The research paradigm that guided most of this research was essentially interpretive, although aspects of naturalistic inquiry have been blended into the study. Elements from both of these theories resonate strongly in the researcher’s world view and beliefs about the value of educational research. The researcher’s theoretical orientation was further challenged as a shift occurred from interpretation to action as an ethical commitment. The general stance taken cannot therefore be categorized in an absolute way. In practice, it will be revealed that there was no perfect correspondence between the chosen research strategy and matters of epistemology, ontology, and methodology. It is also important to acknowledge an awareness of the influence of the researcher’s values, for those values, to some extent, influenced for example, the choice of topic. Moreover, the method, conclusions, and implications are all grounded in the researcher’s moral and political beliefs.
The study was designed using a case study approach and predominantly qualitative methods. Results from two questionnaires provided some quantitative data. The project began with a pilot study followed by two phases of data gathering. These phases are summarized in Table 4.1.
Table 4.1
Data Gathering Methods used in the Study
Research Question Phase One (2005) Phase Two (2006)
1. What are the characteristics of
mathematical giftedness recognized by the school policies, students, teachers, and parents? Student interview Teacher interview Parent interview Parent questionnaire Documents Student interview Teacher interview Parent interview Documents 2. How are mathematically gifted and
talented students identified?
Student interview Teacher interview Parent interview Documents Student interview Teacher interview Parent interview Documents 3. What provision for the students’ education
in mathematics has been made within the classroom and school contexts?
Student interview Teacher interview Parent interview Parent questionnaire Documents Observations Student interview Teacher interview Parent interview Documents Observations 4. What are the characteristics of an effective
teacher of mathematically gifted and talented students? Student Interview Teacher interview Observations Student Interview Teacher interview Observations 5. What roles have parents played in their
child’s mathematical development?
Parent questionnaire Parent interview 6. How is a school transfer managed for a
mathematically gifted and talented child?
Student interview Parent interview Teacher interview Documents Student interview Parent interview Teacher interview Documents 4.2.1 Case Studies
Case studies were chosen as the most appropriate method for answering the research questions. The case studies of this project have evolved as both descriptive and
interpretive. Merriam (1988) supported this situation and acknowledged that case studies become interpretive when they move beyond description and that “in reality most case studies are a combination of description and interpretive or description and
evaluative” (p. 35). Through the case studies, the multiple complexities of the education of a group of mathematically gifted and talented students were explored from a variety of perspectives and over a sustained period of time.
4.2.2 The Researcher
From the outset of the study, the researcher’s intentions were made explicit to the gatekeepers and participants. Background information about the researcher’s credentials and experience was provided. Initial contact was made through phone conversations and prior to visits, before any formal data gathering took place. A positive relationship and rapport with participants was established and a professional manner maintained throughout the study. All participants were treated with respect and their contributions acknowledged.
4.2.3 Research sample
Choices were made for the cases and the ‘cases within cases’ based on a purposive sampling process. The criteria for the cases were firstly, that the participants would be Year 6 and Year 8 students from schools that identified and provided for mathematically gifted and talented students. Secondly, the aim was to include at least one school that provided for these students within a regular class programme, one that was streamed for mathematics, and one that provided for the students through a fulltime class for gifted and talented students. The 15 participants (four girls and 11 boys), type of school, and organizational provisions are outlined in the following three tables (Tables 4.2, 4.3, & 4.4). Table 4.2 provides a few details about the students. Table 4.3 outlines the three Phase One schools, teachers, and types of provisions. This is followed by Table 4.4 detailing the Phase Two schools, teachers and provisions. The Phase Two sites were determined by the schools that the 15 students in the study transferred to in the following year. The students in Phase Two moved from three schools to eight different schools across a much greater geographical area than the Phase One schools.
Table 4.2
Participants
Code Pseudonym Gender Age* Brief Description of the Student
A1 Lily Female 11 Twice-exceptional: Asperger’s Syndrome
Attended one-day-a-week programme in Year 6
A2 Bob Male 11 Articulate student, enjoys mathematics
A3 Nardu Male 11 Maltese, educated overseas and in N.Z.
A4 Jarod Male 11 British, educated overseas and in N.Z.
