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L L L L L L L ` 8 ! < Smith200625425425427Smith, CathyeNRICH Mathematics Project Evaluation: Interim Report October 20062006NRICHhttp://www.nrich.maths.org/content/id/2719/eNRICH-maths-Interim-evaluation.doc(Smith, 2006), but in this paper we describe the aspects of the project that we consider relevant to other enrichment mathematics projects with aims of social justice.
In the next section we discuss the intended aims of eNRICH Maths, and how they relate to social justice. We then move to describing how we adapted the original programme design. This happened in response to the involvement and feedback from the participating communities, and also in response to our own reflections on what was happening in the sessions and questioning whether and how we were or were not meeting our goals. We end by describing some outcomes for students, and relating these to aims for empowerment.
One of us has been centrally involved in organising and teaching on the project, the other became involved as an external evaluator. This paper arises from conversations between us, and with other team members, school teachers and assistants, representatives of the funding charity and LEAs, and the students themselves. For clarity in this paper, we have had to merge these separate voices that contributed. The institutional viewpoints that we present do not reflect the thoughtful arguments and varied positions of individuals, but we have selected them to illustrate the bold outlines of our debates.
Aims of eNRICH Maths
We see the eNRICH programme as related to social justice in two main ways: firstly through the aim of widening access and participation, and secondly through the aims inherent in its pedagogy. Ernest ADDIN EN.CITE Ernest200224124124117Ernest, P.Empowerment in Mathematics Education
Philosophy of Mathematics Education Journal Philosophy of Mathematics Education Journal152002internal-pdf://Ernest empowerment-1419454505/Ernest empowerment.pdf(2002) describes three forms of empowerment that are concerns of mathematics education: mathematical empowerment, concerned with students mastery of domains such as school mathematics; social empowerment, concerned with students ability to exercise mathematics for social goals, and epistemological empowerment, being able to take responsibility for the creation and validation of knowledge.
Participation and aspiration
In its initial choice of students and localities, the eNRICH programme most clearly focussed on social empowerment. Mathematics has a history of selection and of exclusivity. It is historically aligned with white middle-class male hegemony, and it still serves to police barriers in society that disproportionally separate some groups, including women and BME groups, from power ADDIN EN.CITE Harris19972442442446Harris, M.Common Threads: Women, Mathematics and Work1997Stoke-on-TrentTrentham BooksGates20012482482485Gates, PeterGates, PeterWhat is an/at issue in mathematics education?Issues in Mathematics Teaching7-202001LondonRoutledgeFalmer(Harris, 1997, Gates, 2001). Funding for the eNRICH project was given specifically to provide mathematics enrichment in ethnically diverse and socially deprived areas of London. Participation in any form of optional out-of-school events is seen as having positive effects for social inclusion and educational outcome, perhaps through widening students social groups and purposeful relationships with adults ADDIN EN.CITE Eccles200324524524517Eccles, J.S. Barber, B.L. Stone, M.Hunt, J. Extracurricular activities and adolescent developmentJournal of Social Issues Journal of Social Issues865-88959 2003(Eccles et al., 2003). In addition, the particular cultural status of mathematics itself increases the likelihood of participation, and suggests valued outcomes.
The eNRICH programme aimed to raise students aspirations and increase their opportunities to engage in mathematics, with the expected benefit that they could eventually enter higher status forms of education or employment. It was designed for students who were identified by their teachers as potentially highattaining in mathematics or with an interest in problem solving. It thus sought to address a specific cultural hypothesis: that insufficient numbers of 16-year olds are continuing to study mathematical subjects at A level despite having a good GCSE grade. This isnt a hypothesis that can simply be confirmed or refuted it depends on what is meant by insufficient; for whom, and for what purpose? However we follow many, including Ofsted ADDIN EN.CITE Ofsted200620420420427Ofsted Evaluating Mathematics Provision for 14-19 year olds.2006London Ofsted(2006), in asserting that mathematics teaching effective for gaining good examination grades at GCSE is not necessarily helpful for later mathematical achievement or indeed motivation. A further aim of eNRICH Maths, therefore, was to introduce students to practices associated with higher-level mathematics that they may not have experienced in school. In particular, students should experience collaborative problem solving which lies at the heart of the NRICH pedagogy. In the next section we discuss the principles underlying this pedagogy and how it is related to mathematical and epistemological empowerment.
Before doing so, we note that this emphasis on new practices is a different form of social empowerment than simply affording participation. Here, we were not facilitating students in achieving goals established in school, but focusing on more distant goals such as mathematics degrees, and suggesting new ways of achieving them. A focus on endpoints, on becoming not being, is seen as central to the cultural construction of adolescents as in crisis and in need of guidance ADDIN EN.CITE Lesko20012782782786Lesko, NancyAct your Age! a cultural construction of adolescence2001New YorkRoutledgeFalmer(Lesko, 2001). Through this practice we claim privileged knowledge of mathematical goals in order to justify our involvement in school practice, but this in turn reinforces the positions and practices that keep adolescents powerless.
