## Introduction

The concern over recognition, identification and provision for gifted and talented children in society has always been muddled in myriad of controversies. For instance, there are those who perceive this category of children as invaluable resources whereby they are admired and honoured. On the other hand, they are perceived differently with some kind of suspicion (Johnson, 2000). Cockcroff (1982) explains that these mixed feelings and attitudes could be equated to ‘love-hate’ relationship whereby human beings perceive giftedness as precious. The latter author accentuates that the issue of mixed feelings does not only reside in society.

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It also extends in academic institutions. The effort to promote recognition and provision in academic institutions has faced several setbacks due to deep rooted doubt among some teachers and policy makers on whether gifted students really exist (Johnson, 2000). There are those who view gifted students as normal learners who are slightly high achievers than other students. On the other hand, there are those who claim that special provision will lead to undue advantage over ordinary learners bearing in mind that giftedness is already a natural privilege (Koshy & Robinson, 2006). NCTM (1980) underscores that the kind of negative perception which persisted in academics was an enormous hindrance to practice aimed at exploiting special talents and gifts that gifted children possess. Koshy and Robinson (2006) also accentuate that educational institutions are biased towards making special provisions for underachievers in academics and therefore, they do not want to treat gifted children differently (Gerrish & Anne, 2006).

As epitomized above, talented learners have always been viewed with skepticism both in society and in learning centers. However, it is imperative to note that mathematically talented children seem to be the most affected since they are usually neglected in society (NCTM, 1980). The myth that this category of learners are able to excel in class without assistance has led to further neglect against them. However, this deep rooted and misinformed attitude started taking new course due to the numerous innovations in education and psychology (NAGC, 2005). The movement towards positive perceptions and recognition was accelerated by publications into this phenomenon by pioneer researchers like Lewis Terman in 1916 & 1925 (NAGC, 2005; Fox & Zimmerman, 1985). The above researches provided significant base upon which growing interest in education of this particular group of students was conceived in the United States of America and other parts of the world.

Nevertheless, challenges that were associated with identification and provision in education for gifted learners did not end there since researchers discovered another enormous setback owing to the various cultural biases and ideological differences. According to Bevan-Brown (2003), giftedness cannot be viewed independently from the social constructs elements that shape its perception in society because social construct dimensions are shaped by values, customs, attitudes, beliefs and practices of that society. This implies that the concept of giftedness is understood differently from one cultural group to another. In addition, cultural differences have crucial role to play in shaping perception, identification and nurture of gifted learners in society (Bevan-Brown, 2003). As a result, researchers mainly dwelt on the development of theories to enhance recognition. Scholars who study giftedness are always faced with the challenge of developing an all rounded theory since their theories have to show recognition of broader philosophies that are embedded within diverse cultures.

On the same note, political perceptions of giftedness on whether the same should be given precedence in education have hindered development of relevant programs to support the talented in society. Up to now, very few countries have provisions for special education as in some countries particularly developing ones do not have the capacity to provide education to gifted and average students separately. Apart from the financial challenges, some of the hindrances to the development of education policies to cater for the gifted can be blamed on the negative perception of the phenomena among the political policy makers (Phillipson & McCann, 2007). For instance, policy makers in the USA are still caught up in the quagmire of whether to promote excellence or egalitarianism while addressing the needs of gifted students. Phillipson and McCann (2007) elucidate that excellence view in the USA seeks to provide gifted children with opportunities to maximize their talent while the egalitarianism view perceive the provision to be a violation of philosophy that view all men are equals. The disparities and diversities in both cultural and political perceptions of giftedness have lead to the development of varied and complex educational provisions around the world.

Despite of the challenges cited above, the significance of educational programs to cater for the gifted and talented students cannot be overemphasized. It imperative to mention that, giftedness can occur at various aspect of life, however, this paper will lay more emphasis on mathematically gifted and talented. On this perspective, past literature from authentic sources will be reviewed with an aim of identifying how the issue of identification and provision are conducted around the world and whether the provision therein promote excellence among the mathematically gifted (Jankowicz, 2005). A critical evaluation of available literature presented in this paper is timely to question whether the treatment of the mathematically gifted attracts positive implications in education (Cranton, 1996; Daloz; 1986).

### Background to the study

Empirical and theoretical researches have indicated that gifted children operate on a higher level than their peers in specific aspects of life. On the same note, the mathematically gifted are not different and research indicates that for them to excel in their talent they require a curriculum that is tailored to meet their needs (Johnson, 2000). Moreover, Johnson (2000) emphasize that educational programs are crucial to the development of those children since when they are integrated with average students they are likely to lose ,motivation and consequently, they lose interest in academic activities. Correspondingly, scientific experiments indicate that if gifted children are not exposed to challenging tasks, the level of their brain development is slowed down. This further implies that challenge is important to the gifted children and integrating them with the average students is undermining their talent (NCTM, 1980). The same view is highlighted by (Johnson, 2006) whereby he point out that mathematically gifted students in a mixed-ability are hindered form attaining effective learning since they the tasks are such setting are too easy to achieve fully engagement.

