Teaching Mathematics to Gifted Students in a Mixed-Ability
Classroom. ERIC Digest.
by Johnson, Dana T.
Mathematically gifted students have needs that differ in nature from
those of other students. They require some differentiated instruction,
defined by Tomlinson (1995) as "consistently using a variety of instructional
approaches to modify content, process, and/or products in response to learning
readiness and interest of academically diverse students." Yet recent studies
have found few instructional or curricular modifications in regular elementary
classrooms (Archambault et al., 1993; Westberg, Archambault, Dobyns &
Salvin, 1993). In grades 9-12, students may be able to select honors, advanced,
and AP courses; however, even in these more homogeneously grouped classes
there is a range of differences that need to be acknowledged.
WHY SHOULD WE DO ANYTHING DIFFERENT FOR MATHEMATICALLY GIFTED STUDENTS?
Gifted students differ from their classmates in three key areas that
are especially important in mathematics. These are summarized below.
How Gifted Learners Differ from Classmates:
1. Pace at which they learn
2. Depth of their understanding
3. Interests that they hold (Maker, 1982)
Relationship to Mathematics Learning
1. The sequential nature of math content makes pacing an issue.
2. Deeper levels of understanding and abstraction are possible for most
mathematical topics, so differentiation becomes important.
3. If the interest is snuffed out early, the talent may not be developed.
Mathematically gifted students differ from the general group of students
studying math in the following abilities: spontaneous formation of problems,
flexibility in handling data, mental agility of fluency of ideas, data
organization ability, originality of interpretation, ability to transfer
ideas, and ability to generalize (Greenes, 1981). No list of characteristics
of the mathematically gifted includes "computational proficiency," and
yet at levels prior to Algebra I, this is commonly used as the criterion
that determines who gets to move on to more interesting material. Furthermore,
there is a myth that gifted students don't need special attention since
it is easy for them to learn what they need to know. On the contrary, their
needs dictate curriculum that is deeper, broader, and faster than what
is delivered to other students.
Mathematics can be the gatekeeper for many areas of advanced study.
In particular, few gifted girls recognize that most college majors leading
to high level careers and professions require four years of high school
math and science (Kerr, 1997). Students may drop out of math courses or
turn toward other fields of interest if they experience too much repetition,
not enough depth, or boredom due to slow pacing.
An Agenda for Action: Recommendations for School Mathematics of the
1980s (NCTM, 1989, p. 18) says, "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." By 1995, when the NCTM created
a Task Force on the Mathematically Promising, not much had changed (Sheffield
et al., 1995).
WHAT DO THE CURRICULUM STANDARDS OF THE NATIONAL COUNCIL OF TEACHERS
OF MATHEMATICS (NCTM) SAY WE SHOULD DO ABOUT MATHEMATICALLY GIFTED STUDENTS?
The NCTM Standards do not mention gifted students explicitly but recognize
that students are not all the same. For all students, the Standards place
a greater emphasis on areas that traditionally have been emphasized for
the gifted. All students are now expected to complete a core curriculum
that has shifted its emphasis away from computation and routine problem
practice toward reasoning, real-world problem solving, communication, and
connections. "The Standards propose that all students be guaranteed equal
access to the same curricular topics; it does not suggest that all students
should explore the content to the same depth or at the same level of formalism"
(NCTM, 1989, p. 131). At the high school level, additional topics are suggested
for "college-intending" students. The Report of the Task Force on the Mathematically
Promising recognizes that there are special issues relating to the education
of the mathematically promising student (Sheffield et al., 1995) and has
made recommendations that include the development of new curricular standards,
programs, and materials that encourage and challenge the mathematically
WHAT SHOULD BE DONE TO DIFFERENTIATE CURRICULUM, INSTRUCTION AND
ASSESSMENT FOR THE MATHEMATICALLY GIFTED IN THE REGULAR CLASSROOM?
Historically there has been debate about the role of acceleration versus
enrichment as the differentiation mode for mathematics. Most experts recommend
a combination. The following are suggestions for differentiating for the
mathematically gifted by using (1) assessment, (2) curriculum materials,
(2) instructional techniques, and (4) grouping models. These opportunities
should be made broadly available to any student with interest in taking
advantage of them.
* Give pre-assessments so that students who already know the material
do not have to repeat it but may be provided with instruction and activities
that are meaningful. In the elementary grades, gifted learners still need
to know their basic facts. If they do not, don't hold them back from other
more complex tasks, but continue to work concurrently on the basics.
