Abstract
There is a paucity of research investigating aspects of provision for gifted children within primary schools, particularly in mathematics. This study aimed to address this topic by illuminating issues arising from classroom practice and the experience of both teachers and children. Based on in-depth case studies in four primary schools in which different methods of provision were being implemented, this study found that although there is an interest in addressing the needs of mathematically gifted children and schools have taken measures in pragmatic terms to cater to them, the needs of these children are not always addressed in classrooms. Evidence from classroom practice showed that the effectiveness of any method depends on teachers’ expertise and self-confidence, the level of focused attention given to gifted children, the size of the class, and the nature of the work set. Additionally, this study identified issues relating to organizational strategies of classroom provision which need consideration and further research.
During the past three decades, there has been a growing interest in the recognition and development of giftedness in different academic domains within schools in England and Wales. In mathematics, in particular, this interest started when the Cockcroft (1982) report refuted common views that considered children with higher abilities in mathematics as students who do not need attention because they can cope with their studies independently, arguing,
The statement that able children can take care of themselves is misleading, it may be true that such children can take care of themselves better than the less able, but this does not mean that they should be entirely responsible for their own programming, they need guidance, encouragement and the right kind of opportunities and challenges to fulfil their promise. (para. 332)
Experts such as Straker (1983), who later directed the National Numeracy Strategy in the United Kingdom, added their voices to the argument, contending that “Gifted pupils have a great deal to contribute to the future well-being of the society, provided that their talents are developed to the full extent during their formal education” (p. 7).
Radical and fast changes relating to the education of the gifted, however, started in 1999 when the “Gifted and Talented” policy was launched with the Excellence in Cities (Department for Education and Employment, 1999) initiative. This initiative required schools within a significant number of inner-city Local Education Authorities to identify 5% to 10% of their pupils as “gifted” and provide them with a distinct teaching and learning program. Although this was initially launched to provide help for gifted disadvantaged pupils in the most deprived cities, towns, and rural areas, it has, since then, raised awareness of the need for the education of gifted and talented children within all mainstream schools and given impetus to this area.
The gifted and talented policy now encourages all schools to identify their gifted and talented pupils in any domain (e.g., academic, sports, arts); maintain their own register, with no limit on the percentage of the gifted (Department for Children, Schools and Families [DCSF], 2008a); and organize provision for them mainly within classrooms through grouping and enrichment, and outside the classroom by engaging the families and communities (DCSF, 2008b). Further guidance and teaching materials are provided for practitioners through an electronic version of the National Curriculum (www.education.gov.uk/schools/teachingandlearning/curriculum). The question is whether and how the needs of mathematically gifted children are actually addressed in real classrooms.
Theoretical Perspectives
The literature relating to giftedness and talent suggests that mathematically gifted children have a special ability to effectively use mathematical facts and perform mathematical tasks which is a natural gift or acquired talent, but which is further developed through suitable experiences and opportunities. For instance, according to Gardner’s (1983) theory of Multiple Intelligences, specific types of intelligences, such as logical–mathematical intelligence, are connected with distinct centers in the brain and nervous system, and are correlated with specific types of giftedness. However, the extent to which each type of intelligence will be developed for different individuals depends on the environment in which the individuals live, and the experiences and opportunities that they will have (Gardner, 1999). Similar views about the existence of a natural ability and its development into a specific talent have been presented by Gagne’s (1985) Differentiated Model of Giftedness and Talent and by the Munich Model of Giftedness and Talent (Heller, 1990, 1991; Perleth & Heller, 1994). Also, Renzulli’s (1978, 1986) Three Ring Conception of Giftedness and Sternberg’s (1985) Triarchic Theory of Human Intelligence highlight both heredity and experiences as factors for the existence of a particular ability such as mathematical ability.
A special program for educational provision from early stages at school can provide suitable experiences for the development of mathematical giftedness, if it is specifically designed for this reason; if it is a “subject-specific” provision, according to Koshy (2001), or a “content-specific” provision, according to VanTassel-Baska (1992). A subject- specific provision program for gifted pupils must ensure a differentiated curriculum for them, and must involve differentiated instructions, ongoing assessment, and flexible grouping (Tomlinson, 1995; VanTassel-Baska, 2007). A method to differentiate the curriculum may be the “curriculum compacting” technique, developed by Renzulli and his colleagues (Renzulli, 1994; Reis, Burns, & Renzulli, 1992). Alternatively, Gagne (2007) suggests that “enrichment” should be implemented in schools in combination with “acceleration” and “full-time grouping.”
Educational researchers with experience in studying methods of teaching mathematics to high-ability children (e.g., Casey, 1999; Koshy, 2001; Koshy, Ernest, & Casey, 2009; Sheffield, 1999, 2003) mainly emphasize the role of teaching rather than the organizational structures for provision (e.g., acceleration, enrichment, and differentiation). They argue that the ideas of grouping by ability or doing different and more difficult work by themselves cannot meet the needs of children with higher abilities in mathematics. They agree with other researchers from the wider field of mathematics that teachers’ instruction is crucial for the effective teaching of mathematics to all students (Ball & Bass, 2003; Ball, Lubienski, & Mewborn, 2001; Ernest, 1998, 2000; Hill, Rowan, & Ball, 2005; Hill, Schilling, & Ball, 2004; Schoenfeld, 1992).
An effective mathematics instruction for gifted children should focus on learning mathematics at higher cognitive levels, on generating positive beliefs, attitudes, and motivation, and on extension. Teachers must be prepared to engage their students in higher level cognitive activities (Casey, 1999, 2002; Ernest, 1998, 2000; Koshy, 2001; Schoenfeld, 1992; Sheffield, 1999, 2003). These activities must offer opportunities for challenging pupils’ higher order thinking skills, such as reasoning, hypothesizing, communicating, decision making, refining ideas, creating new and original thoughts, problem solving, and metacognition (Ernest, 1998; Koshy, 2001; Schoenfeld, 1992). The latter involves skills for self-assessment and self-regulation (Koshy, 2001) needed for planning, monitoring, and evaluating the learning activity (Schoenfeld, 1992). To help teachers plan and teach mathematics at higher cognitive levels within either the regular classroom or special groups for gifted students (e.g., pullout groups or top sets), some models and frameworks of provision for mathematically gifted children have been developed. Sheffield’s (1999) model for “adding depth and complexity” in mathematics, Casey’s (1999) Key Concepts Model, and Koshy’s (2001) framework for teaching mathematics to able children based on Bloom’s (1956) Taxonomy of Educational Objectives, are some examples. These frameworks are not based on the existence of a structured grouping program or on a large number of difficult puzzles and problems but on teachers’ instruction and questioning, which must challenge pupils’ higher order thinking skills. Sheffield’s model, for example, suggests that pupils should be encouraged to find the answer in as many ways as possible, posing related questions, investigating possible patterns, making hypotheses, evaluating their observations, and discussing their findings. Sheffield contends that there are examples of pupils who were taught to explore problem patterns and connections, posing new problems and creating their own solutions, starting to approach mathematics differently than others. When they used to work by following only the one way that the teacher or the textbook proposed, the most able pupils started feeling frustrated by the “one right method.”
