pepin2014

Policy Futures in Education Volume 12 Number 2014 www.wwwords.co.uk/PFIE Using the Construct of the Didactic Contract to Understand Student Transition into University Mathematics Education BIRGIT PEPIN Sør-Trøndelag University College, Trondheim, Norway ABSTRACT In this article the concept of the Didactic Contract is used to investigate student ‘transition’ from upper secondary into university mathematics education The findings are anchored in data from the TransMaths project, more particularly the case of an ethnic minority student’s journey from his school to a university mathematics course taught at a large inner-city university in England Results show that there is a transformation (or rupture) of the Didactic Contract from school to university mathematics, and that this is likely to have serious consequences for students’ success, or failure, at this crossroads of students’ mathematical development, in particular if students are left to ‘bridge the gap’ from one contract to the other Further, the author argues that in order to help to smooth the passage, a refinement of the Didactic Contract is helpful, a re-conceptualisation in terms of Normative Didactic Contract and Personal Didactic Contract, and each needs to be considered at each level of development This is likely to raise awareness about the necessary conditions for success at both sides of the transition junction, and providing appropriate support for students at both levels is likely to provide access for more students to successfully stay in higher education mathematics Introduction Transition from School to Higher Education in Mathematics Education There has been widespread concern over the lack of preparedness of students making the transition from upper secondary to university mathematics It appears that students experience different difficulties at different stages, and develop different strategies to make these transitions successful At the same time, institutional practices afford, or hinder, students developing a mathematical disposition and an identity that supports their engagement with mathematically oriented subjects in upper secondary and tertiary education This links to most European governments’ concerns about student participation, and success, in mathematics In the United Kingdom (UK), the Smith Report (2004) stressed the need for more young people to continue to study mathematics, which, it was suggested, could be achieved by wider recognition of the importance of mathematics, improved teacher supply and professional development for teachers, and changes in the curriculum and qualifications pathways, so as to provide appropriate progression for all students (Brown et al, 2007, p 18) Internationally, the European mathematics education community has argued that in developed countries the interest in mathematics during the pre-age 18 school career has declined sharply in the last 20 years (Academia Europaea, 2007) In addition, analyses of ‘underused STEM (Science Technology Engineering Mathematics) potential’ have been carried out by the Organisation for Economic Co-operation and Development (1999, 2003), and these provide worrying reading for most European countries, which aim to increase the number of STEM graduates in order to further develop a dynamic and competitive knowledge economy (European Commission, 2002, 2003) 646 http://dx.doi.org/10.2304/pfie.2014.12.5.646 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015 Student Transition into University Mathematics Education There have also been concerns about ethnic minority students’ participation in higher education, and more generally, a debate about ‘cultural diversity and learning’ (Ogbu, 1992), that is, whether different ethnic minorities have greater or lesser difficulties crossing cultural boundaries at school and university to learn Furthermore, there is an argument that transition to higher education involves ‘identity shifts concomitant with increasing participation in the valued practices of the institution’ (O’Donnell & Tobbell, 2007, p 316) and that shifting institutions demand that students’ existing knowledge may be resituated in the new context (Lave & Wenger, 1991) The literature proposes different models for secondary–tertiary transition in mathematics (e.g Gueudet, 2008) or describes particular, sometimes innovative, practices (e.g Croft et al, 2009) However, there are few studies (beside the Manchester projects [1]) that directly address the widening participation agenda with respect to higher education mathematics engagement, learning and identity development The literature generally shows that transition is often a ‘threat’ to progress, especially for certain students, and that efforts to align practices on either side of transition can help (e.g Hoyles et al, 2001) However, mathematics at the secondary/tertiary interface is believed to be particularly problematic for pedagogy, especially for proof and mathematical communication (Nardi, 1996; Hoyles et al, 2001), and ‘formal’ mathematical thinking generally This is likely to have implications for students’ success or failure at this stage of their learning mathematics In the next section I explain and develop the proposed theoretical frame: the Didactic Contract (Brousseau, 1997) In the subsequent part the research design is explained and the project described on which this study is based Following that, the findings are presented, and finally the discussion of the findings and conclusions, including the theoretical insights and refinements of previous theories Theoretical Framework: the Didactic Contract The contrat didactique is a French term for the Didactic Contract, a construct created by Guy Brousseau (1997) and defined by him as the set of reciprocal obligations and sanctions which each partner in the didactic situation imposes, or believes to impose, explicitly or implicitly, on others, and those which are imposed upon him/her, or s/he believes