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However, initially, interest in the education of gifted children focused more on other talents and ignored the mathematically talented. However, studies in the former USSR and USA about the significance of the mathematically gifted in technological development and advancement heightened the interest of researchers and policy makers alike towards the development of programs aimed at taping exceptional abilities of the mathematically talented (NCTM, 1980). These pioneer studies can be praised for paving way for numerous researches into the phenomena.

As mentioned above various studies directed towards the phenomena of mathematically gifted and more importantly the research have been interested in identification and provision issues in education. On this note, the US can be praised for the considerable effort to promote the gifted children. Several longitudinal researches have been undertaken mostly in the US as scholars try to develop theories to facilitate the identification of the mathematically gifted students (Benbow & Lubinski , 1996). The effort towards the development of theories to address the needs of talented children have given rise to theories namely, *multiple intelligence theory, Triarchic Theory of Intelligence and differentiated model of giftedness (*Eurydice, 2006*). *Although, have been proved to be relevant in practice, the *multiple intelligence theory *has contributed a great deal to the development of education curriculum for gifted children because the awareness exhibited via the theory inspired policy holders in education to consider provision and implementation of the said curriculums (Eurydice, 2006). Correspondingly Gardner (1983) presented very strong views about the existence of exceptional mathematical skills and we ahead to develop *mathematical intelligence theory, *upon which subsequent studies concerned with mathematical giftedness have been founded.

A critical overview into the US education system indicates the provision for the gifted in education is firmly established. Moreover, the recognition, identification and provisions for the mathematically gifted are clearly established in the US educational system (Leroux, 2000). As matter of fact, owing to the considerable effort by the National Council of Teachers of Mathematics (NCTM) in the US, a harmonized programs known as *Principles and Standards for School Mathematics *was developed to provide guidance to teacher about methods of administering mathematics course to enhance recognition of the mathematically talented and gifted (NCTM, 2000).

Furthermore, the recommendation by NCTM is that all students be exposed to challenging tasks and whenever a teacher identifies a student with exceptional abilities he/she should use additional resources that are over and above the average to ensure full engagement of the gifted. By so doing, the mathematically gifted have an opportunity to excel this area (NCTM, 2000). On the same note, other EU countries have followed in the footsteps of USA to promote and nurture giftedness via the educational system (Benbow & Lubinski 1996). However, a closer look at the methods of provision indicates that they vary from country to country mainly because of the cultural, policies and ideological differences towards the perception of giftedness (Marland, 1972). For instance, Malta and Norway, the gifted are incorporated into the general educational policy whereby a teacher is expected to address their needs using a differentiated approach but within the mainstream class (Eurydice, 2006).

On the contrary, in some countries like Greece, Spain, Slovakia, Czech Republic, Slovenia, Scotland, France and Ireland gifted and talented students are classified under the same category with the academically challenges whereby their ability is seen as a defect that calls for special education (Eurydice, 2006). This perception can be attributed to the rising number of special schools that specialise in different fields of academic (Eurydice, 2006).Such specialized schools are also present in Austria, Netherlands and Romania whereby there are numerous academics, institutions and networks which are only concerned with promoting excellence among the gifted (Phillipson & McCann, 2007). This brief background information further accentuates the fact that the gifted and talented individuals are an invaluable resource in the society and their needs should be promoted through identification and provision of resources within the education system. This present study, therefore, is of great significance for a critical analysis of current literature will shed light on education implications of such provisions in the curriculum.

## Analysis and Findings

### Overview

In analyzing the findings from data that will be collected using secondary methods, it is imperative to evaluate, analyze and establish the authenticity of sources used. As already mentioned, not all information that is acquired from secondary sources is suitable and consistent. As such, it is instructive that in this analysis, information from these sources be carefully reflected with regards to their steadfastness in their contributions to current research. Usually, the rationale of a given piece of secondary material significantly determines the findings of the study (Eurydice, 2006; Fai, 2000). However, as aforementioned, researchers have postulated that data collected to improve the welfare of a given group is not authentic enough to constitute sources of information for a research work of this nature. All the same, the sources that have been used in this study and their level of precision and data collection methods, as will be examined in this analysis, provides principle and purpose for conducting the study. Therefore, the paper will commence the analysis and findings of this research by describing and analyzing three sources that were used for the study. Thereafter, findings will be presented in a highly structured manner (Barnett & Juhasz, 2001).

### Analyses of sources

“**Mathematically gifted and talented learners: theory and practice” Authors: Valsa Koshy, Paul Ernest and Ron Casey Brunel University, Uxbridge, UK; School of Education, Exeter University, Exeter, UK.**

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The above mentioned article by Ron Casey, Paul Earnest and Valsa Koshy emphasizes on the need for early and primary school learners gifted with talents in mathematics to be given special attention. In the article, the authors recognize that there are gifted mathematicians who need to be identified and accorded special support (Koshy, Ernest & Casey 2009, p. 222). Published in 2009, the article reviews the theory of giftedness found in mathematics, conducts current research and reviews development of policies. Besides, it examines some of the factors which are necessary to structure giftedness in mathematics and discusses their nature (Green, 2002).

This source is indeed credible since drawing from a framework crafted by Vygotskian (1986), it explores how identifying such talents at an early stage will provide more than enough room to motivationally and attitudinally enhance mathematical talents and gifts as well as provide apt cognitive challenges that will boost learning experiences among school children (Heller et al., 2000). Moreover, special attention given to mathematically gifted students will develop them as citizens who are informed, create developed world leaders and enable them to be competitive in all aspects of life.