Casey’s (1999) model proposes a teaching instruction, which emphasizes not only on learning “algorithms” and “facts” with “fluency” but also on how to “conjecture,” look for “proofs,” and attempt “generalizations” and applications in similar situations (“isomorphism”), whereas Koshy’s (2001) framework suggests the use of Bloom’s taxonomy for teachers to plan mathematics lessons at higher cognitive levels and ask higher order questions with emphasis on “analysis,” “synthesis” and “evaluation.” Beyond these frameworks, developed specifically for actualizing mathematical promise, in mathematics literature, there are further exemplars of high-quality mathematics instruction that can help primary school teachers engage pupils in higher cognitive activities (Ball, 1990, 1991; Ernest, 2000; Schoenfeld, 1992).
In the research literature on teaching mathematics, it is generally agreed that subject-specific knowledge is an important factor of instructional quality that affects both learning gains and motivational development of pupils (Ball & Bass, 2003; Ball et al., 2001; Hill et al., 2004; Koshy & Casey, 2005; Koshy et al., 2009; Sheffield, 1999). Ball and colleagues named this knowledge mathematical knowledge for teaching and developed a set of measurement instruments for its assessment in elementary (primary) school teachers’ teaching. In the case of teaching gifted mathematicians, Koshy and colleagues, as well as Sheffield, contend that teachers must be highly trained in both higher level mathematics and in recognizing mathematically gifted children.
Mathematically gifted children do not usually have a problem to gain high scores in ordinary schools. What these children need is extension, so that they reach to their fullest potential. For this, Koshy et al. (2009) suggest systematic monitoring of their learning, which should take place within their zone of proximal development (ZPD; Vygotsky, 1978), and emphasis on the development of positive beliefs, attitudes, and motivation, as this is illustrated in Ernest’s (1985) “success cycle” model for learning mathematics (Figure 1).
The success cycle
Vygotsky’s (1978) ZPD is a learning zone between the real level of development and the potential level of development, where the child needs the help of a teacher, peer, or parent in order to accomplish a difficult task that appears to be beyond his or her abilities. Vygotsky suggested a “scaffolding” support approach for teaching problem solving within the ZPD. According to this approach, the teacher helps children accomplish their tasks, but with a view to gradually withdrawing the control and support while the children increase their mastery of the task. In the case of teaching mathematically gifted children, therefore, the teacher needs to plan and provide challenging tasks and instructions within their ZPD in order to extend them.
Ernest’s (1985) “success cycle” (Figure 1) represents a cyclic process (with no real beginning) to successful learning through the positive affect. This is also applied in the case of gifted mathematicians. For instance, pupils who have positive attitudes and beliefs about mathematics, high self-confidence, and mathematical self-efficacy will enjoy challenging tasks and be motivated, which leads to increased effort, persistence, and choice of more demanding tasks. Increased effort and persistence will give rise to continued success at mathematical tasks and overall achievement in mathematics, and this will further enhance positive attitudes and so on, completing the success cycle.
In contrast with the large number of theories and recommendations of good practice for nurturing mathematical ability there is relatively little empirical support from real school-life. An action research study titled “Mathematics Enrichment Project” (Koshy & Casey 2005), which was conducted by the Brunel Able Children’s Education Centre on the ways to actualize mathematical promise within 11 Local Education Authorities in Inner London primary schools, revealed that many teachers felt uncomfortable teaching able mathematicians and were not aware of available supporting resources. The study further showed that the key point with regard to provision is to raise teachers’ awareness of how the needs of able mathematicians can be met. Personal development of the teachers was also found to be a basic factor in raising their self-confidence in teaching mathematics to very able pupils.
Although the most recent reports from the school practice on gifted education (Office for Standards in Education [Ofsted], 2009) and mathematics (Smith, 2004; Williams, 2008)—commissioned by the government—did not focus on the education of gifted mathematicians, they drew attention to the lack of teacher expertise and, also, of providing higher level mathematics lessons that could excite pupils’ interest. Furthermore, Williams and Ofsted reports highlight that gifted mathematicians are not sufficiently stretched in primary schools, even when in-class provision is implemented, recommending, among other measures, better, practical, and ongoing training for the teachers.
The Research Project
This research project aimed to investigate whether and how the needs of mathematically gifted children were addressed within mainstream primary schools in England, along with the perceptions and attitudes of both teachers and pupils identified as more able or gifted mathematicians. The research questions were the following:
What strategies are teachers using, if any, regarding the education of gifted and talented children in general and specifically in mathematics?
What are the teachers’ perceptions of and attitudes toward mathematically gifted children, their education, and the methods used by their schools?
How are the needs of mathematically gifted children met within classrooms in everyday practice?
What is the impact of the schools’ strategies on gifted children’s achievement and attitudes?
The research was carried out in two stages: (a) the preliminary phase, which was conducted in the last term of the 2007-2008 school year and (b) the main study, which was conducted in the 2008-2009 school year.
