The procedure

Figure 19 demonstrates the framework of this research design as a pilot study at Phase 1 and main study at Phase 2.

Figure 19. A framework of this research

Phase 1: pilot study (June 2013). The functions of this pilot study include: (1) to

gain feedback on the validity of the students’ tests; (2) to check the time taken to complete the tests; (3) to identify which questions, if any, were too easy or difficult; and (4) to test the proposed coding system for data analysis (Cohen, Manion, & Morrison, 2013). Although Converse (1986) suggested two pre-tests were necessary for development, evaluation, and polishing tests, the limited time and cost only allowed one trial in this study. Findings of a pilot study would indicate possible category changes to the content of tests and the

procedures of administration. The feedback from the teachers helped to check the testing time with regard to administration and the appropriate complexity of questions in the tests.

In the first two weeks of June 2013, pilot tests in Shanghai were completed. In the final two weeks of June 2013, pilot tests in England were implemented. Two groups of students were both given the tests near the end of the academic year.

Phase 2: main study (February 2014 - May 2014). After analysing the data from the

pilot study, the schedule for data collection in Phase 2 was set up. In November 2014, I convened three brief meetings with the Heads of Maths at each sample school in England to explain the purposes of the Phase 2 data collection in order to gain their understanding and support. For the Shanghai context, Head Teachers at three sample schools cooperated fully in this phase of the study via telephone meetings.

Besides gaining support from mathematics departments, another purpose of the meetings was to discuss three aspects of conducting the research in school level: the arrangement of instruments, interviews with the Heads of Maths, and the details of the proposed classroom observations. During the meetings, I also gave the outline of interviews to each Head of Maths in England and emailed the translated Chinese versions to the selected Shanghai teachers. Before the end of January 2014, the Heads of Maths at the sample schools in both regions gave me a confirmation of an interview date and a time for the students’ tests (at a time linear function/graph would be being taught). For the school visits, each sample school agreed to several classroom observations involving linear function/graph. In England, up to four classes were observed at each sample school for this topic. In addition, the English schools also offered other classroom observations involving other topics in different year groups. In Shanghai, one school offered all nine required and continuous Maths classes for this topic. Due to very similar teaching schedules, I managed to observe three classes at another school, but did not have time to observe more at the third school.

School visits were arranged in order to carry out participant observation. Participant observation aims to achieve ‘intimate knowledge’ of people (Matthews & Ross, 2010, p.

257). In this study, what happened in the classroom played a comprehensive role in the teaching and learning process at each region. Classroom observations provided further contextual information for the study. Meanwhile, teacher interviews following the classroom observation started with a concrete question about the lesson plan. Visiting was also

beneficial in building positive relationships between the teachers and I.

During classroom observations, I acted as participant when students had their own tasks or group activities. During times of teacher input, I remained a complete observer. Expanded notes of the field experiences were made after each classroom observation. The function of these notes was to help me to inspect the different ways in which students would encounter difficulties during their classroom experience, and to be familiar with the teaching process especially in England. Three examples, one from an English classroom and another two from Shanghai classrooms, were used in the following Result part to make teacher’s views clear with more details or to reveal relevant facts.

4.3 Methods

A method refers to an individual technique employed for data collection, while

methodology refers to a set of methods used to inquire into the research questions (Hitchcock & Hughes, 1995) which are often underpinned by certain ontological and epistemological assumptions. Since quantitative and qualitative methods are ‘compatible’ (Tashakkori & Teddlie, 1998, p. 12), the second phase of design is to ‘obtain different but complementary data on the same topic’ (Morse, 1991, p. 122). Findings from qualitative and quantitative data could be combined to ‘produce a general picture’ of the reality (Bryman, 2003, p. 137). Through this triangulation process this research aims to illuminate the interrelationships between the development of students’ understanding and the wider education system. It attempts to do so via interrogation of four sources of information. First, documentary research of the official national curricula would examine the findings from the intended

curriculum. Clarke (2003, p. 156) argued that the similarities and differences in mathematics curricula would be a ‘signature of international comparative’ research. Secondly, document analysis of the textbooks would provide how textbook writers explain the requirements of curricula/syllabi and their hypotheses about students’ learning and thinking, referred to as the ‘hypothetical learning trajectory’ (M. Simon, 1995, p. 133). Thirdly, students’ tests would describe the outcomes of the attained curricula. Finally, teacher interviews would provide insights into the implemented curricula. Table 13 shows the samples in terms of the four perspectives used in the whole study.

