ED 321 487 1990
ERIC DIGEST E482
Author: Miller, Richard C.
The Council for Exceptional Children, Reston, Va.; ERIC Clearinghouse on
Handicapped and Gifted Children, Reston, Va.
DISCOVERING MATHEMATICAL TALENT
Sara, who is 5 years old, listens as her 32-year-old father comments that
today is her grandmother's 64th birthday. "Grandma's age is just twice my
age," he observes.
Although outwardly Sara does not seem to react to this information,
her mind is whirling. A few moments pass, and then the young girl
excitedly replies, "You know Dad, you will only be 54 when your age is
twice mine!"
Sara has been intrigued by numbers and numerical relationships
since she was very small. At first this could be seen in the way she liked
to count things and organize groups of objects. She showed a fascination
for calendars, telephone numbers, dates, ages, measurements, and almost
anything else dealing with numbers. Sara learned and remembered this
information quickly and easily, but what was even more amazing was the
way she played with and manipulated the information she was learning.
She would carefully examine each idea and eagerly search to discover new,
interesting, and unusual relationships and patterns. Although Sara has had
little formal instruction in mathematics, at the age of 5 she has acquired an
incredible amount of mathematical knowledge and is amazingly
sophisticated in using this knowledge to discover new ideas and solve
problems.
Sara is an example of a young child who is highly talented in the area
of mathematics. Like most individuals with this unusual talent, Sara
exhibits characteristics and behaviors that are clues to her ability. Some
mathematically talented people radiate many or obvious clues, others offer
only a few, or subtle ones. Recognizing these clues is often an important
first step in discovering an individual's high ability in mathematics. It is
difficult to believe, but many people with a high degree of mathematical
talent have their talent underestimated or even unrecognized. Their clues
have gone unnoticed or ignored, and the true nature of their ability
remains unexplored. If Sara's talent in mathematics is to be discovered
and appropriately nurtured, it is important that her parents and teachers
recognize the clues.
WHAT SHOULD PARENTS AND TEACHERS KNOW TO HELP THEM BETTER
RECOGNIZE MATHEMATICAL TALENT?
Mathematical talent refers to an unusually high ability to understand
mathematical ideas and to reason mathematically, rather than just a high
ability to do arithmetic computations or get top grades in mathematics.
When considering mathematical talent, many people place too much
emphasis on computational skill or high ability in replicating taught
mathematical procedures. Unless mathematical talent is correctly
perceived, however, important clues can be overlooked and less important
clues can be given too much significance.
Some characteristics and behaviors that may yield important clues in
discovering high mathematical talent are the following:
1. An unusually keen awareness of and intense curiosity about numeric
information.
2. An unusual quickness in learning, understanding, and applying
mathematical ideas.
3. A high ability to think and work abstractly and the ability to see
mathematical patterns and relationships.
4. An unusual ability to think and work with mathematical problems in
flexible, creative ways rather than in a stereotypic fashion.
5. An unusual ability to transfer learning to new, untaught
mathematical situations.
Terms such as mathematically talented, mathematically gifted, and highly
able in mathematics are generally used to refer to students whose
mathematics ability places them in the top 2% or 3% of the population. It
is important to keep in mind the unusually high degree of talent that is
being sought when looking for mathematically talented individuals.
Not all students who achieve the highest test scores or receive the highest
grades in mathematics class are necessarily highly talented in
mathematics. Many of the mathematics programs in our schools are
heavily devoted to the development of computational skills and provide
little opportunity for students to demonstrate the complex types of
reasoning skills that are characteristic of truly talented students. The tests
used and the grades given in such programs usually reflect that structure.
Computational accuracy and conformity to taught procedures may be
overemphasized, and the reasoning abilities associated with high ability in
mathematics may be underemphasized. In this type of environment, test
scores and grades of less able students who are good in computation,
attentive in class, willing to help, and conscientious about completing all
assignments carefully in the prescribed manner will often be as high as the
test scores and grades of students who are genuinely talented in
mathematics. While high achievement in school certainly can be a clue to
high ability in mathematics, additional information is needed. If care is
not taken, students who are simply high achievers in mathematics can be
mistakenly identified as mathematically talented. It is just as important to
avoid such incorrect identification as it is to identify students who are
truly mathematically talented.
Some mathematically talented students do not demonstrate outstanding
academic achievement, display enthusiasm toward school mathematics
programs, or get top grades in mathematics class. It is important to know
that there are students like this, for their ability in mathematics is easily
overlooked, even though they may exhibit other clues suggesting high
ability in mathematics. There are many possible reasons why these
students do not do well, but often it is at least in part because of a
mismatch between the student and the mathematics program. Many of
them refuse, or are unable, to conform to the expectations of programs that
they see as uninteresting and inappropriate. For their part, educators may
not recognize the true ability of these students or see a need for adjusting
the existing mathematics program.
HOW CAN STANDARDIZED TEST RESULTS HELP IN RECOGNIZING
MATHEMATICAL TALENT?
