teaching.

2. They should be flexible, bright, quick-thinking, and alert.3. They should be able to relate well to the needs and per-

sonalities of their students. A good sense of humor ishelpful in promoting teacher-student interaction.

4. They should have respect for students and their feel-ings/opinions.

5. They should desire to see their students reach maximumpotential. This will require diligence on behalf of theinstructor and may require extra time spent workingwith students outside the regular classroom situation.

6. They should be enthusiastic about mathematics.7. They should be willing to teach Mathematics fast, at a

pace appropriate to the students' abilities. This implieschallenging individuals to go one step beyond what theyalready grasp.In addition to these criteria, there are certain skills

that these instructors should exercise while teaching:1. The lectures should be concise and at a quick but appro-

priate pace. This pace may vary as content becomes moredifficult. As mentioned previously, instructors must beflexible during a lecture, allowing students to participatein the development of concepts.

2. They should facilitate interaction between students.3. They should be sensitive to the students' level of under-

39

standing. In a fast-paced, enriched class, they shouldteach at a pace that challenges the top students in theclass. Through wise homework selection and extra helpoutside of class, a skillful instructor will maintain thefast pace without allowing the less able students to fallhopelessly behind. In a class that follows the DT 'PImodel, the instructor must be a master pacer and stimu-lator, knowing the speed at which each student canlearn new material.

4. They should not be threatened by the abilities of theirstudents. They should not be intimidated by the factthat many mathematically gifted youths are capable oflearning mathematics better and more quickly than theydid. Teachers should not be threatened by students'questions. Students frequently ask difficult questionsthat may be beyond the scope of the present topic. Thesequestions should be handled in a manner that furtherstimulates the questioner's curiosity. Inquiries shouldnot be dismissed as irrelevant merely because the in-structor has no answer. In such a situation, it is bestnot to fake an answer ; admit ignorance and then obtainappropriate reference materials or point the questionerin the direction of such materials. A good teacher willallow time for a certain number of questions that maynot relate directly to the current subject. This keepsthe interest level high and expands students' knowledge.In-depth questions that require a lengthy answer may bepursued after class.

5. They should not spoon-feed material to the students.Mathematically gifted youths should learn self-teachingskills. Good instructors will not use class time to explainevery detail of the concepts being taught. Instead, theyshould assign homework that helps pupils learn thosepoints not covered in class. Through diligent effort onhomework, bright students are able to master a conceptthat was covered rapidly or merely alluded to in class.This approach gives the instructor more time to explainand perform examples of more difficult concepts andproblems. It also speeds up the class and reduces bore-dom.

4 140

6. As much as possible, teachers should not answer ques-tions that deal directly and specifically with a topic al-ready covered or reveal how to solve a problem that isbeing assigned for further study. Instead, instructorsshould answer with leading questions that will revealthe answer. These bright students probably have neverhad to work very hard at academics and have alwaysdone well. Therefore, when they begin to be challengedin mathematics, rather than make an extra effort tolearn difficult material, they frequently attempt to usethe teacher as a crutch who predigests the material forthem. Instructors must not fall into a pattern of beingused this way by students. These students can be quiteclz:ver at disguising laziness and lack of effort. Theymay say they are unable to understand a concept, whenin fact they could understand it if they tried a bitharder. Teachers who answer a pupil's question with aquestion force that student to think things through andthus encourage a fuller use of capabilities.From these lists of criteria and skills, it is clear that

the instructor of an accelerated mathematics class shouldbe quite bright in mathematics and possess masterful teach-ing techniques. Mathematically gifted college students havebeen used quite successfully as instructors by SMPY. Par-ticularly those who themselves were students in acceleratedmathematics classes pos-ess a keen understanding of mathe-matics and can identify with their students. Mathematicallygifted students have proven to be especially able as mentorsof other bright youngsters in one-on-one situations usingthe DT PI model. College professors often make goodaccelerated mathematics class instructors (and have beenused as such by SMPY). (32)

TEXTBOOK SELECTIONFrom the discussion thus far, one may have inferred

some criteria for selecting textbooks to be used in accele-rated mathematics classes. Some of these criteria are :1. The textbook should contain explanations sufficiently

clear for a student to be able to learn many concepts

41

without teacher assistance. As has been stated, this isparticularly important when using the DT --> PI model.

2. It should be concise.3. It should contain numerous examples of completely

solved problems.4. It should contain large numbers of problems that can

be assigned for completion in class or for homework.5. It should not provide the answers to all of the problems.

