Raising Expectations and Achievement Levels for All Mathematics Students (REALMS)

Raising Expectations and Achievement Levels for All Mathematics Students (REALMS) FINAL REPORT TO THE ESMﾃ右 FAIRBAIRN FOUNDATION

Professor Judy Sebba, Dr Phillip Kent, Lori Altendorff, Geoff Kent & Claire Hodgkiss University of Sussex Professor Jo Boaler, Stanford University, California 1

Contact: Professor Judy Sebba j.c.sebba@sussex.ac.uk Tel: 07788 724577

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Additional assistance with statistical analysis was provided by Christopher Brown, Associate Researcher University of Sussex and with data collection by David Baker, Visiting Research Fellow University of Sussex.

Rationale for the Research

The UK faces an enduring problem of low and inequitable achievement in mathematics (Boaler, Altendorff and Kent, 2011), high levels of innumeracy among the population and declining numbers of students taking mathematics to advanced levels (Boaler, 2009; Smith, 2004). Examination results (DCSF, 2008) show that students’ performance in the National Curriculum tests for mathematics at the end of Key Stage 2 (11 year olds), Key Stage 3 (14 year olds) and GCSE are lower than those in English and Science. International comparisons (e.g. Gonzales et al., 2008) suggest that 14 year old students in English schools still perform less well in mathematics compared to students in many other countries. Ofsted’s (2008) inspectors summarise the state of mathematics teaching in English schools in the following way: Pupils rarely investigate open-ended problems which might offer them opportunities to choose which app roach to adopt o r to reason and generalise. Most lessons do not emphasise mathematical talk enough; as a result, pupils struggle to express and develop their thinking (Ofsted, 2008, p5).

In this context, the Raising Expectations and Achievement Levels for All Mathematics Students (REALMS) research project set out to conduct research on an approach that starts from the principle that all students can achieve in mathematics, if they are given the opportunity to do so. The approach involves teaching students in ‘mixed ability’ groups and engaging them in collaborative mathematical problem solving. Distinctively, the approach includes pedagogical ‘groupwork’ strategies for teaching students to listen and learn from each other, and to organise their own learning as a group.

Key aims

The research aimed to explore the potential of a pedagogical approach called ‘Complex Instruction’ (CI – see below) that was designed in the US to make groupwork more equal, addressing the status differences that often emerge when students work in groups. The research set out to systematically assess the efficacy of using CI in English secondary mathematics classrooms.

More specifically, the research sought to: • • • • •

understand the learning opportunities afforded by group work drawing on the CI approach; assess the effects on students' understanding of mathematics; assess the effects on students’ enjoyment and attitudes towards mathematics; identify the challenges faced by teachers in implementing a CI approach and the support they need to do so; develop guidelines for schools/teachers to adopt CI, formed in part by ‘video case studies’ to be made widely available for use in teacher professional development.

Complex Instruction

Complex Instruction is a general pedagogical approach (Cohen, 1994; Cohen & Lotan, 1997) that involves collaborative problem-solving in 'mixed ability' classrooms, that has been shown to increase students' engagement and achievement in mathematics (Boaler, 2008; Boaler & Staples, 2008). The approach has been used and evaluated in the USA since the 1980s with students working at different education levels and in a variety of subject disciplines. The Department of Education at Sussex has been a pioneer of research on the use of CI in the UK, and the REALMS project is the biggest UK project to date on CI.

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The CI approach addresses the problems of under-achievement and unequal participation in mathematics by the teaching of students in mixed ability groups, where the focus of teaching and learning is on groupwork-based, mathematical problem solving. It does not assume that students know how to work well in groups, and suggests instead that students need to learn groupwork behaviours, through a ‘skill building’ process (Cohen, 1994). CI consists of instructional strategies that incorporate the use of norms and roles and structured teacher interventions which hold both individuals and groups accountable for learning (Cohen & Lotan, 1997). This is combined with learning tasks that are ‘group worthy’: – tasks that are ‘rich’ in terms of (mathematical) content, require interdependence of the group members in finding a solution, and are ‘multi-dimensional’. Most mathematics classrooms are uni-dimensional in that there is only one way of being successful, which is to follow the teacher’s demonstration of methods and reproduce them. In multi-dimensional classrooms all mathematical ways of working are valued. This involves asking good questions, seeing problems in different ways, representing ideas through diagrams, words, symbols and graphs; connecting methods, reasoning and using logic. In Boaler and Staples’ (2008, p630) research, a key finding was that when there are many ways to be successful, many more students are successful.

What makes the CI approach distinctive from other forms of groupwork is a set of pedagogical methods that are designed to make group work equal. A common objection that teachers give to the use of group-work is that some students do the majority of work and some students are left out, or choose to disengage. In the CI approach, students are assigned roles within a group, so that there can be a responsibility for each others’ understanding and collaborative completion of a task. The schools participating in the research used different versions of the roles (see later), drawing on those developed by US educators as follows (Cohen and Lotan, 1997): Role title Organisation: Resources: Understanding:

Inclusion:

Role functions Keep the group together and focused on the problem; Make sure no one is talking to people outside the group.

You are the only person that can leave your seat to collect rulers, calculators, pencils, etc., for the group; Make sure everyone is ready before you call the teacher.

Make sure all ideas are explained so everyone is happy with them; If you don’t understand, ask whoever had the idea…if you do, make sure that everyone else does too; You must make sure that all the important parts of your explanation get written down.

Make sure everyone’s ideas are listened to; Invite other people to make suggestions.

CI is a systematic approach – the learning tasks chosen by the teacher have to function as a combined ‘system’ with the roles and norms of group-based behaviour (for both students and teachers), which rely on students and teachers having the required skills; roles, norms and tasks cannot each function or be meaningful without all of the other elements. We observed that a major factor for success with CI was how well teachers and schools were able to work with all these elements systematically.

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Methodology

Year 7 (age 11-12), and in several cases year 8 (age 12-13) mathematics classrooms were followed in 6 schools in the South and East of England, through the academic year September 2010-July 2011. Four teachers and their classes were followed in each school, based on an observation visit each term in which lessons were video recorded and interviews were carried out with teachers and students.

The characteristics of the research schools are summarised in Table 1. Previous experience with CI was varied, as shown in the table. Each school was using mixed ability mathematics teaching in Year 7 (and in some cases other year groups), and had made a commitment to using CI with the Year 7 students. There were differences in whether all of the Year 7 students were involved, or a subset and most teachers did not use CI in all lessons. In accord with research ethics procedures, students/parents were offered the option of opting out of video-recorded lessons, which a few students took up.