Pedagogy and practice
Historically, the aims of the NRICH project ADDIN EN.CITE Beardon20033027Toni BeardonA Short History of NRICH2003Faculty of Education, University of Cambridge.To come(Beardon, 2003) were to encourage able youngsters to engage in mathematical activities similar to those provided by local masterclasses, and to make such opportunities available to all young people via a website ( HYPERLINK "http://www.nirch.maths.org" www.nrich.maths.org). The site places an emphasis on good mathematics problems, clear exposition and well reasoned proofs, and publishes work sent in by students. This enrichment programme was part of an initiative widening its focus from presenting problems in book/screen form into guidance for their use in other settings, including practices that could be used in school. At the same time, the team worked on reflection and theorising about its existing practices. A sense of an NRICH pedagogy emerged from these discussions, in terms of the relationships between mathematical tasks, experiences for learners, implications for teachers and extended outcomes in terms of learner aspirations and views of the subject (Piggott, 2005). In our view, pedagogy is not a predetermined science of teaching an identified collection of skills, with an associated emphasis on endpoints, but an art of influencing the social, physical, time, and emotional features of classroom relations between teacher and students so as to promote practices that we value as mathematical.
The Maths project discussed here was premised on the view that enrichment can be built around problem solving. There are many studies that argue for problem solving to be central in mathematics pedagogy. Drawing on this field (notably Stanic and Kilpatrick, 1998; Nunokawa, 2004; and Wilson Fernandez et al, 1993), Piggott (2005) describes four justifications for structuring teaching around problems, each of which can be related to social justice. Firstly, the utility argument: problem solving is a generic skill applicable to other subjects and offers a way of thinking that enables the learner to take a critical view of the world. This description promotes social empowerment, not just through acquiring desirable transferable skills, but in developing a perspective in which mathematics is a tool for critical analysis of ones environment. Problem solving has value and utility in its own right, but it is also valued as a fundamental part of real (ie non-school) mathematical activity. It brings together desirable attributes in both entrepreneurial and academic discourses. Students and parents who deploy such information about what is available and valued in educational discourse support the students identity work on career trajectories ADDIN EN.CITE Ball20002432432436Ball, S.Maguire, MegMacrae, Sheila Choice, pathways and transitions post-sixteen: new youth, new economies in the global city2000LondonRoutledge/Falmer(Ball et al., 2000). The programme actively demonstrated the social value of this new perspective on mathematics through the involvement of universities and commercial sponsors, and most noticeably in providing impressive venues for special events and visits.
Secondly, learners must engage in problem-solving activities in order to learn how to problem solve, ie they learn about problem solving through doing some. The third justification is that problem solving is an effective approach to teaching mathematics more generally - learning and developing understanding of new mathematics through problem solving. In these two purposes, learning about and through problem solving, NRICH aims for empowerment within linked but different domains of school mathematics, sometimes described as metacognitive and cognitive skills. So far as problem solving is seen as part of the mathematics curriculum or giving access to the mathematics curriculum, this is a form of mathematical empowerment. However, metacognitive problem solving skills are often associated with autonomy and critical evaluation ADDIN EN.CITE Schoenfeld19923131315Schoenfeld, A.Grouws, D.Learning to think mathematically: problem solving, metacognition and sense making in mathematicsHandbook of research on mathematics teaching and learning335-370curriculumproblem1992New YorkMacmillan(Schoenfeld, 1992) and so these arguments also promote epistemic empowerment for the students.
There have been historic difficulties in integrating these domains within UK mathematics classrooms, as for example in the contested role of Using and Applying Maths in the National Curriculum ADDIN EN.CITE Hewitt199296969617Hewitt, D.Trainspotter's paradise?Mathematics TeachingmtMathematics TeachingmtMathematics TeachingmtINCOMPLETEassessmentproof1992(Hewitt, 1992). Because we believe that problem solving is integral to mathematics, we did not want to separate learning problem solving from learning mathematics but we were also aware that teaching requires you to prioritise contingently. We therefore decided that teaching about the problem solving processes themselves was to be prioritised over any other mathematical concepts connecting the tasks. The problems chosen for eNRICH Maths were selected to allow some teacher commentary about three recurrent features of problem solving: systematising, generalising and finding analogies. The themes were introduced to teachers at preparation days, and planned as emergent rather than driving the workshops. The outcomes of the workshops were set out in terms of raising students awareness and use of problem solving processes, and not as reaching solutions to the tasks or articulating particular statements.
The fourth justification for using problem solving in teaching is that it is a motivational tool a means of giving relevance to other aspects of mathematics, encouraging learners to ask questions and be inquisitive. Learners are encouraged to develop as confident and independent, critical thinkers, they are encouraged to be creative and imaginative in their application of knowledge. This last purpose fits most closely with Ernests notion of epistemological empowerment as a personal sense of power over the creation and validation of knowledge ADDIN EN.CITE Ernest2002241, p224124117Ernest, P.Empowerment in Mathematics Education
Philosophy of Mathematics Education Journal Philosophy of Mathematics Education Journal152002internal-pdf://Ernest empowerment-1419454505/Ernest empowerment.pdf(Ernest, 2002, p2) We view motivation as linked both with a sense of personal power and with a sense of being permitted by others to occupy such powerful positions.
