P.O. BOX 1586







Dr. James Delisle -- Professor and Co-Director of SENG, Kent State University, Kent, Ohio

Dr. Jerry Flack --Univ. Of Colorado-Colorado Springs

Dr. Howard Gardner -- Professor, Graduate School of Education, Harvard University, Cambridge, Massachusetts

Ms. Diane D. Grybek -- Supervisor of Secondary Gifted Programs (Retired), Hillsborough County Schools, Tampa, Florida

Ms. Dorothy Knopper -- Publisher, Open Space Communications, Boulder, Colorado

Mr. James LoGiudice -- Director, Program and Staff Development, Bucks County, Pennsylvania Intermediate Unit No. 22 and Past President of the Pennsylvania Association for Gifted Education

Dr. Mary Meeker -- President of SOI Systems, Vida, Oregon

Dr. Adrienne O'Neill - Chief Education Officer, Timken Regional Campus, Canton, Ohio

Dr. Stephen Schroeder-Davis -- Coordinator of Gifted Programs, Elk River, Minnesota Schools and Past President of the Minnesota Council for the Gifted and Talented

Dr. Bruce Shore -- Professor and Director, Giftedness Centre, McGill University, Montreal, Canada

Ms. Joan Smutny -- Professor and Director, Center for Gifted, National-Louis University, Evanston, Illinois

Dr. Virgil S. Ward -- Emeritus Professor of Gifted Education, University of Virginia, Charlottesville, Virginia

Ms. Susan Winebrenner -- Consultant, Brooklyn, Michigan

Dr. Ellen Winner - Professor, Boston College


In April I visited Capitol Hill to refresh my memory about one of this nation's greatest institutions, the Library of Congress, and to join in the celebration of its 200th anniversary. Gifted children can increase their understanding of many areas of American society by studying the origins and development of this national treasure. For example, they could learn about how Thomas Jefferson gave it a major boost by selling his personal collection of over 6,000 books to start the LOC. (Today it consists of more than 27 million volumes.) The United States purchased Jefferson's books over the objections of many Federalist members of Congress who thought the range of topics covered was too broad and unfocused. In celebration of its 200th anniversary, the Library of Congress will display about 90% of the original Jefferson titles -- 3,300 books are copies that serve as replacements for originals destroyed in an 1851 fire. It would be interesting for gifted students to study and classify some of these titles in order to draw conclusions about Jefferson's interests, particularly the influence of the Age of Enlightenment on his reading habits.

The architecture of the Jefferson Building of the Library of Congress exhibits brilliant craftsmanship on both the external facade and inside the Great Hall and Main Reading Room. (The style is a mixture of the Classical Greek, Baroque and Federal schools of design.) Particularly impressive are the beautiful domes, marble columns and stairways, and statues and paintings. Along the walls of both of these rooms, one sees statues and paintings of the great thinkers of world history and their quotations on philosophy, law, poetry, religion, science, etc. Of course, another major area of study for all gifted students would be to learn about the Library's collection of books, papers, musical scores, audio recordings, films and pictorial materials. The CD ROM entitled, "Eyes of the Nation" (1997), would help students to delve more deeply into these archives.

In this issue, Paula Olszewski-Kubilius, Director of the Center for Talent Development at Northwestern University and her colleague, Lisa Limburg-Weber present an excellent discussion of resources for high school gifted students. The problem of teaching mathematics to the gifted is ably addressed by Semyon Rafalson, a high school teacher of mathematics who received his doctorate in this field from Leningrad State University. He emigrated to the United States from the former USSR in 1989. This issue also includes a review of the Iowa Acceleration Scale (1998) by Dan Holt who is known for his work on studying humor in gifted children and using MI theory in gifted education. Michael Walters concludes with an article on The Actors Studio and cooperative learning.

Maurice D. Fisher, Ph.D., Publisher

Gifted Education Press





By the time their children begin high school, parents have had a great deal of experience with the educational system. They have probably interacted with teachers dozens of times over many different issues, and received both positive and negative responses. Parents may feel they are quite well versed in the organization and structure of schools, chains of authority and decision making, and communication patterns. They know whom to call for each particular problem. But, just when they have figured things out, parents of adolescents must readjust their thinking and orient themselves to a different organizational structure—the high school.

High schools are generally larger institutions than elementary and middle schools. There are more students and teachers to contend with. Teachers are not organized or grouped by grade, but by departments that correspond to subject areas. Students' schedules are typically more complex, and they have many more choices about the classes they take and the activities they participate in. Counselors assigned to provide academic advising usually have hundreds of students to deal with, with a wide range of abilities and problems. There may or may not be a home room teacher, and usually, counselors provide the link between home and student.

At the same time that students are dealing with a more open, complex environment, there are more decisions to make, and the potential impact of those decisions (i.e. which courses, what extra-curricular activities, which groups of friends) on a child’s future seems much greater.

For parents of a gifted child, there is the added concern of whether the high school curriculum will be adequate, whether the child’s placement will be appropriate, and whether the school climate will support continued achievement. A major worry for parents is the effect of the peer culture on their child’s achievement.

How can parents handle all the changes brought by the transition to high school and insure that their gifted children get the most out of secondary school? Two concepts are critical—knowledge and advanced preparation/planning. Parents need the following kinds of information in order to enable their child to get the most out of high school:

1.Knowledge about their child's specific areas of academic strength and potential; and

2.Knowledge about the various types of programs that can develop their child’s academic abilities and are appropriate for the high ability learner and about ways to get access to such programs, whether through the local high school, through other community institutions or resources; or through other programs nationwide.

Parents equipped with this knowledge will be able to more effectively chart their child's educational course—and avoid shipwrecks along the way.