In addition to the six research schools, a matched sample of four schools having broadly similar socio-economic profiles and academic outcomes to the research schools, but with ability grouped mathematics teaching in Year 7 and Year 8, were involved. The extent of other differences in teaching style and use of problem-solving approaches could not be judged with certainty in these comparator schools. Two of these schools were each matched to two, rather than one research schools, as securing agreement to participate from a full set of six comparator schools proved problematic. The students’ relative progress in Year 7 was measured for all 10 schools by an identical end of year test (QCDA Optional Year 7 test, paper 2), and attitudes were assessed for all Year 7 and Year 8 students by means of a questionnaire developed in our previous research (Boaler, Altendorff & Kent, 2010).

Table 1: The research schools School:

Highfield Quayside

Dean Park

Ridgeway

Location

South, urban

South, urban

South, urban

East, urban East, urban East, suburban

Total pupils (Yr 7)

1680 (293)

930 (127)

1280 (277)

600 (129)

600 (126)

1200 (240)

SES deprivation Medium High measure

Low

MediumHigh

Low

Low

GCSE 2009 performance (5+ A*-C incl maths & Eng)

27%

25%

47%

39%

67%

81%

Mixed ability mathematics teaching

Year 7 only

Year 7 only Year 7, 8, 9 All years (new policy 2010)

All years

Year 7, 8, 9

Developing as standard teaching approach over 5 years

Developing as standard teaching approach over 3 years

Prior No No experience with teaching teaching, CI some CPD with university

No teaching, teachers attended university workshops

Developing as standard teaching approach over 3 years

Sideview

Waverley

Note: Quayside was in ‘special measures’ (regarded as failing to achieve adequate standards) throughout the year and consequently had termly Ofsted inspections.

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Table 2: The Comparator schools School:

Ellen Walker

Waterside

Alan Bennett

Location Total pupils (Yr 7) SES deprivation measure (2009) GCSE 2009 performance (5+ A*-C incl Maths & Eng) Compared with:

South, urban 1660 (315) Low

South, urban 630 (125) Medium-High

East, urban 1060 (160) Low

Cardinal Newman South, urban 1320 (190) Medium-High

66%

35%

62%

33%

Dean Park & Highfield Waverley Sideview (Source: Department of Education website – schools’ data for 2009)

Quayside & Ridgeway

Data sources

Lesson observations, interviews, pupil attitude questionnaires and test results were analysed in order to explore the learning opportunities that were provided by the REALMS approach, the depth of mathematical understanding, students’ enjoyment in, and attitudes to mathematics, challenges faced by teachers and the support that teachers need. The data sources comprised: •

66 lessons observations (53 of Year 7, 13 of Year 8), 59 of which were video recorded (equating to more than the 100 hours of recorded lessons suggested in our initial proposal). Two cameras were used in almost all of the lessons, one camera following the teacher throughout the lesson, the other camera being used to record selected group dialogues in 10 to 15 minute episodes; small ‘Flip’ cameras were used in some lessons to record further data on groupwork.

•

54 student group interviews with a sample of 4 to 6 students (270 students interviewed compared to the 80 suggested in the proposal) after each lesson observation;

•

52 teacher interviews with 23 teachers (20 teachers were suggested in the proposal) after each lesson observation;

•

3870 student questionnaires (see blank form in Appendix 1) addressing attitudes to mathematics, were administered (during class time) to Year 7 and Year 8 students in 10 schools; 2980 questionnaires were returned giving an overall response rate of 77% (4000 was the estimate in the proposal);

•

1980 students’ performance in national curriculum tests in mathematics at age 11 was compared to performance in a Year 7 ‘end of year’ test (Paper 2 of the Year 7 Optional tests from QCDA) provided to schools by the project to assess relative change in mathematical attainment in one academic year. Some Key Stage 2 test data were missing due to the national boycott in 2010 though some schools were able to provide teacher assessment levels as an alternative.

All schools except Waterside did their own marking. Waterside’s tests were marked by two of the Maths PGCE tutors at Sussex. A moderation process was carried out on 10% samples of papers from 6 schools (Highfield, Dean Park, Ellen Walker, Waterside, Alan Bennett, Cardinal Newman); this was done by the same Maths PGCE tutors. No systematic biases in

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marking were found, but several questions were regularly misinterpreted, and corrections of the order of plus or minus 2 marks were found in 50-70% of the sample papers. We considered this to be within the expected error margins of the test paper itself when used for measuring levels of attainment and no changes were made to the test scores returned by schools, except that any corrected scores from the moderation were used in subsequent data analysis.

Statistical methods (using SPSS, Excel, and R software) were used to analyse the quantitative data from the pupil test scores and attitude questionnaires for the research and comparator schools. These data were triangulated against the observational, interview and documentary data.

For analysis, test scores were converted to ‘level points’ according to the table in Appendix 2; 3 points spans a whole attainment level, and 1 point an attainment sub-level.

Findings

The first analysis looked at mean ‘gain scores’ for each school, that is, the points difference between KS2 and Year 7 scores. The results are summarised in the following table.

Table 3: Comparison of ‘gain scores’ (mean level points difference divided by 3), KS2 test to Year 7 test Mean level Mean level Mean level Mean level change KS2 Research change KS2 SAT change KS2 TA change KS2 SAT to Y7 Comparison Schools to Y7 test to Y7 test TA to Y7 test test schools Dean 1.01 Park 0.71 [n=75,27%] [n=132,42%] Ellen Walker 0.39 0.41 0.35 Highfield 0.53 [n=53,18%] [n=268,91%] [n=119,94%] [n=34,27%] Waterside 0.32 0.36 Cardinal Quayside [n=113,88%] [n=171,87%] Newman 0.60 0.36 Cardinal Ridgeway [n=96,74%] [n=171,87%] Newman 0.51 0.72 Waverley 0.59 [n=172,72%] [n=171,72%] [n=65,41%] Alan Bennett 0.81 Sideview 0.87 [n=97,77%] [n=102,81%] Ellen Walker [Numbers in brackets: n=number of students in calculation, % of Year 7 cohort included] Of these comparisons, Ellen Walker (comparator) exceeds Dean Park (research school) by a significant difference, and Ridgeway (research) exceeds Cardinal Newman (comparator).

Other differences are not statistically significant, using 95% confidence intervals.

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The use of gain scores is questionable on several grounds. Firstly , it is widely argued that the levels assigned by KS2 tests are not equivalent with levels at KS3, in that non-equivalent 2

Thanks to Prof Kenneth Ruthven (Cambridge) and Prof Margaret Brown (King’s College London) for advice on this subject.

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mathematical performance is being measured by the tests. Added to this, we found literature on a long-running methodological debate in educational assessment concerning the statistical validity of gain scores in pre- and post-testing for educational interventions (Dugard & Todman, 1995).

Analysis of covariance The preferred alternative to the analysis of gain scores is to use analysis of covariance of the year 7 test results with KS2 score as a covariate (Dugard & Todman, 1995). The results of this for research/comparison school pairs were that no significant difference was found between pairs, using 95% confidence intervals, except for the pair of schools Dean Park and Ellen Walker, where the research school underperformed relative to the comparator.

