Comparing episodes of mathematics teaching for higher achievers in England and Germany

Abstract

To illustrate similarities and differences in lower secondary level mathematics teaching with higher achievers and thereby explore privileging processes, we contrast a teaching episode in Baden-Württemberg, Germany with one in South West England. These have been selected from a larger study as typical within each region for higher achieving students in the school year in which they become 12 years old. These episodes reflect the dominant discourses and recent policy initiatives in each country. Descriptions of the episodes are linked to wider debates about the ways education practices benefit some student groups, and potential areas for further exploration are identified.

Introduction

Interest in those students referred to as educational elites who show the highest mastery of skills and knowledge has increased in recent years (Gaztambide-Fernandez, 2010; Van Zanten et al., 2015). This is not surprising since a concern solely for how educational institutions marginalise ignores privileging processes which are of equal importance in the reproduction of inequality. However, studies are seldom comparative, thereby allowing the similarities and differences to be explored, and few consider the specifics of how elites are taught; this is our focus.

Teaching is shaped by the societies and education systems in which it takes place (Alexander, 2000; Goodson and Lindblad, 2011; Osborn et al., 2003) and is subject to many levels of policy responding to both transnational influences and domestic agendas (in the European context, see Alexiadou, 2014; Nordin, 2014). As such, Alexander (2001) argues that the competing values and ideas underpinning teaching must be taken into account when making cross-national comparisons. He recommends a focus on pedagogy; the act of teaching and the discourses in which it is embedded, yet suggests difficulties in design have led to little comparative pedagogical research (Alexander, 2009).

In this study, we used an approach developed by Kelly (2013) to compare mathematics teaching in England and Germany. As 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 terms of agriculture, industry and tourism and with generally low unemployment. The English teachers all worked in local authority community colleges, and in this article, we report on the experiences of higher sets. The German teachers also taught in public schools within the selective tripartite system and here we consider those in the academic Gymnasien, which focus on preparing students for university entrance. In all we engaged in 16 teaching observations of mathematics lessons for pupils aged 12–13 years in each country, with each participating teacher observing teaching and then interviewing on two occasions. In this article, we limit ourselves to analysing two teaching episodes with higher achievers selected as typical of each country to contrast teaching and the values and understandings underpinning it.

Contrasting policy contexts: England and Germany

National differences in education contexts and cultures mean children in England and Germany can have dissimilar experiences of school mathematics; experiences can also vary considerably within countries, but this is not the focus of this article. Broadly speaking, the English tradition is for individualised teaching (Goodson and Lindblad, 2011) which, in recent years, has focused on promoting high student attainment on national tests because of the importance placed on this by influential bodies such as the schools’ inspectorate, Ofsted (Ball, 2008). As such, testing has become high-stakes with negative effects (Stevenson and Wood, 2013; West, 2010). A British Parliament Select Committee report (House of Commons, 2008) identifies shallow learning, short-term retention of knowledge, and pupil stress and demotivation as resulting from increased teaching to the test. In Germany, a concern with student achievement on PISA 2000 (OECD, 2001) stimulated considerable public debate and, some argue, provided the impetus for significant policy reform (Ertl, 2006; Waldow, 2009). Emphasis on educational outcomes increased and the curriculum was redesigned to focus on developing subject competences. Amongst other things, this has resulted in the introduction of national standards in subject-specific competencies similar to those underpinning PISA and of Länder-administered standardised assessments for student evaluation and school inspections to support school development, although inspection reports are only available to school management. However, compared to England, student assessments and school inspections are both relatively low-stake as neither are published or jeopardise teachers’ reputations or employment. Rather, given its historical impact, it is PISA performance that concerns schools more, something largely ignored by teachers in England but which remains significant to policymakers in both countries. Finally, 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. 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.

Recent reforms increasing parental school choice have meant that 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. Meanwhile, widely reported school improvement in England along with burgeoning school fees have fuelled a move of students including many higher achievers from independent to state schools; and most recently, some politicians have voiced a desire for the limited return to selective state grammar schools where there is parental demand. In both countries, it will be interesting to see how differences between parental expectations, and school capacities and capabilities are managed.