A5 Victor Male 11 Māori student, interested in practical aspects of
mathematics
B1 Mia Female 10 Multiple talents, especially dance
B2 Eric Male 11 Confident, very interested in mathematics and
the research study. He was not identified as gifted and talented in Phase 2.
B3 Martin Male 10 Reticent, lacking confidence in his ability in
mathematics
B4 Ryan Male 10 Very gifted in mathematics, attended one-day-a-
week programme (Years 6 & 7)
B5 Tim Male 10 Did not appear to be gifted and talented in
mathematics
C1 Karen Female 12 Not particularly interested in mathematics but a
capable student in all learning areas
C2 Nina Female 12 Indian, educated abroad and in N.Z. Very gifted
student in mathematics and English
C3 Amir Male 12 Bangladeshi, educated in N.Z.
C4 Lewis Male 12 Enthusiastic, attended after-school mathematics
programme
C5 Paul Male 13 Chinese, educated in N.Z., finds mathematics
easy but with few challenges, additional mathematics provided at home
Table 4.3
Phase One Schools, Teachers, and Organizational Provisions
Students School Class Level Teacher* Provision
Lily, Bob, Nardu, Jarod, Victor
School A Year 6 Mrs N (School AP) Cross-class ability
grouping (mathematics) Mia, Eric,
Martin, Ryan, Tim
School B Year 6 Mrs J (School BP)
Miss S (School BP)
Regular class ability grouping (mathematics) Karen, Nina,
Amir, Lewis, Paul
School C Year 8 Miss L (School CI) Full-time gifted class
(intermediate school) *Code denotes title, an initial letter, school letter, type of school─Primary (P), Intermediate (I).
Table 4.4
Phase Two Schools and Teachers after Students made a Transfer
Student(s) School Teacher* Provision
Lily School D: Intermediate Mrs J (School DI) Cross-class but
within-syndicate ability grouping for mathematics Bob, Nardu,
Jarod, Victor
School F: Co-educational Years 7-13
Mr J (School FIS) Streamed class
Mia School G: Co-educational
Years 7-13 Mrs K (School GIS) Streamed class
Eric School E: Intermediate Miss S (School EI) Regular class
Martin, Ryan, School E: Intermediate Mrs O (School EI) Cross-school ability
(withdrawal) grouping for mathematics
Tim School H: Independent
boys’ school Years 5-8
Mr H (School HPI) Regular class
Karen School I: Co-educational
secondary
Mr P (School IS) Streamed class
Nina School J: Co-educational
secondary
Mrs R (School JS) Streamed class
Amir, Lewis, Paul
School K: Boys’ secondary school
Mr M (School KS) Streamed class
*Denotes title, an initial letter, school, type of school─Years 5-8 (PI), Years 7-8 (I), Years 7- 13 (IS), Secondary (S).
4.3 Data Gathering Methods
The following section outlines the different data gathering methods that were used in the study: interviews, questionnaires, documents, and observations. This data gathering process was also supported by a research diary.
4.3.1 Interviews
The purpose of the interviews was two-fold, firstly to motivate the participants to share their knowledge and secondly to elicit information that related directly to the research objectives. The initial interview questions for students, teachers, and parents were trialled in the pilot study and then minor amendments were made for the final schedules (Appendices A, B, and C). In preparing the interview schedules, the following elements, as suggested by Bryman (2004, p. 324), were taken into consideration:
create a certain amount of order on the topic areas so questions flow; formulate questions that help answer the research questions;
use language that is comprehensible; do not ask leading questions; and
gain relevant background or context information.
The sequence suggested by Charmaz (2005), an initial open-ended question, followed by intermediate questions, and then an ending question was used. For example from the Phase One—Parent interview:
Initial open-ended: Tell me about your child’s early mathematical development.
Intermediate: What do you think the special programme has provided mathematically for your child?
Ending: What are your future hopes for your child, specifically next year and beyond that?
