PISA,National and Regional Education Policies and their Effect on Mathematics Teaching in England and Germany

Abstract

To consider how processes of education governance linking the work of international organisations and national and regional policy-making in two contrasting policy environments affect policy enactment in schools, differences in mathematics teaching between English and German secondary schools were analysed using Bernstein’s account of pedagogic practice. This allowed the opportunities for achievement provided to different groups of students to be identified. The findings suggest that, as a result of strong governance pressures, English higher achievers have more opportunities to make progress than lower achievers, a concern which is consistent with standardised assessment data. Despite policy changes, similarities in the teaching of higher and lower achieving students in Germany remain and these account, in part, for the narrower gap in achievement there.

Soft and hard governance

The days are long gone – if they ever existed – when education policy was the preserve of national governments. International organisations such as the Organisation for Economic Co-operation and Development (OECD) that administer and analyse the Programme for International Student Assessment (PISA) comparative test data, and the European Union (EU) whose education mission has become to support the cross-national development of human capital, are highly influential. Whilst international surveys of student outcomes such as PISA are used in many countries to justify education reform (Lingard et al., 2013), the influence of the OECD has increased with its provision of explanations for national differences in scores linked to policy recommendations (Sellar and Lingard, 2014).

Soft governance approaches rely on information rather than diktat to steer local practice; for example, by publishing research, surveys, guidance and advice, and creating various fora where this can be shared and discussed. Ideally, policymakers or practitioners then make use of this information to effect improvement. In reality, however, accountability measures can conspire to harden governance effects by increasing the status of officially endorsed information, leaving policymakers or practitioners unable to choose alternative courses of action, however promising they might believe them to be, without facing disapproval or sanction. Much has been written about the soft governance role of the OECD and the EU (for instance, Mundy et al., 2016). Alexiadou (2014), for example, provides an analysis of policy learning through the EU’s open method of coordination which challenges more simplistic accounts of policy borrowing, whilst also showing how the European Commission’s governance activities stretch into areas such as education for which the EU has no legislative remit. Such developments have led Lingard and his colleagues (2013) to adapt the policy cycle (Ball and Bowe, 1992) to include a level of global governance. As such, education reform agendas often share ‘performative’ similarities (Ball, 2013).

Our concern here is to elaborate particular versions of this extended policy cycle, identifying how processes of education governance linking the work of international organisations and national and regional policymaking affect policy enactments in schools. Our hope is to understand this better by comparing different versions of the policy cycle. To explore policy enactment, where the cycle meets students, we analyse interactions between the three ‘message systems’ of curriculum pedagogy and assessment (Bernstein, 1975) which constitute pedagogic discourse (Bernstein, 1990). We use an approach which draws on Bernstein to compare mathematics teaching in England and Germany, setting each within their national and regional policy contexts. Because schools in Germany are administered on a regional basis, we focused on one land, Baden-Württemberg, contrasting this with South West England, a region with a similar economic profile in agriculture, industry and tourism and with generally low unemployment. The English teachers all worked in local authority community colleges. The German teachers also taught in public (state) schools within the selective tripartite system, half from the academic Gymnasien which focus on preparing students for university entrance and half from the lowest tier Hauptschulen which provide a more general education. In all we engaged in sixteen teaching observations of mathematics lessons for pupils aged 12–13 years in each country, with each participating teacher observed teaching and then interviewed on two occasions.

This study is novel in comparing extended policy cycles in contrasting education governance contexts and, in particular, analysing policy enactment within each policy cycle – something Ball and his colleagues (Ball et al., 2012) suggested is crucial in policy evaluation but difficult to do well – by comparing pedagogy which Alexander (2009) also identifies as an important but under-researched area. Methodological rigour is provided first by combining the insights of insider researchers in lesson observations with the perspectives of both teachers and students in interviews and from participant validation; and, second, by using Bernstein (1990; 1996; 2004) to analyse pedagogy as an emergent socio-cultural phenomenon. Significantly, the analysis provided here casts light on the complex relationship between hard and soft education governance and education outcomes for different student groups.

Comparing education in England and Germany

Teaching is closely linked to the societies and education systems in which it takes place (see, for example, Goodson and Lindblad, 2011) and comparison can help illuminate how the political economies and the systemic and accountability structures within which teachers work give rise to different conceptions of teaching. While the neoliberal education reform agenda in England is long established, a similar programme has been slow to emerge in Germany for reasons we discuss shortly. However, both countries draw on humanist education traditions which have long been subject to progressive influence. Comparing pedagogy in each of these circumstances provides for an improved understanding of the broad relations between teachers, practices and pupil experiences. Together these can inform an understanding of policy implementation and the policy cycle (Ball and Bowe, 1992). Concluding an extensive review of comparative education policy research, Busemeyer and Trampusch suggested this is much needed: ‘this domain would benefit from theoretical work on the micro–macro problem in understanding outcomes of education policy’ (Busemeyer and Trampusch, 2011: 434).