To refine the analysis, we took note of the fact that in 3 of the 6 research schools (Dean Park, Highfield, Waverley), only some of the Year 7 classes had had experience with complex instruction, whereas the other classes had received a very varied experience of group work and/or problem-solving type activities. We re-ran the analysis of covariance using only the ‘CI classes’; the results of this were no significant difference for Waverley and Alan Bennett (as before), and Dean Park and Ellen Walker, and a slightly better performance of Highfield (research school) over Waterside (comparator).

Pupil Attitudes The questionnaire had been designed and pilot tested in previous research, and was used again without modification. Questionnaires were sent to 10 schools, and the table in Appendix 3 shows satisfactory or better response rates for all of the schools. A total of 2979 responses were manually entered into SPSS software; 1603 for year 7, 1376 for year 8. The administration of the questionnaire was not supervised in any school by the research team.

We suggest some caution in interpreting results, particularly for the comparison schools, where teachers and students had no direct knowledge of the purposes of the questionnaire or the REALMS project. Students were asked to give their names (for purposes of connecting with the test data), and these were obviously going to be visible to the class teacher and the teacher (sometimes the Head of Department) coordinating the data collection. We do not know if there was any (inadvertent) pressure, or perceived pressure, for students to respond in certain ways.

Internal reliability of the questionnaire data Significant numbers of ‘careless’ responses were observed during data entry (for example, ticking all left hand boxes all the way down the page). We therefore decided to check the internal reliability of the data, using pairs of questions which had been deliberately set up as equivalent or inverses of each other – for example ‘I like maths’ (5a) against ‘maths makes me feel stupid’ (5f); ‘only some people can achieve high levels’ (4a) against ‘anyone can be good at maths if they really try’ (4c).

Gamma measures were computed, to measure the degree of association between the responses to pairs of questions. Computing gamma for the whole data set showed moderate (thus satisfactory) associations for the 7 question pairs tested. Gamma scores for individual schools were more variable, and suggested a predominance of unreliable responses from Quayside and Cardinal Newman.

Analysis of the ‘maths is …’ responses The final question in the questionnaire was: If you were asked to describe your feelings about maths, in one or two words, what would you say? “Maths is …...”

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The set of responses was sufficiently rich to warrant a categorical analysis. This offers complementary data to the Likert response items from the rest of the questionnaire.

All the student responses were put through a frequency counter for all 1, 2, 3, ... word phrases (‘Textanz’ concordance software, www.textanz.com). This pays no attention to meaning or grammar so many (indeed the majority) of sequential but meaningless phrases are counted. We then went manually through this phrase frequency list and mapped all the meaningful phrases against 'scores' on 3 dimensions of attitude: appreciation, utility, and difficulty. These dimensions are the same as those used in a similar analysis in the previous research. The coverage of these dimensions to the students' responses was extremely good with a small percentage of uncategorised responses. Some responses only scored on one dimension (e.g. ‘(very) boring’ [appreciation] or ‘hard’ [difficulty]) but many scored on two dimensions (e.g. ‘boring but important’ [appreciation and/utility]).

The score scales were set to -2 to +2 according to: •

appreciation for experience of learning maths: (-2) strong dislike - dislike - neutral - like - strong like (+2)

•

utility of maths for personal goals or 'for the world' : (-2) strong negative - negative - neutral - positive - strong positive (+2)

•

difficulty of learning maths: (-2) very easy - easy - neutral - difficult - very difficult (+2)

The next stage was to calculate scores for appreciation, utility, difficulty for as many individual students in the questionnaire data set as possible, using an automatic process of pattern matching against the score/phrase list (we used ‘R’ statistical software for this). This process had a few flaws as some responses were not counted (due to untypical form of expression), and some were miscategorised from 'very positive' appreciation to 'positive' appreciation (i.e. scoring 1 instead of 2).

Table 4: Summary statistics, all schools Appreciation Utility Difficulty

n 2161 576 773

mean 0.07 0.9 0.76

sd 1.08 0.55 0.81

min -2 -1 -2

max 2 2 2

Table 5: Summary statistics, Year 7 students, CI schools and non-CI schools CI schools Appreciation Utility Difficulty Non-CI schools Appreciation Utility Difficulty

n

mean

sd

min

max

718 174 265

0 0.94 0.77

1.08 0.42 0.82

-2 -1 -2

2 2 2

488 100 160

0.31 0.88 0.78

1.06 0.59 0.82

-2 -1 -2

2 2 2

For utility and difficulty, responses are similar. Whereas for appreciation, the Non CI Year 7

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students are significantly more positive than the CI students (and the separation is stronger than calculating for Year 7 and Year 8 students combined).

Significance testing of questionnaire items Significance testing of students’ responses did not show many significant differences between CI and non-CI students. The notable ones were that students taught in maths classes using CI approaches expressed the view that they made significantly greater use of their own thoughts and ideas in maths lessons than those taught in ‘non-CI’ classes, were less likely to regard the teachers’ methods as important and more likely to learn from each other. However, while the students in classes using CI reported being significantly more likely to like maths when working in groups, in general they liked maths significantly less than those taught in ‘non-CI’ classes (this agrees with the findings for appreciation above).

Looking to the interview data, the lower appreciation for mathematics among CI students could be attributed to several factors. The experience of set maths lessons in Year 7 puts students in a ‘comfort zone’ where habits learnt in primary school can be maintained.

Lessons using CI put students outside of their comfort zones so could create negative feelings. However, this contrasts with the more probing questioning in student interviews where there was a lot of positive appreciation for groupwork. It was also evident that most of the observed teachers were struggling to make CI work in their teaching, and students were experiencing those difficulties: a typical comment in interview was that students liked groupwork if it ‘worked’ but lessons were too disrupted by poor behaviour, or tasks were perceived as unchallenging.

Use of groupwork Reported use of groupwork ranged from Waverley school in which teachers and students claimed that over 95% of maths lesson involved group tasks, to Highfield in which very limited groupwork appeared to be taking place. The student attitude survey suggested that there was significantly more groupwork in four of the CI schools (Dean Park, Waverley, Quayside and Sideview), than non CI schools (p < 0.05). These survey data were supported by both interviews and lesson observations in showing that while the range across schools was marked, groupwork was a frequent feature of maths lessons in two of the schools and a weekly or fortnightly experience in three of the others. In most schools, a typical pattern of teaching was one groupwork lesson per week or fortnight, which might be allowed to spread over 2 lessons if the time was required.