In terms of mathematics achievement at age 15, certainly Germany’s PISA 2012 (OECD, 2014) performance had improved on previous cycles and, with the percentage of both low and high achievers at about 17.5%, was significantly better than England’s in all respects. The mean English performance remained almost the same as in 2006 and 2009, although the gap between low and high achievers widened with the proportion of low achievers, at 22%, nearly twice that of high achievers. Yet in England, the percentage of pupils in state schools achieving five or more GCSE (the principal subject examinations which mark the end of compulsory schooling) or equivalent passes in 2012 increased slightly from 2011, whilst the percentage of pupils making the progress expected of them from the end of primary school in mathematics increased by about 4% from the year before (DFE, 2013). However, 80% of students with higher prior attainment made the progress expected of them, almost four times the proportion of those with lower prior attainment. Although this picture is complex, when considered alongside PISA findings one might reasonably surmise that mathematics teaching in England is much more successful with higher than lower achievers, whilst that in Germany appears more balanced.

Analysing pedagogy

Bernstein (1990, 1996, 2004) is persuasive in explaining the part school pedagogy plays in reproducing inequality. He suggests pedagogic discourse is made up of two elements: an instructional discourse regulating the content, sequencing and pace of teaching, and the approach to and focus of assessment; and a regulatory discourse which manages the division of labour and promotes appropriate conduct in the classroom (Bernstein, 1990). He invokes two principles to examine power relations in pedagogic discourse: classification, being the strength and weakness of borders between categories such as phases of school, subjects, or academic and vocational approaches, any movements in this and the work of boundary maintenance; and framing, indicating who has control over, for example, sequencing and pacing the teaching and learning interaction (Bernstein, 1996). When the control is with the teacher, the framing is strong; in weak framing, the student appears to gain more control.

For Bernstein (2004), pedagogic practice emerges when wider disciplines such as mathematics are adapted into school subjects in contexts where some forms of knowledge and ways of working are valued above others. This occurs at different levels, beginning with the development of national curricula and guidance. But adaptation within schools and classrooms in particular is not straightforward. For one thing, teachers face many varied demands, and classrooms can be considered sites of struggle between competing influences and goals (Ball, 2006; Kelly et al., 2013). ‘Teaching has always involved making decisions within a complex and rich field of contradictions, dilemmas and priorities’ (Ball, 2006: 83). Not only are classrooms contested fields they are also social spaces to which both teachers and students bring their own agendas and identities. And they are also physical spaces which are resourced to allow certain forms of practice and preclude others. In this study, we have sought to explore teachers’ mediations across these various situational factors in schools and classrooms, thereby analysing pedagogy using Bernstein’s frames.

Mathematics and mathematics classrooms

Bernstein asserts that school mathematics is organised as a vertical discourse (Bernstein, 1999: 159) of decontextualized abstract ideas, articulated using precise and specialist language. Success, including access to higher grades in exams, requires students to work with these abstract ideas (Cooper and Dunne, 2000) which, according to Hazzan and Zazkis (2005), are often removed from their everyday experiences and built on prerequisite ideas of increasing complexity. Mathematical ideas also objectify processes (for example, the idea of addition covers the processes of counting and combining groups) with students expected to move readily between the idea and objectified process in mathematical exercises. This aspect of abstraction proves problematic for some students (Mason, 1989), something Sfard (1991) puts down to the epistemological shift required when moving from a concern with objects to relationships. Hernandez and his colleagues (2011) suggest key moments in mathematical development result, where students become stratified; a significant one, beginning at approximately 13 years of age, the age of students taught in this study, concerns algebra (Malisani and Spagnolo, 2009).

Bernstein (1999) recognises that the horizontal discourses of everyday life and the workplace are also adapted in school mathematics, often in numeracy which, for the most part, requires an everyday faculty with calculation. In addition, teachers often mediate lower achievers’ experiences of the vertical mathematics discourse by situating mathematical ideas in commonplace contexts, thereby emphasising their usefulness. This approach, Bernstein (1975) argues, limits student engagement with the vertical discourse to a procedural level and rarely emphasises abstraction, but adds that many pupils, particularly those from working-class backgrounds, favour learning presented thus. Meanwhile higher achievers are more likely to begin with the vertical mathematics discourse when mathematising (making mathematical) routine real world problems.