The English and German policy contexts

Whilst experiences of school mathematics can vary considerably within countries, our concern here is to identify those significant variations between England and Germany which can be attributed to differences in traditions, policies and circumstances, including how these mediate international influences. Amongst the most reported features of education in England in recent years are the regular standardised testing of students, publication of tables comparing results at a school level, and high-stakes school reviews by a national inspectorate (Ball, 2013). Without doubt the consequences of failure for students, teachers and schools, combined with the competition engendered by a long tradition of individualism (Goodson and Lindblad, 2011), have exacerbated the influence of testing and inspection, and many have raised concerns about their negative impact on student learning (Stevenson and Wood, 2013; West, 2010).

Until 2010 the significant ongoing educational reform climate in England meant PISA attracted little media attention or direct policy response, despite moderate student performance. However, this began to change when the UK Labour government of the time announced that performance in PISA 2009 would provide an informal baseline from which to measure the success of their reforms; and, subsequently, Michael Gove’s Education White Paper in 2010, under the new Conservative–Liberal Democrat coalition government, mentioned PISA as justification for further reform. In Germany, however, a concern with student achievement on PISA 2000 (OECD, 2001) stimulated considerable public debate and, some argue, provided impetus to significant policy reform (Ertl, 2006; Waldow, 2009). From the mid-2000s an emphasis on transparency and accountability meant German education reform was increasingly steered by output evaluations. National educational standards in subject-specific competencies, similar to PISA, were developed, with Länder-based comparative standardised student assessments following in 2009. However these standardised assessments have remained relatively low-stake partly because they receive very little public attention compared to the performance of German schools in the PISA tests – something largely ignored by teachers in England. Curriculum and assessment addressed subject competences by focusing on subject application and everyday relevance, and this led to an increase in cross-curricular pedagogy and in-class student differentiation. Testing a representative student sample in each state was introduced to monitor variations in student achievement of the national standards, thus avoiding high-stakes for students and teachers; national monitoring is summarised in a biannual report on education, Education in Germany.

Since 2006, school development has been supported through the regular and systematic evaluation of schools through Länder-based school inspections. Because reports are available to school management and the responsible local school administration alone, and are not linked to rewards or consequences, compared to England these are also relatively low-stake (Kotthoff et al., 2015). However, greater autonomy for school heads in relation to budget, staffing and teaching programmes has given them increased opportunities to respond. Whilst inspections in all schools are now carried out regularly in Germany, the use of such inspections over many years in England has helped build extensive databases of school performance data. This has led to the development in England of school inspections that are targeted more and where the results from previous inspections, school self-evaluations and student achievement data are used to visit schools proportional to their need. This means poorly performing schools are inspected more frequently than other schools. Meanwhile, other systematic change in Germany has sought to increase heterogeneity, including the introduction of comprehensive Gemeinschaftsschulen, ending the tripartite system in some areas, all-day schools which extend the school day, and preschool services which offer increased language support to children from non-German speaking homes. Interestingly, the underachievement of migrant groups on PISA 2000 came as something of a surprise in Germany, whereas such groups had long been identified and monitored using comparative test data in England. It is important to note that at the time of this study schools in England were just emerging from a period of significant prescription, both of curriculum content and teaching approach, through national government strategies. During this period the focus in England has been on skills rather than competences, with mathematics renamed ‘numeracy’ to reflect this, and to some extent this sidelined subject cohesion, allowing concerns about fragmentation to emerge. In Germany curriculum content remains regulated by official textbooks, particularly in maths and foreign languages, which are based on educational standards, whilst teachers are left to decide how best to teach.

There are significant differences in how schools in each country relate to what Ball (2013) called policy technologies. The market focus in England encourages parental choice linked to the performative appraisal of schools through the publication of high-stakes national standardised test and inspection outcomes, as described above. Thus, school management is dominated by the surveillance of teachers and students in an attempt to ensure outcomes are met. This brings with it job insecurity, where teachers are only as good as their students’ results allow, and tightly controlled working conditions which combine teachers’ pastoral and subject responsibilities. These technologies combine to create a climate of relatively hard governance with the emphasis shifted strongly from professional autonomy to accountability.