Understanding Your Gifted Child's Strengths and Potential

The process of planning a child's education and preparing him or her for success—in high school, college, and beyond—begins early. Most parents of a highly able child become aware during their child's primary school years that he or she may require special placement or programs to be appropriately challenged in school. Understanding each child's special constellation of gifts and talents is key to beginning the long-term planning process.

How do parents ensure that they understand their child's area(s) of giftedness? Most parents rely heavily on two sources of information: school grades, and scores on in-grade achievement tests (such as the Iowa Tests of Basic Skills, the California Achievement Tests, etc.). If their child is presenting straight-A report cards and receiving above average scores on achievement tests, parents usually do not see a need for further scrutiny of the child's abilities. Grades and achievement test scores certainly are important ways of evaluating children's current levels of achievement and progress through the years. However, exclusive reliance on these measures can result in an incomplete picture of academically gifted children's areas of strength.

The grades that gifted children receive in school can be deceptive in two ways. First, gifted students who are underachieving in school (because of boredom, a bad match between the child and the curriculum, or other issues) may well show a pattern of poor grades, even if both their understanding of course material and their potential to achieve are very high. For these students, poor grades may be a symptom not of poor ability, but of above average ability masked by other issues.

Second, gifted students who are achieving highly in school may show a pattern of excellent grades across every subject area. Parents, however, may have a hard time discerning whether such a child is actually working up to his or her ability level in every academic area. Achieving all As in grade-level schoolwork can be a sign that the child is well-placed in school and is working hard. But there is another possibility. A child may be getting good grades because, given that child's ability level, the assignments are very easy and the class pace is slow. Such a child may be capable of handling much more advanced or fast-paced school-work.

Similarly, in-grade achievement test scores provide only part of the information parents need to understand their gifted child's abilities. Such tests are designed to measure whether children have mastered grade-level content in specific subject areas; thus, they contain only a small number of questions that would be difficult for the average child at that grade level to answer. What these test scores don't show is how much higher the child might have scored if the test contained a larger number of difficult questions.

For instance, sixth-grade students who score very highly on the mathematics portion of an achievement test (say, in the 95th percentile and above) are essentially demonstrating that they have mastered most or all of the mathematical concepts that the test makers expect sixth-graders to have learned. But there still may be significant differences among the sixth graders who score at this level.

One sixth grader scoring in the 97th percentile in mathematics, for instance, might be well-placed in his current math class. "Alex" might benefit from some additional math enrichment activities, but in general, he is being adequately challenged with grade-level mathematics instruction.

Another sixth grader with an identical score, on the other hand, might have much higher potential in mathematics. "Sarah" may have "hit the ceiling" of the test; in other words, she answered all of the sixth-grade questions correctly, but because the test did not include seventh- or eighth-grade questions, she was unable to demonstrate the extent of her advanced understanding of mathematics. Sarah might be capable of moving directly into an accelerated, honors-level algebra class in seventh grade—two years earlier than expected—but the test does not distinguish between her abilities and those of Alex.

Because grades and in-grade achievement tests are only two pieces to the puzzle of understanding the gifted child's academic strengths, parents must consider other ways of gaining information about their child's potential and needs. Some parents choose to have their children individually tested by an educational specialist experienced with gifted children. This option yields extensive and often very helpful feedback on the child in a number of areas. It is especially important that parents seek individualized professional assistance if they suspect that psychological issues such as depression or anxiety, lack of motivation, or family stress may underlie their child's pattern of poor school achievement.

Unfortunately, expert individual psychological/academic assessment can be time consuming, expensive, and difficult to locate. Off level testing, through one of the national level talent searches, is a much less expensive and readily available option for most families.

An off level (or "above-level") test is any test given to a younger-than-average student for the purpose of assessing above-average ability. In this country, the ACT and SAT (college-entrance exams) are used as off level tests for children in sixth through eighth grades, and the EXPLORE test (developed as a test for eighth-graders) is used as an off level test for fourth, fifth, and sixth graders. These tests, when given to students who are two to four years younger than the average test taker, contain enough difficult items to give a more accurate picture of the gifted child's abilities and potential.

If the sixth graders in our example had the chance to take the SAT, "Alex" may have scored a 300 on the mathematics portion of the SAT, while "Sarah" might have scored a 700. These SAT test scores would make the difference between Alex's and Sarah's abilities much clearer—and would help parents and teachers more effectively plan appropriate educational options for both children. Might both children still be considered "gifted"? Certainly. But their needs, in this particular subject area, are very different.

Off level tests such as the ACT or SAT can be accessed most easily through an established Talent Search program. Currently, four university based centers for gifted education in the US offer a yearly Talent Search for gifted students in fourth through eighth grades: Duke University, Johns Hopkins University, Northwestern University, and the University of Denver (see below for contact information). Students sign up for tests in the fall and take the tests in the winter of each year, and high scoring students are recognized in special Awards Ceremonies each spring. Across the country, over 100,000 gifted students each year use the Talent Search testing process to help them and their families better understand their academic achievement and potential.

Some Talent Search programs send students not only test scores but also extensive counseling materials designed to help them and their families move into the next stage of planning for their educational futures. For instance, the Center for Talent Development at Northwestern University sends each student a personalized Long-Range Academic Planning Form, along with lists of recommended high school course sequences that correlate with students' exam scores. CTD also provides students with contact information for hundreds of special programs. Such material can help parents focus on the next big task after the child's abilities and talents have been assessed and understood: planning for high school and college.

Planning for Your Gifted Child's Educational Future

By the time your child is in middle school, the path he or she plans to follow in high school, including specific courses s/he plans to take, should be articulated in writing. This exercise will make it very clear what classes and opportunities can be provided by the high school and which ones need to be obtained via other institutions and organizations that exist outside of school.

Planning should take place with the following goals in mind:

1.Produce an articulated sequence of courses within each major subject area (language arts, mathematics, social science, science, and foreign language). Include courses at the highest level possible given your child’s interests and ability levels, and areas of talent.

