The mixed-methods

Two approaches in this study. Both quantitative and qualitative approaches are

involved in the study as follows:

Quantitative approach 1. Content and document analysis was used in order to

delineate the official materials in both regions in respect to the basic knowledge contained and the expectations of how to apply this knowledge to real world situations. Here, the study considered national curricula at a regional level and textbooks used at the sample schools level as the official documents. The analysis of documents could be undertaken in both quantitative and qualitative ways (May, 2011). It is worth noting that Bryman (2004, p. 291) identifies content analysis as unable to answer any ‘why’ questions. This study, however, took the quantitative form. Instead, it investigates ‘what’, thereby fitting with this thesis’ own tendencies to examine the ‘what’ over the ‘why’.

The generated theoretical categories of the understanding model identified the requirements from the curricula, and the examples and exercises within textbooks at certain levels of understanding. In addition, the ways in which textbooks present the application questions and solutions was analysed from the perspective of the three main representations for linear function. Thus, each category was transformed into numbers in order for

comparisons to be made between different regions.

Quantitative approach 2. Tests were undertaken in order to identify how well

students understand linear function in each region using self-designed instruments. Student understanding was measured by the model of understanding function that was comprised from existing research and described in Chapter 3. Here, construct validity, which evaluates ‘how well the measure conforms with theoretical expectations’ (De Vaus, 2014, p. 54), needs

to be a focus. Therefore, a pilot study was conducted first. The self-designed tests at the pilot stage were based on the general model. Findings from the pilot tests would not only identify issues of validity, but also indicate necessary changes for the main research (Matthews & Ross, 2010). Later, the specific barriers to understanding basic knowledge and favoured or inferior solutions shown in application for each cohort would also be examined from the modified paper-pencil tests in the main study.

Qualitative approach. Teacher interviews were used in order to answer research

questions about the contextual factors of teaching and learning processes. The results would reveal teachers’ underlying beliefs towards the implemented curriculum, namely the lesson plan based on their perception of barriers to student’s understanding. Harding (2013) described three frequently used interview types: biographical interviews; semi-structured interviews; and unstructured interviews. According to Bryman (2004, p. 439), if there is a ‘fairly clear focus’, then semi-structured interviews are best suited to the task. Thus, semi- structured interviews were chosen in this study. It was expected that the two groups of teachers would express their beliefs and understanding in their own way, using their own words as fully and as spontaneously as possible. Each interviewee was provided with an outline of the interview before conducting the data collection. The content of the interview focused on how teachers understand the official documents and perceive students’ learning in order to gain a better understanding of the implemented curriculum.

Mixed-methods used in other small-scale comparative studies. There are three

distinctive features involved in smaller scale comparative studies: (1) the tendency to employ several different questionnaires/tests for diversified mathematics rather than one assessment for the whole; (2) the importance of a pilot study; and (3) the trends towards systematic explanation as a continuum rather than as discrete categories.

Instead of one test to imply the whole of mathematical learning, many studies have made efforts to develop more targeted questionnaires that assess different types of

mathematical performance, for example, comparing the reality of students’ problem-solving and problem-posing between China and USA (Cai & Hwang, 2002); problem-solving for word problems between China and Singapore (Jiang & Chua, 2010); and problem-solving behaviour between Japanese and USA students (Becker, Sawada, & Shimizu, 1999).

Furthermore, more studies advocate a pilot study to form these assessments and qualitative methods in order to explain the results from tests. Harpen and Sriraman (2013) explored high school students’ mathematical creativity and problem-solving and problem- posing in China and USA. Both the mathematical content test and the problem-posing test were conducted after several pilot phases. Meanwhile, follow-up interviews with students aimed to achieve explanations of performance within these tests.

Another series of studies was also related to Chinese and USA students in respect to their problem-solving performance. For example, Cai (1995) examined how students reacted to three types of problems: computation problem, simple problem, and complex problem. The assessment tasks were used after three pilot studies and were followed by interviews with their teachers. The purpose of the interviews was to find out a causal relationship between what students were taught and what they carried out on different types of problems. Therefore, the following interviews for either students or teachers played a supplementary role to the whole research. Meanwhile, the trend of current comparative education studies turned to investigate the holistic views of participants instead of one aspect so that more than one method of inquiry and more than one type of data were needed.

Furthermore, the multiple approaches used by those researchers also indicate that the findings from mixed methods could be verified with each other as measured by triangulation, which will be explained in a later section of this chapter: Validity of the study. Cai (2004)

investigated the relationship between students’ problem-solving strategies and their use of representations. In order to figure out why students would have a particular tendency, interviews with teachers were used to discover their beliefs. Meanwhile, teachers’ marking for each representation was collected to connect what teachers believe with what they actually did in a practical way. An et al. (2004) proposed a framework for secondary school teachers’ pedagogical content knowledge which will be discussed at Chapter 8: Teacher Interviews. Mixed methods, such as questionnaires, interviews and observations, were applied for views of the Chinese and USA teachers in their study. The data from classroom observations and interviews was used to confirm the feedback from teachers’ questionnaires.

Therefore, my chosen methods approach conforms to the main features of small-scale comparative studies. Two sets of student tests for understanding were devised and piloted: one to test pure knowledge and another to test application of that knowledge. To explore how students’ understandings were shaped, curricula and textbooks provided explanations from a presenting mathematical knowledge perspective, while teachers offered their views to make students’ understanding development clear. The procedure for the whole data collection will be discussed in the next section.