The Preliminary Phase
The preliminary phase involved a questionnaire survey. A total of 224 questionnaires were distributed to all mainstream primary schools within five Local Education Authorities in Greater London. One questionnaire per school was sent. It was addressed to the head teacher asking him or her to make sure that a mathematics coordinator or a classroom teacher involved in teaching mathematics to gifted children would complete it. The aim of using the questionnaire was to help the organization of the main study, which would involve an in-depth look at primary classrooms to find out how teachers address the needs of mathematically gifted children in everyday practice. Teachers from 44 schools responded to the questionnaire. The majority of these teachers were mathematics coordinators (31), most of them (19) with an extra role in provision for gifted children. The analysis of the teachers’ responses gave first insight into what methods of identification and provision the schools use, as well as teachers’ attitudes toward teaching able mathematicians. It also helped in choosing the case study schools for the next stage, the main study.
The Main Study
The main study involved four case studies of classroom provision for children who are able or gifted in mathematics. The reasons for choosing a case study methodology were that it is described as the most appropriate method to explore in-depth classroom activities within a limited time scale (Bell, 2005; Yin, 2003); it can ideally combine different methods of data collection (Cohen, Manion, & Morrison, 2007) so that more information and different resources can be integrated, in this way enhancing the credibility and validity of the research; and it can present to the readers a more accurate and clear picture of what was studied (Cohen et al., 2007; Hitchcock & Hughes, 1995).
The Sample
After analysis of the questionnaire responses, four different schools were chosen as the venues for conducting the case studies (see Table 1). The criteria for selection were that the schools had to implement different methods of provision for their gifted mathematicians, including within-classroom ability grouping, setting (“regrouping” in the USA), pullout grouping, and mentoring—methods that were found being implemented by schools reviewed in the first stage—and be situated in different Local Education Authorities representing a range of socioeconomic backgrounds. The study of different methods would help understand what teachers were doing to meet the needs of children who are gifted in mathematics, illuminating issues that would be of benefit to both teachers and policy makers.
Description of Case Study Schools
Note. G&T = Gifted and talented. Data on this table are based on teachers’ responses to the questionnaire in the first phase of the research.
Within each school, one teacher with her group of children identified as able or gifted mathematicians was chosen. The choice of teachers was based on suggestions from the mathematics coordinators of each school. They were teachers who had undertaken responsibilities in teaching mathematics to able or gifted children. In two cases, these teachers were the mathematics coordinators themselves. Although the intention was to have both female and male teachers as participants, the sample consisted of female teachers only. The selected children were nominated by the case study teachers. The four teachers with their groups of selected children were individual cases; their profiles are presented in Tables 2 and 3, respectively.
Description of Participant Teachers and Case Study Classes
Note. G&T = gifted and talented; LEA = Local Education Authority.
Children’s Profile
Note. 3B, 3A, and so on, are the attainment levels that pupils may be awarded. In primary and secondary education, the levels range from 1 to 7, with 1 being the lowest. Each level is subdivided into three subgroups (e.g., 4a, 4b, and 4c, with 4c indicating the lower place within Level 4). In primary schools (age 5-11 years), according to the National Curriculum, the majority of pupils are expected to work between Levels 1 to 3 in Key Stage 1 (age 5-7 years, Years 1-2) and Levels 2 to 5 in Key Stage 2 (age 7-11 years, Years 3-6), and are expected to attain Level 2 and Level 4 at the end of the Key Stages 1 and 2, respectively.
Methods for the Collection of Data Within the Case Studies
Yin (2009) suggests that the way to ensure construct validity in a case study is to use multiple sources of evidence. This study used multiple sources of evidence, which involved classroom observations, interviews with both teachers and children, and documentary evidence. All the data were personally collected by the researcher. In order to collect the data, five visits to each school were needed: one for discussing the project with the teachers and to make the arrangements, one for the interviews, and three for the observations.
Interviews
The interviews with the teachers and the children were conducted before the observation of their lessons. The children (20 in total) were interviewed one by one to avoid the possibility of them copying each other in a group interview. Therefore, extra care was taken with them to ensure that they felt confident during the interview, which was conducted in a place close to the classroom so that the child being interviewed would not feel isolated. All the interviews were semistructured. As Drever (2003) has suggested for this type of interview, there was a general structure by deciding in advance the area that would be covered and the main questions that would help me focus on the purpose of my study (a schedule of interview questions for both teachers and children is presented in Appendix A), but the detailed structure was left to be worked out during the interviews. The initial questions sometimes changed during the interview according to the answer. All the interviews were taped, transcribed, and then analyzed.
Classroom observations
The observations were arranged in collaboration with each teacher. They were nonparticipant and unstructured observations. The aim was “to be as unobtrusive as possible so that observed behaviour was as close to normal as possible” (Bell, 2005, p. 189). For this reason, the use of any device (e.g., voice recorder or video camera) to record the data was avoided and instead field notes were kept. Keeping field notes through unstructured observations is very difficult. Following guidance from Bell (2005) an observation form was used, which was separated into sections for the researcher to record what was happening every 10 minutes (see a sample of a 10-minute observation note in Appendix B). The interviews with the children before the observations made the research easier because the children became familiar with the researcher as well as easier to be recognized in the class and better observed. The order, in which the children were interviewed, was used to code their names (e.g., P1, P2, . . . for the first, second pupil interviewed, etc.) in the field notes. The time spent for each observation was the time each lesson lasted (1 hour, apart from the lessons in Emma’s class, which lasted 45 minutes each) plus an additional time (approximately 30 minutes) required before and after each lesson to discuss with the teacher the aims of each lesson, the materials and resources that were going to be used, and actions that happened during the lesson which required further explanations or clarifications. All field notes were clearly typed in a word processor (see a sample in Appendix B) and then analyzed.
Documentary evidence
Documents pertaining to the study subjects were collected. They were school documents about policy and planning, tracking sheets of the children’s achievements, records of their assessment and photocopies of their work. These helped to acquire a clearer and more accurate picture of each case. Documents, according to Burns (2000), are important for supporting evidence derived from other sources.
Ethical Considerations
Before the research began, informed consent was obtained from the teacher, the head teacher, and the children’s parents, who gave their consent for both the interviews and the observation of the lessons. Parents also consented to the anonymous publication of samples of their children’s work. All the names (schools, teachers, and children) were replaced with pseudonyms or codes (e.g., School A, Pupil 1) and all necessary measures were taken so that confidentiality and anonymity would be secured throughout the research project. Teachers were given the opportunity to review their interview transcripts and to read the observation notes before the analysis took place.
