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Clarke (2003, p. 149) argued that international comparative studies should be ‘evaluative’, rather than a mere comparison of the similarities and differences. From this point of view, the following intended curriculum comparison and textbook research will document the literature findings related to England or China, or both.

The Intended Curriculum comparisons. Since TIMSS proposed the tripartite model

of curricula (see Figure 1 above), researchers have sought to discover whether the coherence between the intended curriculum and the implemented curriculum matters with regards to successful learning outcomes. Oates (2011) proposed that it should be the foci of education reform in England, developing curriculum coherence and curriculum control as high-

performance education systems did. Curriculum coherence is defined as the process in which ‘standards move progressively towards the understanding of a deeper structure’ (Schmidt, Wang, & McKnight, 2005, p. 529). The meanings of control and autonomy are opposite concepts in public education policy. Curriculum control policies might include ‘textbook adoption, curriculum guidelines and testing’ (Archbald & Porter, 1994, p. 22). Curriculum control is essential to curriculum coherence. Curriculum coherence includes the national curriculum content, textbooks, teaching content, pedagogy, assessment, or incentives which can reinforce one another. For example, Singapore’s success is mainly secured by promoting curriculum coherence through the approval of textbooks and other teaching materials.

Singapore’s textbooks which follow its national framework facilitated students’

understanding development compared with USA situation (Ginsburg, Leinwand, Anstrom, & Pollock, 2005). In Finland, textbooks and teaching materials are also tightly controlled. The curriculum coherence in terms of understanding linear function for both regions will be explored in Chapter 9: Summary and Discussion.

Cheng and Wong (1996) identified the features of conformity which shape the more uniform education systems such as Shanghai’s: the single, consistent textbook and

standardized assessments at the province or municipal levels. The Shanghai education system is highly centralised and controlled by the local municipal government rather than being regulated at the national level. In 1997, Shanghai was allowed to have its own curriculum. The local municipal (metropolis) curriculum and centralisation of textbook production are

both now subject to educational authority in the Shanghai government. Conversely, the English education system tends to be much less centralised ‘in terms of educational experience provided’ (Whitburn, 1995, p. 347).

Secondly, studies have been carried out to compare features of the intended curriculum in different countries. A national curriculum consists of concepts, principles, fundamental operations, and key knowledge. Elizabeth Truss, Parliamentary Under Secretary of State for Education and Childcare as mentioned at first chapter, advises that ‘a rounded curriculum’ is required for the curriculum reform in England (Truss, 2013, September 18). With respect to the content or structure of national curricula, Oates considered that the national curriculum in England has had ‘significant structural problems’ which should have been ‘concept-led and knowledge-led, not context-led’ (Oates, 2011, p. 132). Referring to how to introduce topics in the English curriculum, the intended curriculum has a distinctive attribute for arranging topics, which is separated into different years and becoming more complicated as students are allowed to progress and accumulate knowledge from year to year, which will be discussed at Chapter 5: Curriculum Analysis. A convoluted approach is

adopted in USA as well. In comparison to the ‘spiral curriculum’ in USA schools’ curriculum, the Chinese curriculum and instruction are much more ‘sequential and non- repetitive’ (Moy & Peverly, 2005, p. 253). Burghes (1999) suggested that topics in the mathematics curricula should have been treated more in-depth as well as with better

organisation for the topics. The narrow yet deeper scope of the curriculum in China has also enabled students to gain basic knowledge and skills, but may not be appropriate for

cooperative learning (Cai, Lin, & Fan, 2004).

It has been argued that the requirements of the mathematics curricula in East Asian countries are much more difficult than those in Western countries in terms of mastering the complexity of mathematics knowledge (Bao, 2002). For example, researchers examined the

difference between England and Japan regarding the solution of quadratic equations in junior secondary schools (Whitburn, 1995). The results showed that in England the approach to this topic is too limited, while in Japan it would be taught both algebraically and graphically. The approach towards mathematics in terms of linear function will be compared in Chapter 6: Textbook Analysis.

Thirdly, researchers also looked specifically at how students’ understanding changed during the curriculum reform, especially in terms of algebra learning. For example, in the secondary school mathematics curricula in the USA, there was a fundamental change from the traditional emphasis on symbol-manipulation to a focus on problem-solving and

application of mathematics knowledge. Comparing traditional curricula with the implemented reforms (to a standards-based mathematics curriculum), by using algebra and function as specific cases, Huntley, Rasmussen, Villarubi, Sangtong, and Fey (2000) found the algebra instruction that emphasised the use of graphing technology in order to solve authentic

application problems (AAP), would be of benefit not only for students’ problem-solving, but also for articulating abstract mathematical ideas. That is, the foci of curriculum changing towards solving real world situation facilitate students’ understanding of mathematical topics, while how to balance these two perspectives should be carefully considered. For example, Cai and Wang (2006) have examined the difference between the traditional and reformed curricula in the USA in terms of their learning objectives, definition, and the development of equation-solving abilities in the case of linear function. From a mathematical perspective, they suggested that (1) the USA reformed curriculum should introduce more interpretations of the concept instead of just one explanation; (2) that the idea of ‘variable as an unknown’ in traditional curricula gave rise to confusion; and (3) that the application of linear function in real-world situations, such as weight and height, articulates the relationship between two variables. Therefore, the curriculum reform contributes to a deepening of students’

understanding of mathematics and successfully addresses the drawbacks from the traditional one. The question for policy makers is what will be the most effective changes to improve students’ understanding for their own country, fitting in with different cultures, organisations and other realities.