Classroom activity is constructed by teachers and pupils together (Dowling, 1998). Knipping and her colleagues (2008) identify teachers as contributing to the stratification of achievement using pace, individualisation, low expectations and obedience, which restrict access to vertical development for lower attaining students and exacerbate the gap between them and more successful students. But for Doyle (1983, 1986), it is pupils who use their cooperation or its withdrawal to encourage teachers towards their preferred ways of working. Many pupils favour teacher direct instruction over supervised independent work, and, most likely, tensions in lessons increase when more responsibility is assigned to pupils, or the ambiguity of class work increases allowing pupils greater freedom of choice, or teachers make more abstract and esoteric demands. By such processes, Apple (2012) has argued, low achievers contribute to their continued underachievement and sabotage efforts to help them make better progress.

It is within this frame that mathematics pedagogy in England and Germany are compared. In each context, we look at the nature and development of the vertical mathematical discourse and its relation to both adapted horizontal discourses and mathematical success.

Method

The study from which these teaching episodes are taken is a comparison of two cases – England and Germany – chosen for their contrasting education policy contexts. Mathematics was chosen because, as a core area of pupil learning with wider social significance and status, this has been the site of much reform and contention over the past 25 years in England and 15 years in Germany. Local advisors identified the schools 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. Pupils were aged 12 to 13 years old (year 8 in England, grade 8 in Germany). This allowed consideration of subject teaching beyond basic levels when pupils were beginning to grapple with algebra, but avoided working with teachers focused entirely on school-leaving examination preparation. The profile of the participating teacher group in each country was similar, and all were identified as promoting high student attainment by their school managers.

The teaching episodes described here were selected as representative of those across the sample of 16 lessons observed in each country. Lessons were observed and audio recorded during the summer term, when classroom norms and routines were fully established, by insider researchers who were native speakers of English or German. On each occasion, both the teacher’s planning and samples of the pupils’ work were collected. Following each lesson, the observer’s notes, audio recording of the teacher in the lesson, 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 oriented 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 response to the lesson. Later, participant validation tested the verisimilitude of the resultant descriptions.

Teaching episodes

The two schools where these episodes take place were recommended by education officers as successful in teaching mathematics. Each is situated in a medium-sized town environment of mixed suburban character with a socially mixed catchment. The English community college caters for pupils aged 11–16, whilst the German Gymnasium teaches students aged 10–18.

Mathematics teaching in a higher set, England: fractions, equivalence, ratio and proportion

Pupils are divided into separate classes of about 25 students, called sets. In this lesson, the teacher is male, a mathematics specialist with six years teaching experience. This fairly typical lesson takes place in the spring term and lasts 50 minutes.

On arrival, once they have sat down, pupils immediately engage in a short puzzle or investigation. On this occasion, a relatively open-ended task is written on the whiteboard, ‘3×1½ = 3+1½, 4×1⅓= 4+1⅓, 5×1¼ = 5+1¼, what is the pattern here?’, and pupils have to make choices as to what to do and how far to explore. After about five minutes, the teacher asks two of the children to describe what they have been doing to the rest of the class, asking the others to comment on what they have heard and whether they had done the same.

Although teachers often provide learning objectives for the children, this does not happen this time. Yet even when objectives are provided, many lessons are allowed to wander from them to some extent. When this is raised following the lesson, the teacher insists that pupils are ultimately responsible for their own learning; he promotes this by encouraging them to track their own progress against curriculum targets so that they can see what they need to do next. But he recognises the tension in balancing student freedom to explore mathematics with his responsibility to prepare them for exam success, and finds this a constant struggle.

