Roeper Review, 31:53–67, 2009
Copyright © The Roeper Institute
ISSN: 0278-3193 print / 1940-865X online
DOI: 10.1080/02783190802527372
UROR
DELVING INTO DIMENSIONS OF MATHEMATICAL GIFTEDNESS
Test of the Three-Mathematical Minds (M3) for the
Identification of Mathematically Gifted Students
Three-Mathematical Minds
Ugur Sak
In this study, psychometric properties of the test of the three-mathematical minds (M3) were
investigated. The M3 test was developed based on a multidimensional conception of giftedness to identify mathematically talented students. Participants included 291 middle-school
students. Data analysis indicated that the M3 had a .73 coefficient as a consistency of scores.
Exploratory factor analysis yielded three separate factors explaining 55% of the total variance; however, one-factor solution seems to best fit the data. The convergent validity analysis
showed that M3 scores had medium to high-medium correlations with teachers’ ratings of
students’ mathematical ability and students’ ratings of their own ability and their liking of
mathematics. The findings provide partial evidence for the validity of M3 test scores for the
identification of mathematically gifted students.
Whether giftedness is a single entity or it emerges in different forms has been a subject of numerous theoretical
debates. Ideas related to this issue are abundant (Heller,
Monks, Sternberg & Subotnik, 2000; Sternberg & Davidson, 1986, 2005). The assessment of multiple forms of giftedness in mathematics, using the three-mathematical minds
model, based on an integration of the theory of successful
intelligence (Sternberg, 1997), studies on expertise and theories of mathematicians about mathematical ability (e.g.,
Gould, 2001; Polya, 1954a, 1954b) is the subject of this
study. First, I review theories and empirical research related
to the nature of mathematical ability and mathematical giftedness. Then I present the three-mathematical minds model
and the test of the three-mathematical minds I developed
based on the three-mathematical minds model for the identification of mathematically gifted individuals.
Recieved 1 July 2006; accepted 7 September 2007.
Address correspondence to Ugur Sak, Anadolu University, College of
Education, Department of Special Education, 26470 Eskisehir, Turkey.
E-mail: [email protected]
MATHEMATICAL ABILITY ACCORDING
TO THE PSYCHOMETRIC VIEW
In psychology, mathematical ability has often been studied
using factor analysis. Mathematical ability takes many
forms in factor analytic studies, depending on the nature of
mathematical tasks. Mathematical tasks also vary, depending on the branches of mathematics such as arithmetic, algebra, geometry, numbers, or statistics, and on the cognitive
processes such as induction, deduction, or computation.
Following the advent of factor analysis, at least four or
five factors underlying mathematical ability were found frequently. A numerical factor was found in early factor analytic studies of mathematical ability (Spearman, 1927;
Thurstone, 1938; Werdelin, 1958) consisting mostly of
addition, multiplication, and other arithmetical problems.
Another common factor found was related to visual or
spatial tasks that required the manipulation of objects in
two- and three-dimensional space. Other common factors
found in factor analytic studies related to mathematical
ability were the reasoning factors, consisting mostly of
induction (number series) and deduction tasks (Thurstone;
Werdelin).
Carroll (1993) reanalyzed 480 studies using exploratory
factor analysis. Many of the studies also had datasets
relevant to mathematical ability. His reanalysis indicated a
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hierarchical structure in cognitive abilities. In his reanalysis,
Carroll found quantitative ability under the factor fluid
intelligence. No unique mathematical ability existed according to the reanalysis. However, some first-level factors were
found. These first-level factors were general sequential reasoning, quantitative reasoning, and induction. Carroll
(1996) suggested that fluid intelligence was related to mathematical ability because most reasoning activities under
fluid intelligence were associated with logical and quantitative concepts. However, most tasks used in prior factor analytic studies seem to measure aspects of analytical
mathematical ability and mathematical knowledge, leaving
out essential tasks underlying mathematical creativity.
Mathematical creativity was not a subject in prior factor
analytic studies with the exception of Sternberg’s study of
successful intelligence, which indicated that separate analytical and creative abilities existed in mathematics (Sternberg,
2002).
According to Sternberg (1997), the theory of successful
intelligence has three aspects that underlie intellectual performance in academic domains, including mathematics.
These are creative ability, analytical ability, and practical
ability. Analytical ability involves comparing, contrasting,
evaluating, and judging relatively familiar problems.
Creative ability is invoked when the information processing
factors of intelligence are applied to relatively novel problems or situations to create, design, imagine, suppose,
explore, invent, or discover. Practical ability involves solving real-life problems. Practical ability is related to tacit
knowledge. Sternberg and colleagues carried out a series of
studies to investigate the validity of the theory of successful
intelligence (see a complete review in Sternberg, 2002;
Sternberg, Castejon, Prieto, Hautakami, & Grigorenko,
2001; Sternberg, Grigorenko, Ferrari, & Clinkenbeard,
1999), using the Sternberg Triarchic Abilities Test (STAT).
Their factorial research revealed separate and uncorrelated
analytical, creative, and practical factors in three domains:
verbal, figural, and quantitative. Quantitative analytical
ability was measured through a test of number series. Creative ability was measured by a test of novel number operations. Furthermore, quantitative practical ability was
measured by scenarios, requiring students to solve real-life
problems.
CONCEPTIONS OF MATHEMATICAL ABILITY
Beliefs about the nature of mathematics seem to contribute
to conceptions of mathematical ability. For example,
abstractions and generalizations are viewed to be the
essence of mathematics, and mathematical thinking therefore is deemed mostly abstract and generalized thinking
(Kanevsky, 1990; Krutetskii, 1976; Sriraman, 2003). In
fact, beliefs about the nature of mathematics were found to
influence how mathematicians carry out research (Sriraman,
2004a). Beliefs about the nature of mathematics and that of
mathematical thinking therefore are seen in definitions of
mathematical ability. For example, the mathematician
Poincare (Gould, 2001) believed mathematical ability to
be discernment.
Conceptions of mathematical ability vary not only by
individuals but also by disciplines. For example, conceptions of mathematicians differ from those of nonmathematicians who study mathematical ability. In the psychometric
tradition, working memory and analytical processes have
much value, as can be seen both in individual and in group
IQ tests. In contrast, according to some mathematicians,
intuition is as important as logic and power of memory in
inventions in mathematics. Poincare (Gould, 2001), for
example, pronounced that a strong memory or attention
does not make people mathematicians, but intuition is the
main instrument that enables them to conceive the structure
or relations among mathematical entities. He stated that
people with great memory and attention and the capacity for
analysis also can be gifted in mathematics. They can learn
every detail of mathematics, but they lack the ability to discover. There also is another type of mathematical ability,
the mental calculator, who can make very complicated calculations very quickly. According to Hadamard (1945),
only a few eminent mathematicians possess such talent.
In addition to broad conceptions of mathematical ability
discussed above, researchers have investigated specific
characteristics of mathematical ability, particularly mathematical giftedness. For example, Krutetskii (1976) found
the following characteristics to be essential for high mathematical ability: (a) the ability to comprehend the formal
structure of a mathematical problem; (b) the ability to generalize numerical and spatial relations; (c) the ability to
operate with numbers and other symbols; (d) the ability to
switch from one mental operation to another; (e) the ability
to grasp spatial concepts; and (f) the mathematical memory
for mathematical generalizations and structures. Other
researchers underscored specific features of mathematical
ability. Among these features are (a) the ability to visualize
mathematical problems or relations (Presmeg, 1986), (b) the
ability to think recursively (Kieren & Pirie, 1991), (c) the
ability to carry out analogy and heuristics (Polya, 1954a),
(d) the ability to discern mathematical relations (Gould,
2001), and (e) the ability to make decisions (Ervynck,
1991).