Table 13

The Outlining of the Samples

Perspective England Shanghai

Curriculum analysis

Mathematics programmes of study: Key Stages 3 and Key Stage 4 National Curriculum in England

The Shanghai City Primary and Secondary Mathematics

Curriculum Standard

Textbook analysis

Collins New GCSE Maths for Edexcel Modular: Foundation 1 and Higher 1

Collins GCSE Maths 2 tier- foundation and tier-higher for AQA A;

Foundation and Higher GCSE Mathematics: Revision and Practice;

Shanghai nine-year compulsory education textbook: Mathematics Grade 8 (Volume 2).

Student test Pilot study 96 292

Main study 561 907

Teacher interviews 3 Heads of Maths 3 Heads of Maths and 1 _{research teacher} The detailed method will be addressed throughout this section. Each sub-section is broadly divided into three parts: the brief research design; the selection of the sample; and the data collection process and analysis.

4.3.1 Curriculum analysis

The research design. Recent studies have examined existing literature or official

with previous curricula in the same country; or with other countries’ curricula as the features of national curricula between different countries are compared (Bao, 2002; Cai et al., 2011). This approach to curriculum analysis which this research adopts only traced current official documents, with introduced little for the historical background.

Selection of the sample. Within this study, content analysis is conducted on national

curricula documents from England and the local one for Shanghai. It aims to delineate the requirements of intended curricula in the case of linear function in both regions. Specifically the latest released curricula in England will be analysed. From 1th September 2013, the national curriculum programmes of mathematics at Key Stages 3 and 4 have no longer applied. The new national curriculum will be applied from September 2014. During the academic year of 2013, ‘schools are free to develop their own curriculums for mathematics that best meet the needs of their pupils, in preparation for the introduction of the new national curriculum’ (Department for Education, 2013a). However, the draft programme of study for KS2 to KS4 was provided for the 2013 academic year.

In September 2013 the new mathematics curriculum in England, Mathematics programmes of study: Key Stages 3 National Curriculum in England (Department for Education, 2013c) was released. Later, in July 2014, the Mathematics programmes of study: Key Stages 4 National Curriculum in England (Department for Education, 2014) were

launched. As to content of the new curricula, however, there were minor improvements made to earlier draft programmes (Steers, 2014). Therefore, this new KS3 national curriculum instead of the draft was chosen. In terms of linear function, the requirements in the draft and the formal curriculum were similar as well. Thus, in line with KS3, this new formal KS4 curriculum was chosen as well.

The Shanghai City Primary and Secondary Mathematics Curriculum Standard (Shanghai City Education Committee, 2004) was chosen for this study. In 1997, Shanghai

was allowed to have its own curriculum instead of following the national curriculum in China (See Chapter 2 for a fuller discussion of this). Four years later, Shanghai implemented the second edition of the mathematics curriculum. In 2004, Shanghai modified it again and called it the new mathematics curriculum which is still in use.

Data collection. The statutory guidance including the statutory programmes of study

and attainment targets for mathematics at Key Stages 1 to 4, can be accessed from UK government website (see https://www.gov.uk/government/publications/national-curriculum- in-england-mathematics-programmes-of-study). As a former secondary school teacher in Shanghai, the Chinese curriculum was provided by Maths departments in schools. The analysis for the data will be addressed in Chapter 5.

4.3.2 Textbook analysis

The research design. Due to the commercial textbook market in England, the

textbook research is expected to examine not only the books’ characteristics but also how to use them (Fan, 2013). The textbooks analysis focuses on the features, while its function for lesson plans will be illuminated in Chapter 8: Teacher Interviews.