which are imposed on him/her (Brousseau, 2010, p 6, my translation) This implies another construct, that of the ‘didactic situation’ (and later ‘adidactical situation’), which is defined as a situation where an agent, for example the teacher, organises an intervention that manifests its intention to modify the knowledge of another agent, or causes it to develop The second agent, for example, is the student who is allowed to express him/herself in actions (Brousseau, 2010, p 3, my translation) Thus, the Didactic Contract typically implies the ‘modification’ of knowledge Elsewhere, Brousseau (1997) describes the Didactic Contract as ‘a system of rules, mostly implicit, associating the students and the teacher, for a given piece of knowledge’ (p 15) The concept of the Didactic Contract has been widely used in mathematics education, and mainly in the French-speaking research communities and in Canada, as for a long time Brousseau’s writings were only available in French It has been used to describe the dynamics in the mathematics classroom where students, coming from traditional teaching strategies, were asked to perform problem-solving and investigational activities Sensevy (in Sarrazy, 1995) argues that in such situations new social norms have to be established between the teacher and the students, and thus the Didactic Contract is to be redefined One interpretation of the contract, and which is relevant for this study, relates to the sharing of responsibility between the teacher and the students in terms of knowledge creation According to the theory of didactic situations (Brousseau, 1997), the institutionalisation process is the process by which knowledge built by students in the classroom in the solving of problems is linked to the institutional forms of knowledge that the teaching aims at This process is a fundamental process under the responsibility of the teacher, who is the warrant in the classroom 647 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015 Birgit Pepin of this institutional knowledge (Artigue, 2002) Thus, when a student enters university, it appears that the Didactic Contract is no longer the same (Artigue, 2007) De Vjeeschouwer and Gueudet (2011) use Chevallard (2005) and his notion of ‘praxeology’ to develop a deeper understanding of the institutional framing of mathematical knowledge Combining Brousseau’s Didactic Contract and Chevellard’s perspective leads them to three levels of contract, in a given institution (p 2): • a general contract independent of the knowledge (e.g student responsibilities); • A Didactic Contract for mathematics more generally (e.g requirement of rigorous proof); • a Didactic Contract for a given content (e.g calculus) In order to develop a deeper understanding of the institutional framing of the transition to higher education mathematics, I explore mainly the first and second levels of De Vjeeschouwer and Gueudet’s identified levels As this article aims to focus on the sociological/sociocultural implications of the potential change of the Didactic Contract in higher education, in-depth investigation of the mathematics was seen as inappropriate (for further insights in terms of mathematical content see Gueudet, 2008) Based on this body of research, the main question for investigation in the present study is the following: Does the Didactic Contract change at the transition from school to university mathematics education and, if so, in which ways? Research Design The TransMaths project at the University of Manchester [1] investigated how student experiences of mathematics education practices may interact with various (identified) factors to shape students’ development as learners of mathematics, their dispositions and their decision-making at this crucial point It appears that students experience different difficulties at different stages, and develop different strategies to make these transitions successful At the same time, institutional practices afford, or hinder, students developing a mathematical disposition and an identity (Boaler & Greeno, 2000; Pepin, 2009) that support their engagement with mathematically oriented subjects in upper secondary education and moving into higher education The project studied students’ identity in relation to their experiences of different mathematics learning and teaching practices Based on the work of Cobb et al (2009), I propose that students’ developing identities ‘can be made tractable for empirical analysis by documenting students’ understandings and valuations of their classroom [or institution] obligations’ (p 223) The research design (of the whole project) was based on a theoretical framework of mixedmethodology design involving longitudinal survey of outcomes, student biographical interviews and case studies of practice The data chosen for analysis reported in this article were the following: • individual biographical interviews (with linked case-study data) of students over a period of two years, in this case Simar (a pseudonym) and his friends; • case studies of universities, in this particular case one large traditional university in a large city in the south of England, City University (a pseudonym): these include observations of lectures, tutorials and environments where students met; • document analysis of policy and curricular documents relevant to the cases: in this case, documents related to Simar’s school and university; • interviews with participants, in this case head teachers and teachers in Simar’s secondary school, and lecturers and professors at City University In order to develop deeper insights into students’ experiences at the transition from school to university mathematics education, the study was based on sound methodological principles These included: • the principle of the ‘extended time’ survey: in the case of the students, this involved following their development from school (e.g observations, student biographical interviews, interviews with teachers and