The Analysis of the Data
All the data gathered through both stages of the research were checked using Bassey’s (1999) checklist for trustworthiness, triangulated by comparing evidence from multiple sources of evidence (Burns, 2000; Cohen et al., 2007; Yin, 2009) and thematically analyzed for emerging themes using guidance from Creswell (2009). Thematic analysis of the data involved categorizing the data by initially having in mind the research questions with the main issues investigated and then the emerging themes from the field work. This was a process demanding continuous review of the interview transcripts, observation notes, and documentary evidence to find patterns and themes, and for refinement of the themes. Themes that emerged from the preliminary phase (e.g., “existence of a policy,” “organizational strategies,” “teaching resources and materials,” “teachers’ training background”) also provided guidance for the analysis of data collected through the main study, the case studies. Furthermore, the data collected from the observation of the lessons, which concerned teaching and learning strategies for teaching mathematics to gifted children, were analyzed using Bloom’s taxonomy (e.g., “analysis,” “synthesis,” and “evaluation”), Ernest’s “success cycle,” and Vygotsky’s ZPD and interpreted as an indication of learning mathematics at higher cognitive levels, motivating, challenging, and extending higher ability students.
Findings
Data gathered through the questionnaire survey and the four case studies highlighted a number of issues relating to the identification of mathematically gifted children and the educational provision. In this article, only the issues that arose from the four case studies are presented—particularly those that related to the teachers’ methodologies for organizing and teaching mathematics at higher cognitive levels and for extension for high-ability students—apart from the views and experiences of both the case study teachers and the students. These issues are separated into the following three themes:
Teachers’ attitudes toward teaching mathematically gifted children
Aspects of teaching mathematics at higher cognitive levels and extension for gifted children in classrooms
Children’s progress and attitudes
Teachers’ Attitudes Toward Teaching Mathematically Gifted Children
In their interviews, among other questions (see Appendix A) teachers were asked to explain whether or not they felt comfortable teaching mathematics to gifted children. The teachers’ responses compared with their background (see Table 2) showed a strong correlation between their training background, their subject knowledge, their experience, and their confidence level. For example, Kate, who had no training background or experience, displayed a lack of confidence in teaching gifted mathematicians, admitted the need for further training, and recognized the help of the mathematics coordinator as important to overcome the lack of knowledge:
I could certainly do with some more training, in investigations especially and puzzles and problem solving. . . . I need to set aside the time to teach myself alternative methods of problem solving. There are other ways . . . I know that . . . but I’m not very confident so, I think I need to set some time aside to work on that . . . I do get support from the numeracy co-ordinator. She is always there for me. If I say, “I am struggling with this; can you show me a way and explain that to me?” she will take time to explain it to me . . .
The other three teachers, who had either training background (e.g., in gifted education and in teaching mathematically able children) and experience (Emma and Sarah) or initial training as a secondary-school mathematics teacher and experience in teaching higher mathematics (Claire), appeared very confident in their ability to teach mathematically gifted children:
I trained in New Zealand and in New Zealand the training is very much catered to more able children—Catering for more able children and less able children and obviously the ones in the middle—I think that the training I got in New Zealand was particularly top rate. Also, as the maths co-ordinator, I have got a lot of greater knowledge of the resources available for these children, so, therefore I can put them into my lessons. So, it is a bit of both, my training, my experience as well. . . . So, all of it together means that I am quite able to cater to these children and extend them. (Emma) I like maths. So, that is number one and number two: I have done so many courses and as a maths co-ordinator every year I am going on courses. So, I’ve got a lot of knowledge. I think that’s all. (Sarah) Oh yes, [I feel very comfortable] because I used to teach Year 7 to Year 13. (Claire)
Aspects of Teaching Mathematics at Higher Cognitive Levels, Support and Extension for Gifted Children in Classrooms
In Emma’s class
The lessons were shorter than regular classes (45 minutes). Emma did not do a starter activity, but rather did a quick reiteration of the past lesson before she introduced the new one. She chose activities that were not difficult or complicated, but they involved investigations with triangles, quadrilaterals, and polygons; real-life and open-ended problems about “money and prices;” and a construction of a numeracy trail (see Table 4).
The Lessons Observed, the Materials, and the Resources Used in Case Study Classes
SA = Starter activity.
DA = Differentiated activity for more able pupils.
HW = Homework (from previous lesson), discussed before the main lesson.
DA = Differentiated activity for those who finished earlier.
In all the three lessons observed, Emma gave opportunities for work at higher cognitive levels by engaging pupils in situations, which required analysis, synthesis, and evaluation:
Analysis, through her questioning (e.g., “Why?” questions), which encouraged pupils to explain their answers and their methods
Synthesis, through asking pupils to create their own problems (e.g., “numeracy trail”) or through questions (e.g., “What if . . . ?” “Imagine if . . . ?”), which changed the facts, created new situations, and gave opportunities for further investigation
Evaluation, through asking pupils to check the work of others (i.e., pupils were passing their work around, checking, and assessing each others’ work; for more examples, see Table 5)
Samples of Teachers’ Questioning and Children’s Responses in Relation to the Highest Levels of Bloom’s Taxonomy
Emma was continuously monitoring all pupils’ work and intervening to check their process and offer support when necessary (e.g., “How much so far?” to each child; “Are you sure . . . ?” to Pupil 1 who made a mistake). There was immediate feedback to pupils’ work, helping them correct possible mistakes, very good interaction between teacher and pupils, and evidence of learning new knowledge. The small size of the group helped this, which is something that Emma acknowledged: “When I work with my children in a small group, they learn loads.”
In Sarah’s class
Each lesson lasted 60 minutes, including a starter activity which lasted from 15 to 20 minutes. Sarah divided the lessons into two parts. In the first part, she worked with the whole class, mainly on starter activities (e.g., math games or revision of previous lessons) or to introduce the new learning objective. In the second part, she gave an activity on which pupils worked from their tables in ability groups.
For the first part of the lesson, Sarah mainly used computer-based activities (see Table 4). In two out of the three lessons (Lessons 1 and 3), these activities were for older children and required knowledge and comprehension as well as spatial (e.g., symmetry exercise) and analytical (e.g., reading graphs) skills. In the other lesson (Lesson 2), these activities were for reiteration and reinforcement, and especially for helping children who had not mastered basic mathematical knowledge and skills from past lessons (“adding two and three one-digit numbers,” see Table 4).