An increased market focus in Germany has begun to allow some parental choice of school in some states, leading to an increased movement between schools. In some metropolitan areas of Baden-Württemberg more than half of students now transfer from primary schools to Gymnasien – which have therefore become the main secondary school type. However, standardised testing and inspection results are not made available to support parental choice; these results perform a soft governance role instead. Indeed, the job security which accompanies their civil servant status coupled to teachers’ relative autonomy allows teachers much greater control over their working conditions than in England and provides a restraint on pedagogic reform. In some Länder other structural restraints on teachers remain, including the emphasis on teachers’ academic roles in Gymnasien and combined pastoral and subject roles in Hauptschulen.

In mathematics, the mean PISA performance at age 15 for Germany in 2012 (OECD, 2014) was better than previously. At about 17.5% each, both low and high achievers performed significantly better than English students. Amidst this overall picture, however, despite slight improvements in PISA scores for average and weak pupils in Germany, the strong effect of social class and migration status remained; as yet, the impact of the education reforms described above on classroom activity and student performance is not clear. Since 2006 the performance of English students overall has been fairly consistent, but within this the share of low achievers has increased to 22% in 2012 whilst that of high achievers has declined to just over half of that. This appears to contrast with a slight increase in 2012 in the proportion of pupils in English state schools achieving five or more GCSE (the principal subject examinations which mark the end of compulsory schooling) or equivalent passes. However, whilst four out of every five students with higher prior attainment made the progress expected of them between the ages of 11 and 16, only one in five of those with lower prior attainment did so (DFE, 2013). Thus despite the complexity of comparing different assessment outcomes it seems higher achievers respond much better to mathematics teaching than low achievers in England, whilst the response is more even in Germany.

Mathematics teaching

Rather than seeing pedagogy as an idealisation, a set of recommended teaching practices, for Bernstein pedagogy was a phenomenon which emerges from the activities of teachers and students together, each subject to a number of competing factors and influences as they negotiate and pursue their various goals (Bernstein, 1990; 1996; 2004). However, not only do teachers face many demands – as Ball said, ‘teaching has always involved making decisions within a complex and rich field of contradictions, dilemmas and priorities’ (Ball, 2006: 83) – they also do so in social situations where the consent of students to act in accordance with their teachers’ expectations is not guaranteed: it has long been understood that pupils can withdraw their cooperation in overly challenging lessons (Doyle, 1983; 1986). Hence, we can regard classroom activity as constructed by teachers and students together (Dowling, 1998) in sites of competing influences and goals (Ball, 2006; Kelly et al., 2013). As such, some like Apple (2012) suggest these processes allow low achievers to become complicit in their own marginalisation.

In order to analyse pedagogy within this complex picture, Bernstein (1990; 1996) separated discursive practice working towards instructional goals, largely those associated with the subject being taught, from that concerned with regulatory goals, including activity which promotes students’ willingness to accept responsibility for their actions and to behave in a sociable manner. For Bernstein, power is embodied both in the way boundaries between the different objects of these discourses are established and policed, something he called classification, and in who has control over decision-making which he called framing. For example, the ways in which mathematics as a subject is defined and differs from other subjects is a matter of classification, as are the roles adopted by teachers and students respectively. When classification is high, each of these is clearly separated, but in cross-curricular work, or when teachers and students engage in problem solving together, this lessens. Similarly, framing is high when teachers determine the content, sequence and pace of teaching, but appears to reduce as students become more involved.

Clearly the nature of mathematics as a subject is central to the formation of pedagogy. Bernstein separated subject discourses into vertical and horizontal (Bernstein, 1999); the vertical discourse is concerned with increased subject specialisation and complexity (Hazzan and Zazkis, 2005); the horizontal links subjects to their presence and use in other contexts. It is the understanding and ability of students to work with abstract ideas which ultimately leads to exam success (Cooper and Dunne, 2000). In mathematics, particular difficulties have been identified for the vertical development of some students, especially so for lower achievers, when they first encounter algebraic ideas at around the age of 13 (Malisani and Spagnolo, 2009) – the age of students in this study. Hernandez and his colleagues (Hernandez et al., 2011) pointed to those occasions on which the stratification of achievement widened alarmingly; however, Knipping and her colleagues (Knipping et al., 2008) placed more emphasis on how mathematics teachers contributed to this widening gap by, for example, working too hastily and assuming low expectations.

As a way of helping children better understand mathematical ideas, teachers often link them to everyday examples, thereby adapting the horizontal mathematics discourse. Indeed, this can be in response to student disquiet when challenged. However, such an emphasis can limit students’ vertical subject engagement, supporting Apple’s argument above; especially as Bernstein (1975) identified that many pupils, particularly those from working-class backgrounds, favoured mathematics grounded in concrete examples and which emphasises relevance.

