INTRODUCTION
India has a long history of teaching and learning Mathematics, dating back
to Vedic age (1500 to 200 BC). It has rather been the birth place of mathematics.
Vedas are believed to be the repositories of perfect wisdom. The six auxiliaries of
Vedas compile various mathematical skill and adroitness to sort out complex
mathematical calculations. The sutras lay down the shortest and surest lines
wherein the correct intonation of mantras, the correct configuration of Yantras,
the correct time or astral conjugation factor, the correct rhythms etc. all had to be
perfected so as to produce the desired result effectively and adequately. Each of
these required the calculus of mathematics. It is worth mentioning that the notion
of „shuniyam‟ or „pujyam‟ or zero, the concept that connotes nullity evolved from
this land and gave birth to decimal system using 1 to 9 digits and the symbol 0.
Around 5th
century A.D., a system of mathematics was developed that made
astronomical causation easy. The conventional „Ganita‟ was followed by the
development of „Bijaganitam‟ where the term „bija‟ means seed, symbolizing
origin. This concept later developed as algebra. Even algorithm the process of
calculation based on decimal notions has been deducted from the Indian
techniques of geometric computation. The great Indian mathematicians
Aryabhatta (476 A.D.), Bhaskaracharya (1114 A.D. – 1185 A.D.), Brahmagupta
(598- 665 A.D.), Sriniwasan Ramanujan Aiyangar (1887 - 1920) are some of the
pioneers who had formulated complex concepts of modern mathematics in the
form of simple mathematical sutras much before they were developed in the west.
All the great educationists have accepted mathematics as the symbol
of human development. “Mathematics is the mirror of civilization” (Hogben).
It has made our life more systematic and organized and plays an indispensible role
in shaping our mind and behavior. There are several misconceptions about
mathematics. Many people believe that there are only few "gifted" individuals
who have an element that help them to learn mathematics, and that hard work
cannot compensate for this. Another common misconception that prevails is that
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most of its concepts do not have direct day today applications. That is not true
e.g. every day, one wakes up in the morning to watch the weather forecast which
is calculated using complex mathematical equation. In a football match the
probability of a particular team winning a match is calculated using
mathematics. One may even probably avoid smoking on the basis of statistics
revealing that smoking causes cancer. Thus mathematics governs a person‟s life
in more ways than actually perceived. The National Policy on Education (1986)
suggested that, “Mathematics should be visualized as a vehicle to train a child to
think, reason, analyze and to articulate logically”. Mathematics sets the path of
self actualization and its discoveries are nothing but the union of natural
phenomena with human behavior. It develops scientific behavior which leads the
pupil to function at the highest level of enquiry. Thus, “Mathematics should be
studied in the living context which is meaningful and relevant to the learners,
including their language, culture and everyday lives”(Sharma, J.; 1993).
G.H. Hardy in his book “A Mathematician‟s Apology” while comparing
mathematics with painting and poetry writes, “the mathematicians, like the
painters or the poets must be beautiful; the ideas like the colours or the words
must fit together in a harmonious way. Beauty is the first test”. But this
beautiful subject is often most feared and hated subject in the school. According
to Ravindra, Director of NCERT (2000) there is a huge gap between prescription
and practice of a mathematical curriculum. Most of the time classroom of
mathematics is preoccupied with routine teaching & not much time is devoted to
learning of mathematics (Prasad R. & Sarawat, P.; 2011). Hardly a student asks
a question in mathematics classroom. This implies that the learning rarely takes
place in mathematical classroom. For this, one needs to identify that what is
mathematical learning all about?
Math is usually taught as a right and wrong subject and as if getting the
right answer was paramount. Additionally, the subject is often taught as if there is
only one right approach to solve the problem and any other approaches would be
wrong, even if students got the right answer. Students are encouraged not to try,
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not to experiment, not to find algorithms that work for them, and not to take risks.
It is very essential for a mathematics teacher to understand that the origin of
mathematics is hidden in the evolution of nature. The fundamental premises of
the universe we live in have a basic mathematical structure. Even some species of
lower animals are by instinct gifted mathematicians. For example the migratory
bird which flies thousands of miles off from its nest and after a period,
unerringly returns. This implies the presence of a subconscious
mathematical aptitude that works wonders. Similarly, man works from instinct
talent or even genius but, ordinarily he works as a logical entity by initiating with
a specified data, applying suitable reasoning and arriving at a conclusion. This
process of induction and deduction indicates towards his innate tendency to think
mathematically, which we term as Mathematical Aptitude. Thus, in order to
substantiate learning in mathematical classroom a teacher should help the child to
understand the relationship between complex mathematics concepts and the real
world around him. Authentic learning experiences of this type manifest
themselves in various forms according to each learner‟s mental cognition and
aptitudes. Now, a question arises as to what is the need to identify mathematically
talented students?
It has been observed that while designing the instructional program in
mathematics care must be taken to analyze their individual ability and needs.
Students with lower ability may do better with a program not paced so quickly
and more deliberate in developing the mathematical concepts being taught. Such
program must insist reasoning and develop independent exploratory behavior
which is exemplified through discovery learning, looking for underlying
principles, engaging in special projects in mathematics, problem solving,
discovering formulas, looking for patterns and organizing data to find
relationships and to analyze the concepts. On the other hand it has also been
observed that the students having high aptitude in mathematics are least benefitted
from computational drill work and cyclical reviews. Therefore the scope of
mathematical curriculum should be extensive and flexibly paced on the basis of
the assessment of students‟ aptitude and skill. It is now evident that
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assimilating the concepts of Mathematics, requires a pure mathematical aptitude,
identification of which may not only harness the potentials of young minds to the
fullest extent and channelize them in right direction, but may also cultivate a
critical and rational outlook in them. If care is not taken to diagnose this ability,
the students who are simply high achievers in mathematics can be mistakenly
identified as mathematically talented.
