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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.

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.

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.

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.

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 Caseys (2011) ideas that build on Sternbergs (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

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