All of the interviews were conducted face-to-face and tape recorded. The students were interviewed twice; the second interview occurred in the year following a school transfer. (See Appendix A, for interview questions.) The first student interview gave an opportunity to gain background information about the child’s interests, a historical perspective of mathematics learning experiences, a perspective on the current
mathematics programme, teacher qualities, school choice, preparation for school transfer, and expectations for the following year. The second interview provided a forum for following up on the transfer process and to gain insights into the mathematical teaching and learning experiences in the new school. The timing for these interviews differed for each of the students because of issues related to access, school programmes, and teachers’ and the researcher’s commitments. These second interviews were all completed in the one year.
Each teacher in the study was interviewed once to gain an understanding of the school’s and teacher’s views of gifted education. Questions (Appendix B) focused specifically on the teacher’s understanding of the characteristics of the mathematically gifted student, the identification of and provision for mathematically gifted students, and the topic of school transfer. The teacher also provided background information about their own teaching experience and educational information about the student(s) in the study. In the Phase Two schedule, the questions also focused on what information had been provided by the sending schools about the student and how this information was used.
Parents were interviewed twice. They were interviewed once in the first year of the study (Phase One) and again in the following year. Likewise, the scheduling of these was influenced by a variety of factors including when their children were interviewed (the second student interview preceded the second parent interview) and availability. The Phase One parent interview schedule (Appendix C) was designed to gain information about early mathematical development, signs of special abilities in mathematics, school experiences, parental involvement, and factors influencing school choice for the following year. In the second phase, questions focused on the school transfer process and perceptions of the child’s experiences in mathematics in the new school.
The interviews were transcribed verbatim. Some of the interviews were transcribed by the researcher and the rest by contracted transcribers who signed a confidentiality statement (Appendix D). All of the transcribed interviews were checked by the researcher for accuracy. Subsequent amendments were made because of the transcribers’ inaccuracies. The transcriptions were completed in stages as data were
used for papers submitted for publication whilst the study was in progress (Bicknell, 2006a; Bicknell, 2006b; Bicknell & Riley, 2006).
4.3.2 Questionnaires
Two questionnaires were used in the study—one for the students and one for the parents. The student questionnaire was Rogers’ (2002) ‘Mathematics Interest and Attitudes’ inventory (pp. 458-459). The inventory (Appendix E) is a measure of the student’s attitude towards learning and what motivates them in a particular subject area. Three adaptations were made to the original. The word ‘math’ in each item was changed to mathematics and Question 2: My math teachers are usually the best teachers I have in school was deleted as the students (in Phase One, when the questionnaire was given) had the same teacher for all curriculum areas. Finally, a space was provided for students to record comments; this was optional. As a tool, the questionnaire enabled the students’ interests and motivation levels in mathematics to be measured.
The parent questionnaire (Appendix F) was used to gain specific insights into the types and levels of parental involvement in their child’s mathematics learning. The questionnaire was originally designed by Cai et al. (1999) and then re-used in more recent research (Cai, 2003). It was developed to assess the level of parental involvement in students’ learning of mathematics. The 23 questionnaire items were designed to assess parents’ roles as motivators, resource providers, monitors, mathematics content advisers, and mathematics learning counsellors. There were either four or five questions in each category. A few minor changes were made to the questions that related to contexts not relevant to the New Zealand setting. In Question 14: I am always aware of my child’s mathematics requirements by checking notebooks, using learning line, or through phone calls to school, the term ‘learning line’ was deleted as this is not an approach used in New Zealand schools. In Question 16: I think I know enough about algebra to help my child, the word ‘algebra’ was changed to ‘mathematics’. Although algebra is delineated in the curriculum statement as a strand with number, teachers give the topic less explicit emphasis in the primary school than other strands. Children are often unaware that the mathematics they are doing is algebra and therefore unlikely to talk about algebra
with their parents. Also, given the age of the students in the study, it made more sense to keep the general term ‘mathematics’. The purpose of the questionnaire was to gain an understanding of the roles that parents play in their child’s learning of mathematics and the level of that involvement. The Parent Involvement Questionnaire (PIQ) is recognized as a reliable and valid instrument for assessing parental roles in students’ learning of mathematics. It has been used extensively in both ‘within nation’ and ‘cross-nation’ studies (Cai, 2003). For each of the PIQ Likert scale response items the parents chose from strongly agree, agree, disagree or strongly disagree. To help parents identify tendencies, neutral choices were not provided. Those statements worded in a positive way were scored from 4 to 1. The scoring was reversed for any statements written with a negative connotation, for example, Question 11: I seldom spend time talking with my child about his/her progress.