* Create assessments that allow for differences in understanding, creativity,
and accomplishment; give students a chance to show what they have learned.
Ask students to explain their reasoning both orally and in writing.
* Choose textbooks that provide more enriched opportunities. Unfortunately,
curriculum in this country is mainly driven by textbooks, which are used
about 80% of the time (Lockwood, 1992). Math textbooks often repeat topics
from year to year in the grades prior to algebra. Since most textbooks
are written for the general population, they are not always appropriate
for the gifted. Several series that hold promise for gifted learners have
been developed recently under grants from the National Science Foundation;
they emphasize constructivist learning and include concepts beyond the
* Use multiple resources. No single text will adequately meet the needs
of these learners.
* Be flexible in your expectations about pacing for different students.
While some may be mastering basic skills, others may work on more advanced
* Use inquiry-based, discovery learning approaches that emphasize open-ended
problems with multiple solutions or multiple paths to solutions. Allow
students to design their own ways to find the answers to complex questions.
Gifted students may discover more than you thought was possible.
* Use lots of higher-level questions in justification and discussion
of problems. Ask "why" and "what if" questions.
* Provide units, activities, or problems that extend beyond the normal
curriculum. Offer challenging mathematical recreations such as puzzles
* Provide AP level courses in calculus, statistics, and computer science
or encourage prepared students to take classes at local colleges if the
supply of courses at the high school has been exhausted.
* Differentiate assignments. It is not appropriate to give more problems
of the same type to gifted students. You might give students a choice of
a regular assignment; a different, more challenging one; or a task that
is tailored to interests.
* Expect high level products (e.g., writing, proofs, projects, solutions
to challenging problems).
* Provide opportunities to participate in contests such as Mathematical
Olympiads for the Elementary School (grades 4-6), Math Counts (grades 7-8),
and the American Junior High School Mathematics Exam (grades 7-8) or the
American High School Mathematics Exam (grades 9-12). Give feedback to students
on their solutions. After the contests, use some of the problems as the
basis for classroom discussions.
* Provide access to male and female mentors who represent diverse linguistic
and cultural groups. They may be within the school system, volunteers from
the community, or experts who agree to respond to questions by e-mail.
Bring speakers into the classroom to explain how math has opened doors
in their professions and careers.
* Provide some activities that can be done independently or in groups
based on student choice. Be aware that if gifted students always work independently,
they are gaining no more than they could do at home. They also need appropriate
instruction, interaction with other gifted students, and regular feedback
from the teacher.
* Provide useful concrete experiences. Even though gifted learners may
be capable of abstraction and may move from concrete to abstract more rapidly,
they still benefit from the use of manipulatives and "hands-on" activities.
HOW CAN TECHNOLOGY SUPPORT THE NEEDS OF THE GIFTED?
Technology can provide a tool, an inspiration, or an independent learning
environment for any student, but for the gifted it is often a means to
reach the appropriate depth and breadth of curriculum and advanced product
opportunities. Calculators can be used as an exploration tool to solve
complex and interesting problems. Computer programming is a higher level
skill that enhances problem solving abilities and promotes careful reasoning
and creativity. The use of a database, spreadsheet, graphic calculator,
or scientific calculator can facilitate powerful data analysis. The World
Wide Web is a vast and exciting source of problems, contests, enrichment,
teacher resources, and information about mathematical ideas that are not
addressed in textbooks. Technology is an area in which disadvantaged gifted
students may be left out because of lack of access or confidence. It is
essential that students who do not have access at home get the exposure
at school so that they will not fall behind the experiences of other students.
WHAT IS THE RESPONSIBILITY OF SCHOOLS AND TEACHERS IN
DEVELOPING GIFTEDNESS IN MATHEMATICS?
Classroom teachers and school districts share the responsibility of
addressing the needs of gifted students.
* Teachers need training and support in recognizing and addressing the
needs of the mathematically gifted.
* Teachers who teach mathematics to gifted learners need a strong background
in mathematics content. If the school has only a few students with special
needs and does not have such a teacher, a mentor from outside the school
should be located to work with individuals.
* A coordinated curriculum plan needs to be in place so that the mathematical
experiences for students are not duplicated or interrupted from one year
to the next.
* The school should have an organized support system that includes resource
books, technology, and human resources.
Regular mathematics classrooms that offer sufficiently challenging and
broad experiences for gifted students have the potential to enrich the
learning community as a whole since other students will be interested in
attempting, perhaps with help, some of the more challenging tasks. If math
classes offer diversity in assignments, products, and pacing and monitor
student needs, all students will be able to work at their own challenge
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