Intelligence Tests. IQ test results often yield valuable information and
may provide clues to the existence of mathematical talent. Used alone,
however, these tests are not sufficient to identify high ability in
mathematics. Mathematical talent is a specific aptitude, while an IQ score
is a summary of many different aptitudes and abilities. An individual's IQ
is made up of several different components, only some of which relate to
mathematical ability. Suppose two students have the same IQ scores. One
of them could have a high score in mathematical components and a low
score in verbal components, while the other is just the opposite. The first
student would be likely have to much greater mathematics ability than the
second, even though they have the same overall IQ. Children with high
IQ's--no matter how high the score--cannot be assumed to be
mathematically talented. It could be a clue, but more information is
needed.
Creativity Tests. There are differing opinions on how the results of
creativity tests can be used to help identify high ability in mathematics.
Although mathematically talented students display creativity when
dealing with mathematical ideas, this is not always apparent in creativity
test results. However, high creativity assessments, along with indications
of intense interest in mathematics, do seem to be a significant clue of
mathematical talent.
Mathematics Achievement Tests. Mathematics achievement tests also can
provide valuable clues in identifying high ability in mathematics, but the
results of these tests have to be interpreted carefully. Mathematics
achievement tests are often computation-oriented and give little
information about how a student actually reasons mathematically. Also,
the tests seldom have enough difficult problems to appropriately assess
the upper limits of a talented student's ability or show that this ability is
qualitatively different from that of other very good, but not truly
mathematically talented, students. If these limitations are kept in mind,
the results of mathematics achievement tests can be useful. Students
scoring above the 95th or 97th percentiles on national norms may have
high ability in mathematics, but more information is needed to separate
the high achievers from the truly gifted. It should not be assumed that
there are no mathematically talented students among those scoring below
the 95th percentile; those students will have to be recognized from other
clues.
Mathematics Aptitude Tests. Standardized mathematics aptitude test
results should be used in basically the same way that the results of mathe-
matics achievement tests are used. Aptitude tests have some of the same
limitations as achievement tests except that, because they are designed to
place less emphasis on computational skills and more emphasis on
mathematical reasoning skills, the results from these tests are often more
useful in identifying mathematically talented students.
Out-of-Grade-Level Mathematics Aptitude Tests. Many of the limitations
associated with mathematics aptitude tests can be reduced by
administering out-of-grade-level versions of the tests. This process should
be used only with students who already have demonstrated strong
mathematics abilities on regular-grade-level instruments or those who
show definite signs of high mathematics ability. An out-of-grade-level
mathematics aptitude test is a test that is usually designed for and used
with students about one and one-third times the age of the child being
tested. For example, a 9-year-old third grader would be tested using an
abilities test normally written for 12-year-old sixth graders. This gives a
much better assessment of mathematical reasoning skills because the
student must find ways to solve problems, many of which he or she has
not been taught to do. These tests have many difficult problems that will
challenge even the most capable students, thus making it possible to
discriminate the truly talented from others who are just very good in
mathematics.
The out-of-grade-level testing procedure has been used successfully
in several mathematics talent searches and school mathematics programs
with junior and senior high school students over the past 15 years. More
recently, there have been programs that have successfully used the
procedure in the elementary grades.
WHAT SYSTEMATIC PROCESS CAN BE USED TO IDENTIFY
MATHEMATICALLY TALENTED STUDENTS?
Correctly identifying mathematically talented students is not a simple task,
and there is more than one way to go about it. Some common features of
successful identification processes are combined in the following model.
This model is intended to be implemented with a degree of flexibility in
order to give mathematically talented students every opportunity to have
their talent discovered. This may be especially important when looking for
mathematical talent in minority or disadvantaged populations.
PHASE ONE: SCREENING
The objective in phase one is to establish a group of students suspected of
having high ability in mathematics. These students will be evaluated
further in the next phase. In phase one, effort attempt should be made not
to miss potentially talented students.
Step One. An identification checklist (Figure 1) should be set up to record
the names of students thought to have high ability in mathematics along
with the clues that suggest their talent. Students scoring above the 95th
percentile on a mathematics aptitude test are entered first. Next, those
scoring above the 95th percentile on mathematics achievement tests who
are not already on the list are added. If a student's name is already on the
list, the test score is simply added to that student's record. In a like
manner, students who are mentally gifted; students who are creative and
have high interest in mathematics; and students nominated by par-ents,
teachers, self, or peers can be added.
FIGURE 1. IDENTIFICATION CHECKLIST
---------------------------------------------------------------------
Student/Ability/Ach./Gifted/Creative/Teacher-Parent/Out-of-Grade
Name /test /test /Nominations /Level Test
---------------------------------------------------------------------
John
Jones 97 yes-yes
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Sally
Smith 95 yes
--------------------------------------------------------------------
Step Two. The checklist information for each student should be reviewed.