4.3.3 Documents
A variety of documents were gathered during the course of the research. These documents were provided by the principals, teachers, and students in relation to the research questions. They consisted of school policies related to gifted and talented education; mathematics curriculum delivery documents; teachers’ planning documents; students’ results in mathematics tests, examinations, and competitions; and students’ work books. Some of these documents were personal (teachers’ and students’) and some (in the case of school policy) were open to public scrutiny. The purposes for collecting these documents were to help contextualize the cases; provide background information; ascertain students’ levels of achievement in relation to peers; and confirm, contrast, and validate some of the data gained from other sources. All documents were dated and the original source noted.
4.3.4 Observations
The research questions focused on the mathematical experiences of the students in the classroom, so it was important to include observations in the study. The observations of the students in classes were unstructured, although a guiding rubric from Artz and Armour-Thomas (2002) provided a focus to the observations without
making them formally structured. The dimensions focused on were: tasks (modes of representation, motivational strategies, sequencing, and difficulty levels); learning environment (social and intellectual climate, modes of instruction, and pacing, administrative routines); and discourse (teacher-student interactions, student-student interactions, and questioning). The observations were used as a means of verifying student, teacher, and parent responses and to help the researcher contextualize the study. Each class in the study was visited for four to seven days in which the researcher followed one unit of work (such as a statistics or algebra unit). These were consecutive days of the mathematics unit, but not necessarily consecutive school days because of the variability of school programmes. This was organized in response to the teachers’ commitments and programmes.
4.3.5 Research Diary
The purpose of the research diary was to log school contacts and dates, record anecdotal experiences, classroom observations, and post-interview notes. It was structured according to each site in Phase One and Two with subsections devoted to classroom observations, teachers, students, and parents. The sections enabled the researcher to maintain a chronological sequence and to preserve the sections in relation to each case and cases within a case. After leaving each observation and interview, field notes were made. These included key ideas, reflections, and some of the emerging themes. This personal log assisted in keeping track of the research in progress, how the plan was affected, and also prompted awareness of how a researcher was influenced by ongoing data collection.
4.4 Data Analysis
The data analysis was also an ongoing process that was completed in different stages. As Simon (2004) proffered: “Analyzing qualitative data is much like solving a mathematics problem; each step leads to greater insight as to what might be tried subsequently” (p. 161). This ongoing process is justified in terms of identifying emerging themes that influenced interview schedules in later phases of the study, the rationale for the development of grounded theory, and the desire to publish findings from a study ‘in progress’. After collecting the data from the multiple sources as
outlined, they were sorted. For each school, background information (from the Education Review Office’s web site and the school’s web site) were obtained, policy documents, teachers’ planning notes, and students’ work samples were collected, and recordings were made of classroom observations. Both sets of questionnaires (students’ and parents’) were ordered to correspond with the sequence on Table 4.1. Parent interviews were similarly ordered and student interview transcripts separated into Phase One and Phase Two. The researcher listened to all tapes, checked transcripts for accuracy, and made corrections.
After this organization, the Parent Involvement Questionnaire (PIQ) results were calculated. Firstly, a total score was calculated for each parent and then a sub score was calculated for each parent in each category of motivator, resource provider, monitor, mathematics content adviser, and mathematics learning counsellor. The mean score for each role was then calculated so that comparisons could be made across the roles. A few parents had made additional notes, which were coded according to the role that they supported. These were organized in tabulated form, two outlier results were highlighted, and a separate score calculated for four specific questions (Questions 4, 10, 15, and 19). These questions were isolated as they were worthy of separate analysis in relation to the research questions. The student questionnaire, ‘Mathematics Interest and Attitudes’ inventory was likewise treated quantitatively. Each student was given a score and these were tabulated to see individual scores and also combined to see if there was a difference between Year 6 and Year 8 scores. Each of these questionnaires was analyzed based on the theoretical lens provided by Rogers (2002) in relation to student motivation.