Jennie Bennett, Manuel Berriozábal, Margaret DeArmond, Linda Sheffield
(Chair), Richard Wertheimer
The NCTM Task Force on Promising Students consists of five educators
who represent a number of different constituencies including public
schools, programs for promising students, parents, universities,
and researchers. The task force had an initial meeting at the
NCTM Annual Meeting in Boston, read relevant literature, including
the NCTM publication, Providing Opportunities for the Mathematically
Gifted, disseminated a survey on the internet to relevant news
groups and mailing lists, and met for three days to craft this
report that includes issues, recommendations, and a draft of a
policy statement. After the initial report was written, it was
posted on the internet for comments, and revised. A copy of the
initial survey is appended to this report, and the internet responses
are available upon request from the committee chair or on-line
through electronic mail.
In 1980, it was stated in An Agenda for Action, "the student most
neglected, in terms of realizing full potential, is the gifted
student of mathematics. Outstanding mathematical ability is a
precious societal resource, sorely needed to maintain leadership
in a technological world. (NCTM, 1980, An Agenda for Action: Recommendations
for School Mathematics of the 1980s, p. 18). In the 15 years since
this report, not much has changed other than an even more urgent
need for technological leadership. Therefore, it is imperative
that NCTM act quickly to conserve and enhance this precious resource.
In 1972, Public Law 91-230, Section 806, was passed by the Federal
Government stating:
"Gifted and talented children are those identified by professionally
qualified persons, who by virtue of outstanding abilities are
capable of high performance. These are children who require differentiated
educational programs and/or services beyond those normally provided
by the regular school program in order to realize their contribution
to self and society. Children capable of high performance include
those with demonstrated achievement and/or potential ability in
any of the following areas, singly or in combination:
1. general intellectual ability
2. specific academic aptitude
` 3. creative or productive thinking
4. leadership ability
5. visual and performing arts
6. psychomotor ability
It can be assumed that utilization of these criteria for identification
of the gifted and talented will encompass a minimum of 3 to 5
percent of the school population." (The sixth area of psychomotor
ability was later dropped from the federal definition.)
Although educators have attempted to expand the definition of
giftedness to become more inclusive, the 1972 definition prevailed
for over two decades. Today, many states continue to use some
variation of this definition to define gifted and talented students
in order to provide special programs. Research by Howard Gardner
and others during the 1980s led to an expanded definition by the
federal government. In the 1990s, with the passage of the federal
Javits Gifted and Talented Education Act, the definition was broadened
to the following:
"Children and youth with outstanding talent perform or show the
potential for performing at remarkably high levels of accomplishment
when compared with others of their age , experience, or environment.
These children and youth exhibit high performance capability in
intellectual, creative, and/or artistic areas, possess an unusual
leadership capacity, or excel in specific academic fields. They
require services or activities not ordinarily provided by the
schools.
Outstanding talents are present in children and youth from all
cultural groups, across all economic strata, and in all areas
of human endeavor."
The Task Force on Promising Students believes that the definition
of students with mathematical promise should build upon this latter,
broader definition. Students with mathematical promise are those
who have the potential to become the leaders and problem solvers
of the future. We see mathematical promise as a function of
° ability,
° motivation,
° belief, and
° experience/opportunity.
These variables are not fixed and need to be developed so that
success for these promising students can be maximized. This definition
includes the students who have been traditionally identified as
gifted, talented, genius, prodigy, precocious, etc., and broadens
it to include students who have been traditionally excluded from
previous definitions of gifted and talented, and therefore excluded
from rich mathematical opportunities. This definition acknowledges
that students who are mathematically promising have a large range
of abilities and a continuum of needs that should be met.
Traditional methods of identifying gifted and talented mathematics
students such as standardized test scores are designed to limit
the pool of students identified as mathematically promising. Although
these measures are commonly used to identify gifted and talented
students, it is clear that many students are overlooked and that
additional means of identification must be used.
Identification of mathematically promising students is a difficult
and challenging task. Identification might be done for pedagogical,
legal, or financial reasons, but in all instances it should be
done to maximize the number of mathematically promising students.
Programs should have clear goals that guide the identification
procedures; if there are no services for these students, there
is no need to identify them. To avoid bias in the selection process,
identification procedures should include a wide variety of measures
to identify the broadest number of both females and males from
diverse cultural, and socio-economic backgrounds. Measures might
include any or all of the following depending upon the goals of
the program to be offered:
self-selection
observations of students during the problem-solving process
teacher recommendation
parent recommendation
peer recommendation
standardized tests, especially out-of-level testing
measures of creativity
solutions to problems
grades in mathematics classes
student essays
performance in mathematics contests
tests of abstract reasoning ability
measures of spatial reasoning
Identification should be inclusive rather than exclusive; if a
question arises about whether to include a student in the program,
teachers should err on the side of inclusion.
