Raising Expectations and Achievement Levels
for All Mathematics Students (REALMS)
FINAL REPORT TO THE
ESMÉE 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
1
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 approach to adopt or 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.
1
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.
2
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.
3
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.
The use of gain scores is questionable on several grounds. Firstly2, 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.
5
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
7
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 work with people that you
haven’t worked with before. And: we can learn from each other..share what we learn. And: …more
interesting, more helpful, if you do it on your own you have less workspace, and less brain. Some also
contrasted their current experience of groupwork with earlier teaching approaches:
Well we used to kind of just copy from a textbook, like we read through, say if we were
learning about bar charts we'd learn about bar charts and answer questions about bar charts
and that would be the end of the lesson whereas here we learn about it all through 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 being seamless between groupwork and non-groupwork lessons that is the big
thing that has come out for my teaching, and I think for others. Specifically for year 7, we
teach all the time in that style, I don't think of two types of lesson, all of the time they are
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 for 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 future...
because everyone is trying to teach this way, the whole cohort don't know any different, they
only know this way of learning maths. It has made a huge improvement and the classes I
teach are enthusiastic all the time, which I think is because they are being encouraged to
work as teams, which they enjoy. (Waverley-T, 2011/06)
I think trying to integrate it more into the normal teaching. To begin with it’s been quite
explicit that 'we are doing group work today', we have been working on our group work skills,
that is the focus, so merging that into the normal maths work, because I think they are a lot
more confident working in groups now… (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 get better by helping the lower ability ones
Overall maths at Quayside is OK, good, better than primary because don’t treat 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 same.
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 slows you down a bit because you've got to wait for people who can't
necessarily do it or need your help, so it can be a bit hard 'cos you've got to wait for them,
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 focus on the lowest people in the class, they focused
on the middle upwards, just give the lower people work, 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 not so bright but bringing those roles into the group and really firming up, no that’s
exactly what B’s role is, he is there to explain mathematical 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 ideas back to the group, then that’s progressing him
and actually the rest of the group can’t just say, “No, no, no he’s not gonna do anything”. In
order for them to progress, in order for them to complete the task, they’ve got to make sure
that B has that 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
10
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 that they’re all a something coordinator; so they’ve all got the
same kind of sense of importance. They probably do have their favourites, and you always
get the people which are natural leaders, but I think because I’ve been careful for them all to
have experienced all of the 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 the
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.
11
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 them on the board and then we’ll spend a bit of time going through that, so you’ve got
this idea of sharing what’s happening. One group may have thought to do something which
another group might not have thought of, so then they can spin off from that. So that’s stays
up…
Later in the interview she added:
I never start the lesson as a teacher led thing, there’s either something on the table or
something for them to be doing because I couldn’t get them 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 the habits of mind (thinking, being and doing) that are part of the Learning to
Learn curriculum, which is another whole-school policy. CI particularly fits with the doing skills
that we are developing with Year 7s.... This is why I think it is particularly powerful and Year
7s are particularly good at working in groups, because of these other things promoting the
idea of groupwork. We feel CI is creating more opportunities for success (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 benefits of the group task is there are three other teachers in your group to
help you; you don’t just need me as a teacher, so I should be the last resort that you ask, and
for some groups who it took longer to establish that with I’d say who else have you asked?