From the start, the children are put in pairs, with the teacher asking, ‘why have I put you in pairs?’ and then discussing how they might make sense of the topic together through talking. The teacher begins by asking pairs to find equivalent fractions, giving the example that, ‘a half is two quarters or five tenths’, which the children readily do. After a few minutes’ exploration, he asks them to look at one set of equivalent fractions: ‘what is the pattern here?’ The pairs work on this for five or so minutes and then the teacher calls the class back together. Selected pairs report on what they have found, some suggesting the pattern involves replacing numerator and denominator with their equivalent multiples, but the teacher holds back from revealing whether responses are right or not; rather, after each pair has offered an explanation of the pattern they have found, he reflects this back to the rest of the class to encourage the children to use the structure of the mathematics itself to ‘be convinced’ something is right. Now the teacher introduces the idea of ratio, suggesting that in each case the ratios of numerator to denominator are equivalent.

The teacher now presents a different exploration; ‘250/5, 250/50, 250/500, what is the pattern here?’ and again allows the pairs five to ten minutes working on this before following the same approach to feedback, but asking different pairs. As the pupils work, the teacher does not attend to the presentation of their work; he is happy for pupils to organise themselves so long as it is well organised. Indeed, he takes it for granted that pupils will be organised personally, remembering to bring their own equipment like a ruler.

Next he raises the possibility of including decimal denominators, and asks pairs to explore this further; ‘250/0.5, 250/0.05, what is the pattern here?’ From time to time, he makes suggestions but is happy to tell the children, ‘you might want to listen to me – decide if you need to; you can listen to me or just carry on if you think you are ok’. He then asks the children to consider what the ratios of numerator to denominator are in each case, and together they identify 250 : 5, which is simplified to 50 : 1; 250 to 50, which becomes 5 : 1; and 250 to 500, which becomes 1 : 2 or 0.5 : 1.

After five to ten minutes, he calls for class attention and focuses on discussing direct proportionality, which is where the ratios of numerator to denominator are equivalent. This means, he says, a change in one number is accompanied by a change in the other by a constant multiplier. He asks which of the explorations they have done so far involves directly proportional relationships and which do not. The explorations are now combined in a reciprocal process, undoing the previous activities, as he asks pairs, ‘how many fractions can you find where the ratio of numerator or dividend to denominator or divisor is 0.5 : 1, 5 : 1 and 50 : 1?’

As a final challenge, the teacher puts a GCSE question on the whiteboard: ‘The time, t seconds, it takes a water heater to boil some water is directly proportional to the mass of water, m kg, in the water heater. When m = 250, t = 600. Find t when m = 400’. He allows the children five minutes or so to engage with the question and talk in pairs about how they can tackle it, with some coming to a solution. Here the teacher emphasises the need to ‘work it out’ and the idea that this is part of mathematics is emphasised strongly. Indeed, pupils interviewed later describe the mathematics as ‘fun but hard’ and realise that their teacher, ‘wants us to think and to have our own ideas before he explains it’. They also praise their teacher because, ‘he gives us time to think’ and always ‘asks two or three people’ so that there is a need to think about which response is the right one. At such times, the teacher is showing how the mathematics itself, through its own internal logic, can tell the students whether it is right or wrong; as students explain their responses and solutions he is asking the rest of the class, ‘does this follow mathematically?’

Thoughts on this lesson

In this lesson, the focus is on mathematics for its own sake; a highly classified orientation towards mathematics as a set of assumptions and practices is embraced, rather than solely as a body of subject knowledge comprising procedures and rules. Mathematical knowledge is learnt alongside such processes, then mobilised in application tasks as shown, for example, with the introduction of the GCSE problem at the end of the lesson. Hence, to some extent, vertical development precedes horizontal application. Framing is apparently weak, with the focus of teacher instruction being facilitating student explorations and thinking as well as explaining new ideas, whilst students make choices and are positioned as largely responsible for their own learning; however, student activities are also more strongly framed by the teacher’s appeals to the logic of mathematics itself. Whilst student pairings provide lots of time for them to engage in discussion, explanation and evaluation, it is often in these discussions that students work within this strongly vertical mathematical discourse, exploring the structure and internal consistency of mathematics and being encouraged to generalise, although not always symbolically, and to think mathematically by looking at relationships and reciprocity; both of these prepare the way for mathematical abstraction. Teachers also introduce and expect students to use with precision a range of mathematical terms. An atmosphere of exploration, challenge and success prevails in the classroom. In terms of regulation, pupils are expected to contribute towards monitoring their own progress and development, but the lesson also makes links to examination questions.