In conclusion, curriculum coherence, emphasis on knowledge depth, and approaches to presenting a topic are the features of high performance education systems. The numerous ways in which to introduce the topic and being aware of the students’ confusion raised by the arrangement of the curriculum is also becoming a trend of comparative education.

Textbook comparisons. The intended curriculum in all countries has been supported

by ministry directives, instructional guides, school inspection, and recommended textbooks (Mullis et al., 2004). The importance of textbooks is embodied in several perspectives. First, textbooks not only act as a mediator between curricula policy and classroom instruction (Valverde et al., 2002), but they also link the curriculum and activities in classrooms (Johansson, 2003), though considerable gaps between curriculum standards and textbooks still exist (Fan & Zhu, 2007). For example, Johansson (2005) found that there was an

objective gap of requirements between commercial textbooks and the national curriculum in the case of Sweden. Secondly, as a major conveyor of the curriculum (Fan, Zhu, & Miao, 2013), textbooks are the visible manifestations of the curriculum in most classrooms (Son & Senk, 2010). With regards to the implication of textbooks, they appear to influence teaching strategies (Fan, 2013). For example, classroom lessons have been compared in London, Beijing, and Hong Kong that have shown that teaching was highly influenced by the textbooks in these places (Leung, 1995). Thirdly, textbooks help teachers to learn not only the subject matter, but also the pedagogical knowledge (Collopy, 2003). Nicol and Crespo (2006) also found that prospective teachers felt that textbooks could offer more guidence on the basic teaching requirements, particularly because textbooks differed from the one with

which they were taught as students. The function of textbooks in England and Shanghai will be revealed by teacher interviews at Chapter 8.

Another trend of textbook comparative studies aims to discover what mathematics textbooks actually looks like, for example their layout. Compared with French and German textbooks, the layout of English textbooks has fewer questions and the structure is relatively brief (Pepin & Haggarty, 2001). That is, English textbooks are much concise on structure and the number of questions. Another kind of investigation is related with the content presented in textbooks. Eastern textbooks have focused on pure mathematics knowledge, while Western textbooks emphasise real-life situations. For example, Park and Leung (2006) compared the Grade 8 textbooks of Eastern countries (including China, Japan, and Korea) and Western countries (including England and the USA) and found the Eastern textbooks to be more beneficial for students when conveying an idea, but less successful in motivating students. The Western textbooks are effective in expressing the importance of mathematics in real-life, but unclear about the link between real-life situations and the mathematical concepts.

Furthermore, focusing on characteristics of problems presented in textbooks, Zhu and Fan (2006, p. 614) argued that Chinese textbooks should present more authentic application problems (AAP) ‘whose conditions and data are, indeed, from real-life situations or collected by students themselves from their daily lives’; whereas USA textbooks should consider more challenging problems for students with involving more steps in the solution, as China does. USA textbooks also include more visual information than Chinese ones. Furthermore, after comparing the content presentation of the addition and subtraction of integers between American and Chinese mathematics textbooks, the Chinese textbooks contain ‘more problems with high level mathematics content’ (Y. Li, 2000, p. 239).

The solution strategies in examples of Eastern textbooks such as China and Singapore are also less in number than in Western textbooks such as in the USA. Fan and Zhu (2007)

compared China, Singapore, and USA mathematics textbooks for problem-solving

procedures in terms of two layers: general strategies referring to Polya’s four-stage problem- solving model (understanding the problem; devising a plan; carrying out the plan; and looking back); and specific strategies. Chinese and Singapore textbook series merely presented the ‘carrying out the plan’; while more than two-thirds of problem solving procedure presented in USA textbooks adopted at least two stages. This finding may partly explain why American students perform better than Chinese pupils in more open-ended problem-solving, as observed by Cai (1995).

Furthermore, due to the recent boom in textbook research, Fan (2013) proposed a framework for mathematics textbook research (see Figure 2 below). This framework involves three factors: the subject of the textbooks itself; textbooks as a dependent variable (how textbooks are affected by other factors); and textbooks as an independent variable (how they affect other factors). This framework has launched an appeal for the continuation of

textbooks research. With regards to the dependent variable, the intended curriculum is supposed to be the main influence. In terms of the independent variable aspect of the continuation, studies have demonstrated ‘how mathematics textbooks are used by teachers and students’, ‘how they impact the behaviour of teaching and learning of mathematics’, and ‘what the influences of textbooks on students’ achievement in mathematics are (Fan, 2013, p. 772). Furthermore, Fan et al. (2013) pointed out that, although textbook studies have

provided better understanding in terms of the role of textboooks, textbook analysis and comparison, and textbook use, research into the relationship between textbooks and students’ learning outcome is still lacking. Especially in English situation, the relationship between the use of textbooks in England as discussed in TIMSS and student performance should be investigated further.

Figure 2. The framework of textbook research (Fan, 2013, p. 771)