This last view emphasises strongly the role of social inclusion. It encourages learners to pose as well as solve challenging problems, and uses problems that often allow for different solution methods. When appropriate the mathematics is placed within intriguing contexts which can offer opportunities for initial success and which might also allow learners to observe mathematics within a range of cultural settings. Thus the content element of enrichment reflects a valuing of personal views and approaches, and places the mathematics within situations or contexts that can have meaning to the learner. Although eNRICH Maths was aimed at more able learners, the principle of enrichment and its focus on problem solving is seen as inclusive rather than exclusive, and is underpinned by a view of good problems and the processes associated with problem solving. With this in mind, problems and problem solving are about the experiences of individuals, working and thinking independently but also working as part of a community, sharing and making sense of the mathematics they are experiencing.
The NRICH pedagogy was thus established on the basis of four purposes of problem-solving, and their associations with social inclusion and empowerment. These understandings framed perspectives on practice: our aspirations for what students should experience, and guidance on what tutors should do to promote this. Learners should find problems engaging, and have opportunities for initial success that open doors to challenge; they should think for themselves and apply what they know in imaginative ways. Emotionally, a learner needs to experience both a sense of slight unease and:
The joy of confronting a novel situation and trying to make sense of it - the joy of banging your head against a mathematical wall, and then discovering that there may be ways of either going around or over that wall ADDIN EN.CITE Olkin199471Page 43 05Olkin, IngramSchoenfeld, Alan, H., , Schoenfeld, A, H.A Discussion of Bruce Reznick's Chapter [Some Thoughts on Writing for the Putnam]Mathematical Thinking and Problem Solving39-511994Hillside NJLawrence Erlbaum(Page 43 Olkin and Schoenfeld, 1994)
We believe that these learner experiences are promoted in an atmosphere in which the teacher or mentor engages in dialogue and other interactions, including the use of modelling and metacognition, opening the personal experience of doing mathematics to the view of the learner, and acting as
the master, practised in the art of problem solving, who can model and share in problem solving experiences with the student as apprentice. (Piggott 2005 P147).
In this environment teachers and learners are involved in a community of practice (Wenger et al 2002, Wenger 1998) developing appropriate language to enhance communication as a vehicle for individual and communal sense making. When teachers give explicit guidance about rules of mathematical discourse, this allows outsiders to participate more fully in group activities ADDIN EN.CITE Morgan200723623623617Morgan, C.Who is not multilingual now?Educational Studies in MathematicsEducational Studies in MathematicsESM239-2426422007http://0-ejournals.ebsco.com.emu.londonmet.ac.uk:80/direct.asp?ArticleID=4C1AAFDE66022490B373(Morgan, 2007). The teachers role is in encouraging from behind rather than leading from the front. This depends on students becoming able/ willing to communicate what they know and where they are, rather than relying on being cued by a teachers example. Students differing descriptions of their approaches are made the focus of the workshops. They are used as shared texts through which they can build shared meanings around the mathematics, and also evaluate strategies. Teachers value the individual within the group explicitly, by encouraging students to be creative, independent thinkers, and implicitly by giving them time to explore starting points and alternative routes.
In this description of the eNRICH pedagogical practices, we have focussed on its underlying aims and justifications, its aspirations for the learner experience, then the principles which we used to guide teachers decision making. This doesnt quite give an impression of what happened in a typical workshop, which is perhaps best seen from a students point of view:
Interviewer: So could one of you describe me what you typically do when you come to a shine maths session?
Student: When you first come they introduce you the problem and sometimes you have a really difficult one: so you have to think quite hard and work as a group. Then they give us some clues at some problems - if its really difficult, and high standard, and then we try to crack it, and if we have any difficulties we ask them. And at the end the person thats running the programme they get everyone together and say does anyone know it? And if someone does, they tell them. Then they talk us through how they found a solution and we all talk it over.
It is possible to read different accounts of who they are in this students description, although we find it most likely that it is other students who talk through how they found a solution. A teacherly presence running the programme is quite clearly produced: introducing the problem, giving clues, and directing the groups attention, but crucially for eNRICH it is the students, we, who are trying to crack it, and who, in the end, talk it all over. We see aspects of social empowerment in the emphasis on difficulty, mathematical empowerment in the goals of cracking it, and epistemic empowerment in who acts and when.
Developing eNRICH Maths
The previous section discussed the aims of the eNRICH maths programme and the thinking behind the intended pedagogy. In this section we focus on how the programme developed as it became owned by students and schools. We describe events that just happened and changes we instigated to move towards our goals, and again consider how those effects and changes can be analysed in terms of empowerment.
Supporting Participation by recruitment
The project initially attracted funding to focus on learners in an Apphill school which was in the process of closing. Only Year 11 and Year 10 learners were left in the school to complete GCSEs. It was this Year 10 group that the project targeted, considering that they would benefit from the mathematical input and offering them an experience that might support them in feeling valued. Initial slow recruitment eventually reached around 30 students, but less than half the students completed the year. A second cohort was planned for the next year but it failed to maintain sufficiently high attendance levels and the programme was closed in that area.
