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Teacher enquiry at the heart ofraising attainment in mathematicsTeachers are key to raising achievement in mathematics, but changing practice ischallenging and time consuming. Mathematics-specific teacher enquiry is effective in changing practice since teachers themselves have ownership of any innovative strategies and methods used in their teaching.

Teacher enquiry is at the heart of the work of the NationalCentre for Excellence in the Teaching of Mathematics(NCETM). Over 80 teacher enquiry projects have beenfunded to date with more in the pipeline. The projects haveranged from Key Stage 1 to post-16, covering a widenumber of topics. But in every case, teachers identify anissue as a starting point and from this explore and developtheir practice. All projects involve a group of teachersworking collaboratively; many are supported by a facilitatorfrom a university or local authority.

I hope that this bulletin gives you a flavour of the range ofprojects that have been initiated by the NCETM and thesense of the excitement about teaching and learningmathematics that has been generated. Each project must of course be customised to the local context and to local challenges, but could involve making use of a variety of teaching and learning strategies, such as:

• supporting rich mathematical questions in the classroom

• incorporating mathematics trips into applying mathematics in the primary school

• using mathematical modelling within functional mathematics

• using digital tools for sharing practice (for example, googledocs) or for mathematical learning (for example, graphical calculators).

The projects could comprise of different groups of participants, such as:

• teachers in one school working together to change the teaching and learning of a particular area of the curriculum, for example trigonometry

• a group of teachers from a range of schools working collaboratively on re-energising their classroom practices

• teachers working towards accreditation for a Masters degree

• teachers working within an ITT programme

• teachers working across phases (for example primary to secondary school) in order to identify ways to support the transfer.

So there are many starting points and ways of working.However the key to a successful project is undoubtedlyfostering among students an atmosphere of questioning,reflecting and evaluating the mathematics at stake.Whatever your project, it is also crucial that your ideas areshared with other teachers - to adopt and shape to theirown needs and goals. The NCETM is committed tosupporting this process. It is only in this way that emergingand distributed local knowledge can be injected into ournational system. I hope that this bulletin will inspire you tostart a teacher enquiry project of your own. You can find outmore about the range of projects already funded by theNCETM and new funding opportunities, from the NCETMportal at www.ncetm.org.uk/enquiry/funded-projects

In this issueEasing transfer and transition in mathematicsfrom Key Stage 2 to Key Stage 3 2-3

The economy of teaching mathematics 4-5

The impact of teacher’s sustained collaborative professional development 6-7

Improving primary ITT students’ subjectknowledge of mathematics 8-9

Changing the culture in mathematics lessons 10-11

Improving attainment in the primary school usingproblem solving approaches 12-13

Mathematics trips: why they are worth the hassle 14-15

Learning through mathematical modelling 16-17

Implementing collaborative planning in the mathematics department 18-19

Using ‘masterclasses’ to explore teaching

approaches, with a focus on teaching trigonometry 20-21

Small software for mathematics onhandheld technology 22-23

The NCETM funded projects scheme 24

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Easing transfer andtransition in mathematicsfrom Key Stage 2 to Key Stage 3Fiona Brown and Andy Maytum

Transfer between primary and secondary schools is often difficult for pupils. It isrecognised that pupils need support from schools and parents to overcome the challengesthey encounter, but sometimes schools and parents do not know how best to do this. Themathematics department at Cardinal Newman School tackled the problem by workingtogether with feeder primary schools, parents and pupils, to develop an understanding of the experience of transfer and to devise a cross-phase unit of work to make thetransition in mathematics easier for pupils.

Aims of the studyThe aims of the study were to:

• provide Year 6 and 7 teachers with the opportunity to deepen their mathematical knowledge and knowledge of pedagogy in a Key Stage outside of which they work;

• engage pupils by offering them suitably challenging units of work which were planned to develop across the transfer from KS2 to KS3;

• engage parents.

Dimensions of the study Galton, Edwards, Pell and Pell (2003) carried out researchon transfer and transition in the middle years ofschooling. They suggest that:

‘for many pupils much of Year 6, in the run up to thetests, consists largely of revision with an emphasis onwhole class direct instruction. This narrowing of thecurriculum and the limited range of pedagogyemployed in Year 6 have implications for teaching at the lower end of the secondary school.’(page 7)

They also write that

‘… the Year 7 curriculum is still not sufficientlychallenging’.(page 7)

Further, according to research by Charles Desforges,

‘parents are the single biggest influence affectingchildren’s learning and attainment; parents play a keyrole in providing a smooth transition for their children’.(Desforges and Abouchaar, 2003, page 11)

Cardinal Newman Catholic School and CommunityCollege is a larger than average secondary school. The mathematics department, supported by the Local Authority, thought it was important to addressconcerns related to challenges Year 7 studentsexperience when entering secondary school, asoutlined above. This was a particular concern after a recent Ofsted report for the school, which suggested that:

‘… the good links with partner primary schools arenot widely used to build on existing experiences and achievements.’