Where teachers did groupwork less often, they reported that this was on the grounds that either insufficient mathematical content could be covered in a lesson or that behaviour problems made it impossible to undertake. There was considerable variation in the use of groupwork among teachers in the same school. Some schools (Waverley, Quayside) had a more collegiate way of working which encouraged all teachers to undertake groupwork regularly. In other schools, it was left to individual teachers and there was considerable variability. Some NQTs found regular groupwork particularly challenging whilst maintaining all the other aspects of the classroom. Pupils were mostly very positive in stating a strong preference for groupwork: Groupwork is interesting because you get to wo rk with people that you haven’t wo rked with befo re. And: we can learn from each other..share what we learn. And: …more interesting, more helpful, if you do it on your o wn you have less workspa ce, and less brain. Some also contrasted their current experience of groupwork with earlier teaching approaches: Well we used to kind of just copy fro m a tex tbook, like we read th rough, say if we were learning about bar cha rts we'd lea rn about bar cha rts and answer qu estions about bar charts and that would be the end of the lesson whereas h ere we learn about it all th rough the lesson

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and we talk about it.

Those teachers who regularly used groupwork spoke of progression such that the lessons might no longer need specifically identifying as a ‘groupwork lesson’ but would include some groupwork activities. As two of the CI teachers commented: In terms of b eing seamless between group work and non-groupwork lessons that is th e big thing that has co me out fo r my tea ching, and I think for o thers. Specifically for yea r 7, we teach all the time in that style, I don't think of two types of lesson, all of the time they a re expected to work in those roles, and do any aspect of the lesson in that ethos. You are always working in that way. That is a big change fo r me, using it all the time and I think it works better, it's still not perfect by any means, but it is what we will build on in the futu re... because everyone is trying to teach this way, the whole cohort don't know any different, th ey only know this way of lea rning maths. It has made a huge improvemen t and the cla sses I teach a re en thusiastic all the time, which I think is because they are being encou raged to work as teams, which they enjoy. (Waverley-T, 2011/06) I think trying to integrate it more into th e normal tea ching. To begin with it’s been quite explicit that 'we are doing group work today', we have b een wo rking on our group wo rk skills, that is th e focus, so merging that in to the no rmal maths work, b ecause I think they are a lot more confident working in groups no w… (Dean Park-Y, 2011/02)

Making groupwork a ‘normal’ aspect of maths lessons emerged as a key factor for success and not being discouraged from undertaking groupwork by the lack of an immediately accessible ‘groupwork task’ for the topic currently being studied (see below for a further discussion of tasks).

Allocation of groups There was a range of approaches, from teachers who always assigned the groups (with mixed attainment in each group), typically changing them for each new topic or half term, through to teachers who let students choose their own groups. The larger schools operated a policy in Year 7 of allowing students to ‘settle in’ by being able to choose their own friends as seating partners in most lessons. In such circumstances, for maths teachers to impose groupings as part of CI was not well accepted by the students. Teachers who did not challenge this expectation from the outset experienced problems later in making groupwork acceptable to students. Most students stated that they would prefer to work with their friends but many noted that this may not necessarily lead to the most productive work outcomes. An effective way of dealing with these kinds of expectation was the use of random allocations (as also for selection of groups to give plenary feedback), we suppose because it was not the teacher imposing their decision on the class, but the workings of chance.

As the research progressed, a few of the teachers began to think about having ability groups within their mixed ability class, as a way of promoting ‘better’ groupwork, despite this being contrary to the basic principles of CI. However, in the lessons observed, mixed ability groups prevailed and students were almost always positive about this in interviews suggesting that it enabled greater challenge in the tasks as well as equity: helps the higher ability students g et better by helping th e lower ability ones Overall math s at Quayside is OK, good, better than prima ry because don’t trea t you like a baby, telling you to do level 3, you can do level 4, 5 or 6 I find it better, being mixed ability doesn't make you feel bad because if you knew you were in

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a lower ability group then you'd think, 'oh no I'm not very good at this blah blah' and if you were in the top one you'd be like 'oh yeh I'm really good at that' and if you're in the middle you're like 'oh I'm alright'. But in mixed ability like you're the sa me.

Conflicting views were occasionally expressed by students sometimes seeing both benefits and disadvantages: I think sometimes it can be a bit restraining, if you're quite clever and you can do it, sometimes it slo ws you down a bit because you've got to wait for people who can't necessarily do it o r need your help, so it can be a bit ha rd 'cos you've got to wait for th em, though as [S4] said... it can be good to help someone.

Many students contrasted their current mixed ability maths teaching with that which they experienced in primary school: In my primary school they didn't really fo cus on the lo west people in th e class, th ey focused on the middle upwa rds, ju st give the lower p eople wo rk, really easy work, and then focus on the medium and really good ones.

The teachers in the research schools were predictably more positive about mixed ability teaching than is observed elsewhere in the profession, given they had mainly signed up to be involved in the research (for some, the departmental commitment might have meant they had little choice). Many articulated the benefits of mixed ability groupwork, in particular in conjunction with the roles and gave examples of how individual students benefit: B is seen as no t so bright bu t bringing those roles in to the g roup and really firming up, no that’s exactly what B’s role is, he is there to explain math ematical ideas whether they’re his ideas or not, he is still developing an understanding whether J explains those to B because J is your high ability student. If B has to explain those id eas back to the g roup, then that’s p rogressing him and actually the rest of the g roup can’t just say, “No, no, no he’s no t gonna do anything”. In order fo r them to p rogress, in ord er fo r th em to complete the task, they’ve go t to make sure that B has tha t understanding. (Waverley-C, 2011-06)

Use of roles All teachers began the year with a commitment to using the roles which was strongly identified with the practice of CI. In the first term of observed lessons, teachers would devote 5 to 10 minutes of a 50 – 60 minute lesson on introducing the roles, and finishing the lesson with 5 minutes reflection on the roles. As observations progressed, it was evident that most students had learnt the role descriptions and what each role should involve. They could confidently talk about this when we interviewed them, and reported that roles were useful. However, actual conduct of the roles by students was generally weak, and with the exception of Dean Park, teachers mostly did not pay attention to this as the groupwork was in progress. For example in Sideview, students spoke eloquently in interview about what the roles involved but observation of the lesson showed that they totally failed to enact them; we characterise this as roles becoming ‘rituals’ to satisfy the teacher but not effective tools for practice.

One role, resource manager, was well used by teachers and students because it is very clear: only the resource manager is allowed to leave the group table to bring resources, and also is the only person who can ask the teacher a question. The roles involving inclusion and understanding were in general poorly used by students and teachers. Part of the problem was operationalising the role descriptions. A good strategy developed by Quayside was to

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give out both general role descriptions, and a list of things you will need to do in that role and questions you will need to ask in this lesson. This was backed up by the teacher prompting students to use that information during the lesson.