This brief history of eNRICH maths in Apphill shows that from the start NRICH recognised problems in recruiting sufficient students to justify the constraints of time and expense in running the programme. The focus on a closing school, intended to promote social inclusion, did not engage the students, who may rightly have felt that enrichment was no substitute for school. This early experience resulted in an emphasis on advertising the values of joining the eNRICH programme so that it engaged schools, students and parents.
Over the next three years the programme was run in two other areas, Brunley and Chedham. Here, initial recruitment was helped by the support of their education authorities, and by linking the programme to Gifted and Talented provision, which is required of schools and is widely understood by students. Students have been keen to join eNRICH maths, and mathematics teachers were recruited, trained and paid to lead workshops. We can still ask how successful we have been in our strategies for widening access to enriching activities. Both London areas have populations with a wide range of black and minority ethnic communities, generally with English as one of the home languages, and they experience high proportions of social deprivation. In the eNRICH maths cohorts, the ethnic and social-cultural balances do represent their geographic communities, which does suggest that the programme is attractive and accessible to all. Two exceptions are of interest: one area attracts fewer white British students than indicated by local social statistics, and the other has slightly fewer students from economically disadvantaged backgrounds. These two groups are respectively identified as the main academically under-achieving groups in those areas. Although NRICH had conceptualised the target audience as learners with a potential for engagement with mathematical problems, it appears that schools generally responded by identifying youngsters by their prior academic success. As might be expected, pervasive structural constraints on schools and on groups of students, are not easily overturned by good intentions.
When we recruit for the programme, we necessarily draw upon schools, students and parents understandings of what we mean by problem solving and higher mathematics, and their expectations of how this is valuable to them. If the sessions do not continue to meet these expectations then we can expect attendance to drop off. NRICH have consistently stressed in their recruitment that the sessions will involve unfamiliar problems and are of long-term benefit. The students themselves offer two main reasons for participating: their teacher told them to, and they want to do well in mathematics, both reasons in the domain of school performance. In their feedback, students from new schools entering the programme often remark that they expected more help with GCSE work. The prevalence of the examination discourse is marked, in that schools more familiar with the project have motivated students by suggesting it has beneficial effects immediately before GCSEs. After the programme, although students report that eNRICH maths is very different from school mathematics, they also believe that eNRICH maths practices are close to later mathematics study. For example, from one year 11 student who hesitated to express her A-level choices :
I mean I wanted to do maths anyway, but I think that [eNRICH] probably cos I know that its not just like maths in the classroom, theres more maths to it.
Linking eNRICH to future mathematics is one way that schools and students can place and value the programme in an educational hierarchy. Students explained this belief by calling on their teachers and the NRICH authority. The informal comments from university students who helped in sessions were particularly noted. Another approach used by students in describing its value is to call on utility, associating problem solving with everyday life. Although this places eNRICH maths in an opposition to school life which isnt real, this alternative position is also current in school discourse. Clearly, some students are recruited with an appeal to empowerment in the domain of examination mathematics which is not a true focus of eNRICH, but their subsequent participation gives them other justifications still consistent with existing educational goals.
Supporting Participation by embedding eNRICH in schools
The eNRICH programme was run out of school hours and in a non-classroom setting, at a local university campus and in school sixth-form blocks, with the intention of providing a neutral setting linked to the familiarisation and aspiration raising element of the sessions. We expected participating schools to extend their responsibility in time and place so as to ensure student commitment to the programme both in terms of regular attendance and by supporting learning during the workshops. Negotiating these responsibilities has proved to be an ongoing process that has been dependent, sometimes frustratingly, on school and social cultural and environmental factors.
Outside school routines, teachers had to find new practices of setting and upholding expectations for pupil autonomy. Early discussions focussed on how NRICH, schools, and parents viewed students independence when travelling to workshops. It was necessary to find suitable teachers to accompany some girls from the Bangladeshi community. The location of workshops within the context of the local communities was also influential. Transport to and from Burnley workshops on a Saturday morning needed a commitment from parents, and this ensured high attendance rates. In Chedham, travel by taxi or bus was unreliable and led to some lateness and safety concerns in the evenings, so some teachers found it easiest to organise school minibuses. In this way the level of school surveillance was gradually extended, with improvements for attendance rates.
Later, unexplained pupil absence led to a further request to schools that a member of teaching staff attended all the sessions as an observer and participator. The model of teacher involvement expanded from the initial recruitment of teachers to lead workshops to having required teacher participants with a monitoring role. Moreover NRICH relied almost exclusively on school teachers to pass on the expectations of the community and the school about off-campus locations, and this reinforced their role as administrative rather than mathematical in nature.