Some opportunities for promising students may require no formal
identification process. These opportunities might include such
things as investigating challenging, open-ended problems in mathematics
classes, joining mathematics clubs, entering mathematics contests,
using technology, or accessing mentors on the internet. These
opportunities should be readily available to any student who desires
to take advantage of them.
School or district committees that are deciding on programs and
materials for promising students must decide upon methods of organizing
instruction and assessment for these students. There are many
disagreements over the best programming for these students. Arguments
over enrichment vs. acceleration, homogeneous vs. heterogeneous
grouping, and theory vs. application are constant. Rather than
recommend one particular course of study, this task force is suggesting
that those involved in program development clearly understand
the ramifications of decisions made pertaining to programming.
We reiterate the need to err on the side of inclusion rather than
exclusion. Some of the most popular ways to empower mathematically
promising students are tracking, acceleration, providing opportunities
in the regular classroom, pull-out and magnet programs, and extra-curricular
activities. To open opportunities for the greatest number of promising
students, service providers must be cognizant of the dilemmas
created by any particular program.
Any program for mathematically promising students should consider
the following questions:
- Are programs consistent with recommendations from all three
sets of Standards documents perhaps using recommendations from
a higher grade level of the standards than the student’s current
grade or age level?
- Is technology being used to its fullest extent to enhance the
mathematical power of promising students?
- Are teachers, counselors, and/or other adults available and
capable of challenging, supporting, and guiding these students
appropriately?
- Are there means of evaluating the program as well as the students
in it to determine the level of success?
- Do the opportunities provide for the wide range of abilities,
beliefs, motivation, and experiences of students who have mathematical
promise regardless of their socioeconomic and ethnic backgrounds,
and do the opportunities meet their continuum of needs?
Traditionally, tracking and/or acceleration in many schools has
had the effect of segregating students and locking out many promising
mathematics students. If schools decide to track students, they
should consider the following questions:
- Are students chosen for classes based upon a general IQ score
or because of a specific mathematical aptitude?
- Are there students not identified for these classes who could
benefit from them?
- Are there students who are in top classes who are frustrated
and confused by the high level of the work and who are encouraging
the teacher to present problems at a lower level or are slowing
the progress of other students?
- Will students who are not included in these top classes have
lower self-esteem and decide that they are not capable of becoming
good mathematical thinkers?
- Are methods of identifying the placement of students for tracks
using a variety of measures to identify the broadest number of
both females and males from diverse, cultural and socio-economic
backgrounds ?
- Are students allowed to self-identify for the top classes?
- Are curriculum, instruction, and assessment qualitatively different
and designed to meet the differing needs of promising students?
Is the curriculum more challenging with provisions for individualization,
faster pace, more higher-level thinking, and higher standards
than other tracks?
- Is there flexible movement from one track to another?
The "regular" classroom may take a variety of forms. It may have
a heterogeneous mix of students, it may have mainstreamed promising
students, it may have a cluster group of talented students, or
it may be a "middle track" in a school that is tracked. All mathematics
classrooms need to provide opportunities for mathematically promising
students because these students exist in every classroom even
if they have somehow been tracked into lower groups. In providing
for these students in the regular classroom, the following questions
should be asked.
- Are there resources, projects, problems, and means of assessment
that allow for differences in the level of depth of understanding
and engagement?
- Do teachers have the pedagogical techniques to work with student
populations with diverse learning needs and from diverse backgrounds?
- Are there teachers who have been adequately prepared to work
with mathematically promising students in the role of a facilitator?
- Does the teacher have a contingency plan to determine when students
have mastered mathematical concepts and skills so that they do
not unnecessarily repeat material?
- Are there opportunities for promising students to explore interesting
problems with others of like interests and abilities?
Pull-out programs can be defined as programs that occur during
the school day where students leave their regular classroom to
engage in activities with other students of similar abilities
or interest. Magnet programs are those full-time programs or schools
that have been designed for promising students or for those with
special interests in mathematics. If schools or districts decide
to provide these programs for mathematically promising students,
the following questions should be considered:
- Are there students not identified for these programs who could
benefit from them?
- Will students who are not included in these programs have lower
self-esteem and decide that they are not capable of becoming good
mathematical thinkers?
- Are methods of identifying the placement of students for these
programs using a variety of measures to identify the broadest
number of both females and males from diverse, cultural and socio-economic
backgrounds ?
- Are students allowed to self-identify for the program?
- Are there appropriate opportunities in mathematics that have
clearly-defined, comprehensive, integrated goals and are not simply
isolated activities?
- In pull-out programs, are activities coordinated with the regular
classroom teacher so there is continual support and challenge?