This view point reveals that having high Mathematical Aptitude is
somewhat different from scoring good marks in mathematics. Mathematical
aptitude in real terms cannot be confided to mere computational abilities like
working out a dozen of problems efficiently, mastering the techniques,
remembering formulas, showing familiarity with numbers, manipulating algebraic
expressions or proving theorems/riddles axiomatically.
High mathematical aptitude in children is determined by his unusual
keen awareness, an intense curiosity about numeric information, an unusual
quickness in learning, understanding and applying mathematical ideas, a high
ability to think and work abstractly and ability to see mathematical patterns and
relationships. This self perceived inborn ability of students in mathematics which
we term as Mathematical Aptitude was a critical predictor of success in the field
of mathematics. It was felt that a focus on the concept of mathematical aptitude
and its development is needed to be amplified. Further the identification of
specific factors influencing its generation was also important; so that the best
techniques for intervention stimulus of positive attitude (self efficacy) can be
fabricated.
To understand the concept of mathematical aptitude one has to begin at the
grass root level and speculate the nature of mathematics. “Mathematics is the
study of abstract systems built of abstract elements. These elements are not
described in concrete fashion” (Stone, M.H., 2003). Mathematics is a discipline
that seeks understanding of patterns and structure of the constructs of human
mind. It seeks the highest standards of understanding by demanding rigor in its
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foundation and development. This proposition not only reveals the abstract nature
of mathematics, but also asserts that tackling or comprehending mathematical
Problems requires development of various mental faculties. These mental
abilities pertain to a person‟s Intellectual Ability. Moreover, a mathematical
problem generates a situation of mental tension which an individual tries to
overcome by indulging in an intense abstract thinking and logical reasoning. This
subsequently leads to the development of higher and specialized cognitive skills
which we term as problem solving ability. The speculation reveals that the
proportion of mathematical aptitude in an individual is an outcome of various
psychological correlates of human mind. The first one is General Mental
Capacity of an individual which is also termed as Intellectual Ability and the
second one is a specific ability called Problem Solving Ability. Both of these
abilities seem to play a phenomenal role in determining the mathematical aptitude
in an individual. Now it is a matter of concern to examine as to what extent the
Intellectual ability and the Problem solving ability of an individual determines the
Mathematical Aptitude of a person.
Investigating the concept of Intelligence first, it was found that Intelligence
is the premier attribute of human species. It is a term mostly used by all of us in
our life but even the experts of this field finds it difficult and come up with views
as to what it is. Yet most of us agree that intelligence allow us to profit from our
experience and adapt to our surrounding. Intelligence is a property of mind that
encompasses many related abilities such as capacity to (1) acquire knowledge (i.e.
learn & understand) (2) apply knowledge (solve problem) & (3) engage in abstract
reasoning. It is the power of one‟s intellect and as such is clearly a very important
aspect of one‟s overall well being.
Before Gardner‟s research, many people believed that Intelligence was a
single inherited entity. Binet (1916) was one among them who conceptualized
intelligence as a single, integrated but a complex mental process. Furthermore it
was thought that human beings were born with a blank slate and could be trained
to learn anything, provided it was presented in an appropriate way. But now,
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researchers are more convinced that the complex capabilities of the brain cannot
be measured accurately by the traditional IQ test. During 1980‟s one had to take a
position on whether the evidence was more favorable towards the Spearman view
of one general ability (along with number of test specific abilities); with
Thurston‟s view of 8 to 11 primary mental abilities, or Guilford‟s 120 – 160
ability models. The hierarchical model as developed by Jan- Eric Gustafson and
John Carroll, considered abilities as varying in their generality from fairly specific
ability (e.g. Memory span, associative memory and free recall memory) to broader
ones (e.g. General memory and learning) to the most broad i.e. general
intelligence. Howard Gardner‟s Multiple Intelligences Theory (MIT) (1983,
1999) is an important contribution towards understanding cognitive processes. He
described a total of eight distinct intelligence that were quite independent of each
other. These were identified as linguistic, logical/mathematical intelligence,
musical/rhythmic, bodily/kinesthetic, visual/spatial, interpersonal,
intrapersonal and naturalist intelligence. Among them it is the logical-
mathematical intelligence that gives us the ability to use numbers effectively and
to understand the underlying principles of number system. Mathematicians and
scientists, whose work involves recognizing patterns and explaining the physical
universe, possess a high level mathematical intelligence or mathematical aptitude.