Another aspect of school liaison has become important to eNRICH Maths. We did not appreciate the number of demands that schools themselves make on the time of students, particularly potentially high-achieving year 10s. These have included trips, parents evenings, stewarding at open evenings, revision and career sessions. Students and mathematics teachers have found it difficult to resolve conflicts of attendance within the school domain. For example, one cohort had whole school groups missing in over half the workshops. More recently, a calendar has been produced at the start of each yearly programme, based on known commitments in schools. Schools agree the timetable and the commitment for students to attend all sessions. However, visits and revision sessions in other subjects are still used to compromise the priority at short notice.
The programme has now developed a structure with clearly defined requirements of schools and teachers; close contact between NRICH, the advisory service and schools; and significant administrative overheads. Support of school leaders is evidenced by their availability and encouragement in liaison meetings. However, practice is not always in line with what is described as possible. We find that the management and culture within some schools is such that teachers do not equally share or own responsibility for enabling non-routine activities. For example, some classroom teachers were not in a position to access school attainment records, nor to ask other classroom teachers to release pupils or give feedback on pupil performance. We can make no comment about the reasons for this, but observe that assuring the agreed school involvement has required quite deliberate use of both personal contacts and of school management structures.
School practices do not only exist on an administrative level but also influence the views of the learner participants on how they are expected to participate in learning, and how this will be communicated to them. The project envisaged a long term commitment (in 2006-2007 fifteen sessions over three terms with two optional additional activities). Our evaluation (Smith, 2006) justified this demand in its finding that teachers reported a significant effect on school mathematics when students attend more than fourteen sessions. However, the emphasis in the workshops is on collaborative, informal work, in accordance with our perspective that progress is achieved through changing individual understandings. This is a discourse alien to the school practices of monitoring through frequent recording and comparing performances. In eNRICH maths, it is possible for an individual to pick up and work from where they are; whereas school curriculum structures work to prohibit this. It is probable that eNRICH maths facilitated students in attending less regularly - both positively by providing activities which make it possible to skip sessions whilst still feeling engaged, and negatively by not adopting the recognisable school practices that threaten non-participation.
Pupil attendance emerged as a focal point of our discussions. Planning and early experiences had identified it as an initial concern requiring school support, because we judged that students valued their free time but also that enjoyment and interaction in group problem solving required an initial time investment. During the program, attendance continued to occupy us through its status as an easily visible measure situated at the intersection of concerns of funders, schools and NRICH about resources, responsibilities and pedagogy. Despite its prominence in national performativity discourse, it was not easy to find information allowing us to make comparisons about attendance for similar enrichment programmes; and this absence reinforced the need for our care. The schools and NRICH worked hard to maintain an average attendance level for students at above 70% (at a total of around 20 sessions through a whole year), with a large variance. A concern for empowerment should include opportunities for the students not to participate. Our only practice that acknowledged student autonomy in deciding whether to keep attending was a flexible approach to enrolment: encouraging students to visit for a few sessions before committing to a regular attendance.
In the section above we have described interactions between our intentions of engaging students in order to widen access and some constraints of the socio-cultural environment within which we operated. In developing the project we have found ways to work with the resources of the students environment the teachers practices, school administration, parental support - without which we could not meet our aims. As we have described, we have moved towards a role for teachers in supervising and facilitating the work of small groups, a more structured relationship with school leaders, and a strategic role for the advisory service. As a result there has been more dedicated and pertinent support for the administration of the project on the ground. Despite effective central administrative teams, a substantial input from teachers and schools was still crucial to establishing the programme. This suggests that externally-organised education projects are not an effortless way for schools to enrich learning. Our experience is also that our growing engagement with schools led us into an unexpected model of surveillance.
So far we have largely discussed participation in terms of recruitment and facilitating attendance and have not yet discussed the role of the mathematics programme itself in engaging students to participate. We are uncomfortably aware that the structure required to oversee the eNRICH maths project, with the specific demands for school monitoring, is at odds with the expectations in sessions that independent learning involves the opportunity to take on personal responsibility and a feeling of belonging to a community.
Supporting participation by creating a problem solving community
In our initial pedagogic thinking we envisaged two main forms of classroom management. Workshop leaders would monitor activity time and manage social interactions at a whole class level promoting ways in which individuals could speak and listen to the whole group. The structure of individual or small group work on problems was not explicitly managed but influenced by teachers and university student helpers visiting the groups, questioning and encouraging. During the program we questioned our practices in two areas: how and when we interacted with students when they worked on problems, in the area of social environment; and how we paced the activities within each workshop, in the area of mathematical/ pedagogical environment.
The social environment
Students attending eNRICH maths came from different schools and often different maths classes within schools. The group transport set up as an administrative solution had the benefit that school groups of 6 -10 students arrived together in the, at first, unfamiliar setting. This supported reducing levels of anxiety raised by finding themselves in a different setting and gave the group a sense of cohesion.
The cohesiveness of these school groups, while providing security, undermined our intentions to have students working together as individuals, marking a difference from school lessons and empowering them to learn in a wider community. Workshops worked towards a model where students were seated in groups with a maximum of two from each school. There was a noticeable tension between encouraging students independence and engagement and our insistence on moving them out of the groups within which they felt comfortable. Many students visibly and verbally resented this social control, but some indicated in interviews that, despite their reluctance to work in mixed groups, they valued this opportunity when offered (which matches feedback from other NRICH projects). Such comments reflect the NRICH expectation that to move forward there is a need to experience a level of unease. Leaders insisted longer and more consistently on mixed groupings in some cohorts than others, but the momentum was always to move back to school groups.