With this account of mathematics teaching in mind, the focus of this study is on exploring differences in the teaching of mathematics, as emergent practices in sites of policy negotiation and enactment, analysed using Bernstein’s account of pedagogic practice, between the English and German schools who participated. In part, our analysis of each context concerns the nature and development of the vertical mathematical discourse and its relation to both adapted horizontal discourses and mathematical success. We then relate identified differences to the dominant influences on each context, thereby elaborating two particular versions of the extended policy cycle described earlier: one relatively hard in educational governance terms compared to the other.

Method

To analyse teaching we explored practice by identifying teacher and student goal-directed behaviours, conceptualised as roles. Roles indicate a division of labour and carry an assumption of reciprocation; generally speaking, by acting as a teacher I expect others to act as students. Constructed thus, roles provide a social unit of analysis: they can be assigned, adapted or resisted by the actions of others. Teacher roles are situated within a particular subject, classroom, school culture, and so on, and they are the visible outcomes of mediations across many, sometimes contradictory, influences, including responding to the roles adopted by students. Hence teacher roles characterise the act of teaching whilst acknowledging its situated and reciprocally defined nature.

We categorised roles using Bernstein’s model of pedagogic discourse (Bernstein, 1990; 1996). This comprises an instructional discourse about curriculum content and assessment and the sequencing and pace of teaching, and a regulatory discourse concerned with managing the division of labour and promoting appropriate conduct in the classroom. By linking roles to pedagogic discourse we describe pedagogy, and by contrasting our analysis in and across national groups we compare pedagogy, finally relating this to national socio-political educational debates.

Context of the study

This study, within which this paper is set, is a comparison of two cases – England and Germany – chosen for their contrasting education policy contexts. We recognise that there are many influences on the construction of teaching and that inevitably our account is tied to the contexts in which the study was carried out. However, we acknowledge that our findings might resonate with the experiences of others more broadly, allowing naturalistic generalisations (Stake, 1995) to be made, and indicate worthwhile avenues for future research. An earlier paper, which reported on just two teaching episodes taken from this larger study, used the same method to explore elite mathematics education in both countries (Kelly and Kotthoff, 2016). Mathematics was chosen as a focus for this present study because it has been, as a core area of pupil learning with wider social significance and status, the site of much reform and contention over the past 25 years in England and 15 years in Germany. Local advisors identified the schools involved as those recognised in external evaluations as having been particularly successful in mathematics teaching. We sought such schools to avoid clouding comparisons with issues of competence.

The eight teachers in England worked in four local authority community colleges. There were two in each school, one of whom taught a higher set and the second a lower set. The eight teachers in Germany also taught in public (state) schools in the selective tripartite system; four worked in two Gymnasien, schools more academically oriented and which focus on preparing students for university entrance, and the remaining four worked in two Hauptschulen, schools which provide a general education to students whom it is expected will not attend university. The English community colleges included two city schools, one in a seaside town and one in a rural town; similarly, one Gymnasium and one Hauptschule were located in a city whilst the second of each was in a smaller town. We engaged in sixteen teaching observations of mathematics lessons in each country, with each participating teacher observed teaching and then interviewed on two occasions. Classes for pupils aged 12–13 years old were chosen to allow subject teaching beyond basic level to be considered whilst avoiding a focus entirely on preparation for school leaving examinations. The profile of the participating teacher group in each country was similar, both groups having equal numbers of men and women and containing three teachers with teaching experience of between three and five years, three of between five and 10 years and two with more than 10 years. All the teachers were identified as promoting high student attainment by their school managers.

The lessons observed in each country were divided equally between high and low achieving groups. Two lessons where observed for each teacher (in Germany these lasted 45 minutes and in England 60 minutes) and audio recorded during a three week period (in week 1 and again in week 3), during the summer term (when classroom norms and routines were fully established) by insider researchers who were native speakers of English or German. Each lesson related to a slightly different content area, and on each occasion both the teacher’s planning and samples of the pupils’ work were collected. Following each lesson the observer’s notes, the audio recording of the teacher in the lesson, and the planning and children’s work provided the basis for lesson analysis. Immediately following each lesson a detailed interview was used to explore and illuminate the varied goals and broader expectations which orientated teachers’ work, how they made sense of them and what they did to achieve them. Three boy–girl pairs of students, selected by their teachers as above, at and below average attainment for the class, were also interviewed to explore their understanding of and responses to the lesson.

Data analysis

The analysis presented below results from our categorisation of lesson observations, teacher interviews and student group interviews. We began by looking in our observations of lessons (using the audio recordings alongside the observation notes) for the varied goals and broader expectations (including those implicit in what teachers said and did) which orientated teachers’ work, how they made sense of them and what they did to achieve them. This was also done in the interviews, in response to the researcher’s recalled observations, teacher planning and examples of student work, and related to the student interviews. In the later part of the second interview, teachers’ values and beliefs were explored. The initial coding was carried out individually for each teacher before comparisons were made across teachers. Participant validation tested the verisimilitude of our resultant findings; that is, their resonance with the lived experiences of teachers.