On speculating various theories of Intelligence one settles down to infer
that though there is no conformity regarding the nature of the abilities (which can
either be general or specific, integrated or complex) yet it is explicit that
intelligence is a collection of abilities. Confining our focus on the Thurston‟s
Number factor (N) ability and Gardner‟s logical/ mathematical, we infer that
both of these intellectual abilities accounts for the individual‟s competence in
mathematics. This indicates that an individual‟s proficiency to assimilate the
mathematical concepts or his mathematical aptitude should be influenced by the
intellectual ingredient possessed by him. We also know that aptitudes are
interchangeable with abilities, as aptitude is a psychological trait that includes a
specific collection of independent mental abilities in an explicit cognitive
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domain of an individual predicting his success in a particular area. Similarly,
Intelligence is also a comprehensive cognitive trait that governs all the mental
abilities. Therefore it can be conceived that the measure of Intellectual ability
may be directly proportional to the mathematical aptitude of a person. Contrary to
this, it has also been observed that the children with high Intelligence Quotient
(IQ) no matter however high, are not mathematically talented. Analysis of this
disparity reveals that Mathematical talent is a specific aptitude, while an IQ
score is a combined outcome of many different aptitudes and abilities. An
individual‟s IQ is made up of several different components, only some of which
are related to mathematical ability. Thus, the intellectual ability all alone cannot
be responsible for the entire Mathematical Aptitude of the person. The degree of
mathematical abilities present in one‟s intellect may vary from person to person.
This means that even if individual‟s mathematical abilities are not highly
developed yet his intellectual ability score may be high. This may be due to his
high proficiency in some other domain, i.e. verbal, linguistic, musical, spatial or
so on. The above discussion lead us to believe that an additional distinct factor,
besides the Intellectual Ability exists and it may account for a high mathematical
aptitude of a person.
Now, Mathematics applies directly to abstraction and it enables the child to
handle abstract structure. For this purpose one requires a vigorous thinking and
critical logical reasoning power. Now, reasoning is also an important aspect of
the logical mathematical intelligence. This reflective reasoning generates a
Problem solving skill in an individual. It may be this problem solving ability,
which works behind the disparity stating that people having high intellectual
ability are not always highly abled in mathematics. Thus we conclude that the
I.Q. tests may acts as a clue in identifying the mathematical talent but to be more
particular an individual‟s Problem Solving ability must be viewed as an important
determinant. To understand the latent talents our students possess towards
mathematics, continued research was absolutely essential. (Tapia & Marsh,
2000).
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Patton et.al; (1997) propose that learning to solve problem is a primary
objective in learning mathematics, as problems are an inevitable fact of life. Thus
evidently Problem Solving is a deliberate or a purposeful act on the part of an
individual to realize the set goals by inventing some novel method or
systematically following some planned steps for the removal of interference or
obstacles in the path. It represents the highest category of Intellectual Ability in
Gagne‟s theory of learning. This means that the Intellectual ability plays an
influential role in determining one‟s Problem Solving ability. The conjecture is
also sustained by the Binet‟s (1916) view who defines Intelligence as a general
Intellectual capacity which consisted of the abilities to reason well with abstract
materials, to comprehend well, to have a clear direction of thought, to relate
thinking with the attainment of a desirable end and to be self critical.
This connotes that while using one‟s intellectual ability one indulges in
critical thinking to attain the desired ends or solving a problem. The thought
process is directed in a manner that the solution of a problem can be sorted out.
This implies that in the field of mathematics, one has to resolve his own
intellectual resource to aspire for a highly specialized and distinct ability called
Problem solving ability. But it is not essential that if an individual scores high on
the IQ test then he will possess a high problem solving ability as well, because
problem solving occurs only in a novel or difficult situations in which a solution is
not attainable by habitual methods of applying concepts and principles derived
from past experience in very similar situations (Woodworth and Marquis,
1948).
It now becomes essential for us to know that how an individual develops
Problem Solving ability. It is quite evident that challenges and problems faced by
an individual or by society in general are solved through serious efforts involving
thinking and reasoning. Thinking is a mental activity in its cognitive aspect or a
mental activity with regard to psychological objects (Ross, 1951). Thinking is a
behavior which is often implicit and hidden and in which symbols (images, ideas,
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and concepts) are ordinarily employed (Garret, 1968). These views maintain
that thinking is a process of internal representation of external events, belonging
to past, present or future, or an event or thing which is not actually being observed
or experienced by the thinkers.
Another viewpoint says that thinking is an implicit Problem Solving
behavior (Mohsin, 1967) and thinking is a problem solving process in which we
use ideas or symbols in place of overt activity (Gilmer, 1970). This view is more
concrete because they do not rely on unobservable internal representation and
define thinking as problem solving activity that can be readily studied and
measured (Fantino and Reynolds, 1975). On one hand, Piaget suggests the
development of thinking in man goes through the successive stages of cognitive
development i.e. the sensory motor, pre- operational, concrete-operational and
formal- operational stages. On the other hand, Bruner hypothesized that the
thought process evolves as a result of maturation, training, experience through
series of sequential stage i.e. inactive representation stage, iconic representation
stage, and symbolic representation stage. The latest explanation of thought
process is given by the Information Processing theory, according to which
thinking is connected with the information one receives from the environment
from one‟s senses, and the nature of the thought process depends upon how it is
utilized by the individual from time to time he perceives it, until the time he
processes it at various depth levels in solving his problem or chalking out a
strategy or plan. The salient steps of the processing may be: Registering
information, Retrieving material related to this information from memory
and using both kinds of knowledge purposefully. A close analysis of these
views suggests that problem solving involves higher type of thinking which is
very careful, systematic and organized function.
“Mathematics is a way to settle in the mind a habit of reasoning”
(Locke, J. 2003). In Mathematics this type of critical thinking is indispensible
in order to incorporate prior knowledge, employ reasoning and frame cognitive
strategies so that one may generalize, prove or evaluate unfamiliar mathematical
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situations in a classroom in a reflective manner. It is this stepwise thinking with
a purpose or goal in mind which we term as reasoning (Garret, 1968).