In terms of difficulties, for teachers there is an obvious tension between encouraging an exploratory approach using weak framing and both working towards specific curriculum aims and using examination questions to evaluate learning which demand stronger framing. In other lessons we did find some evidence of teachers asking apparently open questions, but seeking specific answers and positioning students to focus in on these. Work is not differentiated within the class and there are few obvious formal supports for those finding things difficult. However, there are a number of opportunities for pupils to learn from each other.

Mathematics teaching in a German Gymnasium: comparing decimals and fractions

Pupils are taught by a trained Gymnasium tier specialist mathematics teacher, a male with over ten years’ teaching experience. In the lesson here there are 28 children in the class, balanced between boys and girls. This lesson takes place in the summer term and lasts 45 minutes.

Again, despite some differences, this lesson typifies many of those observed. The focus of this lesson is comparing decimals and later fractions with the expectation that students will establish and record rules for this and use these rules competently in mathematical exercises. The teacher does not mention curriculum goals or examination expectations in the lesson. Nevertheless, in his planning, the lesson is linked to Bildung standards or competencies set out by the Ministry of Education in Baden-Württemberg for each of the subjects taught in the different school types. These are short documents; the standards for eighth grade mathematics are set out on one page. When asked, students indicate that they take the study of mathematics seriously, not because of the usefulness of academic qualifications in the subject but because of the benefits it will bring them in later life, for example, in helping them understand tax issues or explain mathematics to their own children.

The lesson has a clear and transparent structure, and transitions between different elements are noticeable and led by the teacher. It proceeds at a reasonable pace and students have adequate time to complete the tasks. Throughout, the teacher asks questions to keep students alert: What is the mathematics we are dealing with here? What have we been dealing with in the last hour? What’s next? For the most part, both roles and division of labour in the classroom are clear; students are responsible for doing their work whilst teachers are responsible for instruction and developing students’ understanding and efficient application of rules in set exercises.

The lesson begins with a two-minute video of a gymnast which the teacher stops when the points are awarded by the judges. He asks the students to explain how the points are awarded and then asks, ‘What is the value of the decimal numbers compared to the whole numbers?’ Following a brief discussion, the lesson moves into a regularly used arrangement where children work, sometimes in groups and sometime alone, to find a mathematical rule. The teacher distributes decimal numbers on cards to student pairs, for example, 15,9 and 15,26 (in Germany a comma is used to separate units from tenths), who have to compare their numbers. After five minutes, together as a class, they are asked, ‘Is 26 more than 9?’ and discuss in pairs place values before coming eventually to the rule, ‘append zero before comparing values’, so, ‘Is 26 more than 90?’ Further examples follow including 1,5 and 1,05. After some discussion, the rule is written on the whiteboard. However, it becomes clear to the teacher from asking individuals that misconceptions still exist. He talks to the children about place values, moving to the right of the comma with tenths in the first column and hundredths in the second column using red-coloured marks for tenths and blue for hundredths with different two place decimals. The teacher then repeats the rule and the students write down the rule in the Regelheft, a book every child keeps solely for recording mathematical rules.

The teacher then asks, ‘Who has the feeling that they have understood everything?’ Not all signal that they have, but nevertheless now the students turn to textbook exercises concerning ordering times and later decimals not linked to time. This lasts about 20 minutes. There is no differentiation at this stage, with all children completing the same exercises. There is a clear structure to presenting their work which students are expected to follow in their books, although some students still express uncertainty whilst recording their answers. Differentiation is provided by one additional task for students who work faster. For the quick workers, the teacher asks, ‘Who wants to tackle the problem of the month?’ This involves comparing fractions and is a prelude to the next part of the lesson. Students work on this alone for a few minutes before presenting their solutions to the teacher and their fellow quick workers on the whiteboard.