Most schools came to adapt the role titles and descriptions in some way, so for example Waverley had a ‘spy’ who towards the end of an activity would visit another group in order to find out what different approaches they had used: I’ve tweaked the roles a bit, so tha t they’re all a something coordinato r; so they’ve all got the same kind of sense of impo rtance. They probably do have their favourites, and you always get the p eople which are natural lead ers, but I think because I’ve been careful for th em all to have experien ced all of th e roles, I don’t think they’d actually mind taking on any of the roles.

When they have been able to choose their own roles, I say make sure you are not doing th e same role as you did last time and have a swap around, so you get a bit of a mixture.

(Waverley-M, 2011-06)

Table 6: Versions of role definitions Cohen et al

NRICH

*Team captain

Sideview school

Waverley school

*Organiser

*Materials manager

*Resource manager

*Resourcer

*Resources & Representing

*Facilitator

*Facilitator *Inclusion *Understanding *Understanding coordinator

*Inclusion *Verbal communication/Spy

*Reporter/ Recorder

*Recorder/ Reporter

*Recording & Publishing

Sources: Cohen (1994), nrich.maths.org/content/id/6966/Roles.pdf The table shows some of the versions of roles used in the schools. Major differences are in the US context having an explicit ‘leadership’ role (team captain, organiser), whereas English teachers have tended to prefer to have two facilitator roles, ‘Inclusion’ (with focus on group participation) and ‘Understanding’ (with focus on understanding the mathematical content); students have found the inclusion and understanding roles most difficult to perform; knowing how to include a group member who is resistant to being included requires substantial skills; also the distinction between understanding the mathematics yourself or helping the group to come to an agreed understanding, sometimes without understanding it yourself at the outset.

In summary, we seem to have clear evidence about what not to do with roles - repetitious rituals at the start/end of lessons without teacher's’ active engagement with students-inroles throughout a lesson. In successful lessons that we have observed, roles do become 'implicit' over time but it was not fully clear what is going on. The central issue is not 'doing roles' but maintaining classroom norms such that equity and group engagement are maintained. If a group has 'good chemistry' this is likely to be the dominant factor over roles.

Roles seem to matter when groupwork is being established, and when groups do not 'naturally' work well together.

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Participation in groupwork and in feedback from groups In general, there was less variation in levels of participation within schools than between them though one or two specific lessons (with the older students) went astray where behaviour appeared to be problematic. The roles are the ‘formal’ route into maximising participation though one teacher managed to get high levels of participation without the use of the roles with the Year 7 students. One student focus group claimed that they were engaged for at least 75% in the maths lessons that were groupwork compared to under 50% when the work was mainly individual. This is consistent with the data from the attitude survey suggesting that students like maths when they are working in groups.

One teacher in Waverley was clear that the groups need engaging from the moment that the lesson begins and had devised some effective strategies for this: …each group has to come up with two or three ideas as to how you would start this task; they write th em on th e board and then we’ll sp end a bit of time going th rough that, so you’ve got this idea of sharing what’s happening. One group may have thought to do something which another group migh t not have thought of, so then they can spin off from that. So tha t’s stays up…

Later in the interview she added: I never sta rt the lesson as a tea cher led thing, there’s either so mething on the table o r something fo r them to be doing because I couldn’t get th em quiet to start off with.

A significant observation comes from Quayside school where Year 7 students were participating in an extensive general programme of ‘learning to learn’ for a full day each week, which included an emphasis on learning how to work cooperatively and in groups. The maths teachers found students very adaptable to working in CI groups, and described a high degree of complementarity between CI and study skill programmes such as ‘learning to learn’: it [CI] promotes th e habits of mind (thinking, being and doing) that are pa rt of the Learning to Learn curriculu m, which is anoth er whole-school policy. CI pa rticularly fits with th e doing skills that we are d eveloping with Yea r 7s.... This is why I think it is particula rly powerful and Yea r 7s are particula rly good at working in groups, because of these other things p romoting the idea of groupwork. W e feel CI is creating mo re opportunities for su ccess (Quayside-H, 2011-

05)

Interdependence Central to CI is the notion that students all contribute and all succeed while acknowledging that success takes many forms. In practice, this means ensuring that through the selection and construction of the task and the use of the roles, successful task completion requires everyone to contribute. Developing this level of interdependence is challenging and few examples of genuine interdependence across the whole of the group were observed. More often two, or sometimes three students undertook most of the task and gave the feedback with one or two students barely participating – a few notable examples did achieve full interdependence. However, in the teacher interviews there were many references to developing interdependence. For example, one teacher who made much use of the roles stated: one of the main ben efits of the group ta sk is th ere are three oth er teach ers in you r group to help you; you don’t just need me as a tea cher, so I should be the last resort that you ask, and for some g roups who it took longer to establish that with I’d say who else have you asked?

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(Waverley-M, 2011-06) Another teacher noted that this was a key priority for future development: …the one area I would want to b e wo rking on is the communication and inclusion bit, they don’t sit and read the task together. One group wan ted to split in half to wo rk, bu t then they realised that th e other two migh t not know what th ey had done…L always does his own thing ; the group find him hard to wo rk with. He had identified the test for divisibility and had it all down on paper in front of him, but when the g roup fed back, it wasn’t him who spoke, and what they fed back was nowhere nea r as he’d got on his o wn. It’s trying to get that wealth of ideas to co me back to the whole g roup. (Waverley-C, 2011-03)

There seem to be several contributing factors in this finding. First, most tasks did not score highly on requiring interdependence, which is a key factor for groupworthiness. A number of 3 well-tested ‘skill builder’ activities can be found in the CI literature (e.g. Cohen, 1994 ) and 4 these have been adapted for UK maths classrooms by NRICH but were used very little by the teachers observed, although effectively when they did so. Establishing similar levels of interdependence within the longer mathematical tasks was harder to achieve. Secondly, when circulating round the groups or responding to a request for help, teachers tended to speak predominantly to one student rather than to the whole group thus (unintentionally) discouraging interdependence. Similarly, this meant that they rarely highlighted contributions made by ‘low status’ students in the group (for example by acknowledging the contribution made by another student “As Jack was saying…”), a key feature of effective CI. Finally, while some teachers reiterated the distinctiveness of the roles and tried not to respond to student requests where they came from other than the resources manager, these examples were in the minority.

Progress of students in groupwork skills There was little evidence of progress in groupwork skills from the recorded lesson observations over time but since the tasks were very different it was difficult to assess this.

One notable exception to this was Quayside, which had adopted CI as a maths department commitment, and it was observable in the final set of observations that general engagement and groupwork had improved. In general, most teachers spoke about progress that they had seen in the students and ways in which they encouraged this. In particular, two teachers from Waverley mentioned the need to move students from a competitive to a collaborative approach to learning: a number of people who came into my class who I hadn’t taught befo re, who were very reluctant to share things, to talk about what they were doing, because th ey saw everything as kind of competitive …I have a multi-dimensional classroom. You know, success takes a va riety of forms and it’s impo rtant that you can sha re tha t but recognise that… I spent time, guiding them on how to co mmen t about imp rovements …And rather than just say ‘oh but we don’t like that bit there’; okay, well why not? What could they have done better? You need to give them so me kind of feedback and I think that was the skill they n eeded to develop.