For the second part of the lesson, Sarah gave the same activity to all groups in two of the three lessons. In these two lessons, the activities were for older children again. Although the activities were common, Sarah differentiated her instructions according to the ability level of each group. For instance, in the first lesson, although the common question was to construct many different cuboids and to find a way of recording the cubes used for each cuboid, Sarah asked the five children in the higher ability group to find number patterns while they were recording their cubes and to attempt generalizations about the numbers that make a cube.
In both parts, in all the lessons, Sarah used higher order questioning, which gave opportunities for pupils to be engaged in situations requiring not only knowledge and comprehension but also analysis, synthesis, and evaluation:
Analysis, through encouraging pupils to explain their answers (e.g., “Can you explain . . . ?” “Why . . . ?”) and their methods (e.g., “What do you have to do to . . . ?”), present different solutions, identify patterns, conjecture, and attempt generalizations
Synthesis, through creating new situations and giving opportunities for further investigation (e.g., “What if you have nine?”)
Evaluation, through encouraging pupils to examine proofs introduced by someone else and evaluate their own work or method (e.g., “Why did you choose 9 + 9?” “What mistake [did you do]?”; for more examples, see Table 5)
Sarah, with the help of her teaching assistant, was monitoring the work of all pupils in the class most of the time, offering support when necessary. She intervened many times during the problem-solving process to help pupils be systematic and checked their understanding by asking questions (see Table 5). She only left the group of able pupils to work alone in one lesson. In that lesson, Sarah gave them a different activity, “shape & space problem-solving” (Lesson 2, see Table 4). This was because, as Sarah explained before the beginning of the lesson, she wanted to work with the low-ability children on that day. Therefore, she chose an activity that, as she said, the able children would be able to do by themselves. In this particular activity, able pupils did not do well, as none of the five pupils in the group managed to successfully complete it. In addition, two of them (Pupil 6 and Pupil 8) stopped trying and started playing instead. Those who managed to answer the questions did not have them all correct, but they did not realize this when the lesson was finished, as they did not have the opportunity to discuss their results with the teacher.
In Kate’s class
Although Kate was teaching a top mathematics set, there was a range of abilities within the set. Because of this, she separated her pupils into three ability groups, who were sitting in designated places in the classroom (“basically, the high ability [group] is on one side of the room and the lower on the other, so the middle is in the middle” [Kate]). Kate organized her lessons by following the Hamilton Plans for Year 5/6 (Hamilton Trust, 2009), from which she chose activities according to the Year 6 plan—activities relating to “areas,” “patterns and sequences,” and “ratios” (see Table 4).
Each lesson lasted 60 minutes, including a starter activity, which lasted from 15 to 30 minutes. As in Sarah’s class, the lessons were separated into two parts: (a) teaching the whole class and (b) teaching in groups by ability.
In the first part, Kate mainly used computer-based activities. In two out of the three lessons observed (Lessons 1 and 2), these activities were math games (see Table 4), which worked as a starter lesson and lasted 30 minutes (half of the lesson time). They looked like commercial electronic programs made for video games and their learning background only involved number calculations. In one lesson (Lesson 3), the computer-based activity was to introduce the new learning objective (“ratios”).
In the second part, pupils worked in groups by ability on the same activities. On these activities, the able pupils worked unattended in all three lessons observed. Kate remained attentive to those in the middle and lower groups who had difficulties.
In the first part of the lessons where activities on the whiteboard were involved—with or without the computer—Kate gave opportunities for all children to display their knowledge and abilities for the following:
Analysis, by encouraging them to explain their answers (e.g., using “Why?” and “How?” questions) and present different solutions (e.g., “[Is there] another way?”)
Synthesis, by asking them to use what they had learnt about calculating the area of rectangles to calculate compound areas and
Evaluation, by encouraging them to evaluate the activities that they did (e.g., “What made it difficult?”; for more examples, see Table 5)
The observations and the study of the children’s written work showed big differences in the able pupils’ performances between the first and the second parts of the lessons. When these pupils did exercises on the whiteboard and had the teacher’s interaction and immediate feedback, they all did a perfect job displaying knowledge, comprehension and, some of them, as aforementioned, skills of analysis, synthesis, and evaluation. In contrast, when they worked on their tables without help on exercises that were similar to what had previously been done, all six able pupils had difficulties completing them correctly. They seemed confused, made mistakes, and did not manage to finish all their work in any of the lessons observed.
In Claire’s class
Each lesson in Claire’s pullout group lasted 60 minutes, including a 20-minute starter activity. Claire systematically used activities she had created, mainly for the starter lesson (see Table 4). These activities involved exercises on the whiteboard and math games with cards. The onboard exercises for the starter lesson were always for reinforcement and connecting the previous with the present lesson, which sometimes included additional exercises that required language and logic skills (e.g., a decoding game created by Claire). The math games with cards were puzzles on advanced mathematics for those who finished early (e.g., an algebra jigsaw puzzle) or for reinforcement as a starter lesson (e.g., a domino game with negative number calculations).
The activities for the main lessons were advanced mathematics from the KS-3 curriculum and involved knowledge of mathematical facts (e.g., “Pythagorean theorem”); investigations with numbers, number patterns, and formulas using “Pascal’s Triangle,”; sequences using “Fibonacci numbers”; and algebra (see Table 4).
In all lessons, Claire provided opportunities for learning mathematics at higher cognitive levels requiring analysis, synthesis, and evaluation:
Analysis, through encouraging pupils to explain their answers (e.g., “Why 5 and 5 . . . ?”), identify patterns, conjecture, and attempt generalizations (e.g., when investigating Pascal’s Triangle)
Synthesis, through using patterns from Pascal’s Triangle
Evaluation, through encouraging pupils to evaluate a method (e.g., “Can you explain how working systematically could help here?”; for more examples, see Table 5).
Claire continuously monitored all pupils’ work, intervening to check their understanding (e.g., “This is not right . . . Careful about this!”).There was immediate feedback to the pupils’ work, helping them correct possible mistakes, good interaction between teacher and pupils, and evidence of learning new knowledge. Sometimes, however, Claire appeared critical of pupils who had difficulties during the lessons (e.g., “She is going to wake up in a minute. It’s all right,” she said about Pupil 17, who seemed confused and unable to follow the others) or impatient, answering some of the questions herself when pupils hesitated, without giving them time to think:
That’s more than a hundred, do you agree?