A further period of group work follows. In the transition to this, the class is put into four groups, but this grouping is not accepted by all students so the teacher refers to the class group work rules which include that boys and girls need to be in groups together and work calmly and quietly. The task involves ordering trios of fractions with a few that are less clear, like ‘2/3, 4/6 and 1/10’, which some students still find very difficult. The children have ten minutes. In group work the students help each other whilst the teacher helps individuals and groups and applying rules. The teacher goes from group to group, helping when there are difficulties by explaining in detail, providing alternative approaches and examples for those who really cannot understand using a pizza model. Indeed, at one stage, a group of students themselves try to change their task of comparing fractions to a pizza comparison. This is acknowledged by the teacher. Interestingly, in the same task, another group of students chooses to use calculators to convert fractions to decimals to compare them, but generally students are encouraged to ‘seek’ conventional strategies already known to the teacher in group work rather than create their own strategies. Minor misbehaviour including interfering with group work is not sanctioned immediately, but students are reminded of the need to cooperate. Similarly, small disagreements are not addressed, and rule violations are tolerated to a certain degree. Throughout, the teacher is appreciative and friendly in his dealings with students.

The teacher stops the work by announcing, ‘One more minute’. Shortly after, the teacher waits until all students are quiet. He then calls on individual students in the plenary phase, demanding they justify their answers and trying to ensure each student is actively involved. The strategy of converting fractions to all have the same denominator is suggested by one student group and illustrated by the teacher using the whiteboard. The teacher asks about alternatives, and another group of students calls to convert fractions to decimals, but this is rejected by the teacher because it is too complicated and requires the use of a calculator. The rule of common denominators is now laid down in the rulebook. To conclude the lesson, a set of textbook problems ordering fractions similar to those considered in group work is given as homework with some additional tasks which are also explained.

Thoughts on this lesson

The curriculum goals and content offered, revised several years ago to better mirror PISA expectations, are similar across Gymnasien within Länder and, although often not shared with students, these are to some extent regulated by the exercises provided in authorised textbooks. Additionally, in the post-lesson interview, the teacher expresses a strongly classified view of mathematics as a unified and true body of knowledge which becomes more powerful as its abstraction from everyday contexts and concrete and enactive representations increases. As such, mathematics is best passed on to students as rules. To do this, highly framed whole class instruction identifies the rules and procedures classified as important which students are then required to think through logically using the structure of mathematics to help.

This high degree of classification and teacher framing continues as students practise using rules and procedures correctly in specific exercises with teacher support. The tasks provided for students by the teacher throughout are closed, demanding unique solutions to problems through the correct application of rules or procedures, with only minor deviations accepted. During the lesson, tasks gradually become more complex, building on previous tasks, and symbolic representations are introduced. In this, teaching is highly oriented towards a vertical discourse; helping students apprehend the fixed world of mathematics for its own sake, regulating and disciplining their use of computational processes and mathematical rules and, through argumentation and challenge, moving them towards increased abstraction. There is some brief horizontal linking of rules and processes by the teacher later in the lesson, using pizza examples to help children to compare fractions. But, in general, rules are only occasionally linked to everyday contexts considered relevant to students as an aid to their understanding, and there is no mention of notions such as transferable skills; more often they are linked to routine application exercises such as using decimals in gymnastics scores or timings.

Discussion

In both the English and German lessons observed, there is a highly classified curriculum focus on mathematics with high achievers. In this, the vertical mathematical discourse for teachers in England to some extent concerns student engagement with processes of seeking and identifying generalities, whereas in Germany the focus is more on students’ facility with rules of ever-increasing generality. Hence, although both have strongly humanist conceptions of mathematics, this is fairly authoritative and disciplined in form in Germany but, in emphasising rationality and choice, more liberal humanist in England.

In both countries, horizontal mathematical discourse concerns, as Bernstein (1999) suggested it might, the application of knowledge first apprehended in a vertical discourse, although on occasion everyday examples are used to contextualise new ideas in the German lessons observed. This discourse is most often found in consolidation and assessment tasks; routine application problems of the form used in PISA tests in textbook exercises in Germany and GCSE problems which are worked through in England.