The aforementioned characteristics of high mathematical
ability also were supported empirically by recent studies.
For example, Sriraman (2003), based on analyses of gifted
students’ mathematical problem-solving, found gifted students spending a great deal of time trying to understand
problem situations and devising plans to solve these problems. In this study, gifted students were reported to utilize
particular cases to make generalizations, to search for similarities among problems, and to employ analogies to explain
similarities among problems. In another study, Sriraman
THREE-MATHEMATICAL MINDS
(2004b) observed gifted students using visualization and
reversibility in constructing mathematical truths. According
to Sriraman, gifted students’ problem-solving processes were
guided by their strong intuition as they made conjectures and
devised constructions to validate their initial conjectures.
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mind. However, mathematical expertise, although more
related to mathematical knowledge than cognitive processes, might be much more related to mathematical analysis and creativity than I consider. That is to say, analysts and
creators differ in their use of cognitive processes, whereas
experts differ in their knowledge structure.
Levels of Mathematical Ability
As with all intellectual abilities, mathematical ability shows
developmental stages and levels within each stage and sublevels within each level. For example, Usiskin (2000) proposed seven hierarchical levels of mathematical ability
based on educational attainment, mastery of mathematical
concepts, and contribution to the domain of mathematics.
At the lowest level, Level 0, are adults who even are
unaware of arithmetic. The first level, basic talent, can be
characterized as the development of ability and knowledge
repertoire to reason about number concepts and arithmetic.
The levels from the first to the sixth distinguish among talent levels that can be found among mathematics students
from high school to the end of graduate school. However,
Levels 6 and 7 characterize adult mathematicians who are
on the peak of the mathematical domain. Level-6 mathematicians are exceptional and can be found in the top few percentage of mathematicians. At the seventh level are those
mathematicians who make great contributions to the
domain of mathematics. They are the geniuses of mathematics, such as Euler and Gauss. Note that Usiskin’s proposition of mathematical ability deals with how mathematical
talent develops into hierarchical levels based on formal and
informal mathematical experience.
Levels of mathematical ability also can be seen in
Poincare’s classification of two types of mathematicians,
analysts and creators (Gould, 2001), although this hierarchical classification is implicit in his conception of mathematical ability. According to Poincare, analysts can do great
analytical work but lack the ability to create, whereas creators
have the potential to analyze and to invent. Hence, creative
mathematicians can be considered a subset of analytical
mathematicians. By the same token, Sriraman (2005) distinguished between mathematical giftedness and mathematical
creativity, proposing that creative mathematicians make up
a small portion among mathematicians, and mathematical
giftedness does not imply mathematical creativity; that is,
all creative mathematicians are mathematically gifted, but
all mathematically gifted mathematicians are not creative
mathematicians.
Types of Mathematical Ability
Among mathematicians, two kinds of ability characterize
two types of mathematical minds; one that is analytical and
one that is creative (Gould, 2001; Hadamard, 1945; Polya,
1954a). The third mind that I believe is important for understanding and assessing mathematical ability is the expert
Analysts and Creators
Analytical mathematical ability and creative mathematical ability are defined differently by those who deal with
mathematics and the psychology of mathematics. Poincare
(Gould, 2001), for example, defined creativity in mathematics as discernment or the ability to choose among mathematical combinations that are useful and analytical
mathematical ability as the ability to dissect mathematical
combinations. Creators make discoveries of theorems. Analysts, on the other hand, usually do microscopic work by
analyzing mathematical rules, axioms, combinations, or theorems the creator already has discovered. The primary
thinking tool of analysts is logic. Deductive reasoning is the
particular case of this logic. The primary thinking tool of creators is mathematical induction by rule discovery, analogy, or
mathematical constructions.
Poincare (Gould, 2001) believed that the nature of mathematicians’ minds makes them either analysts or creators,
and this can be seen in the way they approach a novel problem. That is, not only do the two minds work differently, but
also the ways these two minds deal with a problem make
them different. Analysts approach a problem by their logic,
whereas creators approach the same problem by their intuition. That is to say, the nature of the problem does not
change the nature of thinking of the two minds. Further, the
analyst is weak in visualizing space, and the creator is weak
in long calculations.
Similar to Poincare’s thoughts about mathematical ability, Polya (1954a, 1954b) proposed two kinds of reasoning
underlying mathematical ability. One is demonstrative reasoning. The principal function of demonstrative reasoning is
to distinguish a proof from a conjecture or a valid argument
from an invalid argument; thus, demonstrative reasoning
ensures certainty in mathematics. The other type of reasoning
is plausible reasoning. The primary function of plausible reasoning is to differentiate a more reasonable conjecture from
a less reasonable conjecture by providing logical evidence.
Empirical research also reveals evidence about how creative mathematicians produce creative work. For example,
Hadamard (1945) uncovered the nature of creative mathematicians’ thinking. He reported the prominence of visual
images, as well as sudden illuminations guided by intuition
and subconscious processes during mathematical discoveries. Similarly, Sriraman (2004a) interviewed creative mathematicians about their problem-solving experience and
work habits and identified common characteristics of mathematical creativity as intuition, proof, social interaction,
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imagery, and heuristics, in addition to the four-stage Gestalt
model of preparation-incubation-illumination-verification.
Knowledge Experts
The concept of expertise refers to a well-organized body
of accessible domain-specific knowledge and skills.
Researchers have explored the nature of the skills and
knowledge that underlie expert performance (e.g., Ackerman,
2003; Alexander, 2003; Chi, Glaser, & Farr, 1988; Ericsson,
2003; Hatano & Osuro, 2003; Lajoie, 2003). One of the key
findings is that experts’ knowledge structures differ from
novices (Chi et al., 1988). The kind of knowledge experts
possess is characterized as involving an organized, conceptual structure or schematic representation of knowledge in
memory. Their knowledge is organized around important
ideas of their disciplines, and it includes information about
conditions of applicability of key concepts and procedures.
This type of representation of knowledge, as opposed to isolated facts, enables experts to think around principles when
encountering a problem, whereas novices tend to solve
problems by attempting to recall specific formulas that
could be applicable to the problem (Chi et al.). For example,
compared to novices, mathematics experts are more able to
recognize patterns of information in situations that entail
specific classes of mathematical solutions.
Experts also show some variation. Some researchers
underscored the importance of studying different forms of
expertise, such as routine expertise and adaptive expertise
(e.g., Alexander, 2003; Hatano & Inagaki, 1986; Hatano &
Osuro, 2003). Routine expertise involves the quick and accurate solving of relatively familiar problems. In contrast, adaptive expertise involves processes that lead to innovation and
those that lead to efficiency. Routine expertise is a form of
knowledge expertise, whereas adaptive expertise is a form of
analytical or creative mind. Those mathematicians who only
teach mathematics courses at universities are good examples
of knowledge experts or of routine expertise, whereas mathematicians who publish or review research or theoretical articles for scientific journals are good examples of analytical or
creative mathematicians or of adaptive expertise.