Teaching mathematics in England

At the time of this study in the 2013–14 academic year, the common curriculum in England was principally utilitarian and accompanied by detailed guidance. However, because the primary aim was preparation for employment, the curriculum was not entirely focussed on the concrete and practical; progression towards symbolic representation and manipulation was also included. For the four lower set teachers we interviewed in this study, mathematics represented a box of tools which could be used to solve various, mostly calculation, problems deemed relevant to everyday life. In addition, these teachers generally believed students whose parents or carers’ work was mostly unskilled and manual came to school disposed towards learning things they saw as practical, useful and linked to their everyday experiences rather than abstract and esoteric ideas (Hatcher, 2012). This performance focus on utility, comprising both a notion of numeracy as a set of skills or procedures and an assumed linear route from learning to application, and the need to do well in high-stakes standardised tests and exams, dominated instructional discourse. Here it was assessment rather than the curriculum which drove pedagogy; despite the apparent skills focus, how the subject was tested had a greater effect on how it was taught. In the four higher sets the emphasis was more on mathematics as a highly classified subject, a set of assumptions and specific practices which worked together and were then applied, and particularly on past exam questions (at the end of the lesson). Hence, vertical development preceded horizontal application in past exam questions for higher achievers, to some extent paralleling the linear relation between basic learning and application in assessment exercises for low achievers.

Pedagogy in lower mathematics sets in England coupled this utilitarianism to a strong individualism and, broadly speaking, all four teachers focussed on the progress of individuals in the lessons observed, often supporting them independently of each other and emphasising the importance of test success to all. While for the most part children were taught procedures through whole-class instruction, this was followed with students working individually – albeit grouped with others tackling the same exercises – and practicing using these procedures, during which their teachers supported them. The focus on highly individuated teaching providing differentiated tasks to groups based on mathematics test scores meant that children had to rely on teacher support alone, and peer support was not encouraged. As a result there were few opportunities for students to learn from each other, be scaffolded by more knowledgeable others working alongside them, make links through serendipity, see alternative approaches, hear alternative accounts and explanations, see and use multiple representations or talk about their work. Setting, which collected weaker students together in classes, exacerbated this. Coaching fitted this procedural emphasis well and a horizontal focus on real-world examples attempted to make learning relevant; throughout, teachers emphasised making mathematics relevant and meaningful to students, and in this way learning always led to application. The stress on utility and the focus on relevance meant teachers often started with the familiar to help students make sense of ideas, brought everyday examples into lessons and used concrete representations to support the understanding of abstract ideas with low achievers. However, the main emphasis remained on test achievement and there was reference to this, and what was demanded, throughout. Here, usefulness implied useful in achieving grades, and national test strategies were sometimes emphasised in preference to a focus on mathematical development. As such, relevance was used as an aid to understanding mathematics so that it could then be applied in tests and exams rather than used in students’ everyday lives. However, this limited vertical mathematical development to procedural and instrumental understanding (Bernstein, 1996; Skemp, 1976). Teachers explained students’ dislike of the uncertainty engendered by problem- or learner-centred approaches to teaching (Apple, 2012) by suggesting such work was too challenging and led to frustration. Instead, and because teachers wanted to control pupil learning to ensure they did well, they used highly classified and framed teaching, allowing pupils little freedom to decide for themselves. Finally, groupings meant difficulty dominated the atmosphere, in the absence of the keenness of brighter students, and there was a critical mass of students in each class who resisted when responsibility moved from teachers supporting whole class instruction to students engaged in individual work. Students resisted challenging work, including abstraction and closed down tasks; student resistance took the form of distracting activities, where they misbehaved or created social crises involving disputes with their peers until the teacher reduced or eliminated the demands placed on them.

The four teachers in the higher sets supported student engagement and helped them think about problems mathematically, so that framing was apparently weak. Here, students assumed responsibility for their mathematical work and were expected to monitor their own understanding and progress. However, by emphasising mathematical thinking, teachers aligned students with the structure of mathematics and used this to frame their engagement. In contrast to lessons for low achievers, student pairings provided lots of time for them to discuss their thinking with each other and thus to engage with a strongly vertical mathematical discourse, exploring problems within the structure of mathematics and looking for patterns and generalisations, relationships and reciprocity. Together, these led to increasingly sophisticated levels of abstraction, while teachers scaffolded the precise use of a range of mathematical terms. However, links were also made to examination questions and this presented a difficulty for teachers, that of choosing between encouraging student choice in thinking things through and working towards specific curriculum aims, using examination questions to evaluate learning. The former was weakly framed: the second demanded stronger framing. We often found teachers began by asking apparently open questions but quickly closed these down and sought specific answers. Interestingly, in the higher sets we observed, students all engaged in identical work with little support for those experiencing difficulty, but this was countered by the many openings for pupils to learn from each other.