In all of the lessons observed, pupils are entirely cooperative with their teachers. However, the apparently weak framing of pupils by teachers in England (and the tensions this brings for curriculum coverage) contrasts with the highly framed instruction found in Germany, although pupils in both countries are also strongly framed by expectations that they follow the logic of mathematics. Indeed, despite differences in teacher framing, lessons in both countries are structured to allow students to work as a class, in groups and individually. Pupils are expected to talk to each other about their mathematics and, at times, make decisions. As such, both position students as responsible for their own learning, whilst emphasising the importance of students thinking things through for themselves. But in England, tasks set are often open, with a variety of solutions encouraged. Students are encouraged to seek patterns or generalities and recognise the process by which they have done this, potentially engendering flexibility and creativity in constructing new procedures or strategies each time a new problem is encountered; the teacher facilitates this process in a relatively informal way. Whilst this seems at odds with the introduction of examination questions as application tasks later in the lesson, the teacher works alongside students on these tasks, positioning them as further challenges and opportunities for exploration. It is possible this approach will afford the development of students who see mathematics as the search for patterns and who are able to identify and express these in increasingly abstracted ways; serendipitously, it also seems likely they will be able to solve examination questions similar to those tackled in lessons and possibly less routine ones. Conjectures such as these (and those that follow), however, require further exploration. Tasks used in the German lessons observed are largely closed, and fixed rules or procedures are sought with understanding so that they can be practised and later used in exercises reliably and efficiently. These rules are recorded formally for future use and to provide a basis for building further rules. Again, it seems likely that students will come to regard mathematics as a fixed set of rules and procedures for solving routine problems. Interestingly, application involves identifying the same exercises in contextualised routine problems, and exercises that are based on standards written to reflect PISA expectations probably help improve student performance on these tests. However, we must be careful not to exaggerate the influence of PISA in the contemporary German classroom. Even though PISA initiated a number of far-reaching school reforms, most teachers are not aware of PISA when they teach. Rather, textbooks, based on standards influenced by PISA, work in the background to shape teaching. Meanwhile the teacher acts as subject instructor, ensuring understanding and accuracy, whilst the classroom relations are relatively formal and regulated by various class work rules.

Conclusion

In England, it is GCSE and not PISA results that are important, an outcome of their high-stake significance for pupils, teachers and schools, and the approach adopted by teachers is effective in maintaining expected progress from earlier assessments whilst providing GCSE-specific preparation. However, in Germany, PISA results matter at the national level and the focus on PISA-like exercises in textbooks prepares pupils well as seen in improving national results. Here national standards shape the content of Länder approved textbooks, something which appears to exert a greater influence on classroom teaching than low-stake Länder-administered assessments and inspections, although this relationship is in need of further elaboration.

There are clear differences in pedagogy between the two countries; a weakly framed process focus accompanied by a slower movement to the symbolic in England contrasts with a strongly framed content focus which moves quickly to the symbolic in Germany. But both focus on mathematics as a strongly classified subject and encourage student responsibility for their own learning. Hence the question remains, how do such differing lessons provide advantage? The curriculum for low achievers in both countries is similarly highly framed, so maybe with higher achievers – who can engage actively, regulating their own work on tasks – other aspects of pedagogy such as the balance of content and process, the move to symbolism and the degree of teacher framing matter less than they do with lower achievers; again, this is in need of further exploration. Interestingly in this regard, English liberal humanism reflects a long independent and grammar school tradition of work with elites as does more authoritative teaching in Germany. Both remain influential despite high-stake education reform in each country, possibly because they operate at a deeper philosophical level unaffected by programmatic and policy level changes (Schmidt, 2011), but this conjecture is also one requiring further consideration.

What then of the implications, and what areas might future studies explore? As of yet, these comparisons neither confirm nor explain the view that England is more successful with higher than lower achievers, whilst in Germany success is more balanced. For this, we need also to compare experiences of lower achievers in each country and relate them to those of high achievers. This would benefit from a more detailed consideration of the effects of pedagogy on that algebraic understanding deemed significant in various contexts and examinations. Put another way, it would allow an exploration of the specificity or flexibility of algebraic thinking privileged by different pedagogic approaches; surely an important consideration for those seeking to improve mathematical achievement for all.