THE THREE-MATHEMATICAL
MINDS MODEL (M3)
From this author’s point of view, mathematical giftedness is
the mathematical competence demonstrated in the form of
production, reproduction, or problem-solving in any branch
of mathematics at a given time and is recognized as remarkable by members of mathematical communities (e.g., teachers
or mathematicians). The three-mathematical minds model is
a tool to reconcile various views about giftedness. However,
the three minds differ in three aspects (see Figures 1 & 2). The
first difference is the cognitive components, such as
ANALYST
Creative analyst
CREATOR
Master
Expert analyst
Creative expert
KNOWLEDGE EXPERT
(Routine Expertise)
FIGURE 1 The Three-Mathematical Minds Model and seven forms of
mathematical giftedness or mathematical expertise. This model is based on
patterns of giftedness proposed by Sternberg (2000), research on expertise
and mathematical ability, and teachings of eminent mathematicians about
mathematical talent.
memory, intuition, or logic, to carry out cognitive tasks. The
second difference is in cognitive tasks, such as routine
work, novel work, or analytical work. Finally, they differ in
their end-products as a function of applying certain cognitive components in different tasks, such as knowledge production, reproduction, or solving familiar problems.
In the three-mathematical minds model, knowledge
expertise refers to routine expertise, while adaptive expertise refers to creators and analysts. Knowledge experts
might differ from analysts and creators in their knowledge
representation, amount of knowledge, and experience but
not necessarily in their cognitive ability and styles.
Although their knowledge is specialized, representing
domain specificity or task specificity, their expertise is in
the form of routine expertise. Therefore, their cognitive
end-products can be characterized as the reapplication of
experience to solving familiar problems, which do not necessarily produce creative work.
Unlike knowledge experts, creators and analytical thinkers manifest their work more as a function of their thinking,
such as the way they approach a novel problem; the differences in their attentional ability, logic, and intuition; and the
way they deal with information, such as to search for novelty or ambiguity. Creative mathematicians and analytical
mathematicians, too, are experts, but in different forms.
They not only are knowledge experts but also experts in
how to think mathematically. As pointed out earlier, their
expertise is in the form of adaptive expertise, leading to
innovation or efficiency.
Figure 1 shows that it is plausible to think of mathematical giftedness in seven forms like patterns of giftedness
(Sternberg, 2000). That is, mathematical giftedness may be
THREE-MATHEMATICAL MINDS
57
Mathematical Minds
Knowledge Expert
Memory
Recall
Creative
Routine work
Routine Problem
Solving
FIGURE 2
Intuition
Induction
Analytical
Novelty
Production
Logic
Deduction
Proof
Reproduction
Major instruments of mathematical minds applied in cognitive tasks and their end-products.
conceptualized based on the interactions of the three minds.
For example, the interaction of knowledge and analytical
ability produces an expert analyst, who is competent both in
domain knowledge and in analysis. The interaction of
knowledge and creativity makes a creative expert, who is a
good intuitive free thinker and has remarkable domain
knowledge. By the same token, the interaction of analysis
and creativity gives birth to a creative analyst, who has both
good, logical judgment and an a priori synthetic judgment.
Finally, the interaction of all brings into being a master,
who demonstrates remarkable analytical ability, domain
knowledge, and creative productivity and who, no doubt,
should be very rare. Compared to Usiskin’s (2000) hierarchical classification of mathematicians, expert and analytical minds of the three-mathematical minds model can be
considered among level-5 mathematicians; creative minds
represent level-6 mathematicians; and the interaction of the
three, or the master, can be thought to be the level-7 mathematician.
The three-mathematical minds model should be thought
as an instrument for developing theory-driven tests of mathematical ability to assess students’ three primary cognitive
abilities for production, reproduction, and routine problemsolving in the domain of mathematics. The rest of this article deals with research about the validity and psychometric
properties of the test of the three-mathematical minds (M3)
developed by the author based on the three-mathematical
minds model to identify students with high ability in analytical, creative, and knowledge aspects of mathematical ability.
The specific questions of exploration in this study were as
follows:
1. What is the underlying structural validity of the test of
the three-mathematical minds?
2. What psychometric properties does the test of the
three-mathematical minds have?
METHOD
Participants
The total number of participants was 291 (female = 133;
male = 158), including sixth (n = 63), seventh (n = 117)
and eighth (n = 111) grade students from four different
schools. Schools A (n = 41) and B (n = 27) were located
in one city, School D (n = 152) was in another city.
School C (n = 71) was located in a rural area. All schools
were located in the southwestern part of the United States.
The socioeconomic status of students in each school varied from lower to upper. Students of Schools A and D
came mostly from middle-class families. Students of
Schools B and C mostly came from low-middle-class families. The participants’ ages ranged from 10.5 to 15.5. The
mean age was 13.04. The racial distribution of participants
was as follows: White (72.2%), Mexican American
(14.4%), Asian (3.4%), Black (2.4%), American Indian
(1.4%), and others (6.2%).
Procedures
Development of the Test of the Three-Mathematical
Minds (M3)
A team of experts, with the leadership of the author,
developed mathematics problems according to the threemathematical minds model (M3) and the three-level
cognitive complexity model (C3), as seen in Table 1.
The team consisted of the following members: two mathematicians, one with a PhD in the science of mathematics and the other with a PhD in mathematics education;
two mathematics teachers, who worked at a middle
school and a high school; and the author, who specialized in the assessment of cognitive abilities, creativity,
and giftedness.
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TABLE 1
Item Development According to the Three-Mathematical Minds Model and the Three-Level Cognitive Complexity Model
Cognitive Complexity
Factor
Knowledge
Analytical ability
Creativity
Subtests
Algebra
Geometry
Statistics
Linear Reasoning
Conditional Reasoning
Categorical Reasoning
Induction
Insight
Selective Problem Solving
Level I
Level II
Level III
Factual
Factual
Factual
5 elements
1 condition
2 sets: 1 superset
and 1 subset
Free classification
Team agreement
Selective encoding
Relational knowledge
Relational knowledge
Relational knowledge
5 elements and additions
2 conditions
3 sets: dissections, 1
superset, and 1 subset
1 relation
Team agreement
Selective encoding and
selective combination
Conceptual-schematic knowledge
Conceptual-schematic knowledge
Conceptual-schematic knowledge
6 elements, coefficients, and divisors
2 conditions and double negation
4 sets: 1 intersection, 2 dissections,
2 supersets, and 2 subsets
1 rule and 1 relation
Team agreement
Selective encoding, selective combination,
and selective comparison
First, the author developed 27 sample problems to measure the analytical mathematical ability, creative mathematical ability, and mathematical knowledge at three levels of
cognitive complexity according to theories framing the M.
Then, the sample problems were sent to the mathematicians
for their review prior to the author’s initial meetings with
them. The author met with each mathematician twice to
review, revise, and develop new problems. The first meeting resulted in modifying 4 problems, developing 20 new
problems, keeping 3 original problems unchanged, and
omitting 20 problems. The second meeting ended with modifying 22 problems, developing 5 new problems, keeping 3
original problems unchanged, and omitting 2 problems, thus
yielding a total of 30 problems.
In the second phase, the final 30 problems were sent to the
mathematics teachers to review the content and difficulty
level of each problem according to the level of eighth-grade
students’ mathematical ability. The teachers used the following scale to rate problems: 1 (very easy), 2 (easy), 3 (average), 4 (difficult), and 5 (very difficult). They rated the
difficulty level of each problem by comparing it to other
problems in the same subtest. For example, the problems in
the insight subtest were compared only to each other, not to
other problems in the other subtests. The reason for such a
rating procedure was the author’s conviction that only problems of the same type should be compared in their difficulty.
