CONVERGENT/DIVERGENT COGNITIVE STYLES AND MATHEMATICAL PROBLEM SOLVING
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JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2
CONVERGENT/DIVERGENT COGNITIVE STYLES AND
MATHEMATICAL PROBLEM SOLVING
Hassan Alamolhodaei
Ferdowsi University of Mashhad
Mashhad, Iran
Students’ approaches to mathematical problem solving vary greatly. In
particular, variations appear for those approaches which require
conceptual understanding and visual ways of solutions. The main
objective of the current study is to compare students’ performance with
different cognitive styles (Convergent vs. Divergent) upon mathematical
pictorial problem solving. A sample of 93 third year undergraduate
mathematics students were tested according to Hudson’s test together
with one mathematics examination done. Results obtained support the
hypothesis that students with divergent cognitive styles show higher
performance than convergent ones in pictorial problems. The
implications of these results on teaching and setting problems emphasize
that pictorial problems and the cognitive predictor variable (Convergent/
Divergent) could be challenging and a rather distinctive factor for
students.
INTRODUCTION
In recent years the study of cognitive styles has become a broad stream in
cognitive psychology and mathematics education. Individuals display their
own personal cognitive styles which is a major attribute in what makes an
individual to respond to various situations (Anastasi, 1996).
According to Messick (1976), cognitive styles are information processing
habits representing the learners typical style of perceiving, thinking, problem
solving and remembering. In fact, each individual has his/her own styles
for collecting and organizing information into knowledge which can be of
benefit (Cross, 1976).
102JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2
A large body of researches suggest that students with different cognitive
styles approach processing of information and problem solving in different
ways. Moreover, a close relationship between cognitive style and learning
style has been revealed (Witkin, Moore, Goodenough & Cox, 1977; Witkin
& Goodenough, 1981; Messick, 1976; Kogan, 1976; Johnstone & A1-
Naeme, 1991).
Styles, regardless of their types, are different from ability which some
believe to be a characteristic of intelligence. Whereas ability refers more to
the content of cognition, cognitive styles help one predict how information
is processed by each individual (Messick, 1976; Kogan, 1976; Witkin et al.,
1977; Witkin & Goodenough, 1981). A widely used dimension of cognitive
style in education is the Convergence/Divergence style, which specifies an
individual’s mode of perceiving, thinking, problem solving and visualizing.
CONVERGENT/DIVERGENT LEARNING STYLES
A convergent (Con) learner is one who tends to look for unique methods
and unique solutions. Such thinkers are noted for creativity or lateral
thinking.
A divergent (Div) learner is characterised by lateral thinking, creativity
and capacity to see new combinations of ideas and to examine the
possibilities of more than one way of doing things, leading to several
outcomes (Hudson, 1966, 1968; Guilford, 1959, 1978). For example, in
mathematics, if asked: what is the solution set of the equation x2 - 5x + 6 =
0, in the set of real numbers (IR)? The student should answer that the only
solution in IR is the set {2, 3} with no other choice, or the one and only
[2 x2 ] + sgn(x)
value of lim is 2. It seems that these examples are typical
x→∞ x2 + x
problems requiring convergent thinking. In fact many of mathematical tasks
require a convergent style.