The social element of the workshops was important to NRICH because it supported the problem-solving activities planned, but even more so to young people. For some, being given the opportunity to be in a setting not overseen by family adults was enough of a reason to be at the sessions. Social interactions that make mathematics a secondary activity are not encouraged in school. In the eNRICH sessions a distinction between social and mathematical practices was not clear in practice, was not elaborated by leaders, nor monitored within friendship groups. Instead leaders spent more time explicitly encouraging socio-mathematical conventions concerning how individuals could speak to the whole group, listen and respond to others about mathematics. Similarly, a number of girls from ethnic minority groups reported that they appreciated the sessions precisely because they combined social and learning opportunities in a secure, formal setting.
The mathematical/pedagogical environment
In the initial stages of the project teachers were recruited to prepare and present the majority of workshops. Teachers were encouraged to attend professional development sessions that allowed time to be involved in problem solving and experience the principles of the project at first hand. This model was popular and offered considerable opportunity for teachers personal development, not only in teaching through problem solving but in team-teaching and planning. A first change occurred when all schools were required to send teachers to workshops. The roles of workshop leaders and the supporting teachers were not clearly distinctive when some teachers alternated between the two, and the responsibility for engaging and encouraging pupils became more diffuse.
Over time we observed that some workshops did not reflect the problem solving ethos intended. For example, a teacher might end a session by stating answers to any remaining questions. Drawing together themes and resolving uncertainty is good practice in school teaching, but eNRICH maths intended an emphasis on themes of problem solving processes not problem solutions. With some reluctance, the project moved to all sessions for the most recent cohorts being prepared and presented by the NRICH team. In this decision we chose to prioritise the nature of the problem solving workshops over our intention to involve and eventually devolve control to teachers.
Coincidentally the funding body renewed their support but without funding for professional development. Teachers continued to monitor attendance at sessions, but were only able to engage with the principles of the project through written guidance and watching what was happening on the ground. There was a noticeable effect on teachers participation in sessions, with less skill evidenced in when to encourage and when to leave pupils to work alone. NRICH has now found its own funding to support training sessions focussing on teachers own understanding of problem solving and their role in encouraging student learning.
A third area of ongoing change was in refining the timing and conduct of the activities in the workshops. In Chedham, an initial outline for a 4-6pm session was to set short starter problems on student arrival, and then tackle one or two main problems, with a clean break for refreshments. This planned workshop structure was effectively abandoned, not because it proved unsuccessful for learning but because it was rarely, if ever, achieved. Late arrivals and tiredness often created a slow start, and the break and change of problems caused a loss a momentum that it was difficult to recover. Eventually, most sessions started with students socialising over the starter problems, and then the class tackling one problem together, with refreshments in group work time. In contrast Brunleys 3 hour Saturday morning sessions allowed more time for breaks, the students had more energy, and this permitted more flexibility in combining work on several problems at once and extracting themes. The same management techniques simply did not achieve the same pace in after-school sessions.
In the sessions students often felt challenged but also felt bored: a cause for concern about participation that we felt we must recognise. However this was also part of the learning experience that we decided we could address only in certain ways without compromising our aim of epistemic empowerment. In prioritising students being the sources of ideas, the NRICH leaders themselves could not step in to provide resolution and fresh impetus. Instead teachers could vary pace by changing the form of student activity, for example changing from class work to group work, or from thinking to talking and listening. They could suggest that students looked at different perspectives on the problem, and they could give an audience to individuals good strategies, but couldnt insist on that way of working. They could offer alternative problems, but students often lacked the energy to start something new and were not themselves satisfied by just giving up. Some students articulated a connection between frustration and satisfaction. For example one year 11 describes what mathematicians do as:
They try to solve the problem and if it doesnt work they try again, if it doesnt work then they try again, and they never give up, and they, when they really want to find the best solution to that problem then they never give up.
This tension was again evident when the students reported that the sessions were useful and enjoyable but wished they were more fun. From the beginning, some sessions were included on the programmes precisely because they were more unusual and active, and we have considered strategically timing such sessions over the year to keep up morale. Apart from this sequential planning, the management of each workshop is still demanding for leaders who have to sustain momentum at a pace slow enough to encourage individual thinking and respond flexibly to students varied inputs.
Observed outcomes of the programme.
An evaluation of eNRICH Maths ADDIN EN.CITE Smith200625425425427Smith, CathyeNRICH Mathematics Project Evaluation: Interim Report October 20062006NRICHhttp://www.nrich.maths.org/content/id/2719/eNRICH-maths-Interim-evaluation.doc(Smith, 2006) considered the effects of the program on students problem-solving and school mathematics, their aspirations and attitudes to mathematics. The evaluation drew on observations, interviews, videotaped workshops, student questionnaires, educational data and teacher profiles over three cohorts to give a rich profile of the program incorporating the perspectives of those involved. In this last section, we briefly discuss some of the findings of the evaluation that we consider relevant to social justice.