This teacher noted that the students that had been working at higher levels in a separate group in their previous class had refused to share their ideas and the ones working at lower levels developed more confidence in the small groups to ask questions rather than leaving misconceptions to develop. Two teachers had been working together on the need to model 3 4

http://suse-step.stanford.edu/r esources#skill NRICH site: http://nrich.maths.org/6933

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the roles, using the language that they might use in order to support student progress in groupwork skills.

Another teacher suggested that once the roles were well established they could become less explicit in lessons: if you’re working well as a team actually a lot of these roles are in the background though a teacher at another school explained that over the year as the students developed their groupwork skills she paid less attention to the roles but noticed that they gradually became more off task. She tried different tasks but that had little effect so more recently she: went ba ck to th e roles and brought the roles ba ck into th e group wo rk and I think it kind of gave the stud ents an identity and a purpose beyond just the purpose of co mpleting the exercise. So they had this, I know that everyone talks about it; the group accountability, which I began to think would you know continue without roles but it absolutely didn’t. (Waverley-C,

2011-06)

Progress in mathematical understanding Despite mixed findings from the quantitative test results in the six schools, teachers reported on the pupils’ increased mathematical understanding that they had observed since adopting approaches drawing on CI: I purposefully pick people who have had an understanding, but may not have had belief in that understanding to sta rt off with. They can then share tha t with the class. If th ey can reconstru ct their understanding to explain it to so mebody else, that’s a fo rm of su ccess. A parent commented at paren t’s evening that they couldn’t believe how confident their child was at Ma ths now. (Waverley-M, 2011-03)

And: I am able to p raise, people tha t perhaps I wouldn’t in no rmal mathematical lessons b ecause they would be because either th ey wouldn’t b e producing work or feel very self conscious about their mathematics, unsu re and not willing to give answers in a lesson, o r to get involved in a plenary. But I’ve really noticed so me of those people rise up in the CI lessons

(Dean Park-Y, 2011-02)

One teacher was surprised to find that the younger students made more progress on mathematical skills than the older ones on the same task: The making of th e box and measuring it p roved very challenging for some of them [older students]…I did a similar lesson with my [younger] classes on area , perimeter and volume and every single group produced about 4 boxes perfectly, one inside the oth er. They p roduced reasons for which ones were bigger, and so me of th em had used calculations to prove volumes. One girl got as fa r as the tea cher no tes, she produced the g raph, how it had changed over time, she’d recognised tha t the maximu m point wasn’t a cube. (Waverley-T,

2011-03)

The same teacher went on to identify what it takes to develop mathematical understanding:

…understanding that they actually want to be ma thematicians to understand why things a re happening and to justify it; and tha t’s starting to happen. I think that’s the big thing and it’s working. The challenges are getting the tasks right and getting the feedback right, getting them to understand…(Waverley-T, 2011-03)

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Appropriateness of task Some tasks (e.g. calculating the number of fire exits needed in Wembley Stadium in order to meet the evacuation requirements, using area, volume and scale to undertake an interior design task) seemed much more suited to encouraging groupwork and even interdependence, than others. Pupils recognised this in their comments, for example: Code breaking was good for groups as could help each o ther with that. Boaler and Staples (2008) cite Horn (2005, p22) in suggesting that groupworthy problems are those that: … illustra te important ma thema tical concep ts, allo w for multiple rep resentations, include tasks that draw effectively on the collective resources of a group, and have several possible solution paths.

A general finding here is that effectiveness of a task in a classroom was a function of both the 'intrinsic groupworthiness' of the task and the established groupwork culture of the class. That is, for teachers and students who had become comfortable with groupwork they would tend to tackle any task in a group-oriented way (e.g. Vedic squares at Waverley); this finding resonates with the research done in the USA (Boaler & Staples, 2008). On the other hand, a very groupworthy task would fail to be effective where groupwork culture is poor.

Inclusivity Some of the observed tasks offered multiple entry points, and some teachers commented that this had been intentional to their task design (Highfield paid particular attention to this).

Instances of teachers assigning competence to ‘low status’ students were very rare – although this is suggested as a key method for CI, and we discussed it with teachers at almost every termly meeting. Teachers did however, report particular benefits of the groupwork for lower attaining students or some of those exhibiting behaviour problems: There is a girl in one of the g roups, and she’s got one of the lo west targ et g rades in th e class; and she participa tes really well in the group wo rk and also in the class discussions about it sh e was participating well, which is nice to see. Also one of th e boys in her g roup, who I have had problems with , contributing to g roup work b efore, he seemed to be leading the group and really pushing them forward and he’s one of the lo wer ones in the class as well, so tha t was really good to see. (Dean Park-Y, 2011-02)

Other aspects of teaching Higher order questioning by the teacher seemed to take place most often when they were speaking to a small group, often an individual student in the group. Some higher order questioning occurred in plenaries though this was less frequent. One school had a policy of very clear timescales being given throughout the lesson supported by a timer on the whiteboard, and some teachers did this in other schools. In most schools, there was common use of the ‘3,2,1’ countdown in order to silence the class and get their attention, which was mostly effective.

Wider issues Analysis by individual teacher suggests that length of teaching experience does not seem to be the determining factor in capacity to adopt CI approaches. The teacher (Waverley-C) whose students gave the most positive responses on the attitudes to maths questionnaire was only in her second year of teaching.

Some teachers noted the importance of establishing a balance between good groupwork tasks that maximised learning and the need to meet traditional requirements for attainment

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in tests: …you can take twice as long to cover conten t using an extended task because you have to make sure th ey have the kno wledge and can apply it in a differen t context, b ecause they apply it in a mostly abstract con text in the group wo rk task, it's making sure th ey can apply it … it's horrible to say, but th ey need to be able to do well in th e tests they are set, the GCSEs, functional skills exams, so we have to make sure that they can apply what we a re teaching to that situa tion...(Waverley-C, 2010-10)

It was also noted that focusing explicitly on levels was in conflict with the CI approach but was a requirement imposed in some schools: …a new school initiative for all of yea r 7 says that all students must know where they a re working at, and all a re given a minimum targ et grad e for the end of yea r 9. They have to know that on the spot if asked by a member of senior managemen t. Which goes against our principles in the maths depa rtment – which is that you can't give a student a nu mber, you can't walk a cross the school hall being a 4+, being labelled. (Waverley-C, 2010-10)

Teachers at Quayside reported criticisms by Ofsted of CI teaching which did not in the inspectors’ view ‘guarantee’ learning outcomes for every student in every lesson. Waverley had an Ofsted inspection in term 2, rated ‘outstanding’, and was complimented on its innovation with CI. At an NRICH conference in 2011 attended by the REALMS team, a maths HMI stated that more innovatory pedagogy would be welcomed by Ofsted, but the experience of teachers engaged in the research was that this was not necessarily what individual inspectors expect to see. Publications (e.g Ofsted, 2008) decry the lack of innovative pedagogy, and what it can do to promote improvement, but school staff reported being genuinely fearful of criticism by Ofsted inspection teams and tended to revert to more traditional teaching during inspections.