Although, a variety of responses to stimuli is the unique feature of
divergent thinking, this does not mean that such a way of thinking has no
positive role in the process of reaching a unique conclusion. Hudson (1966)
suggested that convergers are naturally attracted toward one end of the
spectrum and divergers to the other end. He also rejected the belief of
many psychologists that divergent thinkers are potentially creative and
103JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2 convergent thinkers are potentially uncreative. In addition, the convergence/divergence dimension is a measure of bias, not the level of ability. CONVERGENT/DIVERGENT STYLE AND ABSTRACT LEARNING The literature in the convergence/divergence cognitive styles (e.g., Hudson, 1966, 1968; Guilford, 1967 and Messick, 1976) suggests that convergent thinkers prefer formal materials and logical arguments. They may be superior in performance to divergent thinkers on tasks which are well structured and demand logical ability, while divergent thinkers presumably are better in the more opened tasks than convergent thinkers. The convergers enjoy precision and logical conclusions, whereas the divergers’ views are restrictive. Guilford (1967) suggested that generating logical necessities is the critical feature of convergers, whereas generating the possibilities from the given information is the characteristic of divergers. In addition, Hudson (1966, 1968) found that being highly imaginative is a striking feature of divergent thinking learners. He also suggested that convergent learners like to keep emotions apart from studies and that divergent ones prefer studies involving emotions. CONVERGENT/DIVERGENT AND SCIENCE EDUCATION Support for the suggestion that science students are biased towards convergent thinking and that arts students towards divergent thinking may be found in several studies (e.g., Guilford, Hoepfner & Peterson 1965; Hudson, 1966, 1968; Mackay and Cameron, 1968; Field & Pool, 1970; Richards and Bolton, 1971; Sally & Bostack, 1979; Webster & Walker, 1981; and Runco, 1986). It has been suggested by Hudson (1966, 1968) that convergent pupils tend to specialize in the sciences and classics, but divergent pupils in the arts, history and modern languages. He also found that between three and four times as many convergers do mathematics, physics and chemistry for every one tending to go into the arts. Results cited by Field and Pool (1970) indicated that although the majority of science specialists entering university were convergent thinkers, it is 104
JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2
mainly the divergent thinkers among them who finally achieved better
results. These researchers found that there was a relationship between
students’ choice of faculty (arts or science) and their convergent/divergent
learning style in agreement with Hudson’s finding (1966) in this domain.
Al-Naeme (1991) suggested the important role of convergence/divergence
style in tackling the mini-project problems in chemistry, with the superiority
of divergent thinking over convergent thinking in such tasks. An important
question could be raised at this point. Is there any discipline in which
students could cope equally well with a convergent or divergent bias? Orton
(1992) suggested that, biology, geography and economics are subjects which
do not fall into just dimension of divergent thinking or convergent thinking.
It seems that only a minority of learners may cope well with convergent
and divergent styles at the same time.
CONVERGENT/DIVERGENT STYLES AND MATHEMATICS
PROBLEM SOLVING
Guilford et al. (1965) suggested a positive correlation between divergent
thinking and learning mathematics. On the other hand, Kempa and
McGough (1977) found that students with an interest in art (divergers) tend
to prefer the verbal communication mode in learning mathematics, whereas
students’ mathematical biases are found to be strongly associated with
performance in the symbolic communication mode and anti-performance
for the verbal mode.
It may be reasonable to note that the nature of mathematical tasks indicate
that students should cope well with convergent and divergent thinking in
the problem solving situations. In fact, at the beginning of a solution they
need to think openly and converge step by step to the necessary solution.
It was also found in a research work that learning various aspects of
calculus tasks demands different dimensions of cognitive style on the part
of learners. For instance, divergent thinkers favour pictorial thinking, curve
interpretation and calculus word problems (Alamolhodaei, 1996). This
means that divergers achieved high results than convergers in such
questions tasks in calculus.