Because of strong agreement between the author and the
mathematicians on the difficulty level of the problems, high
correlations were expected between item cognitive complexity (ICC) levels of the 30 problems and difficulty levels
of these problems as rated by the two teachers according to
eighth-grade students’ levels of mathematical ability. However, correlational analysis indicated low and nonsignificant
correlations between the ICC levels and the two teachers’
ratings of the difficulty levels of the problems (r = .29 and
.33), contradicting the initial agreements. An examination
of the ICC levels and teacher ratings of the difficulty levels
of the problems showed that ICC levels of the problems in
the induction and insight subtests diverged considerably
from teacher ratings. The main reason for this divergence,
perhaps, was the fact that these problems were not developed according to the C3, but the author and the mathematicians agreed on the difficulty level of each problem.
Therefore, I set out a third phase, during which I revised 7
problems, replaced 3 problems, and omitted 3 additional
problems. At the end of this process, the test included nine
subtests for a total of 27 problems. Each subtest had three
problems. The teachers’ ratings on the problems in the
insight and induction subtests also were integrated in the
final revision. Then, the mathematicians reviewed the final
problems before the test was given to the participants. However, the problems were not sent again to the teachers for
their ratings of item difficulties. As reported, in the results
section, correlation between item difficulties and the ICC
was moderately high, supporting the validity of the C3.
Table 1 shows the contents and the psychological sources of
the ICC levels revised after the final phase.
Item Content and the Use of the M3 in Item
Development
A total of nine subtests were developed to measure three
components of the three-mathematical minds as seen in
Table 1: knowledge component, creativity component, and
analytical component. It was hypothesized that each component of the three-mathematical minds was measured by
three subtests. Problems in each subtest hypothetically were
developed to measure an aspect of one of the minds. Studies
on expertise provided the theoretical background in the
development of knowledge component and problems in this
component. Three branches of mathematics (algebra, geometry, and statistics) were used to develop three separate
classes of problems. The theoretical purpose of these subtests was to measure factual, relational, and schematic
knowledge to distinguish between those who demonstrated
the knowledge of a novice and those who demonstrated
knowledge possessed by experts. Algebra problems
required knowledge of substitution, transposing, and factoring, as well as that of solving an algebra word problem by
translating it into an equation with two unknowns. Geometry problems required knowledge of area, perimeter, and of
angles as well as knowledge of relationships between area
and perimeter. Statistics problems required knowledge of
rate, percent, interest and data tables.
Analytical ability was measured by three deductive subtests: linear syllogism, conditional syllogism, and categorical
syllogism. Information processing research informed the
development of these problems. In linear syllogism, two or
more quantitative relations were given between each of two
pairs of items, depending on the cognitive complexity of the
item. One item of each pair overlapped with an item of another
pair, such as A < 2B and B/2 > C. The task of the problemsolver was to figure out the letter with the smallest or the greatest value. In conditional syllogism, one condition was presented with five conclusions, of which only one option satisfied
the condition. The following is an example of a conditional syllogism problem: If x < 0, then: (a) x2 > 2x; (b) x2 < 3x; (c) x2 < 0;
(d) x2 < x + x; (e) x2 > x2. In categorical syllogism, participants
had to figure out relationships between members of classes.
Group memberships were presented in a table.
Creativity ability was measured by three subtests: induction, selection, and insight. Induction problems can be characterized either by mathematical rule discovery or by rule
production. Selection problems can be characterized by
finding out relevant or irrelevant information, by selective
encoding, by selectively combining encoded information,
and by analogizing combinations to other constructions that
are presented in the problem stem or among answer options,
as related to the solution of the problem. The theoretical
purpose of these problems was to measure selection in problem solving as articulated by Poincare (Gould, 2001) and
Polya (1954a) and as theorized by Davidson and Sternberg
(1984). The insight subtest contained problems of mathematical recreations or nonroutine problems. The theoretical background of these problems came from Gestalt psychology.
Item Cognitive Complexity
Item cognitive complexity refers to psychological
sources of item difficulty, such as levels and kinds of
knowledge or cognitive processes a problem requires for the
solution. The difficulty level of each problem used in the
M3 test, except the insight and induction problems, was
developed according to the three-level cognitive complexity
model (C3) developed by the author (see Table 1). The
three-level cognitive complexity model was developed
according to a performance continuum on which people can
be categorized as novices, developing experts, and experts
(see Figure 3) based on their intellectual performance on
some tasks that are related to the domain of mathematics.
Because the continuum reflects a developmental performance,
Number of People
THREE-MATHEMATICAL MINDS
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Knowledge Domain
Novices
Level 1
Developing Experts
Experts
Level 2
Level 3
Levels of Expertise
FIGURE 3 Continuum of expertise in a domain. People might differ in
their level of expertise in different domains. As the level of expertise
increases, the number of people decreases in a domain. An expert in one
domain can be a novice in another domain.
people may be found at every point on the continuum.
Based on the continuum, therefore, expertise can be measured at different levels. Although I used three levels, novices, developing experts, and experts, other levels might be
found by dividing the continuum at different points provided that these points have psychological meaning.
Item Format, Scoring, and Test Administration
Two types of item formats were used. Most items were
presented in a multiple-choice format consisting of a problem stem and five answer options. Only one option was correct in these types of problems. The second format was of
kind—more than one method and solution or more than one
method but only one solution was accepted as correct. One
point was given for each correct answer in both multiple
choice and open problems. No point reduction was taken for
wrong answers. Mathematics teachers administered the test
during students’ regular mathematics classes in the beginning
of the spring semester of the 2004–2005 school year. The
testing was done in one sitting, taking about 45 minutes. The
teachers read standard instructions before the testing.
RESULTS
Reliability of the Tests of the Three-Mathematical
Minds
Means and standard deviations for student variables and the
M3 subtests by grade are presented in Table 2. The reliability of the test of the three-mathematical minds was investigated through Kuder-Richardson reliability analysis
(KR20). The analysis showed a .73 coefficiency level for
the entire test, slightly exceeding the minimum desired level
(.70) for consistency of scores for psychological tests.
Relationships Among the M3 Subtests
The relationships among the subtests of the M3 were investigated using Pearson product-moment correlation coefficient. Preliminary analyses were performed to check the
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theoretically should correlate. Partial correlation coefficients were computed, while grade was controlled in the
equation, to investigate the relationships between teachers’
rating of students’ mathematical ability, students’ rating of
their own mathematical ability, and their liking of mathematics and the M3 subtests (Table 4). Correlations ranged
from low to high-medium, with the majority of correlations
being statistically significant. Particularly important were the
correlations between the M3 total score and the other variables.
These correlations, medium to high-medium, were as follows:
.45 between the M3 total and teachers’ rating (p < .01), .36
between the M3 composite and students rating of their own
ability (p < .01), and .35 between the M3 composite and students’ rating of their liking of mathematics.