2.Outline outside-of-school educational options and extra-curricular activities, if needed to provide advanced courses or extra opportunities.

3.Indicate experiences and preparation in the form of volunteer work, paid jobs, or course work, that will prepare your child to be able to pursue several different college majors at selective institutions of higher education and that will give your child experience with different careers he/she is interested in.

Seek the assistance of teachers and department chairs from your local high school in devising your plan. Ask specifically about the courses available within a subject area, those recommended for high ability students, and options for taking advanced courses if they are not available in your local school. If you plan for your child's school to recognize or give credit for courses taken elsewhere, it is especially important for you to stay in close communication with school officials about your plans.

As part of this planning process, parents and students must also familiarize themselves both with other programs which may be appropriate for their children, e.g., Advanced Placement (AP) courses, the International Baccalaureate Diploma Program (IB), dual enrollment programs, study abroad programs, distance learning programs, mentorships and internships, contests and competitions, and early college entrance programs. A brief description of each type is provided below, as are lists of programs.

Finally, families should begin to accumulate knowledge about activities within the local community that can augment your child’s high school program, including courses at the local university, courses through a cultural institution such as a museum, opportunities for significant volunteer work, etc. Talk to other parents of very able children to obtain information and referrals and use community listings of programs.

Below is an example of such a plan in the area of mathematics for a student such as Sarah, from our example above, who is talented in math:

8th grade: algebra (provided in school) -- summer after 8th grade: algebra 2 (need to seek a summer program)

9th grade: geometry (taken in school) -- summer after 9th grade: volunteer in community math tutoring program (to explore interest in a teaching career)

10th grade: Pre-calculus that includes trigonometry, graphing and functions (taken in school) -- summer after 10th grade: AP Statistics (need to seek a summer or distance learning program)

11th grade: AP Calculus (taken in school) -- summer after 11th grade: seek a career-shadowing program in the area of engineering or computer programming; research programs available at different colleges and universities

12th grade: College level math course (e.g., Linear Algebra, Differential Equations or Discrete Mathematics)—taken at local college, university summer program or via distance learning program

Extra-curricular activities: participate in school math club and Mathematics Olympiad competition.

Program Options for Academically Gifted Students

1. In School Options

Advanced Placement Courses (AP)—The AP program is conducted by The College Board. AP classes are college level courses which students take while in high school. The courses are typically taught by high school teachers at the student’s school. By completing the course, students earn high school credits. To earn college credits, students must obtain certain scores on the AP exam (typically at least a 3 on a scale of 0 to 5) which is given in May. Students can prepare for an AP exam via independent study with a teacher or mentor and can take AP exams without having completed an AP course. There are currently 32 AP courses available from The College Board. Check with your local high school regarding AP courses offered. AP courses can also be taken through a distance education program, such as the Center for Talent Development’s (CTD) LetterLinks Program, or through some summer programs. For more information you can also contact The College Board at Columbus Avenue, New York, NY 10023-6992, 217.713.8066, or

Dual Enrollment is a program that exists by legislation in certain states (currently 22). It is also referred to as post-secondary option. Dual enrollment allows high school students to take college classes through a local university or college, while they are still enrolled in high school. Check with your local school administrators or with the gifted education coordinator for your state for more information about this option. Dual enrollment legislation varies by states and stipulates when such courses can be taken (typically during grades 11 and 12), how many courses can be taken, eligible colleges and universities, fees (most often covered by high school), credits earned, etc.

International Baccalaureate (IB) is an international program that exists within select high schools throughout the world and consists of courses that students take during the last two years of high school. Students who complete the full program earn an IB diploma. The IB program has its own curriculum which includes a theory of knowledge course, a creativity, action, and service requirement, a research project, and academically rigorous set of courses which emphasize second language proficiency, and a formal examination requirement. The IB program is considered academically challenging and rigorous and is highly regarded by colleges and universities. It may be especially attractive to students who anticipate earning a degree abroad or who are planning an international career. To find out which high schools in your area offer IB, contact your local superintendent or the IB organization at the North American and Caribbean Regional Office, 200 Madison Ave., Suite 2007, New York, NY, 10016, 212.696.4464, e-mail IBNA, website:

2. Options Beyond the Local School

Talent Search programs (described above) are offered through gifted education centers at four universities. All of these centers also offer summer programs for gifted students, as well as a other programs.

Center for Talent Development (CTD)

Northwestern University

617 Dartmouth Place

Evanston, IL 60208

phone: (847) 491-3782

fax: (847) 467-4283


Institute for the Academic Advancement

of Youth (IAAY)

The Johns Hopkins University

3400 North Charles Street

Baltimore, MD 21218

phone (410) 516-0337

fax: (410) 516-0200 Registration

(410) 516-0325 Info Service


Rocky Mountain Talent Search (RMTS)

University of Denver

2135 East Wesley

Denver, CO 80208

phone: (303) 871-2533

fax: (303) 871-3422


Talent Identification Program (TIP)

Duke University

Box 90747

Durham, NC 27708-0747

phone: (919) 684-3847

fax: (919) 681-7921


Distance Learning is defined as any educational situation in which the teacher and instructor are not face to face. There are many different forms of distance learning courses including traditional by-mail correspondence courses; two-way, interactive audio and video classes; classes using the internet; and CD-ROM based courses. The advantage of distance learning courses for gifted students is that they can be done on the students own time. Gifted students can use these programs to take courses that are unavailable in their local schools, to take more advanced courses, to take advanced courses early, and to take additional courses for enrichment. Several Talent Search centers offer some form of distance education. Other sources include universities offering college level courses on-line, as well as commercially available high school level courses.