Learning journeys: The cases of Quayside and Waverley We highlight these two schools as making particular progress in implementing CI. In terms of conventional measures of academic performance they are very different schools: Waverley, high-performing in GCSE and over-subscribed; Quayside, a school rebuilding itself after a period of decline. What they have in common is a systematic implementation of the CI approach with the mathematics teachers working as a team, though through different arrangements.

Quayside The school has been in special measures for several years, which means it is closely scrutinised by the local authority and Ofsted. A new head teacher was appointed two years ago, who has implemented major changes. Significantly for work using CI, the focus of these changes has been on changing the attitudes and expectations of students, teachers, and parents about learning beyond seeking quick solutions to boost academic performance. Year 7 students study one day a week in cross-curricular programmes, particularly ‘learning to learn’, which is teaching them different modes of learning, particularly group work. The maths teachers identified this grounding of skills as particularly important for the students to benefit from CI. The development of CI has been led by the school’s Advanced Skills Teacher (AST) in mathematics, who has engaged with the research team at Sussex over several years, attending workshops, and a short programme of CPD for CI was carried out by the research team in the school in 2009, involving all of the maths teachers. The AST was particularly

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concerned about teacher preparation, as the department was experiencing a high turnover of teaching staff, frequent use of temporary supply teachers, and identified a lot of teaching as weak.

The department gained approval for 2010-11 to teach in mixed ability groups for Year 7 (other years remain set) and all the teachers of Year 7 classes committed to collaborating on a CI approach. The AST led this by developing lesson plans for CI lessons, which were used regularly at least once a week. Some of the teachers adopted CI to all of their lessons including those with other year groups. Teachers discussed lesson plans and gave feedback to each other on how lessons had gone. All this experience is now collected as a resource for teaching Year 7 in the next year. The AST’s role appears to have been critical to success in being the driver for change, with active support of the head of department, and (as an AST) having time and resources to act as an ongoing developer and mediator for CI. The first year has been rated so successful that mixed ability and CI will be extended into Year 8. Teachers reported both above target performance in tests, and significantly changed attitudes in students, from dominantly passive to active learners.

Waverley This is a high-performing school, successful on all external measures of performance, with skilled and committed teaching staff. Its mathematics results are outstanding. The decision to adopt CI began with a (now departed) head of department who wanted to develop better mathematics learning and approached the Sussex team several years ago to find out about CI. The school was a participant in the previous pilot research project. Cross-departmental work with CI showed some variability in approaches. The development is being taken forward by four teachers who have decided to make CI a core aspect of Year 7 and Year 8 teaching (other teachers that we met spoke approvingly of CI, but it was not clear if it features regularly in their teaching). Three of these teachers have been carrying out research projects for higher degrees, directly based on their teaching practice; this seemed to be a strong additional motivation to make the effort to develop CI, although all of the teachers showed a high degree of enthusiasm and mathematical skill.

There was no AST or other additional resources in place to coordinate CI development, develop lesson materials and gather feedback on how lessons progressed. The teachers shared this on top of regular teaching loads, through impromptu face to face meetings and online communication via the school’s virtual learning environment; there was not sufficient time to develop and reflect. The 2010-11 year was the first year that the teachers had committed to using CI on a regular basis, with a view to developing learning materials for a CI-based scheme of work in Year 7 and (eventually) Year 8. All the teachers reported the year to have been very successful (and we observed excellent CI lessons on every research visit). The students all reported appreciation for learning with CI. The main ‘pay off’ which the teachers spoke about was to shape attitudes and habits of learning towards the demands of Key Stage 4. All teachers felt that the school over-relied on ‘cramming’ of students to maintain high GCSE performance. Moreover, they were concerned that students would not be prepared for the ‘functional skills’ element of the new mathematics GCSE examinations by ‘cramming’, but rather it would require the problem solving skills which teachers felt came through use of CI and mixed ability grouping.

The experiences of these two schools show that the challenges of using a CI approach

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demand strong collaborative work across the department, long term commitment from teachers, good subject confidence and competence and a supportive school context.

Reflections on methodology and progress Engagement of schools in research In our proposal we identified six research schools one of which was in the North of England, which did not participate and was replaced by another school in the South, giving a total of three research schools in the South and three in the East. We also identified six comparator schools. Providing incentives to comparator schools is always a challenge and in this project one of the schools withdrew and another failed to provide data. Two of the comparator schools were each subsequently matched to two research schools, which while not ideal, was possible given the demographic characteristics of the schools.

Changes in attainment as a measure of impact A similar approach to quantitative data was adopted as that used in the previous research by the Sussex team (Boaler et al, 2010) with improvements to rectify the problems previously encountered. Since one academic year is not enough time for measurable changes in attainment, we were unsurprised that the test data showed no significant differences between students in the research and comparator schools. Nonetheless, schools welcomed being able to reassure senior managers and governors that drawing on the REALMS approach would not be at the expense of maintaining or raising standards, even if progress in test results took longer to demonstrate.

Adopting CI requires time and effort spent on group-oriented and problem solving ‘process’ skills, progress on which are not measured by conventional tests, but senior managers and governors in some schools needed reassurance that this investment of time was not at the expense of improving mathematical attainment. Several schools were explicit that they did not expect to see major change in attainment in Year 7, but that the changed skills and attitudes that teachers reported in interviews would begin to ‘pay off’ in Years 8, 9 and GCSEs. Ridgeway, Sideview and Waverley had been using CI over a long enough period to see evidence of this in their years 9, 10, and 11 students. However, the engagement of two schools that had very low starting points in terms of attainment and quality of teaching in maths made the demonstration of improvement even more challenging.

A good measure of the students' learning in the different schools over 10 months would have assessed mathematics broadly, paying attention to students' reasoning and the development of complex ideas. Instead we relied upon Key stage 2 National Curriculum tests/teacher assessments and the QCDA Year 7 test paper which is mathematical contentfocussed and does not assess the problem-solving skills which CI aims to develop.