TABLE 2
Means and Standard Deviations for Student Variables
and the M3 Subtests by Grade
Grade
6
Variable
Teacher rating
Student liking
Student rating
Algebra
Geometry
Statistics
Linear syllogism
Conditional syllogism
Categorical syllogism
Selection
Induction
Insight
M3 composite
M
Total
7
SD
M
8
SD
4.00 .95 3.45 1.08
3.06 1.57 2.87 1.30
3.89 .96 3.39 1.00
1.60 .66 1.50 .71
.67 .90 .72 .86
1.11 .86 1.08 .80
.73 .87 .76 .72
.81 .82 .89 .83
.82 .92 .62 .79
.52 .72 .49 .62
.90 .79 .91 .66
.32 .56 .27 .47
7.49 3.99 7.23 3.26
M
SD
M
SD
3.50
2.78
3.36
1.61
1.01
1.21
1.01
1.09
.85
.59
1.11
.53
9.03
1.41
1.43
1.08
.72
.90
.79
.90
.99
.94
.67
.71
.69
3.94
3.59
2.88
3.49
1.55
.82
1.13
.85
.95
.75
.54
.99
.38
7.98
1.23
1.41
1.04
.70
.90
.81
.83
.89
.88
.66
.71
.59
3.77
Grade Differentiation
A one-way between-groups analysis of variance
(ANOVA) was conducted to examine performance differences of sixth-, seventh-, and eighth-grade students on the
M3 total score (see Table 2 for mean performance for each
grade level). Levene’s test for homogeneity of variances in
scores of the groups showed no violation (p = .077)
(Tabachnick & Fidell, 1996). The analysis indicated statistically significant differences among the grades (F [2, 288] =
7.5, p < .001). The effect size, calculated using eta squared,
was .05, a medium effect according to Cohen (1988). Post
hoc tests using Tukey’s honestly significant difference
(HSD) revealed the following mean differences among the
grades. Eighth graders performed significantly higher than
seventh and sixth graders (p < .01 and p < .05, respectively)
and no significant performance difference existed between
seventh and sixth graders on the M3 composite. These findings provided developmental evidence for the construct
validity of the M3 in terms of developmental differentiation.
assumptions of normality, linearity, and homoscedasticity.
Algebra, statistics, and induction subtests distributed normally. Geometry, linear syllogism, conditional syllogism,
categorical syllogism, insight, and selection subtests were
positively skewed. That is, scores were scattered around
low performance. Table 3 shows a number of statistically
significant and nonsignificant correlations among the subtests, with the highest associations among the knowledge
subtests: algebra, geometry, and statistics. Categorical syllogism and selection subtests had the lowest associations with
the other subtests. In fact, the categorical subtest did not have
any significant correlations with any other subtests, and the
selection subtest had a statistically significant correlation
only with the induction subtest (r = .16; p < .05).
Item-Total Score, Item-Subtest, and Subtest-Total
Score Correlations
Convergent Validity
Whether the M3 test items and subtests were homogenous or heterogeneous was investigated to determine the
Convergent validity of the M3 was investigated to examine
whether it correlated with other variables with which it
TABLE 3
Bivariate Correlations Among the M3 Subtests
Variables
1. Algebra
2. Geometry
3. Statistics
4. Linear syllogism
5. Conditional syllogism
6. Categorical syllogism
7. Selection
8. Induction
9. Insight
2
3
4
5
6
7
8
9
M3 Total
.35**
.30**
.36**
.28**
.42**
.25**
.34**
.47**
.26**
.26**
.05
.03
.01
.08
.03
.03
.09
.03
.02
.16**
.10
.24**
.34**
.26**
.25**
.26**
.13*
.06
.21**
.36**
.24**
.17**
.38**
.00
.07
.12*
.57**
.73**
.57**
.59**
.67**
.32**
.30**
.55**
.50**
*p < .05; **p < .01 (two-tailed).
THREE-MATHEMATICAL MINDS
TABLE 4
Partial Correlations Between Student Variables
and the M3 Subtests Correlations
Subtest
Grade
Age
Teacher
rating
Student
liking
Student
rating
Algebra
Geometry
Statistics
Linear syllogism
Conditional syllogism
Categorical syllogism
Selection
Induction
Insight
M3 composite
.02
.16**
.06
.14*
.13*
.03
.05
.13*
.16**
.18**
−.03
.11
−.02
.07
.05
.02
.01
.04
.10
.07
.26**
.33**
.38**
.23**
.31**
.09
.13
.29**
.25**
.45**
.20**
.25**
.24**
.11
.24**
.16**
.07
.19**
.18**
.35**
.21**
.33**
.34**
.08
.27**
.01
.07
.22**
.19**
.36**
Note. The effect of grade was removed in correlational analyses
between the M3 subtests and students variables.
*p < .05; **p < .01 (two-tailed).
degree to which the M3 items measured a unified construct
or multiple constructs. The degree of homogeneity or heterogeneity of a test has some relevance to its construct validity. Because the M3 is a measure of multidimensional
61
aspects of mathematical ability, an investigation of item
homogeneity and heterogeneity in the M3 provides information about its construct validity. Point biserial correlations between the M3 items and the subtests were
computed. As seen in Table 5, correlations between the
items and the M3 total score ranged from low to high, with
all correlations being statistically significant except for item
26. It had a very low and statistically insignificant correlation with the M3 total score (r = .06). Items 2, 6, and 15 had
low correlations with the M3 total, although the correlations
were significant.
The second mode of analysis was point biserial correlation computed between the items and the subtests to examine the homogeneity and heterogeneity of the items. In other
words, I assumed that an item was supposed to correlate
highly with the subtest in which it was located to show
homogeneity and to correlate at a low level with a subtest in
which it was not located to show heterogeneity. Correlations ranged from low negative to very high positive correlations. For example, the correlation coefficient was −.12
between item 26 and the subtest selection in which it was
not located (p < .05). The correlation was .88 between item
7 and the subtest insight in which it was located (p < .01).
TABLE 5
Item-Subtest and Item-Total Test Point Biserial Correlations
Subtest
Item-subtests
Algebra
Geometry
Statistics
L. syllogism
Con. syllogism
Cat. syllogism
Selection
Induction
Insight
Total
1. L. syllo
2. Algebra
3. Con. syllo
4. Geometry
5. Statistics
6. Selection
7. Insight
8. Induction
9. Con. syllo
10. L. syllo
11. Induction
12. Geometry
13. Statistics
14. Cat. syllo
15. Cat. syllo
16. Cat. syllo
17. Algebra
18. Selection
19. Selection
20. L. syllo
21. Insight
22. Con. syllo
23. Algebra
24. Statistics
25. Induction
26. Insight
27. Geometry
.24**
.47**
.24**
.20**
.27**
−.03
.15**
.24**
.14**
.17**
.10
.28**
.18**
.13*
−.03
.00
.80**
.03
.05
.10
.24**
.28**
.54**
.13*
.07
.02
.26**
.38**
.08
.29**
.76**
.34**
.05
.29**
.28**
.28**
.19**
.18**
.75**
.25**
.14**
−.06
−.01
.36**
.09
.01
.21**
.30**
.35**
.13*
.15**
.15**
.08
.56**
.29**
.13*
.19**
.29**
.66**
.02
.22**
.29**
.16**
.12*
.11*
.20**
.67**
.04
−.05
.03
.26**
.05
−.03
.06
.19**
.17**
.14**
.38**
.04
−.03
.27**
.67**
.05
.21**
.30**
.29**
−.02
.13*
.28**
.16**
.63**
.10*
.35**
.15**
.09
.01
.09
.33**
.04
.03
.56**
.20**
.14*
.05
.05
.05
−.03
.21**
.17**
.06
.72**
.35**
.31**
.09
.31**
.29**
.61**
.21**
.23**
.39**
.20**
−.02
−.06
.17**
.34**
.11*
.08
.11*
.29**
.62**
.16**
.07
−.04
.09
.24**
.08
.00
−.06
−.08
.01
.04
−.05
.09
−.01
.00
.13*
.08
−.04
.74**
.79**
.51**
.05
.11*
.03
.08
.07
.12*
.04
.16**
.03
.01
.11*
.01
−.09
.01
.03
.07
.69**
.08
−.01
.10
.01
.08
.06
−.03
.06
.03
.13*
.10*
.54**
.51**
.02
.08
.20**
−.03
.12*
.05
−.12*
.13*
.24**
.11*
.16**
.23**
.23**
.02
.19*
.69**
.08
.09
.51**
.22**
.21**
.11*
.06
.12*
.27**
.05
.04
.13*
.11*
.25**
.02
.13*
.58**
−.06
.30**
.08
.04
.29**
.29**
.30**
−.01
.88**
.12*
.21**
.22**
.15**
.27**
.19**
.05
−.02
−.06
.23**
.11*
.03
.03
.61**
.24**
.09
−.02
−.04
.36**
.18**
.46**
.17**
.44**
.51**
.51**
.16**
.41**
.47**
.37**
.34**
.33**
.56**
.37**
.29**
.15**
.21**
.56**
.22**
.14**
.28**
.42**
.50**
.23**
.25**
.18**
.06
.49**
Parentheses indicate items’ subtests.