Teaching mathematics in Germany

Based on common Bildung standards, originally written to reflect PISA expectations but thereafter subject to ongoing discussion and revision, the core curriculum applied to all schools within Länder and, for those in this study, was Hauptschulen- or Gymnasien-specific: both emphasised the structure of mathematics and, to some extent, logical thinking. Curriculum goals, although often not shared with students, were largely regulated by the exercises provided in Länder authorised textbooks.

For the most part teachers held a relatively formal view of mathematics as an abstract, unified and true body of knowledge, which needed to be passed on to students as rules. In Hauptschulen the focus was mostly on calculation rules and effective procedures needed for everyday life and there was no abstraction beyond calculation to algebraic manipulations. The four teachers there combined their instructional concern for students’ mathematical development to allow them to function in society with a more regulatory concern for their personal development as sociable and consensual citizens, a role which also took teachers into supporting students’ wider social relations. Gymnasium teachers also expressed the strongly classified view that mathematics becomes more powerful as its abstraction from everyday contexts and concrete and enactive representation increases. In both school types, textbooks shaped the content and sequence of pedagogy – although teachers determined the pace of teaching – and included regular test preparation exercises which reflected the Bildung standards. Periodic class tests, set and marked by teachers but based on textbook material, were used to monitor student progress. In this, the curriculum, pedagogy and assessment were closely aligned.

As such, mathematics was considered best passed on to students as rules and pedagogy and was largely teacher-led. Highly framed whole-class instruction, comprising explanation and questioning, led to the identification of clearly classified rules and procedures which students, particularly in Gymnasien, were required to think through logically using their knowledge of the structure of mathematics. The use of these rules and procedures was then practised individually with teacher support, in textbook exercises, which often included routine application problems, until the process was fluent. Mathematically there was a strong emphasis on order, provided by sequential progression through state standards and approved textbooks. In Hauptschulen, relevance came from linking basic calculations to everyday contexts. Here the focus of mathematics teaching was almost entirely on horizontal development as students learnt rules and to use correct approaches accurately. In all this the teacher was very much in control, making decisions about lesson pace and monitoring and correcting student work. Thus both subject and role classification and the framing of the lesson were high. Whilst the expectation was for students to work individually on textbook exercises, in practice they supported each other in their groups as well as receiving individual support from their teacher. As such, in terms of regulation there was some sharing of responsibility between teachers and students within clearly defined roles. Nevertheless, throughout lessons there was a lack of challenge and this seemed to be the price of consensus between teachers and students; students cooperated provided they were not overly stretched. Finally, when there was a higher than usual proportion of students with migrant backgrounds, teachers’ first priority was often to ensure children could understand the language, by introducing and supporting students’ use of relevant vocabulary and linguistic structures, and this tended to sideline the mathematics teaching. For example, in one lesson we saw, the teacher introduced the topic ‘money’ to the whole class for the first 20 minutes, and this had very little mathematical content because she linked money to the everyday experiences of children and their families. The teacher’s aim was to introduce and explain new language content as the children provided suggestions of themes related to money.

Similarly, the high degree of classification and teacher framing continued in Gymnasien following whole class instruction as students practised using rules and procedures correctly in specific exercises, with teacher support. The tasks provided for students by the teacher throughout were closed, with single solutions largely requiring the accurate use of rules or procedures. As lessons progressed, tasks increased in complexity and abstraction, and teaching was thereby highly oriented towards a vertical discourse. Students internalised an unchanging set of mathematical ideas as rules and procedures through argumentation and challenge, and practised using these until they achieved fluency. Simple examples were used to support this process – for instance, cutting cakes to understand fractions – but this was the extent to which a horizontal discourse applied; unlike in England, mathematics was not reduced to transferable skills or similar. Rather, rules and procedures were mostly used in routine application textbook exercises.