head teachers) through the first days at university (data point 1: case observations, individual student interviews, interviews with lecturers and student transition support staff) and throughout the first year (data point 2: observation and interviews – individual and focus group) and into the second year (data point 3: interviews – individual and focus group); 648 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015 Student Transition into University Mathematics Education • the principle of ‘continuity’: selected students were ‘followed’ inside sessions (e.g lectures, tutorials) and outside these sessions (informal group discussions/sessions); • the principle of ‘seeing it through the student’s eyes’: when following the students and observing their ‘work environments’, data collection was conducted as far as possible through the ‘lenses’ of the students’ eyes and their work practices (e.g observing them in lectures writing, listening, discussing with their peers etc.); • the principle of ‘reflective investigation’: this included discussing what they had written down, or submitted or said before, in a reflective discussion In addition, and in order to counter threats to the validity of the data, and to further strengthen the rigorous data collection across different sites, the teams of researchers/investigators changed in the following ways: always one person was responsible for a particular case (e.g the City University case), but the second person changed from data point to data point This allowed each investigator to see different cases in situ, and in turn to reflect on his/her own case and its students In terms of data analysis, I have analysed the qualitative data (e.g student interviews, case study data, documents) on the basis of my understanding of the Didactic Contract (as defined by Brousseau, see earlier) In addition, and using open coding with an ongoing formulation and refinement of categories (Strauss & Corbin 1990), I identified the various aspects of the Didactic Contract that students experienced in the different institutions The results are anchored in particular in the data taken from interviews with Simar and his friends (focus group), and the observations I made in his secondary school and university (City University) This methodology clearly has limitations Each case team consisted of at least two investigators, who were conducting the data collection at each data point and who had experience with studies of this kind: typically one lecturer investigator and one PhD student investigator Whilst this helped in the rapport with students as well as professors/lecturers, it had limitations in terms of scope of the investigation: it was clearly not possible to understand and research the full range of students’ experiences at transition, and only with a selected number of students, typically those who volunteered to work with us Hence, the results cannot be generalised across the UK, or indeed across other sites in the same city The in-depth nature of the investigations could however deepen understandings of what students may experience when transiting to higher education mathematics courses, and in which ways this may influence their choices at university Findings: the Didactic Contract from school to university mathematics learning In the two subsequent sections, Simar’s learning contexts at school and university are described and explained in terms of: the general Didactic Contract of knowledge and the Didactic Contract for learning mathematics (with no specific content) (De Vjeeschouwer & Gueudet, 2011) The Context and Didactic Contract at Simar’s School Simar’s secondary school (ages 11-18) is a high-achieving comprehensive school in a large city in the south of England It has an intake of approximately 300 students in the sixth form (upper secondary education), and approximately half of them are doing mathematics Most of these students have been in this school since year 7, that is, throughout their secondary schooling The student intake comes mainly from the neighbouring areas, and 78% of the school’s population are of Asian heritage (Indian, Pakistani, Bangladeshi, Sri Lankan), in addition to about 10% African Caribbean students, and only about 5% indigenous white students The school is very popular, as students generally get good results and are able to go to the university of their choice One of the head teachers talked about a ‘purposeful learning environment’ and the focus on learning as their main key for success What the school has been very successful is establishing a very kind of purposeful learning atmosphere around the school If you walk around the school, the ethos of the place is fantastic The relationships between pupils and teachers are So yes we have a supportive parent group but that’s built upon the success of the school … 649 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015 Birgit Pepin the learning ethos that we place on the whole way we run the school, I think is absolutely essential, and Ofsted [the UK’s official education inspectors] have said that You know, it’s the ethos and the relationships mean that, in the classroom the focus is on learning (HTeacher, SK) He also emphasised the teacher–pupil relationship as a reason for the school’s success, in particular an ‘atmosphere of trust and mutual respect’ I think it’s one it’s a long-term established ethos, where brothers and sisters have come through the school; it is a feeling of mutual respect There isn’t a ‘them’ and ‘us’ feel, you know the teachers, again referring back to Ofsted, the teacher–student relationships are excellent If you walk around the corridors you don’t hear lots of shouting, you know there is a kind of mutual respect, and little things like, we’ve invested heavily in the fabric of the school because our break times and things because all the form rooms are open to the children all the time, so the classrooms are open, and the kids walk in and