Assessing changes using test data In our proposal we said that we would ‘conduct our own assessments at the end of years 7 and 8, which we would compare with the students’ end of Y6 national assessment levels to give a measure of their learning’. This did not prove possible as it would have required an additional test being imposed on schools already using the year 7 optional tests – hence we took the lead from the schools in accepting the use of the optional tests which they administered themselves. This turned out to be problematic. We are aware that there was some variability in how students were presented the test, and we lack information about how much revision/preparation was involved. Some schools used it as the definitive ‘end of

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year’ test, whilst others treated it as a module test (the end of year test being done in the previous term to provide information for setting in the next year). Timing of the test ranged from mid-May to early July. Quayside used the wrong paper, QCDA Paper 1; the scores for this paper are broadly comparable to Paper 2, however the papers do test a different balance of skills (P1 is non-calculator, P2 is with-calculator) and curricular topics. With that caution, we used Quayside’s data in the comparative analysis.

Changes in the attitudes of students Additional data were sought that would be indicative of changes in capability and attitudes in mathematics. The main source of this was through the administration of the pupil attitude questionnaire used effectively in our previous study. However, in order to minimise disruption in schools, the researchers did not themselves introduce or administer the questionnaire to the pupils, but left it to the research and comparator schools to do so. The data that emerged is both less consistent and in general, more negative than that reported from our previous study.

Level of intervention on CI As indicated in our application, the research team did not provide ongoing intervention on CI though schools had received varying inputs in that some had attended a CI workshop the previous year, while others had involved the research team in a staff development session.

Instead, we encouraged schools to collaborate through termly teacher meetings in each of the two areas where we observed. For some schools (Highfield and Quayside) this might not have been sufficient support given their minimal prior experience of CI. We did encourage all teachers to support each other through online activities, and this was fairly successful for the sharing of task designs, but not for the broader experiences of teaching with the CI approach. Our experience in this research confirmed previous research (e.g. Fielding et al, 2005) that teachers do not find it easy to make time to collaborate with colleagues (even when in adjacent classrooms), although there was willingness to do so. The evidence from the findings suggests that the schools in which CI was less well established might have benefited from more input. There is a tension between intervening directly in teaching and trying to build capacity in schools in order to sustain their development following completion of the research. In this case, the research schools needed individualised intervention to reflect their different starting points on CI but the quasi-experimental design of the project would have been compromised by this personalised approach.

Intrusion of researchers and video in lessons The research observations affected both teachers and students to a varying extent. For some schools, an impending visit by the research team led to a specific lesson being planned and taught by all the participating teachers. There were pros and cons to this: it gave us a more ‘common basis’ on which to judge different teachers’ use of CI, and for the teachers it could be a chance to refresh their collaboration. On the negative side, our data came to lack details about the more ‘natural’, regular patterns of groupwork.

There was some evidence that both students and teachers were affected by the potentially intrusive presence of the two researchers and video cameras, and Flip video cameras sometimes used on group tables. In around half the lessons observed, there was a small amount of ‘playing to the camera’ such as waving or pulling faces though in most this was very isolated and short-lived; more rarely did a small number of students become highly distracted. These disruptions need to be balanced against the excellent quality video material obtained from the lessons which illustrate different approaches and provide a very rich source for professional development.

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Conclusions

The test results showed no statistically significant differences between the pairs of research and comparison schools. This suggests that we do not have a measure that might effectively isolate the impact of CI on attainment either positively or negatively â&#x20AC;&#x201C; although teachers generally expressed their perceptions of improved attainment among students. Our analysis of achievement does show that schools working to develop a new approach in line with national recommendations, in which they engaged students in collaborating, problem solving and reasoning, did not sacrifice performance on relatively narrow content tests.

Overall in the six schools, teachers started the year at very different stages of teaching in terms of their use of CI. In three of the schools, the findings suggest that teachers were at the early stages of taking on a new innovation and in the other three schools that had been developing CI approaches for some time, some of the teachers involved in the research had only recently joined the school and so the approach was new to them. Highfield only engaged minimally with the approach, rarely undertaking groupwork with a more didactic traditional approach to maths teaching being observed and confirmed in student interviews.

A common weakness we believe is not recognising the place of the norms and skills of groupwork, the need to specifically target these within teaching and the time taken for them to develop.

Hence, the data provide rich case studies of maths departments struggling to take on board new challenges in the context of competing tensions from external pressures, which in one case included threats of closure of the school. These case studies provide accounts of teachersâ&#x20AC;&#x2122; journeys of learning with small, incremental steps being taken towards a different approach to teaching mathematics and no sacrifice to student achievement on national tests The reports of students and teachers both suggest that the changed learning environments were important to their developing mathematical behaviours such as reasoning and problem solving and depth of understanding in certain mathematical concepts.

Three specific issues emerged relating to the mathematical tasks employed.

1. The interaction between groupworthiness of the task and classroom culture Groupworthiness, as we have seen, has several dimensions including multiple entry points to the task, drawing on the collective contributions of the group and allowing for multiple representations. However, the classroom culture in which the tasks are being undertaken is also important since we observed examples of tasks that were not inherently groupworthy being presented in a way that enabled high quality groupwork and others that were highly groupworthy that did not facilitate groupwork due to their presentation. Sometimes this was due to the timing of the activities within the lesson, so that for example the students never progressed on to the parts of the task that might have enabled deeper learning to take place. Hence, the ways in which tasks are used is as important as the intrinsic nature of the task though selecting and adapting appropriate tasks clearly contributed to the effectiveness of groupwork.

2. Rich mathematical tasks allow deeper learning Some of the tasks selected were richer in their mathematical content allowing deeper learning and often drawing positive comments from pupils about their levels of interest.

However, several of the tasks observed in lessons seemed mathematically infertile, never moving beyond limited numerical calculations. There was evidence that teachers find this a particular challenge and further work is needed on approaches to supporting them to

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develop and adapt rich mathematical tasks.

3. The challenge of developing interdependence The tasks selected were often not conducive to developing interdependence since they allowed one or two students to complete most of the work without involving the others.

Interdependence emerged as one of the greatest challenges in the CI approach. As with groupworthiness, it appeared to be as much about the classroom culture as it was about the task. The ways in which teachers presented the tasks such that task completion was dependent upon each student contributing rarely achieved interdependence, though there were a few outstanding examples of this, typically in only some of the small groups within the lesson. Recognising strategies that ‘pull the students in’ was acknowledged by two of the teachers, one of whom commented: I actually came up with tasks for them and got them blown up and put it on a piece of paper.

Basically, exercise books a way, because it do es pull them in, and the ph rase ‘working in the middle’ from Cohen’s book really stru ck th e chord with me. (Waverley-C, 2011-03)

Developing interdependence appeared to require a task presented in ways that allocated each student a role, a joint focus for working out and presenting their thinking and findings (such as the ‘paper in the middle’) and where students drifted from the group, reiteration of the function of the roles throughout the lesson.

In the context of national and international concerns about maths teaching, these research findings provide some clear areas for further debate in policy, practice and research in particular, teacher learning, task appropriateness and strategies for developing interdependence.

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