“**Serving the Needs of the mathematically Promising” (from National council of Teachers (NCTM) in Developing Mathematically Promising Students (chapter 4).**

This article by Sheffield (1999) focuses on how talented and promising mathematical students can be recognized and promoted. It seeks to determine the most excellent ways with which an educator can identify, encourage and promote the effectiveness of mathematical students (Belenky et.al., 1986; Bell, 1999). In the article, Sheffield examines possible ways gifted students can be recognized and nurtured mathematically.

As a source for the study, it acknowledges the multi-directionality between giftedness on one side and talents on the other as well as the fact that giftedness can be developed and nurtured since everyday intelligence can always be improved through perform a task. The more frequent an individual does mathematics the better he or she becomes at it (Cassell & Symon, 2004).

“**Teaching Mathematically Promising Children” by Ron Casey (chapter 7 from the book : Unlocking Mathematics Teaching : A David Fulton Book).**

This article by Casey (2011) is an important source since it focuses on the implications mathematical giftedness among primary and early school children has on the future of science and technology, the article argues that to furnish upcoming specialists in science and technology, it is imperative that at an early stage, identification of children talented in mathematics, and making necessary, apposite and appropriate provisions for them is a way of capitalizing on a potential and intellectual resource (Gilheany, 2001).

### Analysis of the three sources

#### Conception of giftedness

The overwheming abilities in individual students in early learning years in mathematics would be conceived as gifts and talents. Koshy, earnest and Casey in the article “Mathematically gifted and talented learners: theory and practice” (2009) point out that gifts and talents that school children possess are seen in form of outstanding abilities in particular provinces. It is imperative to note that giftedness or having talents among school children appear in the spheres such as intellectuality and creativity.

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Casey in “Teaching Mathematically Promising Children” (2011) identifies that talents or giftedness among students springs from the socio-emotional sphere and the sensory motor. He also acknowledges the multi-directionality between giftedness on one side and talents on the other. It is instructive to note that understanding the conception of giftedness requires embracing Casey’s (2011) ideas that build on Sternberg’s (1985) Triarchic Theory of Intelligence (Casey 2011, p. 147). He points out that the features of intelligence among students are experiential, componential and contextual in the sense that some gifted students have the ability to evaluate and analyse ideas, make decisions and solve problems. Others students display features such as creativity, and as such can generate both valuable and fresh ideas. Moreover, those experiencing mathematical challenges possess practical abilities that aid them in solving work-related challenges and everyday problems through personal experiences.

On the other hand, Sheffield in her article “Serving the Needs of the mathematically Promising” (1999), examines conception of giftedness from human abilities that include multiple areas of intelligence surrounding mathematics. The article points out that intelligence in academic fields where some students are exceedingly better than others could be a gift from God. Furthermore, she argues that this concept can be confusing to most researchers who fail to understand the orientations of giftedness such as empirical validations, values and axioms (Sheffield, 1999, p. 277). She further claims that giftedness in children, either in combination or singly, can be demonstrated in potential capabilities, achievements, leadership abilities and productivity. Other manifestations can be seen in their creative thinking, specific academic capacities and general intellectual ability (Benbow & Lubinski, 1996).

#### Mathematically Gifted

The article defines giftedness as a special trait that makes it feasible for a person to execute a given task swiftly and well (Freeman, 2001). Research studies have investigated the development of these mathematical abilities by comparing the problem solving capabilities of different students at different ages over a period of time (Adimin, 2001). The students tested were presented with a range of arithmetic, geometric, algebraic, and logical problems of grade difficulty that would allow for mathematical creativity, would be somewhat familiar, and would also allow a researcher to gain insight into the processes being used in solutions (Adey, 1999)..

From the study, he identifies three key types of mathematical cast of mind analytic among the students which includes harmony that displays both geometric and analytic characteristics, geometric that interprets abstract or solves mathematical relationships problem visually and logics that tends to think in verbal-logical terms (Casey 2011, p. 145). Importantly, it is imperative to note that giftedness in mathematics requires students to undergo pre-testing of new modules of work to judge whether they already know. The article adds that student should tackle more challenging and complex activities and the content of their learning materials should contain contents with higher abstraction level. This provides a difference between those who are gifted and the rest since gifted group will show a higher degree of oddity by attaining a higher mark under these peculiar circumstances.

Koshy, Ernest and Casey in their article “Mathematically gifted and talented learners: theory and practice” (2009) highlight a further dimension of mathematical ability, which is a potential or future-oriented in skills. They argue that individuals possess a capacity to master new mathematical facts and skills and also to solve non-routine and unsullied problems. Indeed mathematical aptitudes such as using mathematical notation, sustaining long chains of reasoning, abstracting general features from mathematical material, and employing mathematical reasoning are indications of giftedness in mathematics (Koshy, Ernest & Casey 2009, p. 222). Students who own qualities of mathematically giftedness and talents are characterized by more than a few traits which make them distinct from those with no components of mathematical gifts and talents (Chisnall, 2004).