out Cos we trust them, it’s their form rooms and the kids very much feel it’s their school and there is an understanding that it isn’t ‘them’ and ‘us’, that we’re actually on the same side Taken years and years and years to develop that, but the ethos of the school I’d say is really key (HTeacher, SK) The school’s sixth-form handbook (for mathematics) provides guidance and help Among other things, students are encouraged to Get involved in the lessons You need to be responsible for your understanding, which means that you need to be brave and have a go at offering your ideas and thoughts during class examples and discussions Don’t worry about getting the answer wrong, your teacher will be able to explain to you any mistakes you may make in your thought process Discuss your work with other students If you can explain how to something, it means you understand it! Ask for help when you need it There can be nothing worse than leaving a lesson and knowing that you are going to struggle with the homework because you did not understand the work in class (Extract from the official Student Handbook, SK) Thus, the general Didactic Contract and environment can be characterised by the development/establishment of an atmosphere of ‘trust and mutual respect’ and a ‘purposeful learning environment’, where students are supported by the teacher to learn to take responsibility for their own learning by discussing their work with teachers and peers and by explaining to others They were also encouraged to seek help from the teacher, for example in terms of pacing their learning and getting ready for examinations The Context and General Didactic Contract of Simar’s University – City University The stated aims of City University were stated in the handbook as: [City University] seeks to teach its students to the very highest academic standards, drawing in creative and innovative ways of its research to ensure that students, when they leave us, have the mathematical skills most likely to be useful to them and their employers In particular, these include fluency and accuracy in elementary calculations, ability to reason clearly, critically and with rigour, both orally and in writing (Handbook, Part 3, p 1) City University has traditionally had an intake of about 70-100 students in their mathematics department, but more recently increased their intake to about 300 The intake of City University is varied, and comparable to that of Simar’s school: many from ethnic backgrounds who have lived in the large city all their lives; and there are also many international students Extracts from interviews showed that students talked about City University as having status, and students generally felt comfortable in an university close to their family or relatives The first-year teaching staff (professors), who seemed to be the ‘influentials’, were mostly experienced professors who had been at City University for 20 years or longer Indeed, the Programme Director for mainstream maths programmes mentioned that it was important to have a first-year team that ‘sings from the same hymn sheet’, so that students learn ‘from day one’ that 650 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015 Student Transition into University Mathematics Education they are not in school but in a university mathematics department This included a clear distance between students and lecturers, which was also mentioned by Simar I think it’s, it’s more like the learning here is more general like in a way, like in sixth form it was more personalised kind of You kind of, you was closer to the teacher, you was, you had constant like, you was talking to them – you was after school you was chatting to them You saw them around, like here it’s so funny ’cos when we see the lecturers walking around it’s like they’re like celebrities ’Cos we haven’t got, we haven’t quite got that personalised you know, thing with them so they’re from a distance you know ‘That’s Professor …, that’s Professor wow!’ You’re like wow, they’re about So I suppose it’s less personal in a way (Simar, DP5) The main, and most ‘esteemed’, pedagogic practices were lectures, usually in halls of up to 300 students Lecturers would typically produce handwritten notes projected onto a screen and talk students through the content Students would copy those notes, it seemed most of the time with little or no understanding Simar talked quite enthusiastically about one of his lecturers, and what he (and his peers) would expect from a good lecture, which resembled very much what he was provided with at school in terms of support (e.g notes) S: Geometry: the feedback we got from geometry is, basically he’s faultless He’s brilliant, he’s excellent; the lecture’s engaging, the notes are available – clear notes You can use the notes for the coursework – Int.: The notes are handwritten? S: Yeah handwritten notes yeah And they actually, you can see the kind of proofs – he doesn’t give too much away, but it’s just enough to get you thinking in the coursework’s, which is excellent Because like, what students are finding is that they can go, because the lectures, they’re not gonna walk around with the lecture twenty-four, seven are they? They need something to take away from the lecture and you know, they’re gonna ready at home, they’re gonna read it, and they understand it And they can go to the tutorial, ask whatever questions and the questions with confidence, knowing that they’ve done well like because everything’s there, available They don’t need to go anywhere else, and if they do, the tutorial’s available or the office hours So really it’s probably one of the best Int.: So you think they understand because in the lecture he explains well, or you think they understand because the, it’s so well prepared and written out? S: I think mainly it’s mostly well prepared, definitely, and then to accompany that, the lectures are brilliant as well Yeah it’s really, really kind of funny He catches