Selection of the sample. All the selected textbooks were officially used by the sample

schools in England and Shanghai. Due to limited time and resources, a convenience sample of chosen schools was selected for this research. The limitation of the convenience sample would be that it would ‘not represent any group apart from itself’ (Cohen et al., 2013, p. 164). As discussed in later sections regarding the limitations of this study, I will claim that the findings do not intend to be generalised into a theory.

In England, there exists a variety of mathematics textbooks used for classroom teaching and learning. The English textbook series were developed for two different ability levels: Foundation level and Higher level, in line with the ‘additional mathematical content’ which is expected to ‘be taught to more highly attaining pupils’ (Department for Education,

2013c, p. 3; 2014, p. 3). Both followed the same national curriculum in England. Each school, however, has a considerable degree of autonomy to choose the appropriate series of textbooks for their students.

Three secondary schools located in the North East of England volunteered to take part in this study. According to the national league table for mathematics based on the

qualification results at the end of secondary schooling in 2012, students’ performance in these three schools were within the top 30% of all secondary state schools in England which will be explained thoroughly at next part. Each sample school, however, had a different series of school mathematics textbooks. Therefore, all of the textbooks used in these schools were included in this study. Thus, although the selected textbooks may provide a typical pattern for presenting knowledge in England, I acknowledge that they are a convenience sample and not representative.

In contrast, the choice of textbook is not flexible in Shanghai. The uniform textbooks were developed based on The Shanghai City Primary and Secondary Mathematics

Curriculum Standard (Shanghai City Education Committee, 2004) instead of the national curriculum in China, but it also remains a centralised education system in Shanghai.

Therefore, mandatory textbooks are widely used by all Shanghai students at state schools as well as at public schools during the compulsory education stage (from age 7 to 15). Each term in the school year has one separate mathematics textbook. Linear function is introduced in the second term of Grade 8 (approx. age 14) and, therefore, the one appropriate Shanghai textbook was selected in the present study.

Therefore, the following seven textbooks containing linear function were examined in this study:

1. Collins New GCSE Maths for Edexcel Modular (Foundation 1), published by Collins, 2010;

2. Collins New GCSE Maths for Edexcel Modular (Higher 1), published by Collins, 2010;

3. Collins GCSE Maths 2 tier-foundation for AQA A, published by Collins, 2006; 4. Collins GCSE Maths 2 tier-higher for AQA A, published by Collins, 2006;

5. Foundation GCSE Mathematics: Revision and Practice, published by Oxford University Press, 2006;

6. Higher GCSE Mathematics: Revision and Practice, published by Oxford University Press, 2006;

Shanghai:

7. Shanghai nine-year compulsory education textbook: Mathematics Grade 8 (Volume 2), 2007.

Data collection. English textbooks were collected at the meeting with the sample

schools in November 2013, and I have already possessed the compulsory Shanghai textbooks from the junior secondary school stage. Again, details concerning the analysis of the content of linear function/graph will be set out in Chapter 6.

4.3.3 Student tests

The research design. Students’ performance will be analysed using paper-and-pencil

tests, including the results from both the pilot study and the main study. This study focuses on how well students understand a certain topic and what supports or constrains the development of this understanding. As both the Year 10 English students and Grade 8 Shanghai students are still teenagers, it might be hard for them to evaluate their own understanding, or to reflect with their barriers to that understanding, or to articulate the development of their

comprehension of a certain concept. Therefore, student interviews were not included in this study.

Selection of the sample. In this research, North-East England and Shanghai were

selected as two regions to be investigated for students understanding (see Chapter 1 for justification). Linear function/graph is covered in Grade 8 (approx. age 14) in Shanghai, and in Year 8, 9, and 10 in England. The detailed arrangements of topic within the curricula will be illuminated in Chapter 5: Curriculum Analysis. Year 10 students in England (approx. age 15) were selected for this research as they are supposed to have covered the knowledge for linear function required by the KS4 national curriculum. I acknowledge, however, that there was an avoidable one-year difference between the two samples which will be discussed at later section: Limitations.