It is helpful if the text gives answers to only the odd-numbered problems. The student would be able to checkthose answers, while being required to submit some ofthe even-numbered problems.

6. It is helpful if the text contains chapter tests that canbe used as diagnostic and evaluative tools.Once an accelerated mathematics class is underway,

there are various ways to suppl tment the instruction. Peri-odically the teacher may wish 1 o introduce more advancedtopics that follow from the material then under considera-tion. However, as already menti lned, a subject should notbe probed too deeply during clasL:, especially if there is noprovision for obtaining credit.

In selecting a textbook it is important to realize that aunified mathematics curriculum presents difficulties rele-vant to articulation. For example, suppose 'an individual isenrolled in an accelerated mathematics class, completesAlgebra I and Algebra II, and then is counseled to go backinto the regular mathematics sequence at his or her school.If the school has a unified mathematics curriculum it isdifficult to place this student in the proper course, sinceAlgebra I and Algebra II are integrated and blended withother subjects in mathematics over a period of several years.The same difficulty exists if the accelerated mathematicsclaos follows a unified mathematics curriculum and the stu-dent's school does not.

OTHER ACCELERATIVE OPTIONSThere are a number of opportunities for educational

acceleration that should be made available to mathemati-

42

tally gifted, youths. (20) Sometimes matheiriatically giftedstudents who are not learning mathematics at an accele-rated rate are instead. able to start the precalculus sequencea year ahead of their classmates. For instance, if Algebra Iis offered in eighth grade, some brie. is seventh grade stu-dents can be allowed to enroll in the class. In addition, theseable students can be encouraged to take two mathematicscourses (such as Algebra II and plane geometry) concur-rently.

Some students laarn mathematics at an accelerated rateoutside of school, either through a special class or with amentor. This often leaves an opening in the daily schedulefor other courses. Thus acceleration in the mathematicssequence may lead to acceleration in other areas. Therefore,those bright individuals who are eager and of high abilityshould be allowed to skip a grade in school. This is donemost inconspicuously if the affected student skips the finalyear in one school and begins attending another. For in-stance, in a junior high school (grades seven through nine),an eighth grade student would skip ninth grade and entertenth grade at the senior high school.

Many mathematically gifted youths are quite able toenter college full-time a year or more ahead of their age-mates. Students who are willing and able should be giventhe option of graduating from high school one year early(after eleventh grade) or leaving high school without adiploma a year or more ahead of their peers. Many schoolsystems will grant these students their diplomas after satis-factory completion of one full year of college.

The Advanced Placement Program (APP) allowshighly able students to earn college credit by examinationbefore entering college on a full-time basis. Many highschools offer special courses specifically designed to coverthe advanced material tested by these difficult examinations.Mathematically gifted students should be helped and en-couraged to prepare for the more difficult APP calculusexamination as well as for exams in any other areas inwhich they are talented, such an chemistry or physics.

Finally, these brilliant youth, s should be encouraged totake college courses on a part-time basis while still enrolledin junior or senior high school. (18) This can be done either

on released time from school, in the evenings, or duringthe summer months. Through college courses and APPcredits, many students just beginning a full -time collegeprogram already possess one, two, or more semesters' worthof college credits.

No matter how successful an accelerated mathematicsclass is, the benefits will be negated if there is no coordina-tion and articulation between grades and between schools.Any program of accelerated instruction should be organizedwith the cooperation of the participating schools. For ex-ample, students in an accelerated mathematics class shouldnot in addition be required to attend the regular schoolmathematics class. Most important, the participating.schools should give credit for mathematics courses com-pleted in a special class. This ensures that students will notbe asked to repeat a course if they are placed back into thestandard mathematics curriculum in a later grade or whentransferred to another school district. For instance, if stu-dents who are enrolled in an accelerated mathematics classin junior high find no continuation of the class in seniorhigh, they should not be forced to repeat coursework thathas already been covered in the special class.

CONCLUSIONIf mathematics is truly the cornerstone for many of

the sciences and for engineering and computer science, thenwe need to motivate and stimulate our mathematically tal-ented youths. Flexibility and alternative education seem tobe the key, based on the experiences of state and local edu-cational agencies and on the insights gained through pro-grams such as SMPY.

Educational acceleration is a critical variable. It hasbeen found to be the method of choice for those youths whoreason extremely well mathematically and are eager tomove ahead educationally. Now is the time to examine theevidence. The experiences to date have shown that programdevelopment for the mathematically gifted is not costly. Itis, indeed, a necessity if we are to maximize the potentialof these individuals.