your interest (Simar, DP5) However, often students did not feel provided with learning opportunities at City University, or strategies on how to learn, and they also compared this to what they had expected at school Yeah, trying to catch up with what he’s saying and he’s just talking, it’s like he’s just, he’s just writing, and writing and writing, and all your focusing is, on trying to write everything down but you don’t have a good Like I know, at first when I started university I thought lectures are like lessons, you learn in the lecture but you don’t really In the lectures you sort of get an understanding but you have to more, but with his lectures I don’t understand anything in the lecture I have to go away and it after he just causes more work ’cos you spend the whole lecture (S5, focus group interview, DP5) Talking about a particular lecture, students realised that they had to identify and seek their own help strategies if they wanted to survive in this kind of environment S3: Yeah ’cos sometimes you know you’re writing so you’re trying to catch up with his speed ’cos sometimes he’s really fast, so you’re trying to catch up with his speed of writing, and if you start thinking of, ‘let me try and understand’, by the time you’re understanding, he’s already moving on So you try to write and understand at the same time, so you can’t both of them at the same time which is a bit difficult So you have to always tend to, so if you have any questions, you’re not too sure when to ask them ’cos if you rise up and you’re like, you know what he’s gonna continue writing so let me just … 651 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015 Birgit Pepin Int: Have you ever asked any questions during the lecture? S3: No, no, ’cos then you always go back and then, the only way I understand to my work is, when I’m doing my coursework and there are help questions to your coursework, and this is how I tend to them more and during the tutorials, and I think the tutorials and the courseworks are more helpful than, the lecture The lecture you just get the notes (Focus group interview, DP5) In particular, coursework and tutorials were expected to help students to give them feedback about their developing learning However, students often felt that there was a ‘knowledge gap’ between what they thought they had learnt at lectures and the expectations from coursework I think it’s really time-consuming yeah Obviously we, I think it’s like even though we say, ‘ok each coursework’s only worth 0.1%, but I find that it takes up a lot of my time remember him saying at the beginning of the term, ‘oh you should take about an hour attempting the questions and then come to tutorial’, and I’m thinking, once I was up till six in the morning and it was on the very first coursework so it should have been the easiest one, trying to it using the lecture notes, using three books, using the Internet, and it makes me think that if I need that much, just to a coursework, what the hell I need just to understand it? No seriously, without having to do, this is learning to the coursework so what about just understanding it in general? If it takes me that much time, there’s not enough hours in the day to that So if you think about it if we had to that for every single module, that’s just trying to the coursework (Focus group interview, Simar and friends, DP5) Because I think like when you’ve, from the lecture notes and then when you start the coursework, the like knowledge gap, there’s a bit of space in it where you have to make the links yourself And that does take time, and because of the time constraints that we have, like it’s on a weekly basis that you get the coursework don’t you? (Simar, DP5) On the basis of video footage of selected lectures and pre- and post-video stimulated recall discussions with lecturers, one could identify meanings that were attached to particular practices Particular lectures reflected the kinds of things that a ‘rigorous mathematician’ may need to learn: • ‘reasoning and proof’-based thinking and practices were expected to be developed through geometry and linear algebra; • ‘procedural fluency’ (methods) was seen to be developed through calculus; • practical and context relatedness was seen to be developed through statistics Thus, the general/pedagogic Didactic Contract and learning environment can be characterised by the ‘distance’ between students and lecturers that was established ‘from day one’ at City University Students were expected to find and build up self-helping strategies to follow and understand the mathematics taught in lectures, and they in turn developed criteria with which to assess the different lecture styles (e.g handwritten lecture notes were best) All students in the study found lectures ‘hard’, and they (at least Simar and his friends) subsequently built up and called upon friendship/working groups to help to understand the tasks required for the coursework In terms of the Didactic Contract for mathematics, it can be argued that there was a clear ‘institutional didactic contract’ at City University, made explicit in discussion with lecturers and students, and mediated by particular practices This contract was about helping students to become a ‘rigorous mathematician’ and attain the ‘very highest academic standards’ Becoming a ‘rigorous mathematician’ included making sense of the mathematics in lectures, and different lectures (different mathematical topic areas) appeared to provide the key to particular competencies (e.g reasoning and proof were developed through algebra) However, how students were expected to learn and develop these was not clear It can be argued that this change of didactic contract from school to university appeared to necessitate students becoming more independent learners Discussion of Findings According to Chevellard’s ‘praxeology’ approach (see De Vjeeschouwer & Gueudet, 2011), the Didactic Contract has several levels, and I argue that