Reasoning is the word used to describe the mental recognition of cause and effect
relationships. It may be the prediction of an event from an observed cause or an
inference of a cause from an observed event (Skinner, 1968). Reasoning is
combining past experiences in order to solve problems which cannot be solved by
mere reproduction of earlier solutions (Munn, 1967). Thus reasoning is a highly
specialized thinking which helps an individual to explore mentally the cause -
effect relationship of an event by adopting some well organized systematic steps
based on previous experience combined with present & orient him towards
problem solving.
Problem Solving Skill is rooted in John Dewey‟ philosophy of Pragmatism.
The entire approach of Constructivism is centered on problem solving wherein the
role of the teacher is to select, organize and direct experience so that learners
participating in the activities gain maximum knowledge and understanding. The
behaviorists considers problem solving as a kind of discrimination learning and
analyze it in terms of chains of S-R associations and pointed out that some kind
of hierarchical organization of S-R chains is crucial in any adequate theory of
verbal behavior. Gestalt psychologists such as Kohler (1925) and Wertheimer
(1945) viewed Problem Solving as a matter of integrating previously learned
responses. For them true Problem Solving was insightful, that is, the organization
of responses occurred relatively suddenly and this organization was both enduring
and could be generalized, but Gestalt psychology was silent about the structure of
insightful organization. Similarly, Behaviorists theory of problem solving made
no efforts to understand what happens between stimulus and response, that is
„mediating responses‟ or anything intervening between stimulus and response.
Of late Newell, Simon and Shaw (1958) introduced a new theory of Problem
Solving based on concepts of: Information Processing and computer
Programming. They believe that humans process the information in the same
manner as computer do and considered this processing of information as the
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influencing step between input and outputs. The Information Processing Theory
states that in a computer program the postulated details of a precise set of
mechanisms are described in a formal programming language to account for the
observed behavior. Similarly, the problem solving process include of what human
subjects do in writing\ orally \ in action under experimental conditions while
solving problems and these written \ oral activities \ actions are hierarchically and
sequentially arranged and organized as in computer program. Today, the
Mathematical Problem Solving has occupied a very important place in the
teaching of mathematics. Rosenbloom (1966) and Polya (1966) assert that the
central activity of all teaching of mathematics is the development of Problem
solving skill in the students. Collier and Lerch (1969) observe that problem
solving is a „major force‟ in the growth of modern mathematics and Barnes
(1959) stress that it should be „major concern‟ of the school curriculum.
The entire discussion till now reveals that the essence of mathematics as a
discipline makes it imperative to identify the innate mathematical potential of
students termed as mathematical aptitude and harness it to its ultimate extent.
This shall be possible only if the implicit factors influencing the Mathematical
Aptitude are explored and treated in a rejuvenating manner. The discussion also
reveals that the Mathematical Aptitude is also influenced by the factors like
Intellectual Ability and the Problem Solving Ability. Thus, it will be interesting
to investigate the structural pattern and the relevance of above two variables in
influencing the Mathematical Aptitude of a person.
1.1 ORIGIN OF THE STUDY:
The secondary education plays a significant role in framing the future of an
individual as it is on this behalf that students are selected for diverse academic
courses. It is commonly perceived that mathematics is a very useful subject for
vocational and higher specialized courses of learning. We need to teach
mathematics not to get marks and degrees but to develop an intellectual
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personality with sharp observation, deep concentration, precise decision making
and scientific approach. It has also been a general observation that the students of
the same class differ in their performance in mathematics, though taught by same
teacher with same objective, methodology and environment. Moreover there is a
general notion that the student who performs well in mathematics, fares good in
other subjects simultaneously. Contrary to this, it is also observed that the
students scoring very high in other subjects often scores comparative low in
mathematics. This indicates that despite having same intellectual level the
students may differ in their performance on mathematics. This signifies that
something other than intelligence known as Mathematical Aptitude is required
to be successful in the field of mathematics.
A mathematical aptitude includes power of abstractness, precisions, the use
of words, logical thinking and skills in calculation. It is apathy that despite this
aspect, mathematics is merely associated with pencil, pen, paper, books and lots
of practice. It is considered as a subject that is synonymous with numbers and
arithmetic. Subsequently dexterity with numbers and efficiency in mathematics is
merited as “Mathematical Intelligence”. Contrary to this, the fact is that many
high school students can‟t even interpret graphs, don‟t understand statistical
notions, are unable to model situations mathematically, seldom estimates or
compare magnitudes, can never prove any conjuncture, and most distressing of
all, hardly ever develop a critical, skeptical attitude towards numeric, spatial and
quantitative data or conclusion. Besides this, another fact persists that students
showing highest test scores in mathematics may not always possess high
mathematical aptitude. High scores may be the outcome of high computational
skills, attentiveness in class, conscientious about completing all the assignments
etc. but such an evaluation may neglect the conformity to the thought
procedure and the reasoning ability associated with high level of mathematical
thinking. Consequently the less able students may often score as high as the
genuinely talented students having a high Mathematical Aptitude.