Summer Programs are a good way to supplement a student’s school curriculum. There are many summer programs across the US, including those offered through the Talent Search centers (see above). Summer programs offer gifted students the opportunity to take advanced classes earlier than usual and to supplement their school course work with enrichment courses. Summer programs differ widely in the challenge level of the courses, the length of the program, and additional features beyond the academics such as recreational and cultural components. Students interested in obtaining school credit for courses taken during the summer should also investigate whether the program is accredited. Currently, among the four Talent Search centers, only the Center for Talent Development has received accreditation for its programs.

Study Abroad can help students acquire first-hand knowledge of another culture and increase facility with a foreign language. Study abroad programs include summer and academic year programs. While most study abroad programs are not specifically marketed to gifted students, if chosen wisely, they can be an excellent educational option. Programs vary on a variety of dimensions such as living arrangements, costs, degree of structure, required classes, opportunities to travel, etc. For a listing of over 200 programs for students, ages 12-19, see the following book: Council on International Educational Exchange, The High School Student’s Guide to Study, Travel, and Adventure Abroad, 5th Ed. New York: St. Martin’s Press, 1995.

Full Time Early College Entrance is a viable option for some gifted students. A great deal of research has been done on early college entrants and has shown that they excel academically in their college studies and do not experience problems socially. There are currently 11 early college entrance programs at various institutions in the US. In many of these programs, students simultaneously complete high school course requirements while taking college classes. Three of the 11 programs admit students as much as 3 to 4 years earlier than usual. Most colleges and universities will admit students as full time students one or two years early.

The Early Entrance Program at California State University

phone: (213) 343-2287



Simon's Rock College

84 Alford Road,

Great Barrington, MA 01230

phone: (413) 528-7312


The Clarkson School

Price Hall, Clarkson University,

P.O. Box 5650, Potsdam, NY 13676

phone: (315) 268-4425


The Early Entrance Program at the University of Washington

Guthrie Annex II, NI-20,

Seattle, Washington, 98195

phone: (206) 543-4160



The Program for the Exceptionally Gifted at Mary Baldwin College

Staunton, Virginia 24401

phone: (540) 887-7039



The Texas Academy of Mathematics and Science

phone: (940) 565-3606


The Advanced Academy of Georgia (AAG) at the State University of West Georgia

Carollton, GA 30118

phone: (770) 836-4449


The Texas Academy of Leadership in the Humanities (TALH) at Lamar University

Dr. Mary Gagne at (409) 839-2995.

The Residential Honors Program at the University of Southern California

Penny Von Helmolt at (213) 740-2961.

Special Residential Math/Science High Schools are special high schools in the US designed particularly to meet the needs of students who are interested and academically talented in science or mathematics. These schools are supported by state education moneys and most often, tuition and room and board are free. Most serve students in grades 11 and 12 only. These schools offer an advanced and highly specialized curriculum, e.g. research opportunities, in math and/or science. They also offer a unique and unusually supportive social environment for gifted students. Currently, there are 10 such schools in the US:

The Alabama School of Mathematics and Science

Executive Director: Dr. David J. Laurenson

1255 Dauphin Street

Mobile, Alabama 36604

Phone: (334) 441-2100

Fax: (334) 441-3290


The Arkansas School for Mathematics and Sciences

Director: Dr. Robert Peters

200 Whittington Avenue

Hot Springs, AR 71901

Phone: (501) 622-5100

Fax: (501) 622-5109


The Illinois Mathematics and Science Academy

President: Dr. Stephanie Pace Marshall

1500 West Sullivan Road

Aurora, Illinois 60506-1000

Phone: (630) 907-5027

Fax: (630) 907-5976


Indiana Academy for Science, Mathematics, & Humanities

Executive Director: Dr. Tracy L. Cross

Wagner Hall

Ball State University

Muncie, IN 47306

Phone: (765) 285-8125

Fax: (765) 285-2778


The Louisiana School for Math, Science, and the Arts

Executive Director: Brother David Sinitiere

715 College Avenue

Natchitoches, LA 71457

Phone: (318) 357-3174

Fax: (318) 357-3299


The Maine School of Science and Mathematics

Director: Dr. Dottie Martin

77 High Street

Limestone, ME 04750

Phone: (207) 325-3303

Fax: (207) 325-3340


The Mississippi School for Math and Science

Director: Mike Neyman

P.O. Box 1627

Columbus, MS 39701

Phone: (601) 329-6118

Fax: (601) 329-7205


The North Carolina School of Science and Mathematics

Executive Director: Dr. John Friedrick

P.O. Box 2418

Durham, NC 27715

Phone: (919) 286-3366

Fax: (919) 286-7249


The Oklahoma School of Science and Mathematics

President: Dr. Edna McDuffie Manning

1141 North Lincoln Blvd.

Oklahoma City, OK 73104

Phone: (405) 521-6436

Fax: (405) 521-6442



South Carolina Governor’s School for Science and Mathematics

President: Dr. Leland H. Cox, Jr.

306 East Home Avenue

Hartsville, SC 29550

Phone: (843) 383-3900

Fax: (843) 383-3903.

Contests and Competitions offer students an opportunity to augment their school programs with extra-curricular programs. Benefits of competitions include an opportunity to work on an independent project, feedback on one’s standing within a field relative to other students, evaluative feedback from professionals who work in the field, and a chance to win awards and cash prizes. Two very good resources for information on compe-titions and contests written specifically for gifted students are:

Mary K Tallent-Runnels and Ann C. Candler-Lotven (1996). Academic Competitions for Gifted Students: A Resource Book for Teachers and Parents. Corwin Press: Thousand Oaks, CA.

Frances A. Karnes and Tracy L. Riley (1996). Competitions: Maximizing Your Abilities. Prufrock Press: Waco, TX.