there is evidence in the present study for the 652 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015 Student Transition into University Mathematics Education ‘transformation’ of (1) the ‘general/pedagogic didactic contract’ and (2) the Didactic Contract for mathematics when students transit from school to university At the general/pedagogic Didactic Contract level, school teachers appeared to provide students with clear and concise instructions as to what they wanted them to (see also the school handbook), and the expectation was that students learned according to teacher guidance, and in the ways shown/taught by their teachers This system of reciprocal obligation resembled a contract, as Brousseau (1997) outlines At school this contract included attending mathematics classes; this is compulsory, and teachers provided particular notes which either students had to copy or were provided as copies There were also clear instructions about the course (e.g the content; expected learning strategies; the modes of instruction; modes of assessment etc.) in the department handbook Moreover, both teacher and students were to adhere to the contract and not ‘break the contract’, which had important consequences In this school’s contract the teacher was supposed to create opportunities for mathematics learning (e.g provide particularly useful strategies and worked examples), and recognise when this was not happening (e.g through assessment), and perhaps offer new possibilities leaning on previous knowledge If the student did not adhere to the ‘contract’, s/he was put ‘on trial’ (e.g whether to be entered for a particular examination), but so was the teacher if s/he had not fulfilled what was expected of him/her (e.g whether the teacher created the ‘purposeful learning environment’ and was to teach the module again in subsequent years) In terms of the Didactic Contract for mathematics, students were expected to learn the mathematics skills to solve the exercises provided and in the tests/examinations Thus, the Didactic Contract obliged both sides to some extent, and in an atmosphere of ‘trust and mutual respect’ However, and perhaps more importantly, the teacher was made responsible for creating a ‘purposeful learning environment’ – the main responsibility rested on the teacher for helping students to learn mathematics, and ‘training’ them to develop the skills to pass the examinations At university, though, the contract was such that the responsibility of the lecturers/professors was to provide the lectures and coursework that could be worked on at tutorials – in short, to ‘deliver the content’ Students were expected to manage the learning processes largely by themselves According to Simar and his friends, students were expected: • to listen to lectures (‘sit quietly and copy’); • to ‘take notes’ and ‘read the lecture notes’; • to go to tutorials and pass tests; • to ‘emulate’ what/how professors mathematics This implies that students were responsible for their own learning, and develop adaptive helpseeking if they did not understand – quite different to school, where the responsibility for learning mathematics was largely on the teachers and the school One of the lecturers pointed out that ‘we provide quite a lot of support for students, but they have to accept it, they have to go and ask the questions’ (Lecturer 1, DP1) Thus, the rhetoric is that they support their students (and they genuinely want to), but, practically speaking, students not know ‘how to ask the questions’ Students would need to learn how to ‘diagnose themselves’ to know what their needs/questions are The message at City University was clear: whilst students preferred ‘recipes’, City University lecturers did not want skills training or recipe-like learning; they wanted mathematical thinking which included ‘rigour and proof’, at least that was what they claimed There was an apparent contradiction in terms of what lecturers said and what they did, according to students’ understandings (see Simar’s and his friends’ understandings of expectations above): it appeared that in practice lecturers wanted students to ‘emulate’ what the former were doing in lectures, which in turn was interpreted as a recipe for passing the tests The tests (and preparation for those) was seen (by the university) as getting students on a similar level, but in practice it was more than that: it gave students the message that at City University mathematics was taught and learnt differently from what they were used to at school, and with a different rigour – all this necessitated crucial changes on the part of the students 653 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015 Birgit Pepin Conclusions From the analyses, it appears that students’ experiences at upper secondary school not match/are not similar to university learning expectations, whether in terms of content or in terms of learning styles, or indeed in terms of autonomously managing the resources From the interview data it was clear that the most difficult aspect for students were the changes in teaching styles and associated styles of learning, besides the difference in the mathematics they were taught at school as compared to the ‘new mathematics’ (including more argumentation and proof) at university This is supported by the literature, and several studies (e.g Ozga & Sukhnandan, 1998) have claimed students’ lack of preparedness for learning in higher education In this study students encountered difficulties because they lacked an understanding of what learning mathematics at university involved, and many tried their ‘old’ methods (following the teacher), but without success The tempo was too fast to ‘emulate the mathematics’, and copy At the same time, lecturers saw it as their responsibility to ‘deliver’ the content, rather than helping students ‘learn to learn’ This is supported by the literature (e.g Fallows & Steven, 2000) which claims that most academics’ concerns