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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