Mathematics anxiety or phobia, a kind of terror that exist in the young
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minds of children, is another major factor that deters them from executing any
mathematical notion beyond their curricular regime. Students often develop
mathematical anxiety as a result of learning from teachers who are themselves
anxious about their mathematical abilities in certain areas. Typical examples of
areas where mathematics teachers are often incompetent or semi-competent
include fractions, (long) division, algebra, geometry "with proofs", calculus, and
topology. In many countries, would-be math teachers are required only to obtain
passing grades of 51% in mathematics exams, so that a math student who has
failed to understand 49% of the math syllabus throughout his or her education can,
and often does, become a math teacher. His or her fears and lack of
understanding then pass naturally to his or her students.
For a long time this mathematical phobia had been associated with the
female students. It is thought that women experience more anxiety in
mathematics as a group than men. Until 20th centuries, in western countries
many people believed that inequality between the sexes could be attributed to
biological differences. Thomas Gisborne argued that women were naturally
suited to domestic work and not spheres suited to men such as politics, science, or
business. He argued that this was because women did not possess the same level
of rational thinking that men did and had naturally superior abilities in skills
related to family support. Contrary to this during the early twentieth century, the
scientific consensus held that gender plays no role in intelligence. In his research,
psychologist Lewis Terman found boys were "decidedly better" in arithmetical
reasoning, while girls were "superior" at answering comprehension questions,
though he concluded that sex plays no role in general intelligence. He also
proposed that discrimination, denied opportunities, women's responsibilities in
motherhood, or emotional factors may have accounted for the fact that few
women had careers in intellectual fields. Thus, the above discussion presents two
contrary viewpoints, regarding the role of gender in determining the mathematical
aptitude of the students. This originated a query in the mind of the researcher
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regarding the nature of mathematical aptitude and the factors that may account for
its generation. This query was further intensified because unless the educators
realize the true mathematical ability of these students, they may fail to adjust the
existing mathematical program in their accordance.
Investigation revealed many discrepancies in the present mathematical
curriculum. It was found that, many of the mathematics programs in our schools
are heavily devoted to the development of the computational skills and provide
little opportunity for students to demonstrate the complex type of reasoning skill
that are characteristics of truly talented students. The factors blamed for these
existing lacunae are: lack of time, lengthy syllabus, short duration of the course,
teachers‟ incompetence, and students‟ missing determination and of course the
inevitable present system of education. Besides this, one needs to tackle the
individual differences that exist in a mathematics classroom. These differences
may owe to the varying levels of intellectual abilities present in them. Though
90% of these Intellectual abilities are determined by the persons‟ heredity yet by
enriching his environmental factors the rest 10% of his intellect can be
channelized in right direction.
National curriculum framework NCF 2005 (developed by N.C.E.R.T.)
have characterized two goals i.e. „narrow aims and higher aims‟ of education. By
higher aims we mean to develop the child‟s inner resource to think and reason
mathematically, derive logical conclusions and handle abstractions. While by
narrow aims we mean that child would have a very good algorithmic practice, by
just remembering the formulas. The narrow aims can acquaint us with short term
success but in order to develop a rational mind, a critical outlook and the
proficiency to deal with the practical and complex problems in future life, the
higher aims are needed to be aspired. But the question is that how far had we
genuinely succeeded to achieve these aims?
Now, the higher aims indicate towards the gradual attainment of problem
solving ability not only in the field of mathematics but even in the other areas of
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life. However, one cannot overlook the contribution of Mathematics in laying
down the foundation of this problem solving ability. We need to understand the
nature of Problem Solving skill, because however high Intellectual Ability a
person may possess, unless he is proficient in Problem Solving skill, the concepts
of mathematics shall always pose a challenge and create a phobia in his mind.
Considering all the aspects discussed above a dilemma originated in the mind of
researcher that inspired her to investigate the essence of this inborn mathematical
abilities or the Mathematical Aptitude and study it in relevance to the higher
cognitive skills identified as intellectual ability and problem solving ability.
1.2 NEED AND SIGNIFICANCE OF THE STUDY:
The importance of any subject is assessed by the fact that to what extent
the subject helps an individual acquire the aims of Education. Mathematics is a
subject that has set the path of self actualization. The present Hi Tech society
calls for the essential Mathematical knowledge in order to sustain its compatibility
with time and welfare of mankind. Thus, Mathematics is not only essential as an
independent subject, but it also plays an indispensible role in developing a
balanced personality with a rational outlook that would help an individual to
function at the highest level of enquiry.
In spite of playing such a vital role in our cultural development as well as
for individual progress, mathematics is not a subject of choice for many students.
A negative attitude puzzles their mind and generates lack of desire, especially
with regard to mathematics. Unfortunately, many students are afraid of
mathematics and suffer from mathematical anxiety. Consequently the students
learn about mathematics under great pressure and immediately forget whatever
facts they were forced to learn. The failures in mathematics are thereby
increasing day by day.
In an attempt to unleash the mystery behind this detrimental fact, it was
found that mathematics is taught like a mechanical subject, using no creativity, no
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useful techniques and methods. The students are hardly trained to develop
mathematical skills of calculations and constructions. They are rarely encouraged
to develop mathematical thinking, a positive attitude and a sound mathematical
aptitude that could nurture and facilitate a problem solving approach in them.
Terms such as mathematically talented, mathematically gifted and highly able in
mathematics are generally used to refer to students whose mathematical ability
places them in the top 2% or 3% of the population. But it is not necessary that
students who score highest test scores or receives the highest grade in
mathematics class necessarily possess high mathematical aptitude, because high
Mathematical Aptitude doesn‟t mean high computational abilities or working with
pedantic logical formalism. Rather it is determined by an unusual ability to think
and work with mathematical problems in flexible and creative ways rather than in
a stereotypic fashion.