Students mathematics and ways of knowing:
Students described the eNRICH problems and teaching as very different from school mathematics and a lot more challenging. Over the course of the year, students felt that they had got a lot better at solving the workshop problems: they had learnt new strategies, were more confident in getting started, and knew what kinds of answers they were looking for. Some went beyond a description of skills acquired and described a new way of learning. Most students thought that eNRICH Maths had also improved their school mathematics, and introduced them to practices of advanced mathematics. However they didnt agree on any specific school practices that had improved. A few students protested that eNRICH had been no use because it did not relate directly to GCSE. This suggests a varied but largely positive response in terms of enabling students to meet their own mathematical goals, and some suggestion that students had changed their ideas about knowledge and learning in mathematics. This new way of learning is of course structured by social practices in the workshops, including those manipulated and validated by NRICH. Our theoretical position is not that student empowerment comes from a more independent or autonomous understanding of knowledge, but from engagement with community practices that allow them to take up relatively more powerful positions.
The pervasiveness of the eNRICH maths emphasis on explanation and the multiplicity of possible ideas as central to problem solving is obvious in this description by three year 8 students:
E When we come we look at problems and we like take them apart and we kind of, we just, oh I cant think of the word ..
J we try to explain every single bit carefully along the way. We dont like, see, we dont like try and do the whole problem
E we investigate like the problem.
G and when we get the answer we try it and see if theres any other way of getting the answer of pushing until like you cant solve it any more
E Until we got, until like we understand completely what the whole question is about, and all the possible answers you can get and how you can get them
J Yeah and we also, and although sometimes we have one answer we also try to get loads of different answers cos its like sometimes there isnt just one answer, sometimes you can um
E there can be lots of answers.
The strong sense of ownership in this shared description of practice is interesting because this cohort had experience a consistently strong pedagogic steer from its leader in terms of controlling who spoke and when, but also being explicit about the community goals of these interactions.
School teachers profiled the students against a range of problem-solving descriptors before and after the project. The greatest and most frequently reported change at school was in students ability to explain their reasoning. This reported improvement seems reasonable being a critical audience for students explanations was the main class activity in the workshops, evidenced by the frequency of requests by teachers for different explanations and the time they allocated to listening to and improving on student explanations. Participation and effort on explanations was explicitly promoted and valued by the leaders. This aspect of the workshop culture was successfully carried back into school practice, even despite the difference that students perceived in the teaching styles.
Widening participation:
Students offer enjoyment as the main reason for choosing whether or not to continue to study mathematics to A level ADDIN EN.CITE Roberts200220520520527Roberts, G.SET for Success: The supply of people with science, technology, engineering and mathematical skills.200220/11/06www.hm-treasury.gov.uk/media/813/2C/chap_02.pdf.(Roberts, 2002). In all cohorts the proportion of students who said that they enjoyed school maths lessons rose after a year in the project. Nationally, the trend in mathematics is that enjoyment actually decreases with age and with attainment ADDIN EN.CITE Sturman200425525525527Sturman, L.Twist, E. Attitudes and Attainment: a trade-off? In NFER, Annual Report 2004/05.2004National Foundation for Educational Research (NFER)www.nfer.ac.uk/publications/pdfs/ar0405/05sturman.pdf(Sturman and Twist, 2004) so eNRICH Maths may have positively influenced students enjoyment of mathematics. Students comments about the workshops themselves mixed both enjoyment and boredom. The aspects of the workshops described as enjoyable were related to social relations in the classroom: working in groups, working through the students own ideas, and working with high levels of teacher-student interaction.
In working closely with schools and teachers rather than individual students, we had hoped that there might be benefits to others in the school. About 70 % of students discussed the workshops in school with other eNRICH students, and with friends who didnt attend. Several students challenged the association with the most able students and proposed that the workshops should be made available to all ability groups in schools. They commented that they primarily appreciated being amongst others who wanted to be there.
Changing classroom culture
Analysis of video and observations of small group problem solving in these early sessions suggested that students were aware that they needed strategies such as systematic reasoning, and had some ideas about possible methods in task contexts. However their own methods were described only occasionally and briefly, and no further explanations were offered or requested. Students worked together on tasks with different understandings of the same strategy, whether this originated from a teacher or another student. These understandings were not challenged or reconciled. In groups, one person made the majority of decisions; others could not lead, but only work as individuals. Suggestions for group activity were evaluated according to who proposed them or by teacher approbation, rather than by mathematical evaluation. These last two points are recognised characteristics of group work amongst secondary students. Cooperative small-group learning is shown to be most effective for problem-solving when students are encouraged to evaluate their range of strategies ADDIN EN.CITE Goos199625625625617Goos, M.Galbraith, P.Do it This Way! Metacognitive Strategies in Collaborative Mathematical Problem Solving. Educational Studies in MathematicsEducational Studies in MathematicsESM229-260 301996(Goos and Galbraith, 1996), and when students understanding of mathematical values is strong enough to support a challenge to the usual social positions that determine the focus of the group discourse ADDIN EN.CITE Barnes200318818818832Barnes, MCollaborative Learning In Senior Mathematics Classrooms: Issues of Gender and Power in student:student interactions.Department of Science and Mathematics Education2003University of MelbournePhD(Barnes, 2003).