Discussion

While the curriculum was highly classified in the teaching observed in both countries, for the lower achievers, those participating in this study in the lower sets in England and in the German Hauptschulen, it was concerned with mostly routine calculations and simple geometric problems that could be solved using known rules and procedures. Thus, the curriculum was located within a largely horizontal discourse with only limited vertical development involving small, incremental movements towards increased complexity or abstraction. However, for the higher achievers we worked with, in the higher sets in England and in the Gymnasien in Germany, the curriculum was almost entirely focussed on students’ vertical mathematical development, although the emphasis differed slightly. A concern that students engaged in processes of seeking and identifying generalities as well as making use of known rules and procedures was combined in England, whereas the focus in Germany was mostly on students’ facility in efficiently using rules and procedures of ever-increasing generality in solving problems. As an aside, the valorisation of theoretical over practical knowledge originated with Aristotle and is central to humanist views about education; that it was taken for granted in the mathematics curricula of both countries underlines the continuing humanist influence. Underpinned by this hierarchy, while also considered socially important, the vertical mathematics discourse therefore provided a legitimate means by which student achievement became stratified, explaining its gatekeeper role for post-school education and employment.

In the teaching observed in both countries, a largely vertical learning discourse preceded a horizontal assessment discourse, as Bernstein (1999) had suggested it might, and nationally important tests in each country provided the focus for the horizontal discourse. In England, the limited vertical classroom discourse with lower achievers quickly proceeded to a horizontal one, where mastering test-like problems became the focus of teaching. As such, the horizontal discourse became, in practice, the testing discourse. This was also partially the case in the classrooms in Germany and the higher sets in England, where assessment mirrored the curriculum and provided the main arena for application, requiring students to use identified rules and procedures to solve formalised problems; that is, routine application tasks of the form used later in tests in textbook exercises in Germany and past GCSE exam questions or similar in England. It would therefore be reasonable to link this explicit and specific assessment focus in mathematics classrooms in each country to improvements in PISA results in Germany, and GCSE and national tests results in England, respectively. PISA results were not important in England at this time: GCSEs were highly significant for pupils, teachers and schools and the teachers we observed strived to adopt approaches effective in maintaining expected progress from earlier assessments, and this seemed to work well with higher achievers. However, PISA results were more significant at the national level in Germany. Here it is possible that PISA-like exercises in textbooks contributed, together with other factors, to improving national results. In this, national standards shaped the Länder standards and the content of Länder approved textbooks, something which appears to have exerted a greater influence on classroom teaching than low-stake Länder-administered assessments and inspections, although this is a complex relationship deserving further study. In any case, care must be taken not to overstate the influence of PISA in the classrooms in Germany, because this was not a consideration for teachers in their everyday practice.

However, while in both the lower sets in England and the Hauptschulen in Germany in this study the focus was explicitly on teaching students how to tackle routine calculations and simple geometric problems set in everyday contexts, in reality pedagogy for all learners in England and with higher achievers in Germany was often more concerned with decontextualised calculations and the like. As noted above, its use was only in the context of past and similar test and exam questions. Meanwhile, for lower achievers in Germany straightforward mathematical exercises were often unambiguously set in commonplace contexts, and for all learners in Germany there was a consistency between teaching and assessment. This highlights one difference between the skills discourse which underpinned thinking about the usefulness of mathematics and took transfer for granted in England and the PISA competencies model which dominated in Germany. The former tended to separate learning from application but assumed this to be unproblematic, while the latter did not differentiate between learning and application. However, while on occasion everyday examples were used in the lessons in Germany we observed to contextualise new ideas and support learning, such as by explaining equivalent fractions using a visual illustration of cake slices, in England, and particularly with lower achievers, slightly more complex and abstract mathematics was often made accessible using concrete similes and relevant examples to aid understanding. One example of this we saw was the use of a visual representation of a balance scales to illustrate how both sides of an algebraic equation should be treated equally. However, tying mathematical understanding so closely to concrete examples could limit opportunities for vertical progression which depended to a greater extent on students’ confidence in moving away from the concrete towards abstraction. This distinction between the skills discourses in England and the competencies discourses in Germany would benefit from further elaboration.

Remaining with instructional discourse, textbooks played a strong role in framing and regulating the content, sequence, pace and assessment in the participating Hauptschulen and Gymnasien and helped ensure curriculum and assessment paralleled and consolidated each other, as described above. This included PISA-like routine calculations for Hauptschulen and more sophisticated and abstracted algebraic and geometric rules and procedures in PISA-like tasks in Gymnasien, some in everyday contexts, in textbooks and subsequent class tests. However, given its weight, in England, again, as described above, assessment more than curriculum framed pedagogy with lower sets, although the curriculum had slightly more influence with higher sets where teachers adopted a less direct approach. Meanwhile, in both Hauptschulen and Gymnasien the focus was on the application of rules and practising procedures. Clearly the hard governance climate in England, which gave the highest significance to test results, meant everything was mobilised towards maximising student performance on the tests. In the soft governance environment in Germany, both the federal separation of powers and continued teacher autonomy mitigated against direct top-down reform of the kind seen in England. Here, textbooks, as guides and resources to support teachers, provided a better way of shaping teaching.