According the article, students with the facilities to formalize mathematical materials and still be in a position to segregate them from other contents alongside with being capable of conceptualizing from concrete numerical forms, is said to be gifted and talented in mathematics (Chan, 2000). However, research studies points out that giftedness are a homogenous group that can at times be misidentified and which may results to undergoing inadequate or wrong curriculum provision that leads to misplaced or wrong grades (Koshy, Earnest & Casey 2011, 227). Even so, gifts and talents are diverse as level of abilities and giftedness are usually different from one individual to another.

Therefore a psychosometric test which measure level or degree of intelligence quotient should be used since they are the best and the simplest taxonomy means of measuring level of giftedness. Intelligent quotient tests are capable of profiling strengths and weaknesses of students therefore able to establish discrepancies that may exist between the sequential and the mental age (Freeman, 2000). Nonetheless, these attributes must be extremely excellent to provide the difference between an average, above average and the gifted and talented student. Moreover, it is imperative to note that students who are mathematically gifted and talented have a flexible mind that enables them to change or navigate from one mental operation to another in a sequential or iterative manner (Koshy, Ernest & Casey 2009, p. 219). This is a major distinguishing factor since good mathematicians are creative thinkers in view of the fact that mathematics demands creative minds.

Sheffield in her article “Serving the Needs of the mathematically promising” (1999) talks about the mathematically gifted students as those with talents and who are likely to have the ability to think in more curtailed structures and to shorten reasoning process (Freeman, 1997; Freeman, 1998). In relation to this, she argues that a student can be identified as gifted and talented in mathematics from evaluation and tests that relate to mental work which requires quick reasoning capacity since such test require one to operate in a curtailed environment. Furthermore, another criterion she establishes that can be used to determine giftedness is through giving students opportunities to demonstrate their high level abilities in several tests (Freeman & Joseppson, 2002).

Thereafter, these students can then be categorized according to their levels in order to enable them reach their potential in a full gear in an area where they have strength. According to Sheffield(1999, p. 316), gifts and talents include the exceptionally high level performance in a limited field or in a range of endeavors as well as those potential for excellence that are not recognizable (Bryman & Bell, 2003). In relation to this, gifts are assumed to be more straightforward way of measuring aspects of intelligent development such as high level of achievement and intelligent quotient (IQ). Consequently talents are taken to be the highest level of performance in measurable aspects like art which is usually discovered by experts in these fields. However, giftedness can be developed and nurtured since everyday intelligence can always be improved through practice as the more frequently individuals do something the better they become at it. Therefore, it is evident that intelligence quotient can be developed and increased through learning and training in any subject or field. In addition, this helps one to attain the highest level of performance in an area which one may be interpreted to have some level of giftedness.

#### Identification

Identifying students with gifts and talents is the initial and most important step in developing future specialists in science and technology. Recognizing them requires focusing on numerous peculiarities which make them dissimilar from those with no elements of mathematical gifts and talents. According to Sheffield in her article “Serving the Needs of the mathematically promising” (1999), a student who has the capacity to formalize mathematical materials and be in a position to detach them from other contents together with being capable of abstracting from tangible numerical figures is said to be gifted and talented in mathematics (Sheffield 1999, p.53; Adams & Wallace, 1991). Nonetheless, these traits must be carefully identified to provide the difference between an average, above average and the gifted and talented student.

Furthermore, according to Koshy, Ernest and Casey in their article “Mathematically gifted and talented learners: theory and practice” (2009), identifying the giftedness of students can be achieved through conducting mathematical pre-tests (Koshy, Ernest & Casey 2009, p. 225). A student who does not possess some attributes of mathematical giftedness and talents will not be in a capacity to abstract relevant mathematical materials students’ capability to generalize mathematics materials and be in a position to establish what is of chief important. As such, abstracting from the irrelevant is enough way of identifying that one is gifted and talented in mathematics.

In addition, Casey in the article “Teaching Mathematically Promising Children” (2011) alludes that a student who is mathematically gifted and talented will have a flexible mind whereby he or she is able to switch or navigate from one mental operation to another in a sequential or iterative manner (Casey 2011, p. 5). This is a major distinguishing factor since good mathematicians are creative thinkers in view of the fact that mathematics demands creative minds. Moreover, he believed that a gifted and talented student in mathematics is likely to have the ability of shortening reasoning process and be able to think in more curtailed structures. In relation to this, a student can be identified as gifted and talented in mathematics from evaluation and tests that relate to mental work which requires quick reasoning capacity since such test require one to operate in a curtailed environment (Corbetta, 2003; Cranton, 1994).

#### Provision

Casey (2011) is of the opinion that giftedness is a unique and specific sphere that requires provision of special education. He claims that provision must take into account aptitudes and special abilities that students possess. Provision is a key factor that should be prioritized since it cannot be taken literally that all gifted and talented children have a strong knowledge framework. Furthermore, provision of education in a classroom is imperative since it necessitates adaption of developing these talents and gifts in mathematics hence foster, nurture and promote them further.

