Characteristics of the mathematically gifted and talented students were described from the multiple sources of data. Some of the parents, through their experiences with other children in the family, or their own knowledge and experience, were aware of some of the early signs. These early signs included aspects such as ‘playing with numbers’, a sense of symmetry in pattern making, and viewing the ‘world through a mathematical lens’ (see, for example, Diezmann & Watters, 2000a; Gardner, 1983; House, 1987; Johnson, 1983; Straker, 1983). Many of the parents gave detailed accounts of the mathematical behaviours they observed in their children’s play and talk. They described situations in both the home and early childhood environments. Descriptions of these atypical instances of mathematical play were passed on to the parents by the early childhood teachers. None of the parents were aware of information about their children’s special interests and abilities in mathematics being passed on, or taken into account, as part of the transfer process to primary school.
The students showed an awareness of their mathematical giftedness when they were able to compare themselves with peers at school. The teachers articulated some characteristics, but no teacher described in depth the range of characteristics associated with the mathematically gifted and talented. Collectively, the teachers mentioned most indicators but their contributions were based on anecdotal evidence gained from teaching experience. None of the school policy documents supported the teachers by providing indicators of mathematical giftedness.
The notion of an appreciation for different types of mathematical giftedness was evident from student, parent, and teacher data. The parents were able to differentiate between early signs of spatial reasoning and arithmetic reasoning. The students, on the other hand, associated mathematical abilities with numeric reasoning and computational ability, although later in their schooling, especially when trialling for mathematics competition teams, they realized the importance of problem solving skills. These different types of mathematical giftedness were most clearly defined in the literature by Krutetski (1976). The focus from the students and teachers in this study was not on these differences. However, these are differences that had been
two of the teachers. Spatial giftedness has often been overlooked (Webb, et al., 2007), so it is important that this view of mathematical giftedness is given attention in the identification process.
The students reported a variety of general interests and leisure activities; computer games were listed by the majority of the boys. Sport also featured in most of the students’ lives and five children listed reading as a leisure activity. A few of their leisure activities, such as origami, codes, and modelling, could be said to have links with mathematical thinking. This information about their hobbies and interests was sought from the students as young children’s interests and leisure activities have been found to be a reliable predictor of future high achievement in that area (Freeman, 2000).
Motivation is a key component in the gifted literature (for example, Renzulli’s concept of giftedness, 1978) and an element included in the operational definition for this study, and so it was decided to examine these students’ levels of motivation in mathematics using a reputable tool. Only two of the students rated on Rogers’ (2002) Interest and Attitude Inventory as ‘highly motivated’ and five students as ‘motivated’. The rubric posed a few challenges, but encouraged the students to think about a variety of factors related to their mathematics learning. The results showed that students’ levels of motivation differed between the two year groups. Many of the students attributed success to hard work, and despite planning on continuing their studies in mathematics, they did not see their future in mathematics. The students’ intrinsic motivation was evident in many of their responses and a case-by-case examination of separate items was interesting. The tool provided information on individual students that would be of interest to a parent or teacher and could contribute to the identification process.
In practice, identification for mathematics relied primarily on test and competition results. This practice resulted in one student in the study not being identified as gifted and talented post school transfer. This particular student did not perform well on a test used to identify students for the withdrawal CWSA mathematics class at intermediate school. Eventually, the result from a competition was used as evidence that he could be moved to this class. This school had not used multiple identification methods. If they had used parent, student, and teacher nomination, and records from
the previous school, then it is highly likely that the student would have been placed in the appropriate class in the first instance. There is also the potential for the ‘geometric’ student or the ‘creative problem solver’ to be excluded from selection if a test focuses on a narrow range of knowledge and skills. The hard working student, who performs well in a test (for the targeted age level and not allowing for the ‘ceiling effect’), may be included in the selection process and may not necessarily be gifted and talented in mathematics, as was the case for two students in this study. The other issue is that test results (such as the PAT13 used for students in this study) may not differentiate among the most able, with the consequence that students may not receive an appropriate mathematics education.
There was no mention of formal identification processes early in a student’s schooling, despite those early signs of giftedness recognized by the parents. This means that potentially a student could miss out on identification. There were students in this study that the researcher believes were misidentified. The evidence showed that multiple methods were not used in the identification process. The students and parents admitted that they did not think that they were gifted and talented in mathematics. The outcome has not been problematic; the students have worked hard in mathematics and experienced mathematics programmes that may have been possibly more differentiated than they might otherwise have experienced. Without knowledge about the characteristics of gifted behaviour or potential, then it is possible that the ‘teacher pleaser’ may be misidentified as gifted and talented in mathematics. Likewise, the gifted underachiever may go unnoticed. Students transferring from class to class, within the same school, will be exposed to the same identification processes which may be reliant on limited methods such as teacher nomination and test results.
The process of identification can be problematic. A school policy may document what multiple sources can be used, but it is evident that what is put in to practice appears to differ given the cases cited here. Multiple means more than one source; you would expect a range of sources that includes parent and student nomination. Given that the school documents examined in this study did not describe the
characteristics of the mathematically gifted, then as Renzulli (2004) questioned, how can you identify a gifted population if you do not know what you are looking for? There was no documented information pertaining specifically to the identification of mathematically gifted students apart from nominated tests. There was also no documented identification processes as part of the transfer process from early childhood education to primary school.
These two topics, characteristics and identification, are interconnected especially if one takes the view that an identification procedure is necessary. The schooling system in New Zealand states that there is a legal obligation to identify gifted and talented students. It is the researcher’s view that this would be untenable without awareness of the associated characteristics. Identification provides the “means to an end and not an end in itself” (McAlpine, 2004b, p. 126) and so once students have been identified, the focus should be on the quality of provisions. The nature of the provisions for students may influence whether and to what degree a student’s mathematical potential develops.
The following chapter explores the qualities of the teacher of the mathematically gifted and the varying components of mathematics programmes as experienced by the participants. This is supported by students’, teachers’, and parents’ views about suitable provisions and related issues.
CHAPTER SEVEN:
THE TEACHER OF THE MATHEMATICALLY
GIFTED AND THE MATHEMATICS PROGRAMME
7.1 Introduction
This chapter addresses the question concerning the qualities of an effective teacher of mathematically gifted and talented students. This is followed by multiple perspectives on the nature of educational provisions for the students in this study. The features, strengths, weaknesses, and challenges of the various provisions are presented. The use of out-of-school provision is briefly described as two students attended a one-day-a-week programme. All of these aspects are then summarized and critiqued in relation to the literature.