*p < .05; **p < .01 (two-tailed).
62
U. SAK
What should be read from Table 5 are correlations between
the items and the subtests in which the items are located.
The results indicated that items had high or very high correlations with the subtest in which they were located, whereas
items had very low to medium correlations with the subtest
in which these items were not located. Only item 24 had a
medium correlation with the statistics subtest in which it
was located, and item 26 had a medium-level correlation
with the insight subtest in which it was located; however,
the correlations still were statistically significant (p < .01 for
both items). As seen in Table 5, the pattern of correlations
provided partial support for both homogeneity and heterogeneity of the M3 items because many items also had significant correlations with the other subtests in which they
were not located; however, these correlations were low.
Item Difficulty and Item Discrimination
As reported earlier, item difficulties were developed
according to the three-level cognitive complexity model
(C3). Item difficulties were determined through computing
the proportion of the participants who answered them correctly. As seen in Table 6, item difficulties ranged from .02
(extremely difficult) to .93 (very easy) with a mean of .29
(very difficult). Correlation between item difficulties and
item cognitive complexity levels was found to be .64 (p < .01),
a high correlation supporting the validity of the C3. The
mean item difficulty of the entire M3 test seems to be very
difficult for the entire population used in this study. Such a
difficulty level results from the fact that item difficulties
were constructed according to eighth-grade students’ level
of mathematical ability, but sixth- and seventh-grade students
also were participants in this study.
TABLE 6
Item Difficulties (ID), Item Cognitive Complexity Levels (ICC),
and Item Discriminations (D)
Item
ID
ICC
D
Item
ID
ICC
D
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
.36
.93
.42
.41
.58
.27
.30
.67
.20
.23
.09
.30
.34
.31
.30
1
1
1
1
1
1
1
1
2
2
2
2
2
1
2
.62
.12
.60
.77
.80
.19
.58
.64
.44
.41
.27
.76
.48
.39
.14
16
17
18
19
20
21
22
23
24
25
26
27
.15
.51
.17
.10
.26
.06
.33
.13
.15
.22
.02
.11
3
2
2
3
3
2
3
3
3
3
3
3
.21
.84
.26
.12
.36
.30
.72
.19
.27
.20
.03
.40
Mean
SD
.29
.20
2
.83
.41
.24
Note. Item difficulty range (ID): .80–1.00 very easy, .60–.79 easy, .40–.59
moderately difficult, 20–39 very difficult, and .00–.19 extremely difficult.
Item discrimination: .50 and above high, .30–49 moderate, .15–29 low,
.00–.15 very low and negative.
Classical item discrimination analysis was used to examine item discrimination. The index for item discrimination
was computed based on a comparison of the performance of
the upper-25th percentile group and that of the lower-25th
percentile group on the M3 total score. Table 6 shows the
discrimination index for each item. Discrimination indices
ranged from .12 (low discrimination) to .84 (very high discrimination). The M3 test had a mean of .41 discrimination
index, which is a moderate level of discrimination.
Factorial Structure of the Tests
of the Three-Mathematical Minds
Preliminary Analyses and Prior Decisions
Prior to performing factor analysis, the suitability of data
for factor analysis was assessed. Inspection of the correlation matrix revealed the presence of coefficients of .3 and
above. The Kaiser-Meyer-Oklin value for sampling adequacy was .81, exceeding the recommended value of .6
(Kaiser, 1974). The Bartlett’s test of sphericity reached
statistical significance (p < .001), which supported the
factorability of the correlation matrix. Moreover, prior to
performing factor analysis, three decisions were made.
First, the type of factor analysis to be used was determined.
Exploratory factor analysis “finds the one underlying factor
model that best fits the data,” whereas confirmatory factor
analysis “allows the researcher to impose a particular factor
model on the data and then see how well that model
explains responses” (Bryant & Yarnold, 1995, p. 109).
Moreover, confirmatory factor analysis is “generally based
on a strong theoretical and/or empirical foundation”
(Stevens, 1996, p. 389). Because, the M3 is a new instrument and no studies have been done before on the instrument’s structural validity and the subtests in the M3 were
only presupposed to measure some aspects of the three
dimensions of mathematical ability, principal axis factoring
was used to examine the factor structure of the M3 in this
study. Second, two criteria were established to determine
the number of factors in the factor analysis: (a) A factor
should have an eigenvalue of 1.00 or above; and (b) at least
50% of the variance in the original data set should be
explained by the number of factors to be determined. Third,
the type of rotation to be used was determined. As recommended by Tabachnick and Fidell (2001), multiple rotations
were examined using varimax, quartimax, equimax, and
direct oblimin. Direct oblimin rotation was chosen for use
as the underlying constructs were also correlated.
Factor Analytic Findings
The principal axis factoring of the nine subtests revealed
the presence of three factors with eigenvalues exceeding 1.
Because only three factors had an eigenvalue above 1.00
and explained more than 50% of the total variance, these
THREE-MATHEMATICAL MINDS
TABLE 7
Three-Factor Solution from Principal Axis Factoring
With Direct Oblimin Rotation
Factor
Subtest
1
Geometry
Linear syllogism
Statistics
Algebra
Induction
Conditional syllogism
Selection
Insight
Categorical syllogism
Percent of variance
explained
Eigenvalues
.68
.58
.53
.51
.51
.40
2
3
.39
29.30
−.35
.31
13.58
12.16
2.82
1.11
1.03
Communality
.40
.22
.20
.21
.18
.32
.04
.20
.03
three factors were investigated further, using direct oblimin
rotation with Kaiser normalization. Factors that had less
than .30 absolute values were not reported in the rotated
solution. All subtests achieved loadings above .30 on at
least one factor. As seen in Table 7, the three-factor solution
explained 55.03% of the variance, with factor 1 contributing
29.30%, factor 2 contributing 13.58%, and factor 3 contributing 12.16%. Factor 1 consisted of geometry, algebra, statistics, linear syllogism, conditional syllogism, and
induction subtests. Factor 2 consisted of the categorical subtest and the insight subtest. Factor 3 consisted of the selection subtest only. The first factor was labeled as the
“knowledge-reasoning” factor, the second as the “analytical” factor, and the third as the “creativity” factor.