English participants came from three state schools as discussed in the methods section for the textbooks analysis. In the three sample schools, all Higher Level and the majority of Foundation Level students in Year 10 took part in the study. The students in the lowest set of Foundation Level students, however, were not selected for this study on the recommendation of the Heads of Maths.

After the English sample was chosen, the Shanghai sample was selected correspondingly, i.e. with schools at a similar percentage of school performance in

mathematics. The Shanghai sample was drawn from the Pudong District which is the biggest district with about one-fifth of total Shanghai students. Shanghai has no uniform league table for the senior secondary School Entrance Examinations (Zhongkao) whose function is similar to GCSEs in England as discussed in Chapter 2. However, students’ academic performance in mock examinations could be regarded as providing a similar ranking of schools, especially in the second mock exam whose timing is roughly two months before Zhongkao. Therefore, according to the district league table for mathematics based on the second mock examination

in 2012, three schools that ranked at around the top 30% among all state schools in the Pudong District were selected for this research. All Grade 8 registered students in the three sample schools participated in this research. Due to mixed-ability classes in the Shanghai education system, the lower ability students were also included in the study.

Data collection. In the main study, students’ tests were conducted once students had

finished learning the topic. The tests were administered by the students’ regular classroom mathematics teachers in both regions.

In Shanghai, two schools participated in the pilot study. The feedback from teachers suggested that the two tests should be combined into one test so that it would be more convenient for students to answer. I adopted this idea and the time given to students was increased to one hour to complete this combined test. Due to the uniform teaching schedule in Shanghai, two Shanghai schools finished teaching this topic on Friday 21st February 2014 and another on the following Monday, 24th February 2014. After that, all three schools

immediately arranged an agreed time for the whole Grade 8 students to complete the test. When back in England, I presented the Heads of Maths with the combined test and they were all happy with the layout. Two of the three sample schools also took part in the pilot study so were familiar with the administration procedure. I had a brief meeting with the Head of Maths at the third school. Due to the different teaching schedule in England, one school finished teaching this topic at the beginning of April 2014, one at the end of April 2014, and one in the middle of May 2014 respectively. Once they finished the topic, each respective Year 10 mathematics teacher then immediately arranged the tests.

4.3.4 Teacher interviews

Selection of the sample. In terms of teacher selection, the Head of Maths in each

within their respective schools. Moreover, their beliefs often represent accepted values from the sample schools with regards the process of the teaching and learning of mathematics.

In general, the Head of Maths in English schools is in charge of the Scheme of Work which every mathematics teacher within the whole department will follow. The Scheme of Work prescribes the arrangement of topics in each Key Stage, for example how long a certain topic should be taught and the different requirements for corresponding levels of students, reported by the selected English teachers at their interviews. The Head of Maths in Shanghai normally pays more attention to difficulties in the teaching and learning in each grade, which is discussed at every departmental meeting usually scheduled every two weeks. Therefore, the views either of English or Shanghai Heads of Maths will be more informed than other mathematics teachers.

In addition, Shanghai has a unique researcher-teacher system that is also in operation in other parts of China. One researcher-teacher in the Pudong District was also interviewed. The researcher-teachers in Shanghai (normally they were in-service maths teachers before this role) are mainly responsible for the teaching and learning of mathematics at the district level. The research topic of linear function was arranged in Grade 8, and therefore, the Grade 8 researcher-teacher participated in the interview. His views about teaching and learning represent another level of expert perspective. However, in Chapter 8, Teacher Interviews, I did not distinguish the views from the expert and Heads of Maths, because the purpose of the interview was not to discern the differences between them. Both the Shanghai sample and English sample of teachers are specialists who only teach mathematics subject.

Data collection. The Head of Maths in each sample school was interviewed

separately. Interview data was collected during the classroom observation period. Before the interview began, all the interviewees signed consent forms (see Appendix A) according to the requirements of the Ethics Committee at Durham University (discussed in more detail at

section: Ethics). Meanwhile, every interviewee was informed that I would send them the transcription of each interview for the purpose of verification. In Shanghai, linear function would be taught from 11th February 2014, so I returned to Shanghai on 7th Feb 2014. The teaching schedule of mathematics is approximately nine classes over two weeks. During