Internships are exchanges of work for learning. Internships are ways for gifted students to learn about career fields and to do substantive, real-life work in an area. Through internships, students can have contact with individuals who can provide advice on educational and career decisions. Internships can be set up locally through your school by connecting a student with a community member. Many internships exist within corporations and businesses; while most of these are for college aged students, increasingly more of them are for high school-aged students. For a list of internship opportunities, see the book:

Gilbert, Sara Delaney (1997). Internships. The Hot List for Job Hunters. Indianapolis, IN: Macmillan.

A good source of information for students, parents and educators regarding all of the program options listed above is the magazine, Imagine, which is written specifically for gifted students of middle and high school age. It profiles specific programs and often includes articles written by students who have participated in them. It also contains information about college planning and selection and a profile of a different college or university in every issue.

Imagine—The Johns Hopkins University, The Johns Hopkins University Press, 2715 North Charels Street, Baltimore, Maryland 21218-4363, 410.516.6857.

More information about these programs can also be obtained from the Center for Talent Development's website (, or by requesting CTD's publication: Designs for Excellence (see contact information, above).

As you arm yourself with knowledge about opportunities for talented students during the high school years, remember that it is critical that parents stay very involved in their child’s education. Your wisdom and knowledge is needed to help with issues such as which courses to take, how to manage competing activities, how to juggle time and commitments, and how to manage stress. Your input into planning your gifted child's education can help him or her make choices that will literally last a lifetime.

“To those intelligent people, it must seem absurd to liken mathematics to music as an art to be savored and enjoyed even in one’s leisure time. Yet that is how it should appear and could appear if it were playing its proper role in our (otherwise) civilized society. Just as an appreciation of music is a hallmark of the educated person, so should be an appreciation of mathematics. . . .” From the Foreword: Mathematics in Our Culture by Professor Peter Hilton, 1996. In Mathematics: From the Birth of Numbers (1997) by Jan Gullberg.





There is no doubt that gifted students (in particular, students gifted in mathematics) need special attention from both parents and educators. What is more, such students need the support of the whole nation. Although every gifted child will not necessarily grow into a famous scientist or other professional, the probability of such an event is substantial. The nation should take exemplary care of gifted children as they are very likely to make future breakthroughs in one or more areas of human endeavor. In sum, gifted children are a national treasure.

I would like to emphasize that the challenge of teaching mathematics to gifted children is a multilateral one that cannot be addressed in one article. The complexity of this challenge is further complicated by the fact that the methods used for teaching pre-adolescents and adolescents have to be somewhat different. Distinctions are necessitated by developmental stages in the thinking process. In this paper, I will focus on teaching algebra to gifted high school students.

The first obvious question that arises in connection with the mathematical education of gifted children is whether current textbooks provide the proper level of presentation of the required mathematics curriculum. In my opinion, and to my deep regret, the answer to this question is no. I cannot discuss here, in detail, current textbooks on high school mathematics. I can only point out their two major defects: (1) textbooks are over-saturated with diverse, disconnected concepts and facts; and (2) proofs of many statements are missing.

During the early fifties in the former USSR (from which I emigrated with my family in 1989) there arose the following situation in teaching high school mathematics. The level of high school mathematics schooling turned out to be insufficient to meet requirements of the leading universities and colleges of the country. In order to rectify the situation, the publishing house of the Moscow State University released a series of books and other instructional materials ( many of them were written by Professor P.S. Modenov) in which many topics of the high school mathematics curriculum were considered in greater depth compared to standard high school textbooks on mathematics. The emphasis was placed on: (1) presentation of many topics in a logical, rigorous manner; and (2) analyses of typical mistakes made by high school graduates on college entrance exams. In addition, these textbooks contained many challenging mathematical problems at the high school level.

I graduated from high school in 1951 and in the same year entered the mathematical faculty of the Leningrad Pedagogical Institute. In the course of my entire pedagogical career, Modenov’s books served me faithfully. In fact, these books made a good substitute for textbooks on methodology of teaching high school algebra (which did not exist in the USSR at that time) for prospective and acting teachers of mathematics. In particular, it was a remarkable accomplishment in the methodology of teaching high school algebra and trigonometry, that P.S. Modenov, working with S.I. Novoselov, developed a new approach to teaching methods of solving equations, inequalities, systems of equations and inequalities. Their approach was based on the theory of equivalency. It permitted the whole solution process to be made logically irreproachable. There is not a shadow of a doubt that these ideas are of exceptional value for students and teachers of mathematics worldwide. The books written by P.S. Modenov contain valuable methodological material related to many topics of the high school mathematics curriculum -- in particular, on systems of linear equations, on generalization of the concept of a power, on polynomials and polynomial equations, on different concepts of logic, on mathematical induction, on the theory of trigonometric and inverse trigonometric functions, etc. It would not be an exaggeration to state that the printing of numerous books by Modenov resulted in a tangible improvement in the quality of knowledge of high school mathematics in the former USSR (and even more so of gifted students).

In summary, the first important direction that deserves close attention from mathematicians and pedagogues (in connection with teaching mathematics to gifted students) should be writing special textbooks and other educational materials in which the topics of the high school mathematics curriculum are presented with consistency, logic and all necessary proofs. Publications of this type would be very helpful for: (1) acting and prospective teachers of mathematics; and (2) students who have a special interest in mathematics.

One of the major concerns for a teacher who is teaching mathematics to gifted students should be profound and multilateral development of these students̓ algebraic skills. Without student mastery of algebraic technique, it is absolutely meaningless to talk about their mathematical education. It is a very well known fact that many problems in geometry, trigonometry, analytic geometry, calculus, etc. can be reduced to algebraic problems, so that proficiency in high school algebra becomes a prerequisite for studying all of those subjects. Algebraic techniques have many aspects. Some of them are: the ability to perform transformations with polynomials, algebraic fractions, power and radical expressions, logarithms, ability to factor polynomials, mastery in solving equations, inequalities, systems of equations and inequalities, and proving identities and inequalities. No later than in grade 9, all gifted children should remember formulas for (a+b)2, (a-b)2, (a+b) (a-b), (a+b)3 , (a-b)3, (a+b) (a 2 - ab + b 2), (a-b)(a 2 +ab + b 2).