The most recent studies in mathematics teaching and problem solving
view pictorial thinking as positive factors, around which to build
105JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2 instructions and learning (Moses, 1982; Mundy, 1987; Vinner & Dreyfus 1989; Vinner, 1982, 1989; Resnick, 1989; Presmeg, 1986; Leinhardt, Zaslavsky & Stein, 1990; Dreyfus, 1992; Moore, 1994; Campbell, Collis & Watson, 1995; and Alamolhodaei, 1996). Students’ misconceptions about functions can be traced logically to pictorial meaning (Leinhardt, Zaslavsky, & Stein, 1990). Research into maths education shows that students generally are very weak visualizers in calculus course materials, which in turn lead to a lack of meaning in the formalities of mathematical analysis (Tall, 1991; Alamolhodaei. 1996). The nonvisual way of teaching has the effect of leading students who are visual thinkers to believe that success in mathematics learning and problem solving depends on rote memorisation of routine rules and formulas. (Presmeg, 1986a). VISUAL/NON VISUAL WAYS OF SOLVING PROBLEMS Visual thinking is a way of thinking and can be viewed as a non-analytic and non-algorithmic mode (Moses, 1982). Mathematical visual thinkers are people who prefer to use visual ways of solutions in mathematical tasks which may be solved by both pictorial and non-pictorial methods (Presmeg, 1986b). As Campbell et. al., (1995) suggested, the most effective mathematics learning style involved the use of visual thinking together with an emphasis on abstraction and generalisation. This dual emphasis could be a beneficial help for students reducing the limitations associated with one way of visualization or abstraction (Alamolhodaei, 1996). Vinner (1989) noted that graphical interpretations and the graphical consideration have a crucial role in understanding the course material in algebra and calculus. Therefore, the visual interpretations of algebraic notions should be taught as well as their applications in proofs and problem solving. The visual considerations explain algebraic moves which, otherwise, look artificial and arbitrary. Presmeg cited Moses’ definition of the visual way of mathematics problem solving that included solutions involving constructions, diagrams, drawings, tables, charts or graphs, whether written down or in the students’ mind. 106
JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2
However, as noted by Dreyfus (1992), students’ reluctance to use visual
reasoning and pictorial considerations are documented widely in the
literature. Many students, especially at the secondary school level, tend to
regard mathematical thinking as being largely verbal in nature. This view
is due partly to the highly algebraic form in which mathematical work is
typically expressed and students often fail to develop the visual, nonverbal,
component of their mathematical thinking abilities (Shear, 1985).
Many students are unable to recognize a healthy match between their
visual thinking and the answers they reach through mathematical
manipulation (Mundy & Lauten, 1994).
It was found that calculus students do not find it easy to use graph
sketching and curve interpretations to recognize that a function has limits
or one-sided limits, ( lim F (x) or lim +F (x) & lim -F (x) ) whether it is con-
x→a x→a x→a
tinuous at a point or on an interval, or is differential on that interval
(Alamolhodaei, 1996). Pictorial and geometrical interpretation of some main
concepts and theorems of calculus such as Rolle’s theorem, the mean value
theorem, multivariable calculus (for example, evaluation of double and triple
integrals) can be important for better understanding. As Eisenberg and
Dreyfus (1986) noted, such materials are highly visual in nature and hence
students fail to handle the relevant pictorial transformations into analytical
thinking.
Many studies confirmed that students prefer functions expressed in terms
of algebraic formula rather than in other kinds of representations such as
pictorial form (for example, Vinner & Dreyfus, 1989; Leinhardt, et. al., 1990;
Mundy & Lauten 1994 and Alamolhodaei, 1996). However, Mundy and
Lauten (1994) suggested that learning about functions can be promoted
through the connections between functions and their graphs. In addition,
Ervynck (1981) suggested that the use of graphical representations may
help to overcome the inherent difficulty of passing from a visual image to a
formal definition of the limit concept. Eisenberg and Dreyfus (1986) noted
that, in calculus, spatial visualization is commonly used for explaining the
main concepts, the derivative and the integral. Theorems which are not
easy to be proved algebraically will often be easy to understand and prove
geometrically (i.e. visually) and vice versa (Shear, 1985). Graphical work
107JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2
has also a great importance on developing concepts of rate of change (Orton,
1983).
THE PRESENT STUDY
The main aim of the current study was to identify students’ difficulties
associated with mathematics pictorial problems. As Moore (1994) suggested,
one of the major sources of students’ difficulties in math proof is their poor
pictorial understanding of the concepts.
The focus of this research was to provide a profile of learners’
performance with different cognitive styles (convergent/divergent) in
tackling pictorial problems. Thus, the main question addressed in this study
is to find how different mathematical behaviour of students would be with
(convergent/divergent) learning styles working on mathematics tasks and
that if there is any interaction between students styles (convergent/
divergent) and their performance in pictorial tasks.