12
(Waverley-M, 2011-06)
Another teacher noted that this was a key priority for future development:
…the one area I would want to be working on is the communication and inclusion bit, they
don’t sit and read the task together. One group wanted to split in half to work, but then they
realised that the other two might not know what they had done…L always does his own thing;
the group find him hard to work with. He had identified the test for divisibility and had it all
down on paper in front of him, but when the group fed back, it wasn’t him who spoke, and
what they fed back was nowhere near as he’d got on his own. It’s trying to get that wealth of
ideas to come back to the whole group. (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
well-tested ‘skill builder’ activities can be found in the CI literature (e.g. Cohen, 19943) 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 before, who were very
reluctant to share things, to talk about what they were doing, because they saw everything as
kind of competitive …I have a multi-dimensional classroom. You know, success takes a variety
of forms and it’s important that you can share that but recognise that… I spent time, guiding
them on how to comment about improvements …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 some kind of feedback and I think that was the skill they needed 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/resources#skill
NRICH site: http://nrich.maths.org/6933
13
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 back to the roles and brought the roles back into the group work and I think it kind of
gave the students an identity and a purpose beyond just the purpose of completing 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 start off with. They can then share that with the class. If they can
reconstruct their understanding to explain it to somebody else, that’s a form of success. A
parent commented at parent’s evening that they couldn’t believe how confident their child
was at Maths now. (Waverley-M, 2011-03)
And:
I am able to praise, people that perhaps I wouldn’t in normal mathematical lessons because
they would be because either they wouldn’t be producing work or feel very self conscious
about their mathematics, unsure and not willing to give answers in a lesson, or to get
involved in a plenary. But I’ve really noticed some 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 the box and measuring it proved 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 other. They produced
reasons for which ones were bigger, and some of them had used calculations to prove
volumes. One girl got as far as the teacher notes, she produced the graph, how it had
changed over time, she’d recognised that the maximum 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 mathematicians to understand why things are
happening and to justify it; and that’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)
14
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 other with that. Boaler and Staples (2008) cite
Horn (2005, p22) in suggesting that groupworthy problems are those that:
… illustrate important mathematical concepts, allow for multiple representations, 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 groups, and she’s got one of the lowest target grades in the class;
and she participates really well in the group work and also in the class discussions about it she
was participating well, which is nice to see. Also one of the boys in her group, who I have had
problems with, contributing to group work before, he seemed to be leading the group and
really pushing them forward and he’s one of the lower ones in the class as well, so that 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
15
in tests:
…you can take twice as long to cover content using an extended task because you have to
make sure they have the knowledge and can apply it in a different context, because they
apply it in a mostly abstract context in the group work task, it's making sure they can apply it
… it's horrible to say, but they need to be able to do well in the tests they are set, the GCSEs,
functional skills exams, so we have to make sure that they can apply what we are teaching to
that situation...(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 year 7 says that all students must know where they are
working at, and all are given a minimum target grade for the end of year 9. They have to
know that on the spot if asked by a member of senior management. Which goes against our
principles in the maths department – which is that you can't give a student a number, you
can't walk across 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
16
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
17
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
18
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.
19
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 – 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
teachers’ 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
20
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 away, because it does pull them in, and the phrase ‘working in the
middle’ from Cohen’s book really struck the 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.
21
References
Boaler, J. (1997) Setting, Social Class and Survival of the Quickest. British Educational
Research Journal, 23(5), 575-595.
Boaler, J. (2008) Promoting 'relational equity' and high mathematics achievement through
an innovative mixed-ability approach. British Educational Research Journal, 34 (2), 167-194.
Boaler, J. (2009) The Elephant in the Classroom. Helping Children Learn & Love Maths.
Souvenir Press: London
Boaler, J. and Staples, M. (2008) Creating mathematical futures through an equitable
teaching approach: The case of Railside school. Teachers College Record, 110(3):608-645.
Boaler, J., Altendorff, L. and G. Kent (2010). Complex Instruction in England: The journey, the
new schools, and initial results. Falmer: University of Sussex. [http://nrich.maths.org/7011]
Boaler, J., Altendorff, L. and G. Kent (2011). Mathematics and science inequalities in the
United Kingdom: when elitism, sexism and culture coincide. Oxford Review of Education, 37,
(4), 457-484.
Cohen, E. G. (1994, 2nd ed.) Designing Groupwork: Strategies for the Heterogeneous
Classroom New York: Teachers College Press.
Cohen, E. G. and Lotan, R.A. (eds.) (1997) Working for Equity in Heterogeneous Classrooms:
Sociological Theory in Practice. New York: Teachers College Press.
DCSF (2008)
http://webarchive.nationalarchives.gov.uk/20081024214124/http://www.dcsf.gov.uk/trend
s/index.cfm?fuseaction=home.showChart&cid=5&iid=30&chid=117, accessed 4 Sept 2011
Dugard, P. and J. Todman (1995). Analysis of Pre‐test‐Post‐test control group designs in
educational research. Educational Psychology 15 (2), 181-198.