The mastery of algebraic skills under discussion cannot be reduced to simply juggling formulas. A teacher working with gifted students must teach them to be logically consistent when solving a problem or proving a theorem. The teacher should be especially attentive when reading a student̓s paper or listening to a student̓s oral presentation. Every (even “small”) flaw in logical reasoning should be detected by the teacher. The teacher cannot afford to be lenient in this respect. Not only geometric proofs require strict logic. Sometimes solving equations, inequalities, and systems of equations and inequalities require strict and even refined logical thinking. For a mathematical education of high quality, it is very important that a student be able to distinguish a conjecture from a proof, and a heuristic method from a rigorous proof. I think that thorough discussion and constant use of the method of mathematical induction will contribute a great deal to the development of gifted students̓ logical thinking.

A very important part in the mathematical education of gifted students should be played by so-called “word problems.” Sometimes a “word problem” represents a real-life situation and students find that interesting. A teacher should teach his/her students how to make up a formal mathematical model adequate to this real-life situation. Thus, modeling is the first stage in solving every word problem. I am convinced that the adequacy of the mathematical model to the original problem is the key question. The second stage of the process of solving a word problem is to solve the algebraic model using algebraic techniques. There is one more aspect to solving a word problem that deserves attention. The point is that when a science or economics specialist has to solve a word problem whose plot comes from his subject area, this person is interested, for the most part, in solving the problem in letters (versus with numerical data). Replacing numerical data with parameters opens many opportunities for a solver. In particular, he/she can, by assigning specific numerical values to the parameters, verify the plausibility of the answer.

When discussing the challenge of developing gifted students̓ logical thinking abilities, I cannot ignore the inexplicable fascination of authors of some textbooks on high school mathematics with so-called “formal proofs” in geometry. When carrying out these proofs, students are urged to refer to numerous postulates regarding equality of numbers, congruency of line segments, angles, triangles, etc. Many of those postulates are completely obvious from the common sense point of view. Some postulates are in fact substantial theorems that have to be proved. This approach to geometric proofs is absolutely irrelevant in a high school setting. It would be appropriate only for mathematics majors studying a formal axiomatic system of geometry. A student̓s attention is shifted away from the search for creative ways of proving a statement towards the time consuming, tedious chore of referring to a proper postulate. I am convinced that formal geometric proofs should be banished from the high school mathematics curriculum once and for all. I will add here that the formality of the proving process interferes in a negative way with the creative solution of more difficult geometric problems. We, teachers of mathematics, must teach students to carry out informal geometric proofs logically and consistently. There is no doubt that the level of difficulty of geometric proofs should be increased.

I would like to direct the reader̓s attention to one more important aspect of teaching mathematics to gifted students. The high school mathematics curriculum contains too many different concepts from different areas of mathematics (algebra, geometry, trigonometry, analytic geometry, logic, probability, statistics). Teachers of mathematics have to resist the temptation of introducing too many different concepts over a short period of time. Even for gifted students, it is impossible to assimilate many concepts unless enough time is provided. The introduction of a new concept requires the utmost circumspection. Motivation is a very important first stage in the process of introducing a new concept. Probably, even more important, is to teach students how to make use of the new concept, and how to prove different statements related to it. In other words, students have to see the new concept at work. Only after this goal is accomplished, can a teacher move on to the next concept. The comparatively high level of students̓ performance in high school mathematics in the former USSR (at least in part) can be attributed to the fact that students have been given enough time to assimilate new concepts.

I will address one more issue of teaching mathematics to gifted students. A teacher of mathematics should teach his/her students to express mathematical ideas by means of very accurate, precise language. In connection with the correct language problem, it is absolutely necessary to conduct oral questioning of the students. Students should come to a blackboard and make a presentation. A teacher should pay close attention to the correctness of the student̓s logical thinking and correctness of language. And, what is more, a teacher should ask the whole class to listen carefully to the oral presentations of their classmates and detect any mistakes. Slovenliness in thinking or language is intolerable when working in mathematics. I also believe that comprehensive oral examinations should be instituted at the end of each term. Though this may cause some financial and organizational problems, nonetheless it is impossible to overestimate the benefits of such exams.

I consider it a very serious defect of the high school mathematics curriculum that solid geometry is not included. As a consequence, college freshmen majoring in mathematics, science or engineering have to study solid geometry in college, whereas they could begin learning solid geometry successfully while in high school.

Another important issue is concerned with avoiding boredom in the classroom. Teaching mathematics to gifted students should not be made boring. (This refers especially to middle school children.) To this end, when making a presentation of some new material, a teacher should highlight the basic, most important ideas of the topic at hand and facilitate understanding of those ideas by the students. Whenever relevant, a teacher can provide some levity in the classroom by offering entertaining mathematical problems, A teacher can stir students̓ enthusiasm towards mathematics by introducing interesting mathematical games. (Some of them are very rich in content.)

Next, I will discuss problems related to different kinds of mathematical contests, as part of the system of working with gifted students. In the former USSR, mathematical olympiads started in the 1930's. The mathematical olympiad became a traditional annual event; it involved gifted students from grades 5 to 10 (the last high school grade). In Leningrad, where I grew up and studied, mathematical olympiads were set up in three stages: school-wide, district-wide, city-wide. Winners of the city-wide contest received special diplomas and were given some privileges on college entrance exams. Long before the contest, the publishing house of the Leningrad State University published collections of mathematical problems to help students prepare for the olympiad. In many cases, high school teachers of mathematics organized after-school sessions to help students interested in mathematics better prepare for the olympiad. Some of the gifted students attended mathematical meetings at the Leningrad State University where mathematical sessions were conducted by highly qualified professional mathematicians. One has to acknowledge that mathematical olympiads in the former USSR enjoyed wide popularity.