According to Koshy, Ernest and Casey (2011), the purpose of identification process of gifted and talented students is to establish whether students require special educational provision, whether an additional or alternative to regular instruction is needed and to diagnose their special needs (Koshy, Ernest & Casey 2009, p. 222). Nonetheless, these processes must be inclusive and flexible to achieve the desired outcomes. Consequently, the main purpose of provision process for giftedness and talented students in mathematics is to enrich and accelerate their potential so that they may reach their highest possible capacity (Sebola & Penzhorn, 2010).

Regarding giftedness and talents among students of mathematics, it is crucial to have some aspects of stipulation for mathematically promising students (Eurydice, 2006). Sheffield (1999) holds that there are several ways which includes provision of learning materials that can be used to enrich students (Kitano & DiJiosia, 2002). In relation to this, House of Commons, UK (1997) holds that education which comprises learning of mathematics should be a good experience for enriching students as it acts as the starting point in talent development. Under normal circumstances, education provides an appropriated avenue for development of gifts and talents in mathematics since it acts as a nurturing platform (Elder, 2000).

#### Findings and views

It is imperative to note that the three sources, besides offering contradicting positions on the importance of creativity, strongly agree that identifying mathematical talents among students at an early stage will provide more than enough room to motivationally and attitudinally enhance their mathematical talents and gifts as well as provide apt cognitive challenges that will boost learning experiences among school children

From these sources, it is evident that the authors agree that mathematically gifted students should be identified and be provided with provisions such as special education (NCTM, 2000). It is necessary to examine the conditions under which the individual processes of mathematics learning takes place and to ensure that learning mathematics is embedded in a social context (Koshy, Ernest & Casey 2009, p. 217). Being that mathematics learning is an essential process of active individual construction and enculturation, the discourses and learning conditions within the group are central elements which must be carefully looked into (Brewerton & Millward, 2001).

Additionally, the sources points out that giftedness in mathematics necessitates that a student goes through a pre-test of new modules of work to determine their abilities. According to the findings, the sources should include challenging and complex activities set for students to tackle (Koshy & Jean, 2011). Their learning materials should also contain contents with higher abstraction level(NCTM, 1980). This, in my view, will provide a difference between those who are gifted and those who are not since gifted group will show a higher degree of oddity by attaining a higher mark under these peculiar circumstances. It is also clear from the sources that intelligence quotient can be developed and increased through learning and training in any subject or field. In addition, this helps one to attain the highest level of performance in an area which one may be interpreted to have some level of giftedness.

### The implication for schools

Mathematics learning in schools is an essential process of enculturation and active individual construction. Internationally, proficient students in mathematics will be those who not only possess a conceptual understanding, procedural fluency, and strategic competence, but who also have an adaptive reasoning and a productive disposition. Better mathematical students will mean a better future of specialists in science and technology sectors (Koshy, Ernest & Casey 2009, p. 222). This therefore means that a lot of emphasis should be put by educators to nurture the gifts and talents that mathematical students possess. Students who are mathematically gifted and talented will have a flexible mind, and ability to shortening their reasoning process and be able to think in more curtailed structures (Sheffield 1999, p. 277).

### A Summary of the chapter

To recap it all, identifying gifted and talented students is the initial step in developing upcoming specialists in science and technology. Identifying them requires focusing on several traits which make them distinct from those with no elements of mathematical gifts and talents. As aforementioned, the purpose of identification process of gifted and talented students is to establish whether a student requires special educational provision, whether an additional or alternative to regular instruction is needed and to diagnose special needs.

## Conclusions

To sum up, it has been observed in this paper that many students are mathematically gifted. As such, it has been indicated that there is need for identifying those with gifts and talents and organizing the school curriculum such that it is differentiated to meet their specific needs. Recognizing students with talents requires focusing on numerous peculiarities which make them dissimilar from those with no elements of mathematical gifts and talents. It is imperative to note from the findings that when these gifted students are not presented with learning experiences that are appropriate for their abilities, they lose motivation and in time can lose their excellence in the subject.

As aforementioned, the purpose of identification process of gifted and talented students is to establish whether they require special educational provision or additional instructions necessary to diagnose their special needs. Importantly, mathematically gifted individuals possess a capacity to master new mathematical facts and skills and also to solve non-routine and unsullied problems. As indicated in the findings, proper provisions will aid in improving their mathematical aptitudes such as using mathematical notation, sustaining long chains of reasoning, abstracting general features from mathematical material, and employing mathematical reasoning. This paper will draw its conclusion from research questions that were addressed, limitations of the study, personal learning and implications, after which it will give further research scope.

### How research questions are addressed

The study whose aim was to examine issues relating to the identification and provision for students identified by their teachers as gifted and talented in mathematics, based its research and findings on the following questions:

- What are the characteristics of mathematical giftedness portrayed in the literature?
- How are mathematically gifted and talented pupils recognized? What provision is made for the mathematically gifted? Is it appropriate?
- What are the features of an excellent teacher of mathematically gifted and talented pupils?
- What roles do parents play in the mathematical development of their children?