The factor analytic findings provided partial support for
the theoretical validity of the test of the three-mathematical
minds. Although three separate factors were found, four
subtests did not cluster in the factors for which they were
developed. Indeed, the findings show that one-factor solution might be a better solution for the dataset used in this
study. The induction subtest was developed to measure an
aspect of the creative mind, but this subtest was found in the
knowledge-reasoning factor. Conditional and linear syllogism
subtests were developed to measure the analytical mind, but
they too were found in the knowledge-reasoning factor.
Finally, although the insight subtest was developed to measure an aspect of the creative mind, it did not cluster in the
creativity factor. These findings mean that new subtests
need to be developed to measure creative and analytical
abilities.
DISCUSSION
In this research study, psychometric properties of the test of
three-mathematical minds (M3) were investigated. The M3
63
test was developed based on a multidimensional conception
of mathematical ability. The findings provided partial support for the reliability, convergent validity, and developmental validity of the M3 test. However, factorial findings
did not show strong support for the factorial validity of the
three-mind model.
Reliability of the M3
The reliability coefficient may be interpreted in terms of the
percentage of score variance attributable to different sources
(Anastasi & Urbina, 1997). The analysis yielded a reliability coefficient of .73 for the entire test. Although the coefficient is above the minimum level, it is not very strong, in
that over 25% of the variance in scores in the M3 is attributable to error variance. Recall that the M3 is not a measure of
a single trait, but a measure of multidimensional aspects of
mathematical ability. That is, the items in the M3 are heterogeneous; therefore, the interitem consistency of the M3
should not be expected to be very high because the interitem
consistency is influenced largely by the heterogeneity of the
behavior domain sampled (Anastasi & Urbina). The M3 has
nine subtests measuring separate aspects of mathematical
ability. Overall, the reliability finding shows a good level of
reliability for this preliminary research. However, test–
retest reliability might be a better reliability indicator for the
Test of the Three-Mathematical Minds.
Factorial Validity of the M3
Although the factor analysis revealed three separate factors
out of nine subtests, most subtests clustered in one factor,
which does not provide strong evidence for the factorial
validity of the three-minds model. The factor analysis
showed that some subtests of creative and analytical minds
clustered in the knowledge factor, contradicting the author’s
theoretical position. The author’s initial assumption was
that geometry, algebra, and statistics subtests cluster in the
knowledge factor; the three deductive subtests cluster in the
analytical factor; and insight, selection and induction subtests cluster in the creativity factor.
Table 7 shows loadings of each subtest on the factor in
which it clustered. According to the analysis, geometry,
algebra, statistics, linear syllogism, conditional syllogism,
and induction subtests clustered in the first factor and
explained almost 30% of the variance. This factor can be
labeled the knowledge-reasoning factor. Note that the thirdlevel problems in the knowledge factor require some reasoning because these problems entail conceptual understanding
of subject matter. For example, the level-three problem in
the geometry subtest requires understanding perimeter and
area relationships for the solution.
What was unexpected was the clustering of induction,
linear syllogism, and conditional syllogism subtests in the
knowledge factor. They have low to medium, but statistically
64
U. SAK
significant, correlations with the other subtests of the
knowledge factor. An inspection of the problems in the
induction subtest indicates that the first-level problem (categorizing numbers), and the second-level problem (finding
out the sum of the internal angles of a shape of 14 sides) can
be solved by using mathematical knowledge, as well as by
inductive processes. However, the third-level problem
(finding out the number of turns of a wheel) does not
require mathematical knowledge. Needless to say, the firstand second-level problems contribute to the overlap
between the knowledge factor and the induction subtest.
Another reason for the overlap can be seen from the finding
that the induction subtest has a high correlation with the
geometry subtest because the second-level and the thirdlevel problems in the induction subtest also require spatial
ability, whereas the first-level problem requires numerical
ability.
The subtests conditional syllogism and linear syllogism, which were supposed to cluster in the analytical factor, clustered in the knowledge factor. Linear syllogism
and conditional syllogism problems require focused attention, as well as the comparing and contrasting aspects of
analytical ability. Therefore, I predicted that the linear syllogism and conditional syllogism subtests would correlate
highly with the categorical syllogism and cluster in the
analytical factor, but they did not. The conditional syllogism problems and linear syllogism problems resemble
algebra problems only in language, as they have algebraic
symbols. Nevertheless, they require little algebraic knowledge. Conditional syllogism problems also require the use
of coefficients and factors. Therefore, the correlation
between algebra and the conditional syllogism subtest is
.34, a statistically significant finding. As a matter of fact,
the conditional syllogism correlates with the geometry
subtest at a higher level, .47. This author believes that the
overlap between the conditional syllogism and the knowledge factor occurs because of the similar nature of the
problems in the conditional syllogism and the algebra subtest. In fact, some students, who were not good at algebra,
might not have even attempted to solve conditional problems because of surface similarities between algebra and
the conditional syllogism problems.
The categorical syllogism subtest has very low correlations with the other subtests. The categorical syllogism
subtest was found to be a separate factor in factor analysis
explaining 12% of the variance. The categorical syllogism
problems, like linear and conditional syllogism problems,
require focused attention, contrasting, and comparing.
Therefore, this factor was labeled as the analytical factor
by this author. The categorical subtest has a significant
correlation, .13, with the induction subtest only. However,
the overlap is quite small, 1.6%. Keep in mind that a significant correlation might exist between two variables with
a large sample size. Therefore, this correlation does not
tell much about the relationship between the inductive
processes and the categorical syllogism processes. Indeed,
the finding might be just an artifact of the sample size.
Furthermore, although the categorical syllogism subtest
itself is a separate factor in the factor analysis, the difficulty level of the categorical problems might have contributed to the distinction of this factor. Categorical problems
have a mean difficulty level of .25, a very difficult level
(see Table 6).
The second factor, which included selective problem
solving subtest, was labeled as the creativity factor by this
author because problems in this subtest requires an unusual
mode of thinking—selective thinking. Recall that, as
pointed out before (Davidson & Sternberg, 1984, 1986),
many creative ideas emerge from sorting out related and
unrelated information and combining them selectively or
through the use of analogies. Problems in the selective problem-solving subtest require selective encoding, selective
comparing, and selective contrasting. Overall, the selective
problem-solving subtest seems to measure a different aspect
of human ability. This difference can be seen more clearly
from point biserial item subtest correlations in Table 5.
Problems in the selective problem-solving subtest correlate
significantly only with the total score of this subtest.
Overall, the factor analysis did not show a clearcut picture about the existence of three distinct mathematical
minds. However, this conclusion can be drawn only from
the dataset used in this study, in which only nine subtests
were used and the participants were middle-school students
only. Different findings can be obtained in another study if
more subtests and a different group of participants were
used.
Convergent Validity of the M3
Campbell (1960) pointed out that a psychological test
should correlate with other variables to which it should be
related theoretically to show construct validity. Convergent
validity can be investigated through the correlation of the
same ability measured by different tests or through the correlation of similar abilities measured by the same or different
tests. In this study, the convergent validity of the M3 was
investigated by correlating students’ M3 scores with teachers’ rating of students’ mathematical ability, students’ rating
of their own mathematical ability, and students’ rating of
their liking of mathematics. Note that the first two ratings
are some kinds of measures of students’ mathematical ability.
The liking of mathematics is not a measure of mathematical
ability, but it should be associated theoretically with mathematical performance.