Nevertheless, I would like to warn mathematical educators against excessive passion for the competitive aspect of mathematical contests. First of all, there is a type of gifted student who cannot concentrate well under the dual conditions of nervous stress and limited time. For these students, failure in a contest can be a serious psychological trauma. Secondly, I caution against problems whose solution is based on a “trick” that may not be familiar to the student. I think this kind of “trick” solution should be carefully avoided, since it can misdirect the proper intellectual development of gifted children.

I have to give due credit to some mathematicians from the former USSR who clearly recognized the danger of trickery. They published books that cultivate a serious approach to the problem of mathematical education of gifted students. One of these books is the excellent: Mathematical Circles (Russian Experience) by D. Fomin, S. Genkin, and I. Itenberg (in the series, Mathematical World, Vol. 7, 1996, ISBN 0-8218-0430-8, published by The American Mathematical Society). It suffices to list some of the topics covered in this book (combinatorics, Dirichlet̓s principle, mathematical induction, graphs, invariant, games, inequalities) in order to realize how seriously and profoundly the authors treated the challenge of educating mathematically gifted students. Even the names of chapters attest to the fact that the first concern of the authors is extending students̓ horizons in understanding mathematics and introducing students to research work in mathematics. At the beginning of each section, the authors introduce basic theoretical concepts and facts to the reader and solve some typical problems. Then, they list some problems for the reader. The book is full of methodological remarks and advice for both teachers and students. In the course of the entire book, a reader does not feel set adrift; he/she can always feel the friendly hand of the authors ready to help in a moment of quandary. At the same time the authors encourage the reader to be thoughtful, patient and persistent. It is especially commendable that all basic mathematical ideas related to each chapter are clearly presented.


I would like to emphasize that the challenge of teaching the mathematically gifted students is far from being simple, even if the following three necessary conditions hold: (1) the teacher has profound knowledge of the subject matter; (2) the students are from the same age group; and (3) the students have access to good textbooks on mathematics. I think that one of the fundamental principles in working with gifted students should be an inductive approach to discovering new statements. An inductive approach will introduce students to research work in mathematics. In search for a general statement, a teacher should encourage students to make transitions from particular cases to more general statements until the statement is formulated in full generality. Then, it must be proven with rigor and logic. One more point deserves the teacher̓s attention. Every mathematician in his or her research work encounters some “statements” that after thorough investigation turn out to be false. We have to teach students to be very critical when considering a new “statement.” In particular, we should encourage students to make up counterexamples intended to disprove the new “statement.” In connection with this suggestion, I recommend the following books of G. Polya to teachers:

Mathematics and Plausible Reasoning, Vols. 1 & 2, Princeton University Press, 1954.

Teaching mathematically gifted children is not “mass production.” Gifted children are all different. They differ from each other by turn of mind, by methods they use to approach a problem and by speed of cognition. A teacher should be sensitive to the needs of different types of gifted children. In particular, a teacher should be concerned with the:

● development of the strong aspects of a student̓s talent; and

● helping to eliminate the weak characteristics of the thinking process, thereby facilitating mathematical development of the student.






The term “acceleration” is a misnomer. The process is, or should really be, one of bringing talented youth up to a level of instruction commensurate with their achievement levels and readiness so that they are properly challenged to learn the new material (Feldhusen, 1989). In other words, there is no acceleration if all we are doing is meeting the already existing needs of the student. If we do not meet those needs, though, we are “decelerating” or hindering their achievement or learning process.

It is fairly well-accepted that the American education system was created in the image of the assembly line in order to produce “worker bees” to enable the great industrial machines of yesterday to produce widgets with the utmost speed and efficiency. The assembly line is fast, efficient, profitable, and a way of life. Graded schools arose in response to and met the challenge of the influx of children to city schools during the early years of the industrial revolution. We in education, having adopted this modus operandi, are still trying to produce well-educated adults with this method. Times and the requirements of our society have changed. We now have the ability and technology to create Individualized Education Plans (IEP) for every student in school…and we should. We continue to discriminate against students because of their age. An interesting law suit, recently filed (U.S. Department of Education, Office for Civil Rights: Levi vs. Santa Monica Community College Docket Number 09-99-2308 and the Los Angeles Unified School District Docket Number 09-99-1422):

“…alleges discrimination under PL 94-142 (IDEA) because some gifted children’s disabilities are masked and/or ignored when those children work at or above grade level. Disabilities which prevent a child from learning at the level and pace appropriate for the child’s mental age should be addressed regardless of whether a discrepancy exists between the child’s mental and chronological ages” (Sheard, 2000).

In other words, if chronological age is the only determining factor, then it is a form of civil rights discrimination just as it would be if a person could not get a job because that person is over the age of 60. If we as a society honor and respect the rights of the individual as much as we would like to think we do, then why do we continue to violate the rights of academically gifted students? We hold them back only because of chronological age while ignoring intellectual age.

Grade skipping is one expedient way to challenge gifted students, if done appropriately. In a ten-year longitudinal study of gifted students identified as mathematically precocious and who had been academically accelerated, it was found that there was no “…support (for) the common concern that gifted students may be harmed by accelerated experiences” (Swiatek & Benbow, 1991, p. 528). In fact, the vast majority of studies conducted on this topic over the past several years have indicated that regardless of the type of acceleration used, positive benefits have been noted for students (Benbow, 1991).

According to research, failing to accelerate students who meet the criteria for acceleration has detrimental effects on their education. There are strong indications that many students who remain in under-challenging educational environments will not fully use their considerable talents. They develop poor study habits, apathy, maladjustment, and may not complete school. We, as a society, seem to be more concerned with “equal” than we are with “appropriate.” But the question remains, “How do I know if my student should be accelerated?”