In addressing the above mentioned questions, the study adopted a secondary method of obtaining data, and from its findings drew the following conclusions under each as follows:

**What are the characteristics of mathematical giftedness portrayed in the literature?**

On the issue of mathematical giftedness and its characteristics, it was realized that the mathematically talented are extremely invaluable resources a society has. Societal assumptions that these children are perfect and do not need any attention may not be justified at all since it is selective and quite discriminatory. Inclusion emphasis that has been laid before in our schools has mainly been laying more emphasis on poor performing students than the gifted ones (Koshy & Robinson, 2006). In an attempt to understand mathematically gifted children, a lot of research has been done in the past (Stewart & Michael, 1993). It has been found out that mathematically gifted children exhibit different learning behaviors at different age groups (Landau, Weissler & Golod, 2001). Lack of challenging experiences in gifted children’s curriculum would automatically lead to loss of motivation and interest at school. Classroom tasks that are too easy for the mathematically gifted children will lead to boredom and thus lack of realization of their full potentials.

Although mathematical giftedness is quite diverse in definition, literature suggests that it refers to special mathematical abilities and qualitative mathematical thinking. Thus, mathematically gifted children are individuals with such abilities and thinking. Kruteski, a Russian psychologist studied various students for a long period of time in an attempt to understand mathematical giftedness and consequently came up with three types of mathematical giftedness. These included analytic, geometric and harmonic. Analytical mathematic giftedness involves mathematical verbal-logical abilities while geometric involves problem solving and abstract mathematical relationships and harmonic involves both analytical and geometrical (Coleman, 1995).

Adams and Wallace (1991) established that students who were academically gifted tended to be different from each other in their complex reasoning abilities, using their above-level testing on 150 students. The results obtained exhibited differences in various mathematical concepts and that these students had proficiencies going up as four grades higher than normal. Bergwal, Ziegla and Cartwright found out that there were two different categories of mathematically gifted students. These included ones who are able to solve challenging problems using qualitative methods and others who were able to complete their mathematical problems with less difficulty (Marchaim, 2001).

Furthermore, past studies have indicated that mathematically able learners are able constitute a class of deviance within the context of popular culture (Daloz, 1999). Corbetta (2003) insisted that gifted children tend to show behaviors that can be described as marching to their own drummers, implying that these children invent their own rules on problem solving. Jensen (1994) argued that gifts and talents include remarkably high level of performance in narrow fields. Gifts are ways of measuring intelligence and talents are simply the highest level of performance in physical aspects like sport, art and music. The mathematically gifted students have great abilities in formalizing mathematical materials. They are able put it in a position and isolate real mathematical materials from one that is not mathematical by extracting concrete numerical forms from such materials. They also exhibit ability to generalize mathematics material and are in a position to establish what is extremely important by recognizing the most relevant from the irrelevant. Their mental flexibilities have also been an attribute to their mathematical giftedness. They are able to switch from one mental operation to another without any difficulty. They also exhibit shortening of reasoning processes by thinking through more curtailed structures.

Some academicians believe that giftedness constitutes of homogeneity that can be misidentified and be given the wrong curriculum and thus lead to attainment of wrong grades. It is therefore imperative to understand giftedness in order to allocate the right curriculum and absolutely explore the potential of gifted children.

**How are mathematically gifted and talented pupils recognized? What is the role of teachers and parents?**

Some researchers have proposed models for identification and recognition of giftedness. Among the propositions of identification include psychometric tests which measure the level or degree of IQ and this has been described as one of the best ways of measuring levels of giftedness. IQ tests are able to make establishment of strengths and weaknesses of students. Although there are discrepancies in chronological and mental ages, the strengths and weaknesses established are accurate to a large extent. In relation to mathematical giftedness, Adey (1999) argues that the gifted should undergo pretesting to establish whether they already know and that the testing material should be of higher abstraction level in order to test their correct abilities. The differences obtained from the normal and gifted students were key determinants in finding out gifted children. Besides IQ testing, a lot has been done in determining the mathematically gifted students. For instance, Gardner (1983) sought to find out the distinctiveness of the mathematically gifted students. He found out that students may take part in curriculum activities by applying mathematical skills and predictive logic as he described in his theory of multiple intelligence. Quantitatively, these students presented information using their mathematical skill and a sustained long chain of reasoning.

Parental or peer nominations can be highly helpful in determination of the mathematically gifted since they are immediate identifiers. The ability of tackling mathematics is many times shown during the early stages of development and thus parental notice is usually inevitable before schooling years. In this regard, it is in order for teachers to make consultations with parents in regards to mathematical abilities of the mathematically gifted. Schools should involve parents in identification of mathematical giftedness in children by describing interests of children as soon as they commence school. However, there have been reports of biasness in parental reports or information due to ambition and thought that their children are mathematically gifted. Other well-educated parents underestimate abilities of their children rather than overestimating. Chan (2000)found out that approximately more than 61 percent of parents made accurate reports on their children’s’ abilities as regards mathematical giftedness (Newman & Benz,1998). The remaining percentages were not considered absolutely wrong since they showed advanced descriptions. Teacher nominations have also been confused for biasness since teacher nominations do not provide a clear line between giftedness, training and acquisition of skill (Cockcroft, 1982).

Cognitive checks and IQs are highly helpful in determination of an individual’s potential but they have limitations in one dimension. The verbal and analytical skills of the individual involved in the test are not examined in those tests. Standardized tests require that children should be involved in the tests in giving verbal and quantitative responses while ignoring the fact of non-verbal aspect and generality. Naglieri and Ford argue that those children would have shown better performance should the left aspect have been explored. Despite this fact, these tests have continuously been used in the indication of intelligence in children. Achievement tests that have been used previously in the United States and the UK have been majorly a gauge for entry into universities. They help teachers make decisions on the gifted since teachers’ initial judgment greatly determines student abilities through mathematical exhibitions.