44'I 5

APPEND: ''T ACommonly used in-grade nationally normed first-stagescreening measures that have been used to identify themathematically talented. (Listed in alphabetical order withthe name of the publisher.)1. California Achievement Tests (California Test Bureau)2. California Basic Skills Test (California Test Bureau)3. Cognitive Abilities Test (Houghton Mifflin)4. Iowa Tests of Basic Skills (Houghton Mifflin)5. Metropolitan Achievement Test (larcourt, Brace, and

World)6. Science Research Associates Achievement Series (Sci-

ence Research Associates)7. Stanford Achievement Tests (Harcourt, Brace, and

World).

APPENDIX BMeasures commonly used at the second level of the two-stage screening method for the mathematically gifted.

1. The Preliminary Scholastic Aptitude Test (PSAT)2. The Scholastic Aptitude Test (SAT)

3. The School and College Ability Test (SCAT)

4. The Differential Aptitude Test (DAT)

5. The American College Testing Program (ACT)6. The College Board Mathematics Achievement Tests

Levels I and II.

45

SELECTED REFERENCES

1. Cohn, S. J. "SMPY's Fifth Talent Search : A Time of Changes."ITYB (Intellectually Talented Youth Bulletin) 4, no. 10: 9-11,1978.

2. Daurio, S. P. "Educational Enrichment Versus Acceleration: AReview of the Literature." Educating the Gifted: Acceleration andEnrichment. (Edited by W. C. George, S. J. Cohn, and J. C. Stan-ley.) Baltimore, Md.: The Johns Hopkins University Press, 1979.pp. 13-63.

3. Fox, L. H. "A Mathematics Program for Fostering PrecociousAchievement." Mathematical Talent: Discovery, Description, andDevelopment. (Edited by J. C. Stanley, D. P. Keating, and L. H.Fox.) Baltimore, Md.: The Johns Hopkins University Press, 1974.pp. 101-125.

4. George, W. C. "The Talent Search Concept: An Identification Strat-egy for the Intellectually Gifted." Journal of Special Education 13,no. 3: 221-237,1979.

5. . "The Third D: Development of Talent (Fast-MathClasses) ." New Voices in Counseling the Gifted. (Edited by N.Colangelo and R. T. Zaffran.) Dubuque, Iowa: Kendall/Hunt, 1979.pp. 119-132.

6. George, W. C., and Benbow, C. P. "SM!?Y's Sixth and Last TaleutSearch: It Broke the Barrier." ITYB 5, no. 10: 2-4,1979.

7. George, W. C., and Cohn, S. J. "The 1976 Talent Search ResultsAre In !" ITYB 3, no. 6: 1-2,1977.

8. George, W. C., Cohn, S. J., and Stanley, J. C., editors. Educatingthe Gifted: Acceleration and Enrichment. Baltimore, Md.: TifeJohns Hopkins University Press, 1979.

9. George, W. C., and Denham, S. A. "Curriculum Experimentationfor the Mathematically Talented." Intellectual Talent: Researchand Development. (Edited by D. P. Keating.) Baltimore, Md.: TheJohns Hopkins University Press, 1976. pp. 103-131.

10. George, W. C., and Solano, C. H. "Identifying Mathematical Talenton a Statewide Basis." Intellectual Talent: Research and Develop-ment. (Edited by D. P. Keating.) Baltimore, Md.: The Johns Hop-kins University Press, 1976. pp. 55-89.

11. Keating, D. P. "The Study of Mathematically Precocious Youth."Mathematical Talent: Discovery, Description, and Development.(Edited by J. C. Stanley, D. P. Keating, and L. H. Fox.) Baltimore,Md.: The Johns Hopkins University Press, 1974. pp. 23-46.

12. Montour, K. M. "Charles Louis Fefferman: Youngest AmericanFull Professor." ITYB 2, no. 8: 2,1976.

13. "William James Sidis: The Broken Twig." AmericanPsychologist 32, no. 4: 265-279,1977.

14. Pegnato, C. W., and Birch, J. W. "Locating Gifted Children inJunior High Schools : A Comparison of Methods." ExceptionalChildren 25, no. 7: 300-304,1959.

15. Renzulli, J. S. ,The Enrichment Triad Model: A Guide for Develop-ing Defensible Programs for the Gifted and Talented. Wetherfield,Conn.: Creative Learning Press, 1977.

16. Solano, C. H. "The First D: Discovery of Talent, or Needles in aHaystack : Identifying the Mathematically Gifted Child." NewVoices in Counseling the Gifted. (Edited by N. Colangelo and R. T.Zaffrann.) Dubuque, Iowa : Kendall/Hunt, 1979. pp. 89-106.

