The field of elementary school mathematics tends to be viewed
as a sequential advancement of specific mathematical skills, occasionally
resulting in a mindset that young students can accelerate their mathematical
learning by “racing” or “flying” through checklists of specific mathematical
skills. Indeed, I occasionally hear gifted education colleagues describing
elementary mathematics as an “arms race” mentality, in which the checking off
of specific sequential math skills such as single-digit or double-digit
addition become the sole focus of math learning.

Mathematics is indeed an undertaking far more than a simple
progression of mathematic skills and operations. The definition of mathematical
giftedness may indeed help us pinpoint what we believe to be essential in
developing mathematical talent.

Surprisingly, there has been “little research conducted on
what constitutes mathematical giftedness,” according to M. Katherine Gavin and
Jill L. Adelson, authors of a chapter entitled “Mathematics, Elementary,”
published in the comprehensive gifted education handbook “Critical Issues and
Practices in Gifted Education.” Most of the research focuses on the traits that
mathematically gifted children display.

In the late 1960s and most of the 70s, Russian psychologist
V. A. Kruteskii in a Piaget-like manner observed students, aged between 6 and
16, whom he labeled “not capable,” “capable,” and “very capable.” His research
has been divided into the four major giftedness categories of “flexibility,
curtailment, logical thought, and formalization:

- Flexibility: Students switch strategies in solving a problem with ease and numerous times to help them make sense of the problem.
- Curtailment: Students skip several steps in the logical thought process because they see the solution as one whole thought as opposed to linearly connected logical steps. This phenomenon may help us understand why some gifted math students cannot explain their reasoning in finding a solution as they just cannot retrace any step-by-step process that are required for less capable math students.
- Logical Thought: These are students who think in mathematical symbols such as “less/greater than” or “plus/minus” when filtering data that is being presented to them. These thinkers “look at the world from a logical perspective.”
- Formalization: Based on just very few examples, students can see the overall structure of a problem and thus make generalizations very quickly.

Agreeing on a universal definition of mathematical
giftedness is further compounded by the sub-sets of algebra and geometry.
Kruteskii spoke of students with an “algebraic cast of mind,” characterized by
very abstract thinking, while “geometric” minds tend to visualize problems pictorially.
Kruteskii actually observed that especially elementary-age students who
displayed both minds, culminating in what he termed a “harmonic” mind, are
highly capable mathematicians. One final important observation that Kruteskii
contributed toward the idea of defining giftedness roots in attributes that
actually are not “obligatory.” Specifically, he singled out “swiftness,
computational ability, and memory for formulas and other details” as
characteristics that do not necessarily contribute to mathematical giftedness.

J.E. Davidson and R.J. Sternberg in a 1984 edition of Gifted
Child Quarterly article entitled “The Role of Insight in Intellectual
Giftedness, reported on work with fourth through sixth grade students that
mathematically gifted students use three progressive “insight” processes:

- Selective encoding: These students can “sift out” relevant information from a problem situation.
- Selective combination: These students synthesize the relevant information.
- Selective comparison: Students compared the information that had been synthesized together to other relevant information.

While speed is highly valued in mathematical competitions
such as the “Final Round” in a MATHCOUNTS contest, it is important to note that
Davidson and Sternberg pointed out that ”speed in doing mathematics is
important but is secondary to insight.” The remaining research in identifying
characteristics of gifted math students points toward math students’ “focus on
conceptual understandings,” “ability to abstract and generalize,” and
“persistence and ability to make decisions in problem-solving situations.”

In making the leap from defining mathematical giftedness to
identifying mathematical giftedness, the practitioners of gifted education
frequently – if not solely – rely on norm-referenced standardized intelligence,
aptitude, and achievement tests. It is particularly the use of standardized
achievement tests that is questionable, as those tests tend to focus on
“low-level tasks that require students not to think and reason in ways that
Kruteskii observed as defining attributes of mathematical giftedness,” as
research by L.J. Sheffield of the National Research Center on Gifted and
Talented points out. According to this research, the vast majority (up to 62 to
82% of the items) of questions dealt with the topic of number and operations,
of which the clear majority focused on computation. The most common method in
identifying mathematical giftedness is the practice of using out-of-level tests
such as SSAT-L, PLUS, or EXPLORE (the test that both Quest Academy and the
Center for Talent Development at Northwestern University employ). There is
research that suggests these tests eliminate the ceiling effect (students
reaching the highest level of their mathematical ability) for 98 percent of the
students.