The project worked with pupils from three main primary partner schools, from January 2008 (pupilswere in Year 6) until November 2008 (now the pupilswere in Year 7).

References

Desforges, C. and Abouchaar. A. (2003) The impact of ParentalInvolvement, Parental Support and Family Education on PupilAchievement and Adjustment: A Review of Literature. London: DfESGalton, M. Edwards, J. Hargreaves, Pell, L and Pell, A (2003) Transferand Transitions in the Middle Years of Schooling (7-14): Continuitiesand Discontinuities in Learning. London: DfES

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Year 6 and Year 7 teachers working togetherThe project brought Year 6 and Year 7 teachers togetherto discuss mathematics, approaches to teachingmathematics and issues related to transfer between Year 6 and Year 7. To develop their understanding oftransfer, they:

• visited one another in their different schools to develop a sense of what it might feel like to be a pupil in the different environments;

• used questionnaires to probe the different ways students experienced mathematics in Years 6 and 7;

• surveyed parents to gain some insight into parents’ confidence in mathematics and their general feelingsabout teaching and learning mathematics.

The teachers collaborated to plan a unit of work basedwithin the citizenship area of sustainability, focusing onthe ‘shape and space’ and ‘data handling’ area of themathematics curriculum. The idea was that the pupilswould start work on this unit in Year 6 and then bring aportfolio of completed work to the new school whenthey began Year 7.

Between meetings, the teachers communicated withone another via the NCETM portal.

What could this study mean for you and your students?You may be interested in easing cross-phase transfer for pupils in your school or college.

Consider getting together with teachers from theschools or colleges the students are coming from or moving to. You might like to visit these otherinstitutions to get a sense of how they operate.

Planning some work to stretch across both phasescould provide a useful way of providing continuityfor the students.

Try to find out more about the mathematics thestudents do in other phases. This could help youprepare them for transfer or provide work at a suitable level of challenge after transfer.

What has been learned?The teachers found that coming together toevaluate and listen to accounts of each others’classroom practice was very valuable.

The secondary teachers learned how to usediscussion and group working in themathematics classroom, which has helpedthem begin to address process skills in the newKey Stage 3 curriculum.

The primary teachers extended theirmathematical knowledge and understandingby working with subject specialists, and havebrought this learning to their classrooms.

The use of the portal was a powerful way tocommunicate after the lessons and shareexperiences although it was not used as muchas had originally been envisaged.

The pupils were presented with the question‘How much waste do we create?’ They exploredthe amount of waste collected from lunch boxesin their school. They compared this with otherstatistics and explored the issues surroundingrecycling. Pupils also looked at packaging. Theyexplored different sized packaging of the samebrand of cereal and raisins and madecomparisons about the amount of cardboardused per 1g of product. They also compared costper unit of packaging.

OutcomesEvidence suggests that this project has had a positiveinfluence on classroom teachers in both primary andsecondary phases. One of the biggest impacts has been in the collaborative planning of staff from acrossthe two Key Stages. It seems that this collaboration hashelped to build a shared responsibility and trust betweenall the staff. It has provided teachers with insight into thedifferent teaching approaches used in both Key Stages.All teachers were involved from the start of the planningand developed shared vision, experience and language.

The project appears to have had a positive impact onpupils. This has been seen in their enthusiasm in theclassroom and the way in which they have workedcooperatively and discussed their mathematical work.

Many of the parents appeared to lack confidence in their ability to support their son or daughter inmathematics, and some seemed to think that it isacceptable not to succeed in mathematics. However,they were generally positive about the presentationevening, with many commenting that it was interesting, enjoyable and stimulating.

Fiona Brown is mathematicsadvisor for Coventry LocalEducation Authority. AndyMaytum is head of mathematicsat Cardinal Newman School andCommunity College. The teacherstaking part in the study werefrom: Christ The King, Holy Familyand St Osburgs as well as thewhole department from Cardinal Newman School.

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The following activities were developed bymembers of the group based on Gattegno’s ideasof making mathematics visible and tangible. Theseactivities include;

• using Cuisenaire Rods to help solve equations;

• using four or two images that are similar, and asking what is the same/different about them;

• using visualisations such as‘arms’ using Geometer’s Sketchpad;

• using games such as games with calculators, and ‘fraction golf’.

What does a teacher need to do when workingwith an image to become more comfortable with using it?

Dimensions of the study Caleb Gattegno, founder of the Association of Teachersof Mathematics (ATM), was concerned with researchingteaching strategies for minimising the cost of learning forstudents so that the effort they made to learnmathematics was economical.