Yeah.
Twenty by twenty is four hundred, do you agree?
Yeah.
That number must be 21 squared, 22 squared, 23 squared or 24 squared, agree?
Yeah.
[ . . . ]
What is one times one? . . . One . . . So, 21 squared ends in one.
Children’s Progress and Attitudes
Children’s achievement
All able children in Emma’s, Sarah’s, and Claire’s classes progressed, according to their results in two consecutive formal assessments. In Kate’s class, the achievement of two children remained at the same level after two assessments, whereas the achievement of another child moved to a lower level (see Table 3). The formal assessments in all the cases were based on standardized achievement tests referenced by the National Strategy (e.g., Qualifications and Curriculum Authority tests).
Children’s attitudes toward mathematics lessons and the grouping structures
All nine children in the pullout groups displayed positive attitudes, through their interviews, toward the grouping structure and the lessons that they had, as the following samples show:
[I]f I have a partner, sometimes this distracts me and I don’t get any work done, that’s why I like working alone. . . . At the other class [the regular one] we sometimes work with partners, and sometimes work alone. In this class, we work alone. (Pupil 1, School A) I feel popular. . . . It [the work in the special group] makes me more focused on what I am supposed to do. (Pupil 2, School A) [I]n the Year 6 class, we kind of just stay on the format, while with Miss [Claire], we are just going above the level. I like it better . . . I look forward to Fridays and Mondays. (Pupil 17, School D)
The most positive comments for the work that they did and about the subject were expressed by children from Claire’s Year 6 pullout group. These comments concerned work on new topics “that we’ve never heard of” (Pupil 18) and work that involved investigations in number sequences:
I like maths and I am good at it and it’s good to be challenged. . . . In the regular class we have been learning . . . we have just started Fibonacci, but Miss [Claire] is showing me . . . like . . . ratios and how to make our own one and different numbers than Fibonacci. (Pupil 16, School D)
In Kate’s set, although all pupils appeared happy being in a top mathematics set, and seemed motivated to learn more mathematics, some of them (i.e., Pupil 10 and Pupil 15) appeared unhappy about the lessons that they had, complaining that they were “too easy” (Pupil 10) and sometimes “boring” (Pupil 15). However, in the lesson about “ratios,” in which an activity with real “Smarties” sweets was involved, Pupil 15 was heard saying “This is fun. The best lesson we have had in ages!”
Pupil 10 in particular appeared unhappy and disappointed by the teacher, expressing doubts about her knowledge in mathematics and complaints about her method, as the following samples show:
I could have a better teacher. Do you know Mrs . . . ? [He says the name of another teacher] because she is quite good at maths. I was hoping to have her, because this teacher Miss . . . [Kate] is not. I am working quite alone because the teacher is choosing other people that need help . . . with the answers, so I feel like shutting it out . . . [W]e have gone to a part where Miss [Kate] has to help other people and she really gives . . . simple work and she just says, “Do your own work . . . Help other people a bit . . .”
In addition, the six children from Kate’s class, albeit appearing to enjoy the computer-based games, expressed their wish to have more difficult mathematics. Only one child (Pupil 11) said that she would like to have more work with the computer, while there was another (Pupil 12) who said that she would prefer to play mathematics games against other people instead of the computer.
In their interviews, the five able children from the regular classroom (Sarah’s class) appeared stressed when they were talking about the hard work that they do. One of them, in particular, referred to an activity as a bad experience:
One time, I felt a little bit sad when Mrs. Sarah gave me some other people’s work that I thought that I might be able to do, but the work she gave me was quite hard. . . . I had to have her coming to me all the time. I didn’t like it too much. (Pupil 8)
Discussion
The findings of this study are based on a relatively small sample and this is clearly a limitation, despite the triangulation that occurred through the use of different methods of data collection. However, the case studies in four schools, each using a different method of provision, provide useful insight into how primary schools address the needs of mathematically gifted children, including teachers’ classroom practices and their impact on these children’s progress and attitudes.
The evidence suggests that primary schools in the United Kingdom are taking measures in pragmatic terms to address the needs of mathematically gifted children. These measures mainly involve grouping-by-ability arrangements and differentiated work from higher levels of the mathematics curriculum. Teachers also are aware that mathematically gifted children need different work from the usual and are aware of the available resources (printed publications or online databases). The latter, compared with the findings of a previous research (Koshy & Casey, 2005), which showed the opposite, indicates a progress in this specific field of education in the past few years.
However, the existence of an interest in meeting the needs of mathematically gifted children, as well as the use of ability grouping and extra teaching materials, does not mean that the needs of these children are actually addressed in classrooms. By studying in depth four different programs for gifted children in mathematics, this study found out that differentiated learning for gifted mathematicians through differentiated work and instructions for adding depth, complexity, and extension, as suggested by the literature (Casey, 1999; Koshy, 2001; Koshy et al., 2009; Sheffield, 1999; Tomlinson, 1995; VanTassel-Baska, 2007), was not always achieved in practice. The case studies showed that, although there were different teaching materials from the usual, and aspects of teaching mathematics at higher cognitive levels in all four classes studied, their effectiveness in practice depended on teachers’ expertise and the attention given to gifted mathematicians, as well as the nature of work used and the size of the class.
Although they did not use any specific framework of provision for mathematically gifted children, all four case study teachers used questioning that gave opportunities (more or less) for “analysis,” “synthesis,” and “evaluation,” which are considered higher educational objectives in Bloom’s (1956) taxonomy. This approach, which is suggested for teaching mathematics at higher cognitive levels (Koshy, 2001), worked well with high-ability children in all class types as it engaged their higher order thinking and helped them display their abilities. In mixed-ability classrooms, higher level questions, such as those requiring generalizations or evaluations, were answered by high-ability students. This was because either the rest remained silent or because some of these questions were particularly asked to the able students, both as a group or individually. The latter, as a practice of using Bloom’s taxonomy for differentiating instruction according to students’ ability (e.g., lower level questions for lower achievers and higher level questions for higher achievers) has been criticized by Tomlinson and McTighe (2006) as indefensible, in that all students should have the opportunity to answer higher level questions. The presence of the researcher, who was interested in the education of gifted mathematicians, might be the reason why the teachers addressed higher order questions directly to higher achievers, but this needs to be further investigated.