According to education matters, (2008) the students with extremely high
ability and motivation may benefit from a program that promotes rapid and
relatively independent movement through the instructional content. Therefore,
while designing the instructional program in mathematics care must be taken to
analyze their individual ability and needs. This is possible only if we can identify
the level of Mathematical Aptitude in them. The knowledge of this specific
ability shall enable us to generate an interest towards the concepts of mathematics
in the manner as demanded by the child‟s psychology. Our aim is to make the
scope of mathematical curriculum so extensive as to provide adequate foundation
for students who may become mathematicians in future.
In order to diagnose this specific mathematical ability we need to
determine the observable correlates that facilitate in cultivating mathematical
aptitude. The achievement in mathematics is profoundly affected by the
Mathematical Aptitude of an individual. A significant correlation between
mathematical aptitude and achievement of students in mathematics exists (Sethi,
N. 2011). Soman, (1977) traced that mathematical aptitude (specifically the
numeric reasoning and numeric ability) has occupied a prominent place among
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the five cognitive functions studied in relation to the mathematical achievement.
High intelligence, numeric ability, abstract reasoning and adjustment were some
of the characteristics of a mathematically gifted children (Kalra , 1975; Kabu,
1980). Kalra also found that mathematically gifted were quite high on creativity.
In another study on mathematical creativity, aptitude and attitude, Tuli (1979)
found aptitude for mathematics and achievement in mathematics were
significantly and positively related to mathematical creativity.
Attitude towards mathematics is another influencing factor that may alter
the results in mathematical performances beyond expectation. In one of the action
research studies on (HAL) High Ability learner of sixth grade students in
mathematics, it was found that the students despite having a negative attitude
towards mathematics were able to perform well in the mathematical portion of the
Terra Nova test. This indicates that an inborn ability exist in the gifted students
which they harnessed to score high on the Terra Nova test, even though there
score on attitude towards mathematics was very low. The results also indicated
the need for the gifted students to be shown important connections between
mathematics and its utility outside the context of the school academia. Contrary
to this, researches have been reported saying that attitude towards mathematics
does affect achievement in mathematics. Roslay, A. 1984 & Singh, 1986,
Gakhar, 1981 found that the attitude of high school students towards
mathematics and their acquisition of mathematical concepts were related. Mishra
(1978) revealed that children from better homes had a favorable attitude when
compared with the children from poor homes.
There are many possible reasons to trace for the prevailing negative
attitude in students and their underperformance in mathematics. One such reason
revealed was that, there is often a mismatch between the students and the
mathematics program usually undertaken. Many of the students refuse, or are
unable to conform to the expectation of the program that they see as uninteresting
or inappropriate. Sharma (1975) in his analysis of the textbooks, prescribed for
the delta classes in Rajasthan stated that the syllabus in mathematics was highly
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defective, outmoded and wanting in a proper process of evaluation. There was no
relationship between course content prescribed in the syllabus and that presented
in the text books. Krishna Kumari and others (1980) revealed that only a small
percentage of teachers go through the textbooks thoroughly and assimilate
concepts and methods included in them. The project, SCERT, Andhra Pradesh
(1981), while evaluating the textbook prescribed for class VI and VII revealed
that the parents felt unable to help their wards in solving problem included in
mathematics textbooks. This indicates that there is a dire need to analyze
textbooks at all levels of schooling with a view to finding out their relevance to
objectives content, methodology, feedback etc. Moreover with the recent
introduction of computers in the school, educational computing and emergence of
learning through the understanding of Cause and Effect relationship and the
interplay of variables, the teaching of Mathematics has to be suitably redesigned
to bring it in line with the modern technological devices.
Studies in the past have speculated a probable correlation between factors
generating mathematical aptitude and the intellectual ability of an individual.
Mondkar, 1984 analyzed the Numerical aptitude and some of the abilities of
Guilford‟s structure of intellect were positively significant on mathematics
achievement. Dubey (1987) found that the Intellectual ability and the family
background were also found to be the major contributors of mathematical
achievement (Singh, 1986; Nilima Kumari, 1984; Rajput, 1984; Gakhar, 1981;
Jabbal 1981; Kabu, 1980; Nalinidevi, 1976). In his study Srivastava, J.P.
(1992) revealed that intelligence contributed significantly and positively to the
development of learning outcomes in mathematics in terms of knowledge,
understanding, application and skill. Singh, R.D. and Verma, S.C. (1992)
found that students of high intelligence have more favorable attitude towards
mathematics than the students of low intelligence. Uchat, D.A.(1981) found that
students with high intelligence level possessed high aspiration level while
Rajput, A.S. 1984 found that intelligence affected the achievement of students
significantly at all the three levels (High, Average, Low). Contrary to this,
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Chitkara, M. (1985) found that various strategies like lecture, discussion,
inductive drill, auto instruction group discussion were equally effective in terms
of achievement in mathematics disregarding level of intelligence, sex and
personality type.
The fact indicates that despite possessing a sound intellectual mind, a
specific critical thinking is needed for furnishing the solution of a problem. It is
this specific ability that leads to the development of mathematical aptitude.