Framing by teachers in the participating higher sets in England was apparently weak because they, the teachers, relied on students’ reasoning to provide structure for their thinking, directing this towards curriculum outcomes. This contrasted with the greater emphasis on instruction rather than exploration in the lower sets in both England and Germany, although reasoning was also central to teaching in Germany. The difference for higher achievers in England was that tasks were often open, with a variety of solutions pursued; students were encouraged to seek patterns or generalities and recognise the process by which they had done this, while teachers facilitated this process in relatively informal ways. In this sense, the introduction of examination questions was also done in an exploratory way because often a number of alternative approaches and solutions were sought. In contrast, teachers employed direct instruction with closed questions to frame work strongly in the lower sets. In the lessons observed in Germany this was also the case, the object being reliable and efficient use of fixed rules or procedures in textbook exercises. Rules were built systematically, and the emphasis on them may have contributed to an unchanging but also somewhat routine view of mathematics as a subject by students. Teachers were instructors, ensuring understanding and accuracy in similarly structured and regulated classrooms. Hence there were clear differences in pedagogy between England and Germany, although with higher achievers the focus was on mathematics as a strongly classified subject, and students were encouraged to take responsibility for their own learning, in both countries. Interestingly, teaching in English higher sets reflected a long tradition, as did the more authoritative teaching across student groups in Germany. Both practices endured in each country, regardless of reform, perhaps because they afforded strong affiliations and had deep roots (operating, as they did, at the philosophical level described by Schmidt, 2011); this is an area which would be an interesting topic for further research. In contrast, English lower sets appeared to have been those most affected by two decades of policy reform.

With this in mind, and turning finally to the regulatory discourses seen, roles were clearly defined for the lower sets in England observed in this study, with teachers often assuming responsibility for student learning within a highly framed pedagogy. This was an inevitable consequence of evaluating teachers on the basis of their students’ exam success. Students also received highly individuated teaching, affording few opportunities for peer support and the like, perhaps reflecting a long-held suspicion in English schools that collaboration amongst lower achievers rewarded laziness and, in any case, constituted a form of cheating. The resultant lack of opportunities for social support was exacerbated by the division of students into sets because this created a climate of difficulty and dependence; it also allowed small groups to gain enough influence in classrooms to resist challenge, denying this challenge to the whole class and thereby privileging mediocrity and preventing any engagement with vertical discourse. There was also a lack of challenge in Hauptschulen which helped to maintain student cooperation, and lessons in Hauptschulen were sometimes very contextual and social, particularly when there was a large proportion of migrant students. For the higher sets in England and the Gymnasien in Germanylessons in both countries were structured to allow students to work as a class, in groups and individually, despite differences in teacher framing, which provided for informal peer and teacher support. Pupils were expected to talk to each other about their mathematics and, at times, make decisions. As such, both positioned students as responsible for their own learning whilst emphasising the importance of students thinking things through for themselves, thereby providing cultures of challenge and student choice, promoting student independence and facilitating their engagement with vertical discourse.

To summarise, then, the analysis above is congruent with standardised assessment data indicating that higher sets in England provided more opportunities for and fewer obstacles to student achievement than is the case with lower sets: the approaches used were quite different for each of these two groups. In this regard, the similarities we observed in teaching higher and lower achieving students in Germany might in part account for the narrower gap in achievement between these two groups. However, it is acknowledged that both of these conjectures are in need of a fuller exploration.

Conclusions

The relationship between education governance processes resulting from the work of international organisations, national and regional policymakers and elsewhere and mathematics teaching in English and German secondary schools is complex. However, our analysis of a relatively small number of rich and detailed qualitative cases has allowed us to explore the outcome of this process in terms of the opportunities for achievement offered to different student groups. We contend that, as a result of hard governance pressures, PISA data was appropriated to defend national policy initiatives. As a result of these pressures, higher achievers in England were better supported in their vertical mathematical development than lower achievers; this was reflected in the widening gap between these groups in standardised assessment data. However, the softer policy change environment in Germany was shaped in direct response to PISA data, and afforded a number of similarities in the teaching practices, some established and some new, across groups of students. These similarities provided some indication of why the achievement of lower achievers was closer to that of higher achievers. However, given the limitations of this research, it is clear that these contentions would benefit from further substantiation.

What is also clear is that intricate interactions between international, national and local influences are not served well by simple representations such as that of a policy cycle. We suggest a somewhat looser model is in order; one which recognises the fluid, emergent and socio-cultural nature of social activity and which thereby accounts better for processes of governance and the formation of pedagogy.