Evidence of extension for high-ability pupils was found only when these pupils had the teachers’ attention and “scaffolding” support every time they faced difficulties with an exercise. This method has been suggested by Vygotsky (1978) for learning within the ZPD and by the literature from the field of mathematics as suitable for teaching mathematics to gifted children (Koshy et al., 2009). This type of focused support for gifted mathematicians, however, requires highly trained teachers, who have both the subject expertise and the confidence needed to work closely with gifted children. Evidence from this study, for instance, showed that teachers with training backgrounds and higher confidence spent more time with able pupils, helping them solve problems, talk about their methods, conjecture, hypothesize, and attempt generalizations and evaluations of others’ work or their own work. In contrast, the teacher with no specific training and no confidence in teaching able mathematicians did not stay with the able pupils and, thus, did not help them extend themselves when they faced difficulties. This finding also highlights the correlation between teachers’ confidence and their training background and adds more evidence to the arguments for the need of a better, more practical and subject-specific training for teachers (Ball & Bass, 2003; Ball et al., 2001; Hill et al., 2004; Hill et al., 2005; Koshy & Casey, 2005; Koshy et al., 2009; Sheffield, 1999; Williams, 2008).
The use of an expert as a coordinator by each school, as the Williams (2008) report recommends, cannot solve the problem of lack of knowledge and confidence by teachers, if these teachers have to face alone the challenge of teaching gifted mathematicians in a classroom. In the case of Kate, for instance, although there was an experienced mathematics coordinator with subject expertise who was successfully organizing the provision program, helping Kate when needed, in the classroom Kate failed to extend higher ability pupils and, what is more, her lack of knowledge and confidence did not go unnoticed by the students but rather caused complaints (e.g., Pupil 10). The situation is different and there are positive effects on pupils’ attitudes and progress when an expert undertakes the responsibility to teach the gifted children, as the other cases showed. This agrees with Sheffield’s (1999) recommendations for the use of a specialist, who should work with gifted students in the classroom in collaboration with the classroom teacher or outside in small groups or in one-to-one lessons.
The nature of the work was found to affect the children’s motivation to learn mathematics and their attitudes toward the subject. More able children who had the opportunity to be engaged in nonroutine and complex problems (e.g., in Claire’s class) expressed high motivation to do advanced topics, both as main activities and as extra work for those who finished earlier. High motivation is considered a factor of positive affect for learning mathematics and the progress of gifted pupils (Ernest, 1985; Koshy et al., 2009; Sheffield, 2003). Also, the nature of materials and the way that the teachers used them were found by Stein and Kaufman (2010) to be able to improve the teaching of mathematics in terms of maintaining higher cognitive demand. In contrast, computer-based activities, which placed emphasis on fun through games involving number calculations only rather than on meeting the lesson objectives, did not seem to satisfy all the children, in the same regard as work that was too difficult when it was not supported by teacher instructions. The latter findings suggest that issues exist relating to the choice of the right work between too easy and too difficult ones and the effective use of computers, and they need consideration.
The level of homogeneity and ratio between pupils and teachers in a class affect the level of focused attention given to able children, their extension, and motivation. Although they were in ability groups, able children in large-sized classes with a range of ability—both Sarah’s class and Kate’s set—did not do more challenging work than the other children and, what is more, they often were left to work by themselves. According to what both the teachers and the children said, the latter happened because teachers in these classrooms wanted to offer support for pupils in middle and lower ability groups who had difficulties. Therefore, they left able pupils to work on tasks which they were expected to be able to accomplish unattended. Such a use of ability grouping, however, has been criticized in that it serves the administrative convenience of the teachers rather than the different needs of the children (Fielker, 1997). Particularly for higher achievers, this study found negative effects on both performance and attitudes when these children were left to work in an ability group by themselves. The benefits of group working and group problem solving, which mainly occur through students’ interactions and discussion described by other studies (Howe et al., 2007; Johnson & Johnson, 2002; Slavin, 1996; Webb, 2009) were not observed, as these children generally did individual work. There were only a few cases, where on-task discussion was observed and only when the teachers mediated and asked questions.
The failure of able pupils to successfully complete their tasks unattended, as well as some complaints expressed in their interviews about working alone (e.g., “like shutting it out,” Pupil 10) suggest that able children need their teacher’s attention as much as the other children in the classroom. Furthermore, the lack of both success and persistence at mathematical tasks, which was observed in the cases in which able pupils worked unattended, suggests, according to Ernest’s (1985) “success cycle,” that the self-esteem and motivation of these students are at risk, and consequently, their attitudes toward mathematics and their future success. The interviews with high-ability pupils also revealed negative attitudes (e.g., Pupil 10) toward practice, which wants them to help low-ability peers to do easy exercises. This particular practice is not an effective use of peer tutoring, which under certain conditions can benefit all students (Fantuzzo, Riggio, Connelly, & Dimeff, 1989; Slavin, 1996; Webb, 2009). In addition, it shows that peer tutoring is an issue that needs more consideration and future research.
The evidence presented here, therefore, suggests that teachers in large-sized classrooms of ranging abilities, even if they have the knowledge and self-confidence and also the presence of a teaching assistant (e.g., Sarah’s case), face difficulties in meeting the diverse needs of their students and, because of that, they choose to use peer tutoring and ability grouping to mainly serve their administrative needs. When they decide to focus on a particular group to offer support, they often do it for the lower achieving students, leaving in this way the more able or gifted children to work by themselves, usually on uncomplicated exercises. This does not agree with studies derived from cognitive and educational psychology, which suggest that high ability needs suitable opportunities, instruction, and continuous challenge in order to be developed to its fullest extent (Gagne, 1985; Gardner, 1999; Heller, 1990, 1991; Krutetskii, 1976; Perleth & Heller, 1994; Renzulli, 1978, 1986; Sternberg, 1985). A practical issue, therefore, that teachers of mixed-ability classes must address, is how to offer support for those who always need help and, at the same time, extension for those who can do more. A solution may be the collaboration with a specialist (not just a teaching assistant) who, as mentioned earlier (Sheffield, 1999), would work with gifted individuals either inside or outside the classroom. It is crucial for schools and policy makers to accept that, as Sheffield contends, gifted mathematicians are exceptional students who have special needs that cannot be effectively served by the classroom teacher alone, or in some cases even with the collaboration of an expert, if this takes place in a regular classroom.