The answer to that and many other questions regarding whole-grade acceleration or grade skipping is contained in the Iowa Acceleration Scale: A Guide for Whole-Grade Acceleration K-8 (Assouline, Colangelo, Lupkowski-Shoplik, & Lipscomb, 1998). Presented in a format that is user-friendly and logical, the Iowa Acceleration Scale (IAS) is the first instrument to provide an objective procedure with which to address the questions related to grade skipping a student. A readable manual is provided with case studies, examples, detailed instructions for completing and interpreting the results, and a complete list of references. In addition, a Summary and Planning Sheet and questionnaire form are provided separately for individualized, cost-effective use with many students.

The IAS is comprised of eleven categories, each with a series of questions. Those categories include the following: Student Information, Family Information, Child Study Team Information, Critical Items, School History, Prior Ability and Achievement Test Results, Academic Ability and Achievement, School Factors, Developmental Factors, Interpersonal Skills, and Attitude and Support. The answers are assigned numerical values, and in the final category, the subtotals are combined to yield a “final score” which indicates the advisability of acceleration for that particular student.

Comprised of six sections, the IAS Manual covers the following information:

Section I covers background on the IAS, its purpose, and the advantages of using it. Those advantages include the following:

● A more objective view of the student

● An analysis of the major factors to be considered in making a decision

● Guidelines for weighting the relative importance of the major factors

● Documentation of the student’s strengths and concerns

● A numerical range to guide the discussion and decision of acceleration

● A standard of comparison with students who have had successful accelerations

● The support of the Belin and Blank International Center for Gifted Education and Talent Development which is just a phone call away.

Section II describes the “top ten” issues regarding acceleration, giving case vignettes to assist understanding of the issue.

Section III provides specific instructions on completing and interpreting the resulting scores of the IAS.

Section IV consists of examples of completed forms from actual cases.

Section V provides research documentation on whole-grade acceleration.

Section VI lists references for additional research on this topic.

Grade skipping is an emotional area, and therefore can be very difficult to accomplish in an objective manner. The IAS provides the uniformity and objectivity necessary in making such an important decision, while maintaining sensitivity to the student. I would suggest an objective third party be employed to question the various parties involved, and complete the IAS when the emotions are running high regarding the decision of the school and the desires of the parents and student. The IAS is a major step in eliminating the continuous lack of regard for the individual rights of students in America, especially those who are intellectually more mature than their peers. The final question has to be “Why would you hold anyone, at any age, back in terms of advancing his or her knowledge and educational goals?” The IAS provides those of us concerned with appropriate education for individuals the tool with which to determine the most challenging placement of the student in our educational system.


Assouline, S., Colangelo, N., Lupkowski-Shoplik, A., & Lipscomb, J. (1998). Iowa Acceleration Scale: A guide for whole-grade acceleration K-8. Scottsdale, AZ: Gifted Psychology Press, Inc.

Benbow, C.P. (1991). Meeting the needs of gifted students through use of acceleration. In M.C. Wang, M.C. Reynolds, and H.J. Walburg (Eds.), Handbook of special education (Vol. 4, pp. 23-36). Elmsford, NY: Pergamon Press.

Feldhusen, J.F. (1989). Synthesis of research on gifted youths. Educational Leadership, 46(6), 6-11.

Sheard, W. (January 13, 2000). A civil rights action for gifted children. [On-line]. Available:

Swiatek, M.A., & Benbow, C.P. (1991). Ten-year longitudinal follow-up of ability-matched accelerated and unaccelerated gifted students. Journal of Educational Psychology, 83 (4), 528-538.





Recently, I was stirred by an experience that demonstrates intellectual synergy when I attended a revised version of Edward Albee's play -- Who's Afraid of Virginia Woolf? (first performed in 1962) -- at The Actors Studio in New York City. As I sat in this small and famous theatrical setting, I felt the presence of ghosts from the cultural past. Theatre in the United States from the 1930's to 1950's had a world-wide impact through such performers as Marlon Brando, Montgomery Cliff, James Dean and Marilyn Monroe who appeared on this studio's stage. Many famous directors and dramatists also perfected their craft there, and later they would go from theatre to the motion pictures. It wasn't just theatre but also the American cinema that stirred the world. The play was directed by Arthur Penn who was the film director of such classics as Bonnie and Clyde (1967). The theatre is purposely small so that the audience can achieve an intense experience. After each performance, there are discussions among the director, performers and audience.

The inspiration for modern drama comes from the pre-revolutionary Russian theatre (1890-1917) under the leadership of Konstantin Stanislavski (1865-1938) of the Moscow Art Theatre. Through his direction, it became the forerunner of modern theatre and cinema. In 1898, he produced the play by Anton Chekhov (1860-1904), The Seagull. This was the start of Stanislavski's Method Acting which stresses psychological realism. The performers define a role by saying, "If I was this character, I would. . . ." In order for performers to achieve this realism, they must recall their own feelings and behaviors in a similar situation. Lee Strasberg (1901-82), a highly effective educator of gifted actors and actresses, taught Method Acting to several generations of performers at The Actors Studio from 1948-82.

The best examples of cooperative learning are experiences with intellectual synergy involving the interaction of gifted individuals working on a specific project in a holistic manner. Theatre and cinema are examplars of cooperative learning. Educators of the gifted need to study examples of these forms of intellectual synergy to truly understand what cooperative learning means for the gifted individual. First, it must take place in a context such as a project. Second, it includes the interaction of related disciplines such as directing, lighting, stage design and costume design. Third, it displays itself through a product such as a play or film. Members of The Actors Studio achieve results showcasing their giftedness by using their sensibility and productiveness. The ancient Greek theatre and Shakespeare’s dramatic productions are also examples of giftedness via synergy. Let the show begin!




E-Mail Address: