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In this subsection, I will review PISA results about the relationships between students’ performance and process factors in terms of opportunities, and teaching practices and quality.

3.4.2.1 Opportunities to learn

Students’ opportunities to learn (OTL) is suggested as the core of schooling (Schmidt and Maier, 2009). It is commonly considered that the concept of OTL was originally proposed by John Carroll in the early 1960s (Schmidt et al., 2013; Lafontaine et al., 2015). In his theoretical model of school learning, Carroll (1963) argues that individuals could successfully learn a given task as long as they spend enough time, and introduces OTL as a measure of allocated learning time. The concept and measures of OTL are developed along with early ILSAs (Lafontaine et al., 2015; Suter, 2017). Besides the early defined concept with regard to the time exposure which is suggested by Carroll, it has been also interpreted as exposure to or coverage of

specific content. For the First International Mathematics Study (FIMS) which was launched by the IEA in 1964, Husén (1967) defined OTL as “an

opportunity to study a particular topic or learn how to solve a particular type of problem presented by the test”. Schmidt et al. (2013) suggest OTL as content coverage in classes and the time spent on the specific content. Regardless the inconsistent interpretations of OTL amongst researchers, OTL is used in ILSAs for explaining the variation in students’ performance in

assessments within and across countries, and informing educational

policymaking in relation to national curriculum and its implementation (Yang Hansen and Strietholt, 2018). In PISA 2012, in which mathematics was assessed as the focused domain, OTL measurement was particularly included in the student questionnaire. In PISA 2012, the concept of OTL seems aligned with the definition given by Husén (1967), since it was defined as “coverage of content categories and problem types” (OECD, 2013a, p.186). It is measured by indices of students’ perceived experiences with specific mathematical tasks (e.g. exposure to pure mathematics

problems, exposure to applied mathematics problems), and students’ perceived familiarity with mathematical concepts (OECD, 2013a). Previous research shows evidence that OTL is related to students’

performance (Schmidt et al., 2001). In terms of the OTL measures in PISA, by fitting data with multilevel linear models, the official results of PISA 2012 show that students’ exposure to pure mathematics at school had significant and positive relationships with their mathematics performance across and within education systems (OECD, 2014b). Amongst the OECD countries, on average, increasing one unit of exposure to pure mathematics could bring about 50 score points (i.e. 0.5 SD on PISA international scale) higher in students’ mathematics performance (OECD, 2014b). In contrast, the relationship between exposure to applied mathematics and students’ mathematics performance is curvilinear across education systems that mathematics performance increases along with more exposure to applied mathematics and then after a certain point it decreases (OECD, 2014b). As these differential relationships suggest, it seems that mathematics principles themselves are more crucial than embedding real-life contexts in

mathematics problems (Schmidt and Burroughs, 2015). Although it is not revealed that the relationship between exposure to applied mathematics and mathematics performance is as straightforward as that for exposure to pure mathematics, the official PISA report highlights that good performance in PISA mathematics is also associated with the opportunities to learn applying mathematics in real-life contexts (OECD, 2014b). The OECD suggests that both pure mathematics and application of mathematics in real-world contexts should be taught (OECD, 2014b). Complementing this view, Schmidt and Burroughs (2015) argue that, with regard to applied mathematics, “more is not necessarily better” (p.31), and caution over concentration on embedding real-life contexts in mathematics instruction.

Despite the strong relationships between OTL and students’ performance officially claimed by PISA results report, some researchers argue that their relationships are rather modest (Carnoy et al., 2016; Yang Hansen and Strietholt, 2018). By reviewing OTL measures in ILSAs from 1959 to 2011, Suter (2017) concluded that the ways that OTL is measured considerably influence the magnitude of the relationships between OTL and students’ performance. In PISA, OTL measures are based on students’ self-report. For one of these measures, familiarity with mathematics concepts, one response option is “Know it well, understand the concept” (OECD, 2014b). Yang

Hansen and Strietholt (2018) consider that this measure contains unintended construct besides of OTL, since it involves students’ self- evaluation of their mathematics proficiency. They adjusted for students’ mathematics self-concept for examining the OTL effect, and found that its effect on students’ mathematics performance turns to be smaller.

Regarding the importance of OTL for students’ learning and their achievement, a few more points maybe worth considering as well for furthering the understanding of their relationships and improving OTL for students. For example, Carnoy et al. (2016) find that the OTL effect of exposure to pure mathematics is stronger for the student subgroup who have low family academic resources (e.g. books in home, mother’s educational level) and high mathematics proficiency. Hence, they caution that the various effects of OTL across student subgroups should be taken into consideration for policymaking toward improving OTL (Carnoy et al., 2016). Moreover, it is found that the variation in OTL is mainly within schools rather than across schools (Schmidt and Burroughs, 2015). This suggests, compared with educational inequality in terms of OTL across schools, variation of OTL within schools is more effective for explaining students’ performance. Since PISA student population is 15-year-olds who may not definitely be enrolled at the same grade, to some extent, the within-school variation may be brought by different years of schooling, as suggested by the above reviewed grade effect (see Section 3.4.1.4).

3.4.2.2 Teaching practices and quality

In terms of teaching practices, it seems that a student-oriented (also called student-centred) approach has been widely promoted in education reforms around the world (Westbrook et al., 2013; Cheung et al., 2018). However, PISA results provide a mixed picture about the association between the frequency of employing this approach in teaching and students’ performance.

Participating countries and regions ranked high on the league table do not necessarily also report high frequency of using student-oriented approach (OECD, 2013d; Schweisfurth, 2013). By employing data of 62 education systems, Caro et al. (2016) find inconsistent relationships between this factor and mathematics performance across systems. As to the picture within systems, for example, based on PISA 2012 data of Turkey, Demir (2018) finds negative effect of student-oriented teaching practices on students’ mathematics performance. It is suggested that student-oriented teaching approach could explain student's motivation and self-concept in mathematics rather than their mathematics performance (Caro et al., 2016; Scherer et al., 2016). Caro et al. (2016) suggest that this teaching approach may have differential effectiveness for students in different contexts of classrooms, schools, and education systems, and for students of different characteristics. Regarding the influence of students’ characteristics on the effectiveness of this kind of teaching practices, students’ own learning strategies can matter. For example, through interviewing secondary school students, Campbell et al. (2001) find that compared with students with surface learning strategy, those with deep learning strategy tend to have better recognition and understanding of the potential learning opportunities offered in student-oriented teaching practices. Hence, it is suggested that, along with the employment of student-oriented teaching practices, the skills about learning from the student-oriented classroom activities are necessary to be taught to students as well (Campbell et al., 2001). Researchers argue that the artificial division between student-oriented and teacher-centred approaches, and advancing one over the other, oversimplify the dynamic and complex classroom contexts (Cuban, 1983; Noyes, 2012). Cuban (1983) suggests that it is hard to assess the effect of teaching practices in this

regard on students’ performance, since they are usually dynamic rather than stable over time, and often embody diversity of strategies.

As to teaching quality, Klieme et al. (2001) suggest cognitive activation as one of its dimensions, and find cognitive activation positively associated with students’ mathematics performance. Cognitive activation is featured with cognitively challenging and highly structured tasks (Baumert et al., 2010; Keller et al., 2017), in which students are encouraged to apply their acquired knowledge to different contexts, seek different solutions, communicate and reflect on their thoughts and methods (Klieme et al., 2001; OECD, 2014a). It is suggested that the extent to which students can benefit from cognitive

activation instruction is subject to the pedagogical content knowledge of their teachers (Baumert et al., 2010; Kunter et al., 2013). Pedagogical content knowledge allows teachers to be able to make instruction content accessible to students (Shulman, 1986; 1987; Kleickmann et al., 2013), anticipate and respond to the problems that students would encounter (Keller et al., 2017). However, the provision of the potential learning opportunities in cognitive activation does not mean the actual use of these opportunities by students. It is argued to distinguish these two conceptions and proposed that students’ use of the opportunities mediates the effect of cognitive activation on their performance (Praetorius et al., 2018). By comparing video observation and students’ reported perception, researchers find that students may not necessarily perceive or recognise that they are cognitively activated (Hugener et al., 2009). Mayer (2004) writes that teachers need to provide focused goals and enough guidance to students to allow them build

knowledge through appropriate cognitive activation. Additionally, supporting and scaffolding students is considered to be essential for motivating

students to be engaged in the process of cognitive activation (Baumert et al., 2010).

Demir (2018) finds that Turkish students’ perceived frequency of their teachers’ use of cognitive activation in mathematics classes was positively associated with their mathematics performance in PISA 2012. Similarly, the positive relationships are also found in Korea and Singapore (Yi and Lee, 2017). However, Caro et al. (2016) find that the relationships between the extent to which students are cognitively activated and students’ mathematics performance are positive yet curvilinear across 62 education systems in PISA 2012. They propose that there is no longer association between the level of cognitive activation and students’ mathematics performance when cognitive activation is of quite high level (Caro et al., 2016).

Researchers argue that it is problematic to use a single indicator for the complex construct of cognitive activation (as PISA does) to predict students’ performance, since the possible level of cognitive activation can be different in different types of lessons (e.g. introduction lessons vs practice lessons),

depending on the content of lessons and how it is implemented (Baumert et al., 2010; Praetorius et al., 2014; Praetorius et al., 2018).