As read from Table 4, correlations between the M3
scores and the ratings range from medium to high-medium,
with all correlations being statistically significant at the .01
level. What is interesting in the table is the pattern of correlations between the M3 subtests and the ratings. Both teachers’
ratings and students’ ratings correlate with the pure
THREE-MATHEMATICAL MINDS
knowledge-reasoning subtests, such as algebra and statistics, much higher than their ratings with the other subtests,
such as selective problem-solving and insight. These correlations might mean that teachers and students associate mathematical ability more with the amount of mathematical
knowledge and achievement in math classes than with creative and analytical ability. Perhaps some creativity and analytical problems also were unusual to the students. Overall,
these findings provide evidence for the convergent validity of
the M3. Moreover, further research is needed to investigate
associations between scores on the M3 and scores on other
tests of mathematical ability for a more clearcut picture of the
convergent validity evidence. An investigation of the relationship between scores on the M3 and grades in mathematics
classes or performance on an achievement test also is needed
to provide criterion-related validity evidence for the M3.
Developmental Differences Among Students
of Various Grade Levels
A major criterion used in the validation of a number of
intelligence tests is age differentiation (Anastasi & Urbina,
1997). However, the use of age in the validation of aptitude
tests, such as the M3, is not appropriate because they measure ability that is influenced largely by school learning.
Therefore, the major criterion for aptitude tests should be
grade differentiation, which is what the author used to check
the M3 against grade to determine whether the scores in the
M3 show a progressive increase with grade during middle
school (sixth through eighth grade). The findings provide
partial validity evidence about whether the M3 discriminates among different grade levels. The partial evidence
means that eighth-grade students scored significantly higher
than seventh- and sixth-grade students; however, the sixth
graders scored slightly higher than the seventh graders.
Needless to say, the latter finding contradicts the former,
even though the difference between the sixth graders and
the seventh graders is not significant. The discussion of this
contradiction follows.
An inspection of the sixth- and the seventh-grade sample
sizes shows a significant difference. The size of the sixth
grade is almost half that of the seventh grade. This difference might have contributed to the performance difference
between the sixth and seventh graders, favoring the sixth
graders. Another reason for the contradiction might come
from the statewide achievement difference between sixthand seventh-grade students. Further, developmentally, some
peaks and slumps might exist in students’ performance during middle school (Charles & Runco, 2000; Sak & Maker,
2006). Briefly, the findings show partial, developmental
evidence for the validity of the M3. Therefore, the author’s
future research agenda includes administering the M3 to a
different sample to see if the same or different results are
obtained.
65
Item-Total Score, Item-Subtest, and Subtest-Total
Score Correlations
Internal consistency correlations, whether based on items or
subtests, show the homogeneity and heterogeneity of the
test items used to measure the ability domain sampled by
the test. In other words, the internal consistency correlations
might provide evidence related to whether items in a test
battery measure the same construct, similar constructs or
completely different constructs. Anastasi and Urbina (1997)
stated that correlations among items, subtests, and the total
score are expected to be significant if the test battery is constructed to measure a single, unified construct. The author’s
assumption is that low correlations exist among items and
between items and subtests that are developed to measure
separate constructs even though they are parts of the same
test battery. For example, the M3 was designed to measure
three aspects of mathematical ability; therefore, low correlations should be expected among items measuring different
aspects of mathematical ability, such as the creative mind
and the analytical mind. On the other hand, high correlations should be expected among items measuring the same
aspect of mathematical ability, such as the analytical mind.
As seen in the point biserial correlational matrix in Table
5, 26 of the 27 items in the M3 test battery correlate significantly with the total score. Indeed, most correlations are
within the range of medium to high, providing evidence of
strong internal consistency of item-total score relationships.
In other words, the 26 items differentiate among the respondents in the same direction, as does the entire test battery.
However, item 26 has a low and nonsignificant correlation
with the total score showing low discrimination. Therefore,
this problem needs to be revised or removed from the test
battery. Furthermore, bivariate correlations between the
subtests and the M3 total score are significant, ranging from
medium to high. The correlations between items and the
total score and the correlations between the subtests and the
total score provide evidence of internal consistency related
to the construct validity of the M3.
The correlation pattern in Table 5 also provides additional psychological evidence for the internal consistency
of the M3. What is most important in this pattern are the
correlations between the items and the subtests in which
the items are located and the correlations between the
items and the subtests in which the items are not located.
As seen in the table, the items, such as 1, 10, and 20
located in the linear syllogism subtest, have high and significant correlations with the subtests in which they are
located. In other words, each linear syllogism problem differentiates among individuals in the same direction, as
does the entire subtest linear syllogism. This pattern
repeats itself for the other subtests as well. However, the
only items that do not have high correlations with their
associated subtests are items 24 and 26, although they
have moderate and significant correlations.
66
U. SAK
To this author, more interesting than the high correlations are low-positive and low-negative correlations
between the items and the subtests in which these items are
not located. In other words, these items and the subtests are
not developed to measure the same constructs. The correlation between item 25, an induction problem, and the subtest
conditional syllogism, for example, is −.04, meaning that
they do not measure the same ability and they differentiate
among different abilities. Low negative or low positive and
nonsignificant correlations exist between most items of the
M3 and the subtests that measure different constructs. These
findings show the internal consistency of the M3 at the item
level and subtest level.
beginning of the spring semester; therefore, the participants
had not completed their associated grade level. As a result,
they had not mastered mathematical knowledge and had not
developed skills taught at the end of their respective grade
level. The time of the school year in which the test was
given might have contributed to item difficulty because
problems were developed according to eighth-grade students’ level of mathematical ability as rated by mathematics
teachers. Another limitation is related to the methodology.
The M3 items were hypothetically assigned into their subtests; therefore, a subtest-factor analysis was used instead of
an item-factor analysis. An item-factor analysis is recommended for future research.
Item Difficulty and Item Discrimination
A moderately high correlation was found between item cognitive complexities (ICC) and item difficulties (r = .64).
A careful examination of Table 6 shows that 8 items have
high-level discrimination indices, 5 items have moderatelevel discrimination indices, and the rest have low discrimination indices. Note that the level-three items, the
most difficult items, have low discrimination powers.
These problems have low discriminations not because they
do not discriminate between high ability and low ability
but because they are just too difficult; even some of the
individuals with the highest abilities could not solve them.
Keep in mind that the M3 problems were developed
according to eighth-grade students’ general-mathematical
ability level, but sixth- and seventh-grade students also
were participants in this study. The problems might have
been too difficult for these groups of students. If the comparison percentile is changed from upper 25% to upper
3%, discrimination indices of the items would change
meaningfully.
In conclusion, this study showed the strengths and weaknesses of the test of the three-mathematical minds. The M3
test is a new instrument and in need of improvement, particularly in item difficulty, item discrimination, and item content. As reported and discussed in the previous sections, it
particularly has some weakness at the factorial level. It is
clear from the findings that new subtests should be developed to measure creative and analytical abilities.
LIMITATIONS
A number of limitations exist in this study, most of which
pertain to the sample. First, the sample is small and it is not
an exact representation of the U.S. population because it
was drawn only from the southwest region of the country.
Second, the number of sixth-grade students is much less
than that of the seventh- and eighth-grade students. This
difference might have contributed to the performance variation among the groups. Third, the test was given in the
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AUTHOR BIO
Ugur Sak, PhD, is associate professor and director of the graduate program on gifted education at the Department of
Special Education, College of Education, Anadolu University, Eskisehir, Turkey. He is the founder of the Education
Programs for Talented Students at Anadolu University. E-mail: [email protected]