Fielding, M., Bragg, S., Craig, J., Cunningham, I., Eraut, M., Gillinson, S., Horne, M., Robinson,
C. & Thorp, J. (2005). Factors Influencing the Transfer of Good Practice. Nottingham: DFES
Publications.
Gonzales, P., Williams, T., Jocelyn, L., Roey, S., Kastberg, D., and Brenwald, S. (2008).
Highlights From TIMSS 2007: Mathematics and Science Achievement of U.S. Fourth- and
Eighth-Grade Students in an International Context (NCES 2009–001). National Center for
Education Statistics, Institute of Education Sciences, U.S. Department of Education.
Washington, DC.
Ofsted (2008). Mathematics: Understanding the score. London: Ofsted.
Smith, A. (2004). Making Mathematics Count. London: DfES.
22
Appendix 1: Blank Questionnaire
Mathematics Questionnaire for Years 7 and 8.
The University of Sussex
Your replies will be kept completely confidential.
Name:
School:
Maths Teacher:
Maths Group:
Are you Male or Female? (please circle one)
1. In maths lessons how often:
Never
Once in
a while
For
some of
each
lesson
For a lot
or all of
each
lesson
Do you work in groups
Does the teacher talk from the front
Do you get really interesting problems to
solve
Do you get to use your own thoughts and
ideas
Do you learn from another student
2. Which of these statements do you agree with MORE?
School maths is based on things that happen in the real world
OR
School maths is very different from things that happen in the real world
3. Which of these statements do you agree with MORE?
Success in maths is mainly about learning and remembering facts and rules
OR
Success in maths is mainly about thinking for yourself
4. How much do you agree with these
statements about maths?
Maths is a difficult subject
Only some people can achieve high levels
Anyone can be good at maths if they really
try
It is important to use the teacher’s method
Mistakes in maths are helpful for learning
strongly
agree
agree
disagree
strongly
disagree
5. How much do you agree with these
statements about maths?
strongly
agree
agree
disagree
strongly
disagree
strongly
agree
agree
disagree
strongly
disagree
agree
disagree
strongly
disagree
Once in
a while
For
some of
each
lesson
For a lot
or all of
each
lesson
I like maths
I am good at maths
It is good to look clever in front of others
No matter how hard I try, some maths is too
difficult for me
I can’t achieve my potential in this class
Maths makes me feel stupid
6. I particularly like maths lessons when:
The work is easy
We work in groups
I can see the usefulness of the maths
The problems make me think really hard
I get the opportunity to use my own
thoughts and ideas
7. When I try hard in maths it is because:
The work is interesting
strongly
agree
I want to get a good grade
I know it will be useful in the future
8. How often in maths class…
Never
Are you expected to explain your answers?
Do you talk about maths in the world?
Do you get excited about mathematical
ideas?
9. If you were asked to describe your feelings about maths, in one or two words,
what would you say?
Maths is …………………........................................................
Thank you for completing this questionnaire
Appendix 2: Conversion of test scores to ‘level points’
Level
points
KS2 (whole
levels)
KS2
(sublevels)
Year 7
L34 paper
Year 7
L46 paper
1
2
3
4
5
6
2c
2
2b
7
2a
8
3c
9
3
3b
‘below3’
3
10
3a
‘below4’
11
4c
4c
12
4
4b
4
4b
13
4a
4a
14
5c
5c
15
5
16
5b
5
5a
5a
17
18
19
20
5b
6c
6
6
6b
6a
Appendix 3: Questionnaire response counts
School
Ridgeway
year
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
7
8
n
Response rate %
107
83
101
78
Ellen Walker
285
90
234
74
Highfield
234
80
214
72
Sideview
108
86
121
96
Quayside
103
81
0
0
Alan Bennett
96
60
79
48
Waverley
220
92
156
65
Cardinal Newman
161
83
190
90
Waterside
78
62
96
76
Dean Park
211
76
185
66
Total
2979
77
Note: Year 8 student numbers were not available for Ridgeway,
Ellen Walker, Sideview or Waterside; Y7 number was used