According to Delisle and Galbraith (2002), although gifted students grasp
mathematical concepts readily, they often have little or no patience for regular
math lessons or homework. The superior reasoning power in them makes them
grow impatient. Thus such student needs to be engaged with the activities of
mathematical interpretation and creative Problem Solving. On the other hand the
students who have difficulty in mathematics are often those who cannot use
critical thinking to improve their reasoning. It is evident from the above
discussion; that the mathematical aptitude is significantly influenced by the
thought process or the reasoning ability. The development of sound reasoning
ability in individuals has been shown to be correlated with a multivariate of
variables namely, prior knowledge (Resnik and Gelman, 1985); processing
capacity (Finegold and Mass,1985); cognitive styles (Stuessy, 1989); sex
(Hernandez, Marek and Benner, 1984); IQ(Lawson, 1982); SES (Acuna, 1983)
as well as number of aptitudes (Owen, 1987). Thus willingness towards
problem solving is believed to play a significant role in mathematics
achievement. Besides intellectual and problem solving ability the other factors
responsible for underachievement in mathematics have been some personality
variables , namely self reliance, sense of personal freedom, feeling of
belongingness, withdrawing tendencies, nervous symptoms, social skill, general
anxiety, and test anxiety, parental profession and parental education (Iyer, 1977).
Besides this, our society is suffering from gender bias in mathematics and
other science subjects. This gender gap is a serious blow that results in the under
representation of women in courses and in carriers related to mathematics and
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sciences. Research has shown that males and females do excel in different
abilities in math and science while some researchers have also concluded that men
have slightly higher IQ than women. In his study Sian, 1994 revealed that the
gender gap in higher education has tended to widen. A large gender difference
was found in stereotyping of mathematics as a male domain (Hyde et. al, 1979).
Regarding gender difference in Problem Solving abilities, a study
conducted by Moreno and Mayer, (1999) suggested that males perform better
than female when tasks involve the cognitive skills used in mathematics.
Contrary to these studies, Mokhtar (2000) found that there was no significant
difference in the mean problem solving ability between male and female students.
Similarly, Katiyar‟s (1979) also revealed that boys and girls did not differ in
mathematics achievement. The current theory supporting this view says that
mathematical anxiety and gender difference in all the areas of mathematical
achievements is a product of cultural learning, and if children are treated equally
in earlier stages most of the observed gender difference will disappear (Fennema,
1990). Another category of research work found that females outperform males
on mathematical aptitude and its influencing factors. Pal, G.C. and Natrajan,
C. 1977 found that girls not only perform significantly better in mathematics but
also have a more positive attitude towards mathematics than boys. Similarly, Fall
1993, Sheehan & Gray, 1992 have shown that female outperform males in their
high school average in mathematics. Thus it appears that the young students
especially female are equally capable of solving mathematics problems. In the
early years of schooling so problem solving skill could be nurtured equally in
boys and girls.
A close analysis of the nature, type and quality of studies till now throw
adequate light on the status of research in mathematics education. It is very clear
that the quality of researches completed in different areas delineated in the
preceding pages does not seem to be encouraging and leaves much more to be
desired. The mathematical constructs like numerical ability, reasoning, attitude,
creativity and aptitude in mathematics had been time and again studied in context
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with numerous other factors like Intelligence, problem solving ability, Socio
economic status, parental and teachers‟ attitude, motivation, sex and so on. The
evidences has been traced which are equally supporting as well as contradicting
the role played by various factor in determining the mathematical aptitude of a
person. Today the major issues before teacher educators in mathematics are in
depth study of mathematics curriculum, curriculum renewal, refining teaching
methods in mathematics in the light of the advances in science of pedagogy on
one hand and educational technology on the other. Currently the scope of the
study remains limited. What is needed is proper selection of problem in the area
of method of teaching mathematics, planning long term strategies, trying out
various methods of teaching and measuring multidimensional outcomes, among
students as a result of teaching exercise. Since each method has a limit of
developing a specific abilities and skills with respect to various branches of
mathematics.
It was felt that the majority of the studies had focused their emphasis on the
mathematical achievement and the factors governing it. Factually it is evident
that achievement in mathematics is merely a superficial portray, or an outcome of
an individual‟s prudent mathematical aptitude. Rather it may or may not judge
the overall essence of mathematical ability present in him. The concept of
mathematical aptitude is much more profound and comprehensive to determine
the true mathematical ability. Unless a quest for mathematical aptitude and its
determining factors is pursued the research work in mathematics shall remain far
from extremity. The purpose of this study was also strengthened by the lack of
comprehensive data available in literature review on the topic mathematical
aptitude. The literature unanimously supported the importance of a positive
learning attitude, Intelligence and Problem Solving Ability towards mathematics,
but it did not explicitly describe the underlying factors that enhances or obstructs
the generation of this favorable attitude and how far are the factors like
Intelligence and Problem Solving ability correlated in the generation of a sound
mathematical aptitude.
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The studies have also suggested that General intelligence and Problem
Solving Ability both seems to have a considerable impact on the traits that
constitute the mathematical aptitude of an individual. Moreover Problem Solving
Ability is found to bear a functional resemblance with one of a trait of Intelligence
i.e. logical/ mathematical intelligence, which according to Gardner, calls for a
critical thinking, logical reasoning and Information Processing. It is this, thinking
reasoning and Information Processing that forms the fundamental steps of
Problem Solving Ability. Thus, it is evident that these two factors must play a
significant role in determining the mathematical aptitude of an individual.