Gattegno believed in using resources (e.g. geoboards,number charts) and images (e.g. rotating arm tointroduce trigonometry, www.mathsfilms.co.uk) to makemathematics visible and tangible to students. Studentswould then use their powers of discrimination anddescription to develop concepts. The teacher’s role is in focusing students’ attention and supportingmathematical discussion.

The economy of teaching mathematics Laurinda Brown, Alistair Bissell, Alf Coles, Louise Ordman, Barry Orr,Jan Winter and Tracy Wylie

Aims of the studyThe study aimed to:

• form a collaborative group to work together on developing practical ways of using Caleb Gattegno’s ideas;

• adopt a range of strategies for continuing professional development (CPD);

• evaluate the CPD strategies adopted.

The theoretical and practical influence of Caleb Gattegno runs through much of thecurrent thinking about the teaching and learning of mathematics, and is particularlyevident in ‘rich tasks’ and ’active’ learning. In this study, a group of mathematics teachersand university tutors, with different levels of experience, worked together to studyGattegno’s ideas further. They were particularly interested in exploring links betweenGattegno’s ideas and their own classroom practice.

What are the powers ofchildren?• Extraction, ‘finding what is common among

so large a range of variation’;

• Making transformations, based on the early use of language ‘This is my pen’ to ‘That is your pen’;

• Handling abstractions, evidenced by learning the meanings attached to words;

• Stressing and ignoring, without which ‘we cannot see anything’.

How can functionings be used in the teaching ofmathematics?

• Students can notice differences and assimilate similarities;

• Students can use their power of imagery: ask students to shut their eyes and respond with mental images to verbal statements;

• Students can generalise given that ‘algebra is an attribute, a fundamental power, of the mind’.

Laurinda Brown and Jan Winter are senior lecturers in education(mathematics) at the University of Bristol.

Alf Coles and Tracy Wylie are experienced teachers at Kingsfield School.

Alistair Bissell taught as an NQT at Churchill School; Louise Ordmanat Malmesbury School and Barry Orr at Ashton Park School.

Reference: Brown, L., Hewitt, D. and Tahta, D. (Eds) (1989) A Gattegno Anthology Derby: ATMwww.atm.org.uk/buyonline/products/rea011.html

The ‘Economy of Teaching Mathematics’ groupThe motivation for establishing the group came from aseminar on Gattegno’s work led, as part of her PGCE year, by Loiuse Ordman. For a period of two years teachers ofmathematics (three NQTs, two university lecturers, twoexperienced teachers and an evaluator) explored Gattegno’sideas within the context of their own classrooms. Theyworked together to plan and teach mathematics lessons,observing and discussing these lessons, with an emphasison reflecting on classroom practice.

There were six day-long meetings of the group over the twoyears. In each meeting all members of the group observed alesson taught by one of the group either in their classroomor on video. In addition, the two university tutors observedthe NQTs regularly over the life of the project.

The NQTs began a Masters course in MathematicsEducation. They took a unit that supported the members of the group in documenting their development throughworking in a collaborative group on an issue in mathematicseducation. This unit was tutored by one of the lecturers and one of the experienced teachers. In some day meetings,the NQTs led discussions on their work.

Members of the group attended conferences related to mathematics education, such as day meetings of theBritish Society for Research in Mathematics Education(BSRLM).

Everyone was encouraged to write throughout the project,the more experienced members of the group initiallyencouraging the NQTs to write about their lessons.

The group developed a ‘googlegroups’ shared web space.Here they published work relating to this project, such asPowerPoint presentations, research papers, reflections on lessons and lesson ideas. Seehttp://groups.google.co.uk/group/economyNCETM.

The group found that discussions relating to Gattegno’swork were enriched through contributions from the variedinterests of individual teachers. So, rather than a modelwhere Gattegno’s work is ‘explained’ and participants needto try to find something which is relevant for them fromwhat is offered, contributions from each person were valued and ‘spoke to’ the focus of the group. It was throughthat common focus that others found relevance for theirown interests.

Many of the activities focused the students’ attention on a goal such as finding a square of area 10 on a geoboard. In working towards this goal they used valuablemathematical ideas.

“The activities made the learning of mathematics ‘easy’.”

Pupils were asked to write downsimilarities anddifferences betweenany of the shapes.These could besimilarities/differences betweenall four shapes, orbetween just two of the shapes.

What could this study mean for you and your students?If you want to try out activities that are visible andtangible, have a look at the following web site:www.southwellweb.co.uk/Kingsfield/

Have a look at A Gattegno Anthology for more information about Gattegno and his ideas.

If you are planning to set up a collaborative CPDgroup of teachers, consider a model that – whilesupporting NQTs – also values their voices.