The method of “setting” as used in Kate’s school was different from the models suggested by the literature. Setting must ensure for each set (“top,” “middle,” and “lower”) as much homogeneity as possible by selecting pupils according to their ability independently of their age (across age groups), so that teachers can manage differentiation more easily by selecting and providing work suitable for each set (Koshy, 2001). In Kate’s “top set,” the children were only from a one-year class (Year 5) and had a range of different abilities. Therefore, the top mathematics set looked like a regular mixed-ability class rather than a homogenous group of high-ability students and this was probably the reason why this study did not find the benefits of setting on pupils’ achievement described by other studies (Harlen & Malcolm, 1999). The way setting is used by this school raised a question about how this practice is implemented by other schools and about teachers’ knowledge on this practice, which needs further research.
In the pullout groups, the teachers had small-sized classes without big differences in pupils’ abilities and, thus, no problems in monitoring all the pupils’ learning and providing support when necessary. Consequently, able pupils had more opportunities for differentiated learning and extension. They received more focused instruction on problem solving, open-ended and challenging activities, and lessons at their own pace. The small size of the groups ensured continuous monitoring of each pupil’s work, plenty of time for interaction with the teacher, and time for thoughtful work without distractions. All these had positive results on pupils’ performance and their attitudes toward both the lessons and the grouping practice. Teachers’ interactions particularly helped those who faced difficulties to successfully overcome them, using “scaffolding” methods for learning within the ZPD, and extend themselves, as the relevant literature suggests (Casey 1999, 2002; Koshy, 2001; Koshy et al., 2009; Sheffield, 1999, 2003). Work without distractions by other pupils, especially of lower ability, was also highlighted by most pupils from all cases as essential for completing their tasks successfully. In addition, pupils in pullout groups did not have to help others, something which was also highlighted as a problem by many of the able children in all the case studies.
The positive effects of pullout grouping presented here add more evidence to the arguments that gifted pupils should be separated from their peers of the same age at least for part of their schooling day (Kulik, 1992; Kulik & Kulik, 1992), that the allocation of pupils with the same intellectual peers is a criterion for the success of a program for gifted children (Belcastro, 1987), and that pullout grouping programs can be a feasible programming option for gifted pupils (Vaughn, Feldhusen, & Asher, 1991).
Problems in relation to communication and articulation between the pullout groups and the regular classroom, mentioned by other studies (VanTassel-Baska, 1987), although not investigated in depth by this study, seemed not to have occurred, as there was progress in all pullout pupils’ achievements.
The idea of using an external secondary-school teacher as a mentor to teach the pullout group of Year 6 children seems to work well in practice for the selected children, who appeared very keen to attend the lessons and enthusiastic about the idea that they were taught by a secondary-school teacher. It also seems to work according to what is suggested by experts: that a teacher with a background in higher mathematics can provide promising students with more challenges (Sheffield, 1999) and that a mentor with subject expertise can act as a role model and bring forth enthusiasm for the subject and inspire the pupils (Koshy, 2001). Therefore, it is a method that schools should consider using. However, in the lessons observed, some negative comments made by the teacher against those who seemed confused or made mistakes, and her impatience regarding waiting for an answer when pupils hesitated, are some issues for consideration. Negative comments, for instance, may have negative effects on pupils’ motivation and attitudes, which are important in the learning process, as described by Ernest’s (1985) “success cycle.” Schools may need to organize training courses for external mentors to familiarize themselves with teaching styles in primary schools before they are sent to teach primary children.
Conclusion
The education of mathematically gifted children is not an easy matter that can be addressed simply by separating students into ability groups and giving more difficult work to more able ones. Gifted mathematicians are exceptional students who have special needs, and because of this they need teachers’ attention and continuous support through focused instruction and work at higher cognitive levels in order to develop their potential to the fullest extent. Likewise, teachers of the gifted also need continuous support and specific training in making provision for mathematically gifted children so that they are able to offer suitable differentiated work and instruction to gifted mathematicians within the curriculum.
Footnotes
Appendix
A Sample of a 10-Minute Observation Note From the First Lesson Observed in Sarah’s Class
| 21-30 | The teacher then says that they are going to use small cubes (plastic cubes for construction games) as “bricks,” in order to build their own houses. She asks them to make as many different cuboids as they can and then find ways to count the bricks that they will use. She writes their task on the board and reads it loudly: |
| How many different cuboids can you and your partner make? | |
| How can you record them? | |
| The teacher divides the children into groups by ability, keeps the five children of my focus group on the carpet with her and sends the rest with her assistant to their tables to start working. The teacher gives some directions to the five children and then she leaves them to sit together in a table in a front row. I am moving a little closer to them in order to be able to observe them and hear more clearly what they say, but not too close in order to avoid distracting them. | |
| The teacher and her assistant are circulating among the children and offer help. My focus children work mostly on their own. Sometimes they are speaking to the person who is sitting closest, such as P5 with P6, P6 with P8 and P8 with P9, while P7 seems to work alone. P9 walks to another table and speaks with other children too. | |
| P5 has done a cube. He shows it to the others and says: | |
| I’ve done a cube! | |
| He shows it to the teacher also and the teacher asks him to find how many “bricks” he has used. He says eight. The teacher asks him to try more and think about a pattern. P5 suggests that numbers like eight, six and four (even numbers) make a cube. The teacher asks him to try it and goes to P9. | |
| P9 shows his own construction to the teacher, which is a rectangular parallelepiped. The teacher asks him to count the “bricks” and P9 counted them correctly (twelve). She asks him then if he can transform it to a cube. P9 now starts adding more rows of bricks. |
Acknowledgements
The research reported in this article was based on a doctoral study carried out within the Brunel Able Children’s Education (BACE) Centre of the Brunel University West London: Uxbridge, UB8 3PH, UK. It would not have been possible without the participation of the teachers and pupils. I would like to acknowledge their help and thank them for this.
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
The author received no financial support for the research, authorship, and/or publication of this article.