It seemed to the author, as a main hypothesis, that divergent students
would be expected to show higher results than the convergent ones in
pictorial mathematics problems and curve interpretation. In other words
the mean scores of (divergent) students could be higher than (convergent)
students working on mathematical tasks comprising pictorial problems.
In fact, such questions tend to make the task rather distinctive and
challenging on the part of students.
METHOD
Sample
This study was conducted on third year undergraduate mathematics
students at Ferdowsi University of Mashhad in the north east of Iran. An
attempt was made to select a sample of 93 students, while they were doing
the maths courses as part of higher education requirements. They were
mainly female students.
108JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2
Research instruments and procedures
Cognitive style measure: This research was based on the Hudson’s (1966)
original study in this dimension of learning style.
The researcher used a version of convergent/divergent tests that have
been designed and applied by Al-Naeme (1991) and Alamolhodaei (1996).
The test comprised six short tests for which a limited time for completion
of each test was allowed. Students were required to write as many answers
as possible for every question they were given. One mark was given for
every single correct response (Hudson, 1966). The highest possible score
that could be gained in these six tests was 130. Under this situation, a
normal distribution of performance was obtained. A slice of one quarter
of a standard deviation on either side of the mean scores was classified as
“Intermediate” and excluded from the hypotheses testing in this study to
obtain two contrasting groups (convergent/divergent). The quantity of
the mean score +0.25 SD was regarded as a crucial point between moving
from convergent thinking style into divergent thinking one or vice versa.
Therefore, moving up from the mean score +0.25 SD of sample population
was classified as divergers, while moving down from mean score -0.25 SD
was grouped as convergers. Table 1 represents statistical information of
tests carried out to evaluate the (convergent/divergent) scores obtained
by students while Table 2 shows the number of students in each one of
the styles (convergent/intermediate/divergent).
Table 1
Statistical information of (Convergent/Divergent) tests
Group Mean Score SD Maximum Score Minimum Score
N=93 54.53 11.30 32.00 92.00
Table 2
The distribution of cognitive styles over the sample
Group Convergent Intermediate Divergent
N=93 48.6% 22.58% 36.55%
109JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2
Math task: The effectiveness of the convergent/divergent style should
be investigated by the students’ performance in pictorial mathematical
problem solving. Thus a question task with ten pictorial and curve
interpretation problems was designed (see Appendix). Students were asked
to explain the reasons for their choices in the answer sheets. The maximum
score for this task was 20. In addition, students took part in this exam
without any previous readiness for doing such pictorial mathematical tasks.
Scores obtained by all three groups of cognitive styles (convergent/
intermediate/divergent) represent a normal distribution.
DATA ANALYSIS
The data analysis procedure was mainly done by using the mean score.
A parametric statistical test (One-way ANOVA) was used in order to find
out whether the differences in students’ performance in mathematical
problem solving activities are statistically significant or insignificant. Results
are given at the .05 level. The reliability coefficient (Cronbach’s α) for the
(convergent/divergent) tests was estimated to be 0.65.
RESULTS
In order to examine the hypothesis of this study the performance of students
with thinking styles (Convergent/Divergent) in the math exam had to be
investigated.
The mean scores and standard deviation (SD) in this assessment related
to (convergent/intermediate/divergent) learning styles in the sample are
set out in Table 3. According to one-way ANOVA, on mean scores of the
math exam, a significant difference was found in performance among three
groups of styles (F = 4.23, P < .05). Figure 1 represents the superiority of
students with different styles, (convergent vs. divergent) based on their
mean scores in the pictorial math exam.
Table 3
Mean scores and SD in the math exam
Group Convergent (N=38) Intermediate (N=21) Divergent (N=34)
Mean SD Mean SD Mean SD
Math
exam 9.63 3.1 9.76 3.01 11.75 3.6
110JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2
Math (Mean)
15-
11.76
10- 9.63
5-
Con Div
Figure 1. The students’ achievements with different styles in the math exam
DISCUSSION
According to the mean scores of students with convergent/divergent
cognitive styles in Table 3, the divergent learners performed and achieved
higher results than convergent learners in pictorial questions. The difference
between mean scores in groups of students (convergent vs. divergent) was
found to be significant owing to p < .05.