Question arises as to what relationship exist between the mathematical aptitude,
Intelligence and Problem Solving Ability and how far are they correlated in
determining the Mathematical Aptitude of an individual. Being the student of
mathematics, the investigator attempts to conduct the research, in order to study
the Mathematical Aptitude in relation to Intellectual and Problem Solving
Abilities of Secondary Level Students.
1.3 STATEMENT OF THE PROBLEM:
The problem of investigation is entitled as: “A STUDY OF
MATHEMATICAL APTITUDE IN RELATION TO INTELLECTUAL
AND PROBLEM SOLVING ABILITIES OF SECONDARY LEVEL
STUDENTS”
1.4 DELIMITATION OF THE STUDY:
In order to concretize the problem following controls have been applied:
(1) The study will be conducted in secondary schools of Dehradun District
only.
(2) Both male and female pupils studying in Class XI and XII of Government,
Government Aided, recognized by Uttrakhand Board, Public or Private
secondary schools situated in the rural and urban areas of Dehradun
District will be included in the survey.
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1.5 OBJECTIVES OF THE STUDY:
1. To find out whether different levels of Intellectual Ability have any impact
in the generation of Mathematical Aptitude of secondary level students.
2. To assess whether different levels of Problem Solving Ability have any
impact in the generation of Mathematical Aptitude.
3. To assess whether students differentiated on the basis of sex have any
impact in the generation of variables like Mathematical Aptitude,
Intellectual Ability and Problem Solving Ability.
4. To find out whether the type of school have any impact in the development
of Mathematical Aptitude, Intellectual Ability and Problem Solving
Ability.
5. To find out whether different streams have any impact on the development
of Mathematical Aptitude, Intellectual Ability and Problem Solving
Ability.
6. To find out the relationship of Mathematical Aptitude with the variables
like Intellectual Ability and Problem Solving Ability.
1.6 HYPOTHESIS:
1. There exists no significant difference between Mathematical Aptitude of
students of different Intellectual Ability level.
2. There exists no significant difference between Mathematical Aptitude of
students differentiated on the basis of Problem Solving Ability.
3. Intellectual Ability and Problem Solving ability together have no impact on
the generation of Mathematical Aptitude of the secondary level students.
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4. There exist no significant impact of gender on variables like Mathematical
Aptitude, Intellectual Ability and Problem Solving.
5. Stream has no significant effect on the Mathematical Aptitude, Intellectual
ability and Problem Solving ability of secondary level students.
6. There exists no significant difference between Mathematical Aptitude,
Intellectual Ability and Problem Solving ability of students when
differentiated on the basis of school.
7. There is no significant correlation between Mathematical Aptitude,
Intellectual Ability and Problem Solving Ability.
1.7 FUNCTIONAL DEFINITION OF THE KEY TERMS:
The terms used in the present study are defined below.
1.7.1 Mathematical Aptitude:
“Aptitude is a condition or a set of characteristics indicative of ability to
learn or to acquire proficiency in certain skills provided the necessary
environment is made available for the same” (Warren‟s Dictionary). The innate
nature of aptitude is in contrast to achievement, which represents knowledge or
ability that is gained.
In this perspective Mathematical Aptitude is also an innate ability possessed by
an individual that would determine his competence and proficiency in handling
abstract mathematical concepts. It also determines the probability as to what
extent a person would succeed in the field of mathematics and related areas.
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Thus “Mathematical Aptitude is the specific ability as represented by
the mathematical aptitude tests that have been standardized to prove its
effectiveness in predicting success in mathematics and allied areas”.
1.7.2 Intellectual Ability: Considering the concept of Intelligence, Wechsler
views that Intelligence is an “aggregate or global capacity of the individual to
act purposefully, to think rationally and to deal effectively with his
environment”. Woodworth, in a similar manner considers Intelligence as
“mental capacity to deal with the novel situations effectively”. The concept of
mental capacity is indicative that Intelligence can be rationally represented by the
measure of Intellectual Abilities.
In this context Intellectual ability is an (i) ability to make successful and
rapid adaptation to the new situation and to learn from experiences. (ii)
capacity to integrate experience and (iii) as commonly used in measurement
and testing a degree of ability represented by performance on a group of test
selected because they have proved their practical value in the prediction of
success in academic work and in some vocation” (Good, C.V.).
Thus we conclude that, “Intellectual Ability is a comprehensive measure
or the degree of one‟s potential in varied areas of mental cognition as
assessed by a standardized test on cognitive abilities that has been developed
to prove its effectiveness in categorizing students on the basis of their
intellect”.
1.7.3 Problem Solving Ability: Problem Solving is that structure or pattern
within which creative thinking and reasoning takes place. A problem stimulates a
thinker and thinking solves the problem. Orton (1987) defined problem solving
as “a process by which a learner discovers a combination of previously learned
rules to achieve a solution in a novel problem situation”. Thus, Problem
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solving ability involves a right thinking and proper reasoning which ultimately
depends on the level of person‟ Intelligence. In that perspective Problem Solving
can be considered as a speed test to assess one‟s Intelligence. Moreover,
Problem Solving is the essence of teaching learning mathematics.
Thus, “Problem solving ability is the ability as measured by the set of
mathematical problems, selected because they have proved to be valid to
predict the superiority of an individual in resolving mathematical and other
related problems”.
1.7.4 Secondary Level Students: “Boys and Girls who are studying in
class XI standard of various Government and Public schools imparting formal
education in Dehra Dun”
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