This thesis reports data gathered through case studies of ten of the students who took part in a survey of secondary school students who had been invited to camps at which the New Zealand teams were chosen for the International Mathematical Olympiad (IMO) in the first five years of this country�s participation in the IMO. The case studies data gave individuals� narratives that were captivating yet complex, unique yet universal, clear yet not easily described all of apiece. I read widely in response to the information they offered and found that reflection and a narrative style of presentation assisted the grasping of nuances and implications of the students� narratives. Few of the parents of the students were particularly competent in mathematics or able to account for their child�s curiosity, concentration and skills in pursuing a fascination with number. In most of the families, all members were encouraged to follow their own inclinations and interests, to respect the maintenance of a balance of cultural and physical activities, to regard books and play as normal life supports, and to believe that discovery, enchantment and pleasure were both goals and accomplishments of everyday life. Most of the students experienced less encouragement at school than they might have expected, and unpleasant experiences could be linked with a teacher�s apparent lack of appreciation of a student�s mathematical ability. Both the case studies and the initial survey suggested that most teachers, at any level of formal education, were doing all they were capable of doing in mathematics, and that the students responded to opportunities to self-select subjects and topics that interested them and to the help and company offered by mentors and peers who had flair and competence in appropriate subject areas. Few of the case studies students were motivated by strategies dependent on a high level of competition or a 'sorting' of that offered in formal education (through attitudes and practical organisation such as timetabling) into either Arts or Science subjects. Most were attracted to the study of languages and/or philosophy and some to that of computer science. Most showed interest and some prowess in individual cultural and physical activities requiring perseverance. Largely, they were motivated by finding fresh or novel ways of integrating diverse knowledge, and by associating with peers. They enjoyed and valued self-awareness, intellectual independence, chances to empathise with ideas and people, and tasks that were in harmony with the dictates of their own volition. Evidence of differences among the case studies students - even though they had all been identified as very able in mathematics - led me to Vadim Krutetskii�s theories of the components of mathematical ability and their functioning and thus to new views, first, of the interplay between aptitude and languages of perception, inner comprehension and outer expression and, second, of the relationships between giftedness and other attributes of human abilities and endeavours. These appreciations suggested that the models of education and support commonly exhibited in the case studies students� families and in the environments of their extra-school activities had been more encouraging of their gifts, talents and personal growth than those often exhibited in the schools they attended.

This thesis reports data gathered through case studies of ten of the students who took part in a survey of secondary school students who had been invited to camps at which the New Zealand teams were chosen for the International Mathematical Olympiad (IMO) in the first five years of this country�s participation in the IMO. The case studies data gave individuals� narratives that were captivating yet complex, unique yet universal, clear yet not easily described all of apiece. I read widely in response to the information they offered and found that reflection and a narrative style of presentation assisted the grasping of nuances and implications of the students� narratives. Few of the parents of the students were particularly competent in mathematics or able to account for their child�s curiosity, concentration and skills in pursuing a fascination with number. In most of the families, all members were encouraged to follow their own inclinations and interests, to respect the maintenance of a balance of cultural and physical activities, to regard books and play as normal life supports, and to believe that discovery, enchantment and pleasure were both goals and accomplishments of everyday life. Most of the students experienced less encouragement at school than they might have expected, and unpleasant experiences could be linked with a teacher�s apparent lack of appreciation of a student�s mathematical ability. Both the case studies and the initial survey suggested that most teachers, at any level of formal education, were doing all they were capable of doing in mathematics, and that the students responded to opportunities to self-select subjects and topics that interested them and to the help and company offered by mentors and peers who had flair and competence in appropriate subject areas. Few of the case studies students were motivated by strategies dependent on a high level of competition or a 'sorting' of that offered in formal education (through attitudes and practical organisation such as timetabling) into either Arts or Science subjects. Most were attracted to the study of languages and/or philosophy and some to that of computer science. Most showed interest and some prowess in individual cultural and physical activities requiring perseverance. Largely, they were motivated by finding fresh or novel ways of integrating diverse knowledge, and by associating with peers. They enjoyed and valued self-awareness, intellectual independence, chances to empathise with ideas and people, and tasks that were in harmony with the dictates of their own volition. Evidence of differences among the case studies students - even though they had all been identified as very able in mathematics - led me to Vadim Krutetskii�s theories of the components of mathematical ability and their functioning and thus to new views, first, of the interplay between aptitude and languages of perception, inner comprehension and outer expression and, second, of the relationships between giftedness and other attributes of human abilities and endeavours. These appreciations suggested that the models of education and support commonly exhibited in the case studies students� families and in the environments of their extra-school activities had been more encouraging of their gifts, talents and personal growth than those often exhibited in the schools they attended.