Nava Livne
GIFTEDNESS IN
MATHEMATICS AS A BIDIMENSIONAL PHENOMENON: THEORETICAL DEFINITION
OF LEVELS OF ACADEMIC
ABILITY
Research Goals
The research focused on identifying young people with special ability in
mathematics. The research was based on the Milgram 4 x 4 Structure of
Giftedness Model, as applied to mathematics, and had two specific goals:
1. To define giftedness in mathematics theoretically as a bidimensional
phenomenon, i.e., four types of abilities and four ability levels. Among the abilities
are one specific academic and one specific creative type of ability, each at
four different levels.
2. To develop reliable and
valid psychometric instruments to assess the two types of specific abilities in
mathematics, each at four levels.
Methodology
The research was conducted in two phases. In the first phase, specific
academic and creative abilities in mathematics at four levels were defined by
detailed conceptual descriptions in two Guttman-type mapping sentences. On the
basis of these mapping sentences, two instruments were developed:
1. The Multiscale Academic
and Creative Abilities in Mathematics (MACAM), which has two parts, one
yielding an index of academic ability, and one an index of creative ability,
each at four levels, in solving actual mathematical problems. A Guide for
Scoring the Multiscale Academic and Creative Abilities in Mathematics (MACAM)
was developed, which provided detailed criteria for defining each score on each
item operationally, according to the degree of completeness of its solution on
a continuous scale.
2. The Tel-Aviv Activities
and Accomplishments Inventory: Mathematics (TAAI:M)
assesses four levels of creative ability in mathematics, operationally defined
as the quantity and quality of challenging out-of-school talented activities
and accomplishments in mathematics.
A pilot study was conducted, in which the reliability and construct
validity of the two instruments were examined. Research participants were 487
high school students (364 males and 123 females), aged 16-18, drawn from two
urban public schools, representing a wide range of intellectual ability. The
findings provided strong evidence of both the reliability and the construct
validity of the two instruments.
In the second phase of the research, a sample of 1,090 students (565
males and 525 females) in the 10th and 11th grades, representing a wide range
of intellectual abilities and socio-economic statuses, was drawn from 22 public
schools. The schools were selected from a list of 571 schools and constituted a
nationally stratified and representative sample of students in urban and rural
schools.
The convergent, discriminant, and concurrent aspects of the construct
validity of the 4 x 4 Structure of Giftedness Model as applied to mathematics
were investigated. The goodness-of-fit of the Model for the data was examined
using Structural Equation Modeling technique.
Findings
The findings provided strong evidence for the validity of the
bidimensional ability type/level conceptualization of mathematical ability. The
main results were as follows:
1. The results provided
strong empirical support for the conceptualization of giftedness in mathematics
as a bidimensional phenomenon. One dimension consists of four types of ability,
two general types, i.e., general intelligence and general original thinking
ability, and two specific types, i.e., specific academic ability and specific
creative ability in mathematics. The
second dimension consists of four hierarchical levels of mathematical ability.
The findings indicated that the measures of each dimension, that is, types and
levels, represented different aspects of the abilities and, therefore, can be
used separately to identify mathematical ability.
2. The results provided
strong empirical support for the conceptualization of two specific, distinct
components of mathematical ability, one academic and one creative. The findings
indicated that the academic or standard-logical and the creative or
nonstandard-creative types of thinking are two inherent components of mathematical
ability, which are not only theoretically distinct but are empirically
distinguishable as well.
3. Four distinct
hierarchical levels of both academic and creative abilities in mathematics were
conceptualized, operationally defined, and empirically supported. These
findings about the nature of mathematical ability constitute an intricate
perspective on giftedness in mathematics, which leads to an innovative approach
toward identifying talent.
4. General ability types,
i.e., general intelligence and general original creative thinking, were clearly
distinct from the specific ability types in mathematics, i.e., academic and
creative. The findings indicated that IQ measures predicted academic, but not
creative, achievement in mathematics, and that creative thinking measures
predicted creative, but not academic, achievement in mathematics. Accordingly,
in addition to measures of general intelligence and of academic ability in
mathematics, measures of general original creative thinking and creative ability
in mathematics should be used to identify mathematical talent.
5. General original creative
thinking was related to specific creative ability in mathematics. The findings
indicated that general original creative thinking, often referred to as divergent
thinking, is undoubtedly important for mathematical ability, and may constitute
a necessary but not a sufficient component of creative ability in mathematics.
6. No sex differences were
found in the cognitive processes of mathematical abilities. The findings
indicated that when the cognitive processes involved in understanding
mathematics are carefully defined and measured, there are no sex differences in
the ability to learn mathematics. These data indicate that boys and girls are
equally capable of reaching similar achievement levels in mathematics. It is a
challenge for educators to assure that boys and girls have equal opportunity to
realize their abilities.
Contributions
Three additional contributions emerged from the current study, two
psychometric, and one methodological. The psychometric contributions consisted
of two sets of findings. First, the structural and concurrent validity of two
new psychometric instruments were demonstrated. These findings indicate that
the two instruments constitute promising tools for identifying the levels of
both the academic and creative abilities in mathematics. Second, the
combination of a mapping sentence for defining specific conceptual components
and a holistic scoring rubric for operationally defining the corresponding
assessment scores on a continuous scale was demonstrated. This provided an
example of an innovative approach for the development of psychometric
instruments for educational research and practice.
A particularly important methodological contribution was the
demonstration of the advantage of using of the Structural Equation Modeling
approach over traditional analyses for the validation of a bidimensional
hierarchical model. This technique was successfully used to investigate: 1) the
distinctiveness between two dimensions measured simultaneously by the same
indicators; 2) the validity of distinct hierarchical causal relationships based
on both within- and between-level measurements on a continuous scale, and 3)
the formulation of an actual, data-based estimate of the indirect causal
relationship. The
Individualized, computerized, curriculum units in mathematics, matched to
each student's type and level of mathematical ability, were suggested to
advance the mathematical achievement of both boys and girls in their regular or
mixed-sex classes.