Similar analyses in the later sessions suggested that individual students became able to start problems with their own line of enquiry and make some progress. They didnt usually explicitly aim to discuss each others strategies, but when they did they had already engaged with the problem. They used language to describe strategies that they had heard in class discussion. Students repeated explanations of strategy or reasoning several times as teachers visited the group and engaged with individuals; others listened and acted on what they heard. These explanations became progressively more full and fluent. Students could produce, explain and check their own strategies and their discussions could challenge normal group roles. They spontaneously evaluated a method against mathematically relevant criteria.
We see this comparison (explored more fully in ADDIN EN.CITE Smith200625425425427Smith, CathyeNRICH Mathematics Project Evaluation: Interim Report October 20062006NRICHhttp://www.nrich.maths.org/content/id/2719/eNRICH-maths-Interim-evaluation.doc(Smith, 2006) as illustrating how the model of mathematics offered in whole-class discussion is reproduced between individuals. Students have adopted the discourse of the whole-class interaction. In particular they have internalised the attention to different methods, and the restatement of explanations as a working practice, with the explanations being questioned as to whether they lead to an acceptable solution. The practices illustrated here underpin the metacognitive skills important in problem solving. They give detail to the most significant change noticed by teachers that students were more able to explain their reasoning.
In the above descriptions social and individual practices are discussed together. The social aspects of eNRICH Maths were more visible than some individual practices and so were highlighted by observation. However the account illustrates more than this individual practices take place in social settings, and the awareness of others mathematical thinking gave a questioning and critical perspective to the students own thinking. The social aspects of working in eNRICH sessions are repeatedly mentioned by students, whether pleasantly surprised at being encouraged to collaborate, or wanting to be allowed to work with friends. Their most common comment is that they have learned from others thinking. This could be equally be possible in school although not only curriculum time pressures but social norms operate against it:
Here you cant just judge someones answer by who they are in school, - you have to listen to it. You have to think why, why must you do this, do that. Its really good to listen to other people. (Interview Brunley)
We suggest this change is epistemic in that it extends the possibilities for who is recognised as having knowledge to include not only the teacher and the student but also other students. It challenges the idea that there are a restricted number of mathematical ways of knowing.
Concluding Remarks
This paper describes a process of reflecting on and adapting our practices in eNRICH mathematics, a process that is still current, still responding to student, teacher and funding contingencies and does not lend itself to conclusions. We are however struck by three themes that emerge from the development of eNRICH maths and which concern social justice.
The first is that the procedures by which we have strengthened our control of the eNRICH programme are to some extent in contradiction with our intention to provide a learning environment that was distinct from school and gave learners responsibility and authority over their own actions and thinking. Despite the close involvement of school in administration, many students do describe eNRICH maths as a very different way of working based on giving space to individuals mathematical ideas. Our attempts to empower teachers through professional development have been laid aside in favour of the established skills of NRICH leaders in creating the desired environment for students, and therefore the project has fewer opportunities for extension. The apparent contradiction between close school monitoring of participation and ownership of problem solving knowledge may not be produced in practice at the student level but is present at the school and teacher level.
Our second observation is the importance of managing the social environment as part of creating a problem solving environment. Workshop leaders controlled whole-class features of space, time, speakers and audience; with individuals, they questioned and listened but did not direct in detail the strategies to use. The significant outcome for students during the programme has been in their willingness and ability to communicate their mathematics to others as an integral part of the problem solving process. We find it interesting to think that social justice in the wider community has roots in empowering individuals to speak within groups in the classroom.
Lastly we acknowledge how much we have debated and learnt about planning and running an enrichment programme. In writing this we have aimed to be frank about experiences, naiveties and uncertainties, in the face of a natural inclination to write the story as a predicted journey towards outcomes that clearly justify an investment in resources and efforts. We have described details of the programme and our decisions about what to promote, and what to ignore. We take this approach partly in response to the paucity of existing literature describing how enrichment initiatives fail and how they succeed. It is our view that social justice is best served by honest reporting of such programmes intentions, constraints, outcomes and processes of change. The possibility of comparing enrichment programmes is undeniably helpful in finding and justifying funding in a field of competing educational initiatives. An equally important reason is that detailed reporting allows an analysis of structural features that might guide all such attempts. A recurrent theme of this paper is that planned actions aimed to empower individuals have been affected by practices in schools, communities and in our own setting as a funded educational institution. These are not merely inconveniences to be overcome but help to understand how social injustice is inscribed in education and enrichment. For us the ongoing goal is to be constantly creating ways of building not our own original vision of an enriching mathematical community, but an actual problem solving community that has value by belonging to all its participants.
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