If the information collected for a particular student suggests that out-of-
grade-level testing is not advisable, that student's name should be
removed, because phase two testing may damage the egos of students who
do not really excel in mathematics. However, caution should be exercised
not to eliminate talented students in this process. Parent involvement in
these decisions is rec-ommended.
PHASE TWO: OUT-OF-GRADE-LEVEL MATHEMATICS ABILITIES
ASSESSMENT
The objective in phase two is to separate the mathematically talented
students from those who are merely good students in mathematics and to
begin assessing the extent of the ability of the mathematically talented
students.
Step One. Students who are scheduled to take the out-of-grade-level test,
along with their parents, should be informed about the nature of this test
and the reason it is being given. The out-of-grade-level test would then be
administered with student and parent consent. Figure 2 provides a sample
schedule for such testing.
FIGURE 2. TESTING SCHEDULE
-----------------------------------------------
Current Grade Out-of-Grade-Level Test
(Fall)
-----------------------------------------------
1st 3rd grade - Fall
2nd 4th grade - Fall
3rd 5th grade - Spring
4th 7th grade - Fall
5th 8th grade - Fall
6th 9th grade - Spring
7th 11th grade - Fall
8th 12th grade - Fall
Step Two. The results of each student's out-of-grade-level test should be
evaluated in conjunction with the results of phase one screening.
Generally, the student's out-of-grade-level score will be an indication of
degree of mathematical talent. Scores above the 74th percentile represent
a degree of mathematical talent similar to that of students identified in
regional talent searches such as the one conducted by Johns Hopkins
University. This level of talent places the student in the upper 1% of the
population in mathematics ability. Scores above the 64th percentile
denote a level of talent that most likely places the student in the upper 3%
of the population. Students in these two groups would be identified as
mathematically talented.
WHAT INSTRUCTIONAL APPROACHES BENEFIT MATHEMATICALLY
TALENTED STUDENTS?
Students identified as mathematically talented vary greatly in degree of
talent and motivation. No single approach is best for all of these students.
The design of each student's instructional program in mathematics should
be based on an analysis of individual abilities and needs. For example,
students with extremely high ability and motivation may profit more from
a program that promotes rapid and relatively independent movement
through instructional content. Students with less ability or lower
motivation may do better in a program that is not paced so quickly and is
more deliberate in developing the mathematical concepts being taught.
There are some common features, however, that seem to be important
ingredients in the mathematics programs of mathematically talented
students.
The program should bring mathematically talented students together to
work with one another in the area of mathe-matics. Students will benefit
greatly, both academically and emotionally, from this type of experience.
They will learn from each other, reinforce each other, and help each other
over difficulties.
The program should stress mathematical reasoning and develop
independent exploratory behavior. This type of program is exemplified by
discovery learning, looking for underlying principles, engaging in special
projects in mathematics, problem solving, discovering formulas, looking for
patterns, and organizing data to find relationships.
The mathematics program should deemphasize repetitious computational
drill work and cyclical review. This type of work in mathematics should
be minimal for all mathematically talented students. As ability in
mathematics increases, the benefits to be gained from this type of activity
decrease.
The scope of the mathematics curriculum should be extensive so that it
will provide an adequate foundation for students who may become
mathematicians in the future. In many programs the mathematics
curriculum will have to be greatly expanded to meet this need.
The mathematics program should be flexibly paced. Flex-ibly paced means
that students are placed at an appropriate instructional level on the basis
of an assessment of their knowledge and skill. Each student is then
allowed to progress at a pace limited only by his or her ability and
motivation. Flexible pacing can be achieved in the following ways:
* Continuous progress. Students receive appropriate instruction daily
and move ahead as they master content and skill.
* Compacted course. Students complete two or more courses in an
abbreviated time.
* Advanced-level course. Students are presented with course content
normally taught at a higher grade.
* Grade skipping. Students move ahead 1 or more years beyond the
next level of promotion.
* Early entrance. Students enter elementary school, middle school,
high school, or college earlier than the usual age.
* Concurrent or dual enrollment. Students at one school level take
classes at another school level. For example, an elementary school student
may take classes at the middle school.
* Credit by examination. Students receive credit for a course upon
satisfactory completion of an examination or upon certification of mastery.
CONCLUSION
The fate of Sara and other mathematically talented students will be
determined largely by the ability of their parents and educators to
discover and nurture their special ability. The notion that these students
will achieve their potential anyway is constantly refuted. For too many
students like Sara, lack of appropriate mathematical nourishment seems to
be the rule rather than the exception. At risk are the benefits that these
children might gain from early advancement and the attitudes that these
children will have toward mathematics, school, learning in general, and
themselves. By discovering the mathematical talent of these students and
using that knowledge to provide appropriate academic nurture, we have
the greatest chance to help these individuals reach their gifted potential.
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ERIC Digests are in the public domain and may be freely reproduced and
disseminated. ------
This publication was prepared with funding from the U.S. Department of
Education, Office of Educational Research and Improvement, under contract
no. RI88062007. The opinions expressed in this report do not necessarily
reflect the positions or policies of OERI or the Department of Education.