Schools, universities, individuals and organizations should provide
meaningful experiences for mathematically promising students outside
of the regular classroom. These opportunities might include summer
and after-school programs, mathematics contests, correspondence
programs, mentoring, resources on the internet, and mathematics
clubs and circles. In arranging for these opportunities, the following
questions should be considered:
- Are there barriers to becoming involved in extracurricular activities
- monetary, criteria for entry, time, parental interest and involvement,
busing, etc. that have not been addressed?
- Do extracurricular activities for mathematically promising students
enhance opportunities in the regular classroom?
- Are the opportunities available to all interested students?
If America is to be number one in math and science by the year
2000, our society must value learning, particularly in mathematics.
Issues that pertain to cultural influences include the following:
- What is being done to counter the negative influence of peer
and cultural pressures?
- Is diversity celebrated and encouraged rather than merely acknowledged
and tolerated?
- What kinds of reinforcement and rewards are available for students
who are interested in learning mathematics?
- How does the culture communicate the value of mathematical literacy
to students?
- Have you engaged all stakeholders -parents, teachers, administrators
and counselors, community members, churches, media, business,
government, and the students themselves - in the process of supporting
the development of the mathematically promising?
- Are there serious efforts being made to ensure that mathematically
promising students from all groups, especially those from groups
traditionally underrepresented in mathematical fields, have the
constant, sincere, and deep commitment from a community of adults
and peers to help them succeed?
Because teachers are the primary means of access to learning,
it is very important that both in-service and pre-service programs
for teachers at all levels include information on dealing with
promising students. Regardless of the type of program being offered
to promising students, teachers should have access to professional
development, research information, and resources to deal with
the following issues:
- Are teachers trained to identify or recognize students with
mathematical promise?
- Do teachers have high levels of expectations for all students
and do they continue to challenge top students to even higher
levels of success?
- Do teachers have the pedagogical and questioning techniques
to extend students’ thinking?
- Are teachers able to select and/or develop appropriate assessment
tools that provide opportunities for students to create problems,
generalize patterns, and connect various aspects of mathematics?
- Do teachers themselves have the mathematical power to make connections
and the mathematical sophistication to see the big picture?
- Are teachers able to make appropriate instructional decisions
for these promising students?
- Are teachers aware of, can they use, and do they have access
to technological and other tools?
- Do states require a strong mathematics background for all teachers
of mathematically promising students?
- Are teachers modeling the joy of lifelong learning of mathematics
and are they learners of mathematics themselves?
- Do teachers know how to act as facilitators giving students
support in areas that might be beyond the teachers own expertise?
- What mechanism is used to recommend that teachers who are incapable
of encouraging and challenging promising students be re-educated
or replaced?
The following is a non-prioritized list of recommendations from
the Task Force to the Board of Directors. We realize that some
of these recommendations will take time and money for planning
and execution. However, some of these recommendations can be implemented
immediately to begin to address the needs of the mathematically
promising students.
1. The NCTM should develop a clearinghouse for information on
programs, research, and exemplary practices for promising students.
This should be available on the internet, in printed form through
NCTM publications, and at conferences.
2. The NCTM should provide an area on the internet or information
about such areas for promising students. Students may use this
to find interesting problems and solutions, talk to mathematics
mentors, talk to students across the world about mathematics,
find information on mathematics contests, see examples of exemplary
work from other students, and provide information on undergraduate
and graduate programs and scholarships available in mathematics.
3. The NCTM, the Mathematical Association of America (MAA), The
American Mathematical Society (AMS), the National Association
for Gifted Children (NAGC), the Association for the Supervision
and Curriculum Development (ASCD), and other concerned organizations
should produce and publicize a joint position statement concerning
the development of promising mathematics students.
4. The NCTM should establish a national advocacy group that includes
industrial, political and community leaders to support such actions
as securing funding for programs for promising students, encouraging
stronger mathematics preparation for teachers of mathematically
promising students, and providing incentives for good mathematics
teachers of promising students at all levels.
5. The NCTM should provide a theme or strand at the annual meeting
and regional conferences on mathematically promising students.
6. The NCTM should include a problem section in each journal with
problems to which individuals or groups of students can respond
with solutions for possible publication and/or recognition in
the journal.
7. The Educational Materials Committee should consider the production
of a video, a book, pamphlets, and an executive summary that can
be used for the advocacy and support of mathematically promising
students. This information can be disseminated to the media, state
and local boards of education, school personnel, parent and community
organizations, mathematics and mathematics education departments
in institutions of higher education, and church groups.
8. The NCTM, MAA, and AMS should form a joint task force to examine
means of facilitating the lifelong mathematics education of promising
students from pre-kindergarten through graduate school and into
careers. This task force might consider developing critical and
cooperative linkages among K-12 schools, institutions of higher
education, industries, and professional organizations.
9. New curricular standards, programs and materials should be
developed to encourage and challenge the development of promising
mathematical students, regardless of gender, ethnicity, or socio-economic
background.