focus on content, rather than learning skills However, students were aware, they knew and expected that a new institution meant a ‘new contract’, but they did not know what this would entail It can be argued that there was a distinct transformation, or perhaps even ‘rupture’, of the Didactic Contract from school to university mathematics education, both in ‘atmosphere’ and requirements; as well as in mathematics learning and responsibilities for such learning At City University there was a clear distance between students and lecturers (addressed as ‘Professor’ and ‘Doctor’, and not by first name), whereas at school there appeared to be an atmosphere of ‘mutual respect and trust’ Moreover, the school mathematics did not fit what was wanted at City University; students were told to ‘forget the mathematics you have learnt in school’ At school, students needed to acquire the mathematical skills; at university, there seemed to be a different kind of learning needed, including reasoning and a deeper mathematical knowledge It appeared that City University lecturers wanted to ‘enculturate’ students into a different way of thinking which included rigour and proof, but there was little support of how students could get to that level, except by emulation Although the university department routinely provided different kinds of support (tutorials, peer group teaching etc.), the support students needed most was not provided: ‘learning to learn mathematics’ The question remains of how this ‘transformation’ into a ‘rigorous mathematician’ was expected to happen Neither was it clear how lecturers wanted students to change their learning practices Moreover, it can be argued that there was the need for a common understanding (between academics and students) of what ‘mathematical knowledge’ meant and which practices it entailed, in order that students could understand what ‘learning mathematics’ implied at particular institutions The results of this study showed that there were also particular ‘cultures’ of mathematics and mathematics learning at particular institutions and departments (e.g mathematics in engineering versus mathematics in ‘pure’ mathematics courses) This means that the Didactic Contract would have to be made explicit (for each institution and department/subject area) Further, I contend that the transformation of the Didactic Contract at the transition from school to university mathematics education has implications for students and their mathematics learning strategies With the change from school to university, teachers ‘transform into’ lecturers, lessons into lectures, homework into coursework, textbooks into course materials, tests into examinations and school mathematics into university mathematics However, students often receive little support in how to manage these transformations, and how to steer and direct their learning processes There are different ‘sources’ at school and university level in mathematics education that could be regarded as potentially enriching for and supportive of student learning (e.g texts and textbooks) However, these sources are contextualised according to student perceptions and beliefs, and prior knowledge, and thus cannot be seen to be uniformly ‘useful’ It is perhaps when students look out for these sources, that is, when adaptive help-seeking becomes part of student learning, that these are most effective – it is at this moment that they become most relevant and ‘problematic’ for students 654 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015 Student Transition into University Mathematics Education For Simar and his friends, who were all ethnic minority students, this transformation, or rupture, of the Didactic Contract was as painful as it was for the other white-European/British students However, and despite the difficulties, it appeared that Simar and his friends were ‘ready’ for shifts in practice, within or between communities: as minority students in Britain they had learnt to negotiate or align practices in light of past or present memberships of other communities (e.g home and ‘outside’ communities) In addition, it could be argued that the particular ‘context’ (e.g Simar’s school had a high Asian population intake, and so had his chosen university) helped him to build relationships in a group and community around aspects that were related to his study of mathematics Theoretically, in terms of refinement of the notion of the Didactic Contract, and referring to its social norms, it can be argued that within each institution (school or university) there are particular classroom/lecture expectations and obligations and specific mathematical expectations and obligations which students have to learn how to work within, and perhaps to identify with Of course, it is quite possible that some students in a particular class come to identify with their classroom obligations, whereas others merely cooperate, and still others resist, that is, not ‘oblige by the contract’ Linking to work by Boaler and Greeno (2000) and Cobb et al (2009), I call this the Normative Didactic Contract Holland et al (1998) describe identification as a process by which communal activities ‘in which one has been acting according to the directions of others [e.g teachers or lecturers] becomes a world that one uses to understand and organise aspects of one’s self’ (p 121) In that sense, the expectations and obligations, originally normative ways of acting and perhaps prescribed by the institution, become expectations and obligations to ‘oneself’, that is, the student/s to themselves Therefore, they are not mere conventions that can be modified at will, but they imply accepted ways of acting by students who (have learnt to) identify with these expectations and obligations, or are willing to cooperate within them In addition, and as Horn (2006) notes, students may encounter significantly different classroom obligations as they move from one mathematics class to another (e.g