As recent as the year 2007, Feng, Evans and Van-Tossel carried out a study in South Carolina in an attempt to make identification of the gifted students in areas of low income earning abilities. The methods involved included dynamic assessment which involves a test-intervene-retest process. Non-verbal tests involved focus on spatial and logical organizations of individuals through the use of geometrical designs. Utilization of shapes and designs and not the end results of the tests were of great influence in determination of giftedness of individuals.

As suggested by Gardner (1983), this assessment overcomes problems connected to tests and thus is very helpful in assisting a researcher to find accurately, giftedness in children. It also involves classroom observations and characteristic checklist. The author continues to suggest that portfolios can provide sufficient evidence of attainment of the child to the teacher for addition in the gifted list. Through the three dimensions of testing which include abilities, learning styles and interests, it will be possible to provide substantial information for the teacher to make identification of gifted children in classroom.

Continual and systematic observation by teachers on students has been identified as one of the most effective ways of giftedness identification in classroom. This is however, not very effective since mathematical abilities are sometimes not observable in classroom or in normal behavior. Some characteristics help teachers in understanding some types of behavior in a given classroom environment. By understanding this, teachers are able to determine mathematical giftedness in students. This is because teachers feel confident to have recognized such students by use of different sets of characteristics which can be categorized as a checklist.

Although recognition of mathematical giftedness is clear and precise, barriers to identification of the same exist. To begin with, some students may have learning disorders and thus fail to show their full potential. This will greatly hide the potential of a child or learner and is thus considered a barrier. Secondly, enthusiasm may influence the result of recognition since capacities and preferences are totally different. According to Gardner (1983) on the theory of multiple intelligence students may be gifted in one way and not the other. For instance, he argues that giftedness in mathematics does not necessarily imply bodily-kinesthetic giftedness (Johnson & Duberley,2000). So in an attempt to make an assessment of the child in regard to the immediate environment may not be accurate (Blakemore, 1999). For instance, a child with logical-mathematical intelligence might be influenced by an environment to be enthusiastically involved in sport and thus achieve the wrong results. This has been identified as the greatest barrier in identification of the mathematically gifted students since environmental forces have failed to recognize the giftedness in mathematics that the child has.

**What provision is made for the mathematically gifted? Is it appropriate?**

In order to make provisions for gifted children, there is need for identification of these children for curriculum designers to take their needs into consideration. These provisions must be inclusive to cater for positive outcomes. They should consider some aspects of the mathematically gifted children since they necessitate the adaption of developing talents and gifts in mathematics, hence proving helpful in nurturing them (Denzin & Lincoln, 2003; Daloz, 1999).

Teaching and learning mathematics is diverse but the conventional view that mathematics is exploratory and dynamic still holds. Examination of mathematically gifted students’ capacities for learning, reasoning and natural thinking is very wide and might require future research. The conditions for the learning environment also prove vital for the facilitation of learning process since the process involves individual construction and enculturation (Bryman, 2001). Learning being a cultural product which takes place in a social environment, the discourses and conditions for learning become elements of the learning process. Within any theoretical tradition, the fostering conditions must be helpful and thus learning conditions should be made favorable (Miller & Brewer, 2003).

Sheffield (1999) holds that there are several ways on how this can be achieved such as stipulation of learning resources that can be used to augment students. Similarly, House of Commons (1997) holds that schooling which comprises learning of arithmetic should be a good experience for enriching students since it acts as the starting point in talent development (Potter, 2002). Under normal situations, education offers an appropriate avenue for growth of gifts and talents in mathematics since it acts as a cultivation platform. In order to make the right and appropriate provisions for mathematically gifted learners, recognition of learning arithmetic within a classroom environment should be made.

**What are the features of an excellent teacher of mathematically gifted and talented pupils?**

Teacher identification of mathematically gifted students in the school environment is an absolute and most accurate source of identification of these learners. Thus it is the teacher who proves to be the biggest asset in identification of giftedness. Being in this position, there is high need for containment of special skills and abilities to make such recognitions. It is therefore required that the best teacher for the mathematically gifted be able to understand that there are mathematically gifted students within the school environment (Shi, 2003).

To begin with, an appropriate teacher for the mathematically gifted students should be highly observant in order to make gifted recognitions (Bernard, 2006*)*. Through these observations, teachers realize these children and tapping of their potential commences. The teacher should be keen in observing since there are bound to be illusions in observation that will create a similar situation between the mathematically gifted children and the normal children (NAGC, 2005). Secondly, the teacher should use the Van Tassol approach of the test-intervene-retest method of recognizing mathematical giftedness to find out if students are mathematically gifted or not. In mathematically gifted students, the intervention in the testing process will lead to a fresh thought process but in normal children, the intervention will obscure the intended solutions and make them exhibit cramming tendencies by trying to recall what they initially intended to say (Koshy, Ernest & Casey 2009).