Therefore, this finding could support the hypothesis of this study. It
seems from the results attained that mathematical pictorial problems could
be a rather distinctive and challenging task on the part of learners. Instances
of misconception found in their answers to the questions proposed in the
math exam, may be regarded as a support to this. Particularly questions
No. 2, 4 & 5 represent this occuring (see, Appendix). Table 3 indicates that,
for students with divergent cognitive style the mean scores in math task is
11.75, while for convergent ones it is 9.63. This means that being divergent
in thinking style could be more beneficial than being convergent in tackling
pictorial and curve interpreting problems. Moreover, this result supports
the previous research findings indications (e.g., Alamolhodaei, 1996) that
convergent students experience more troubles when handling the
complexity of curve interpretation and pictorial problems even in school
and higher education.
111JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2 CONCLUSIONS AND EDUCATIONAL IMPLICATIONS The present study showed a positive correlation between convergent/ divergent cognitive styles which are built upon individual differences and students’ mathematical performance in pictorial question tasks and could have implications for maths education. It was found that, one way of thinking (i.e. divergent) would enhance the achievement in pictorial and curve interpreting math tasks compared with another way (i.e. convergent). This means that divergent bias favour visual thinking and more visualization than convergent bias. It is fair to suggest that teaching styles and mathematical tasks should be planned to benefit both cognitive styles (convergent & divergent) of learners. As Dreyfus (1992) suggested, more than a balance in various forms of mathematics concepts, i.e., the integration of algebraic, verbal and visual thinking should be intended. Balance is to be an aim for integration and to achieve this, visual reasoning needs to be given parity along side algebraic and analytic reasoning if mathematics instructors wish to improve students’ understanding. However, it may be reasonable to note that the nature of many mathematical tasks indicate that students should cope well with convergent and divergent thinking in the problem solving situations. In fact, at the beginning of a solution they find they need to think openly and then converge step by step to the necessary answer. Textbooks and current teaching methods in mathematics in schools and higher institutions favour analytical and non-pictorial ways of thinking. The balance between them is not often valued in teaching and learning math by many teachers. Therefore, with respect to the curriculum, the results of the previous researches and the present study favour a parallel development of math that is visual in nature as well as it is analytical. Visual considerations and graphical interpretations have a crucial role in learning calculus. Moreover, teaching methods should match Convergent/Divergent learning styles. However, a pictorial approach to teaching and problem solving is not often valued by a lot of math educators. They are, unaware of the fact that meaningful learning in math and problem solving activities could be easier if both pictorial and analytical modes of 112
JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2
thinking are used. As a ground for future research the following questions
need to be attempted.
To what extent should a pictorial approach in math be taught to students?
To what extent can pictorial considerations become a natural part of
students’ mathematical thinking? The answers to these questions will
undoubtedly be useful research areas for the future according to the present
researcher.
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APPENDIX
Some samples of exam questions are given below:
1. Let f: IR → IR be defined by f (x) = sin x - cos x.
Determine a sketch of the graph of f around the point x =
π
2
π π π π
2 2 2 2
2. For each of the following pairs of relations R1 and R2 on IR, sketch R1 R2
and .find its domain and range.
x2
{ } {
R1 = (x, y): x2 + y2 < 25 and R2 = (x, y):y > 4
9
}
y
y=P(x)
x
3. This is a sketch of the graph of a function y = P(x) which of the sketches
below could be the graph of y = -P (x), y = P (-x)
A B C
y y y
x x x
116JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION IN S.E. ASIA Vol. XXIV, No. 2
4. Each equation mathes one of the following graphs. Write down its matching
equation.
k
y = kx2 y = kx2 + P y= y = qxk y=k x y = pqx
x
(p, q and k are constant).
A Y B C
Y Y
X
X
X
X X
X
D E F
Y Y
X
X
X
X
X
X
5. Which of the following statement are true of the function f defined for
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