from the school mathematics class to the university mathematics class, or even from a calculus class to a linear algebra class within the same institution, or from one first-year mathematics class to a second-year mathematics class) Thus, a transformation (or rupture) of the Didactic Contract implies a personal aspect What is perceived as mathematical competence in a particular mathematics classroom has implications for students’ perceptions of themselves, and their understandings and valuations of the ways in which they can purposefully exercise agency I call this the Personal Didactic Contract In Simar’s case, the Personal Didactic Contract was likely to be connected to his environment: he lived at home with his parents, and his family and friends fully supported his choice of studying mathematics at this particular university The subject was highly esteemed by his family and peers, he had many friends in the faculty and it offered him the prospect of a good job with high esteem – all incentives which helped Simar to be ‘confident’ and determined to succeed in his study of mathematics He described ‘learning as a journey’: it’s like a, I’ve said this to you before it’s like a kind of journey and when you’re passing the coursework it gives you more confidence and confidence is quite important with Maths Like if you’re, if you get it right you think hang on you enjoy it more don’t you, and so as you’re going on you enjoy it more, you learn more things (Simar, DP5) In summary, the concept of the Didactic Contract has been useful in investigating the transition from school to university mathematics education, and I have argued that a Transformation of the Didactic Contract has serious consequences for students’ success (or failure) at this crossroads of their mathematical development, in particular if students are inappropriately supported, or left on their own, to ‘bridge the gap’ from one contract to the other Further, I have argued that in order to help to smooth the passage, a refinement of the Didactic Contract is helpful, a reconceptualisation in terms of the Normative Didactic Contract and the Personal Didactic Contract, and each needs to be considered at each level of development This is likely to raise awareness about the necessary conditions for success at both sides of the transition junction, and providing appropriate support for students at both levels is likely to provide access for more students to successfully stay in higher education mathematics 655 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015 Birgit Pepin Acknowledgement As author of this paper I recognise the contribution made by the TransMaths team in the collection of data, the design of instruments and the project, and discussions involving analyses and interpretations of the results I would also like to acknowledge the support of the ESRC-TLRP award RES-139-25-0241, and continuing support from ESRC-TransMaths award(s) RES-139-25-0241 and RES-000-22-2890 Note [1] http://www.education.manchester.ac.uk/research/centres/lta/ltaresearch/transmaths/ References Academia Europaea (2007) http://www.tsm-resources.com/doc/lisbon07.doc Artigue, M (2002) Learning Mathematics in a CAS Environment: the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work, International Journal of Computers for Mathematical Learning, 7, 245-274 Artigue, M (2007) Teaching and Learning Mathematics at University Level, paper presented at the conference ‘The Future of Mathematics Education in Europe’, Lisbon, December http://dx.doi.org/10.1023/A:1022103903080 Boaler, J & Greeno, J.G (2000) Identity, Agency, 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Adult Education Quarterly, 57(4), 312-328 http://dx.doi.org/10.1177/0741713607302686 Ogbu, J.U (1992) Understanding Cultural Diversity and Learning, Educational Researcher, 21(8), 5-14 http://dx.doi.org/10.3102/0013189X021008005 Organisation for Economic Co-operation and Development (1999) Classifying Educational Programmes Manual for ISCED-97 Implementation in OECD Countries 1999 Edition Paris: OECD Organisation for Economic Co-operation and Development (2003) Education at a Glance 2003 Edition Paris: OECD Ozga, J & Sukhnandan, L (1998) Undergraduate Non-Completion: developing an exemplary model, Higher Education Quarterly, 52(3), 316-333 http://dx.doi.org/10.1111/1468-2273.00100 Pepin, B (2009) The Role of Textbooks in the ‘Figured World’ of English, French and German Classrooms – a comparative perspective, in L Black, H Mendick & Y Solomon (Eds) Mathematical Relationships: identities and participation London: Routledge Sarrazy, B (1995) Le contrat didactique, Revue franỗaise de pộdagogie, Note de synthốse 112, 85-118 Smith, A (2004) Making Mathematics Count London: The Stationery Office Strauss, A & Corbin, J (1990) Basics of Qualitative Research: grounded theory procedures and techniques Newbury Park: Sage BIRGIT PEPIN is Professor of Mathematics Didactics at Sør-Trøndelag University College in Trondheim, Norway She has worked and researched in mathematics education since 1990, in different European countries and at various levels: from lower secondary grades to university and tertiary mathematics education She has published widely, e.g on transition into higher education mathematics; on ‘resources’ (e.g textbooks) in different European countries’ mathematics classrooms; and on teachers’ use of curriculum resources Moreover, she has led several large research and development projects in mathematics education and teacher education/professional development (e.g Norwegian TransMaths project; EU PRIMAS and MaSciL project) Her recent book From Text to Lived Resources, co-edited with French colleagues, has received considerable attention At European/international level she has initiated and led several networks in mathematics education research (e.g ICME, CERME, ECER) Correspondence: birgit.pepin@hist.no 657 Downloaded from pfe.sagepub.com at HEC Montreal on June 17, 2015