Thirdly, the best mathematics teacher for the mathematically gifted should always have and update a checklist for the mathematically gifted. This will enable the teacher to use multi-dimensional approaches in identifying the gifted and thus reduce the error in accuracy. It is vital for accurate identification since students might be given the wrong types of materials as regards their mental abilities (Maxfield & Babbie, 1995). The teacher thus should always seek to keep the mathematically gifted involved in order to sustain their motivation and keep them challenged for a continual schooling.

**What roles do parents play in the mathematical development of their child?**

Parental role in the development of the mathematically gifted cannot be ignored since they present the first social circle for these children (May, 2001). Initial identification from parents is very vital since that recognition will help in determination of the type of materials and learning environment the child needs (Guenther, 1995). For a mathematically gifted child, there is need to provide with high abstraction materials since their level of thinking and abilities are way above their normal counterparts. Through this, the child will need to be presented with challenges by the immediate parent or guardian for further development of the child as regards mathematical giftedness (Blakemore, 1999).

Besides, parental encouragement and appreciation of children with mathematical giftedness plays a crucial role in the development of the gifts and talents of such children. Appreciation of effort and excellent results is usually encouraging to any individual and can thus be considered as a motivational factor for the mathematically gifted in excelling and standard maintenance (Landau, 1990). Thus by sustaining continual appreciations and encouragement from parents, it will greatly influence achievement of the mathematically gifted students.

Besides, reporting to teachers from parents for further development of the mathematically gifted is a crucial step in the development of these children. Past research has indicated that if parental notice is not met with teachers, then recognition of mathematically gifted is not made early for molding and further development (Weinhardt, Stean & Jochen, 2008). This step plays vital role since when teachers become aware of the mathematically gifted students within school environment, correct materials are identified for the needs of such students. When such identifications are not made, their needs will also not be met.

### Limitations of the study

The limitation that this study had is that it only used secondary methodology to collect data. As such, it was not able to gather enough evidence as it would have had it also used primary methods of collecting data. Besides, since its focus was only on mathematics and not sciences in general, researchers were limited and could not explore other key disciplines that students are talented in.

### Personal learning from the research

From the research, I have come to learn recognizing mathematical talents amongst students at an early stage will offer more than sufficient room to motivationally and attitudinally improve their mathematical talents and gifts as well as give apt cognitive confronts that will increase learning familiarity amongst school children.

Additionally, I have learnt that there is need to monitor and improve the abilities of the talented students. As such, it calls for teaching curriculum to be differentiated to meet their precise requirements. Ideally, when these students with both mathematical gifts and talents are not met with learning experiences that are suitable for their capacities, they will drop motivation and attention required for them to do extremely well. It is essential to scrutinize the circumstances under which the entity processes of mathematics education takes place and to make certain that learning mathematics is entrenched in a societal context (Koshy, Ernest & Casey, 2009, p. 217). Being that mathematics learning is an essential process of active individual construction and enculturation, the discourses and learning conditions within the group are central elements which must be carefully looked into.

In addition, the basis points out that giftedness in arithmetic abilities necessitates that a learner goes through a pre-test of new components of work to establish their abilities. According to the results, the sources should include testing and multifaceted activities set for learners to undertake. Their erudition materials should also include contents with superior concept level (Alvesson & Deetz, 2000). This, using my view, will offer a dissimilarity among those who are gifted and those who are not since gifted group will show a higher degree of peculiarity by attaining a higher mark under these strange conditions. It is also clear from the sources that intelligence quotient can be developed and increased through learning and training in any subject or field. In addition, this helps one to attain the highest level of performance in an area which one may be interpreted to have some level of giftedness.

#### The implication for schools

Mathematics education in schools is a necessary process of enculturation and lively individual creation. Globally, expert students in mathematics will be those who not only hold a theoretical understanding, technical fluency, and tactical competence, but who also have an adaptive logic and a productive character. Enhanced mathematical students will mean a better expectation of specialists in science and technology sectors (Koshy, Ernest & Casey, 2009). This consequently means that a lot of stress should be put by instructors to take care of the gifts and talents that arithmetical students possess. Students who are accurately gifted and talented will have a supple mind and ability to cut their reasoning process and be able to reflect in more condensed structures (Sheffield, 1999).

### Further research scope

The scope of this research was giftedness and talents among primary and early school children. It is imperative therefore, that future research study be conducted in a more comprehensive way so as to include the challenges and setbacks facing teachers and learners of mathematics. It was noted that the method of data collection used in the research study was probably not sufficient especially when considering the wide array of factors that may the research questions demanded. Therefore, there is need that further research on this subject area be conducted in order to conclusively determine all the underlying and latent factors that may have not been brought into the surface.

Furthermore, future studies should be narrowed sciences in general as this will assist in coming up with finding that are more elaborate. This is because the current research did not allow tackling the topic appropriately due to the limited area of the subject.

### Chapter summary

In summing up, this chapter has made conclusions on aspects that have been tackled in previous chapters. It has addressed research questions, implications of giftedness and future research scope. Most importantly, it has brought to a close the subject on giftedness in mathematics by asserting that recognizing mathematical talents amongst students at early stage offers more than sufficient room to motivationally and attitudinally improve their gifts as well as give apt cognitive confronts that will increase learning familiarity among school children.

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