17. "Teacher and Pupil Stereotypes of Gifted Boys and

47

Girls." Talents and Gifts 19, no. 4: 2-15,21-26,1977.18. So lano, C. H., and George, W. C. "College Courses for the Gifted."

Gifted Child Quarterly 20, no. 3: 274-285,1976.19. Stanley, J. C. "Identifying and Nurturing the Intellectually Gifted."

Phi Delta Kappan 58, no. 3 : 234-237,1976.20. . "Rationale of the Study of Mathematically Precocious

Youth (SMPY ) During Its First Years of Promoting EducationalAcceleration." The 'Gifted and the Creative: A Fifty-Year Perspec-tive. (Edited by J. C. Stanley, W. C. George, and C. H. Solano.)Baltimore, Md.: The Johns Hopkins University Press, 1977. pp.75-112.

21. "The Second D: Description of Talent." New Voices inCounseling the Gifted. (Edited by N. Colangelo and R. T. Zaffrann.)Dubuque, Iowa: Kendall/Hunt, 1979. pp. 107-118.

22 "SMPY's DT 0 PI Mentor Model : Diagnostic TestingFollowed by Prescriptive Instruction." ITYB 4, no. 10: 7-8,1978.

22, "Special Fast-Mathematics Classes Taught by CollegeProfessors to Fourth- Through Twelfth-Graders." IntellectualTalent: Research and Development. (Edited by D. P. Keating.)Baltimore, Md.: The Johns Hopkins University Press. 197n pp.132-159.

24. "The Study and Facilitation of Talent for Mathematics."The Gifted and the Talented: Their Education and Development( The Seventh-eighth Yearbook of the National Society for the Studyof Education). ( Edited by A. H. Pasdow.) Chicago: The Universityof Chicago Press, 1979. pp. 169-185.

25. "Test Better Finder of Great Math Talent Than TeachersAre." American Psychologist 31, no. 4: 313-314,1976.

26. "Use of Tests to Discover Talent." intellectual Talent:Research and Development. (Edited by D. P. Keating.) Baltimore,Md.: The Johns Hopkins University Press, 1976. pp. 3-22.

27. Stanley, J. C., George, W. C., and Solano, C. H., editors. Educa-tional Programs and Intellectual Prodigies. Baltimore, Md.: TheStudy of Mathematically Precocious Youth, The Johns HopkinsUniversity, 1978.

28. Stark, E., and Stanley, J. C. "Bright Youths Dispel PersistentMyths about Intellectual Talent: Panel Discussion with Parentsand Educators." Gifted Child Quartei-ly 22, no. 2 : 220-234,1978.

29. Terman, L. M. "Mental and Physical Traits of a Thousand GiftedChildren." Genetic Studies of Genius, vol. I. Stanford, Calif.: Stan-ford University Press, 1925.

30. Tuttle, Jr., Frederick B. Gifted and Talented Students. Washington,D.C.: National Education Association, 1978.

31. Tuttle, Jr., Frederick B., and Becker, Laurence A. Characteristicsand Identification of Gifted and Talented Students. Washington,D.C.: National Education Assn., 1980.

32. Program Design and Development for Gifted and Tal-ented Students. Washington, D.C.: National Education Assn., 1980.

33. Zuckerman, H. Scientific Elite: Nobel Laureates in the UnitedStates. New York : Free Press, 1977.

The program ideas presented in Teaching the Giftedand Talented in the Mathematics Classroom have been usedsuccessfully by various state and local agencies as well as byprivate organizations. The ideas should not be consideredrigid guidelines, but flexible methods of facilitation. Sinceeach school and each student is unique, the models shouldbe shaped to fit each individual situation. il

The ideas are developed through eight focal aspects :Identification, Approaches, Teaching Strategies, Implemen-tation, Evaluation, Teacher Selection, Textbook Selection,and Other Accelerative Options. A carefully prepared groupof Selected References provides excellent sources for furtherstudy.

The authors, Kevin G. Bartkovich and William C.George, are with the Study of Mathematically PrecociousYouth (SMPY) at The Johns Hopkins University, Balti-more, Maryland, where Mr. Bartkovich is an Instructor andMr. George is Associate Director. The consultant for thispublication is Dr. Julian C. Stanley, who is Professor ofPsychology at Johns Hopkins and Director of SMPY. Theeditor of NEA's Teaching the Gifted and Talented in theContent Areas series is Dr. Frederick B. Tuttle, Jr., StateUniversity of New York, College at Brockport.

KeaStock No. 0738-7-00