“…in the first place I’d come toget some ideas to try out andthat was it, whereas now I feelI sometimes bring things to thegroup and maybe people goaway thinking aboutsomething that I’ve said.”

What has been learned?It is important to take the specific interests ofindividual teachers into account when working incollaborative CPD groups, but equally importantnot to lose focus. Strong chairing of groupdiscussions was important.

Writing and talking about experiences areeffective ways to bring about deep reflection.Pupils responded positively to the activities andteaching strategies introduced.

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The impact of teachers’sustained collaborativeprofessional developmentJulie-Ann Edwards

Although there is anecdotal evidence of the positive effects of post-graduate study, thereis little research providing evidence of the influence of a Masters level qualification inmathematics education. Julie-Ann and her students undertook a project to providerigorous and substantial evidence of the benefits of undertaking such study, and found considerable impact.

Aims of the studyThe research project aimed to provide an evidence base which addresses a lack of research about teachers’professional and personal development whileundertaking postgraduate research study. It examined:

• the personal and professional impact of research and study on teachers on a Masters course;

• the impact this learning and research has on the curriculum and pedagogical development in classrooms and mathematics departments;

• the influence these developments might have on increased collaboration between schools, given the close proximity of some of the participating schools.

Dimensions of the study A Masters degree in mathematics education draws ontheory and research to extend and develop personalperspectives on mathematics education. A Mastersdegree is not a ‘quick fix’ or ‘top tips for teachers’, andsome teachers might question how useful it is to takea Masters degree. There is little systematic research inthe UK about the impact that academic study (such asMasters degrees) might have on wider professionaldevelopment.

However, there is an abundance of anecdotalevidence that undertaking Masters coursessignificantly changes how teachers view aspects ofteaching and learning in their classrooms. Over manyyears, Julie-Ann observed Masters students and shebecame convinced that their classroom practice waschanging, and, further, that these changes wereinfluenced by their Masters courses. She set up thisresearch project to investigate further.

Seven teachers agreed to be involved in thisresearch project. Six were enrolled at theUniversity of Southampton on the MSc inmathematics education and one was enrolled on the MSc in computer-based learning andtraining. They are all secondary mathematicsteachers. They also mentor students in InitialTeacher Education (ITE).

Individual and collective Masterslevel study

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What was doneAn initial focus group interview, which took place inNovember, 2007, provided evidence about the teachers’motivations to undertake postgraduate study in the formof a Masters degree.

Participants wrote reflective logs, recording momentswhen ‘shifts’ in thinking occurred in relation tomathematics, curriculum, pedagogy, or shared practice in schools. Some teaching sessions were audio-recorded,as a means of enabling the researcher to both teach thesession and record field notes. Some field notes werewritten after the finish of the taught session. Participantswere interviewed, both formally and informally. They alsocompleted assignments for the Masters degree whichprovided some insight into the impact of learning atMasters level and of research activity on personal andprofessional development and on classroom practice.

The audio-recordings were transcribed and an iterativeprocess of categorisation provided evidence of impact.

Research findingsThe data from this research strongly suggests that takingthe Masters course has influenced the teachers in anumber of ways.

The teachers said that they were now more confidentabout themselves as professionals and as classroomteachers. They were also developing awareness aboutchanges in themselves.

“I’m aware of a shift in my previous thinking!!”

Some teachers commented that their love ofmathematics had been rekindled. One gave an example of what she had done:

“Went to the British Museum at the weekend to lookat the mathematics in the roof design.”

Teachers reported that they had made some changes in their approaches to teaching mathematics. Forexample, one teacher told Julie-Ann that she hadintroduced more group working in the classroom.Another reported that she had changed the way shestarted new topics:

“I have begun starting all topics with problems. A good introduction to 10M1, linking Pythagoreantriples with the radius of a circle, means they are nowall well acquainted with Pythagoras.”

It seems that the Masters course was also influencingteachers’ professional lives beyond the classroom.Some reported they had made some changes in the waythe department worked. For example, one reported thatideas from one of the readings from the course weregoing to be added to the scheme of work. Anotherbecame involved in leadership in the school moregenerally and was asked to join a team to look at CPD.

What could this study mean for you and your students?You might be considering taking a Masters course,and questioning whether you have the time or energy to do so. This study shows that teachers feltstimulated by the course and that they enjoyed, andbenefitted from, engaging with research literature.

You could be wondering whether taking an academiccourse will contribute to your classroom practice. Thisstudy shows that the Masters course does seem tohave made a difference to the classroom practice ofthe teachers involved.

You could find out about Master s courses offered atyour local university by checking their web sites.

What has been learned?This research has provided evidence of changes inteachers. The changes can be seen in differentareas. Some changes relate to the teacher and canbe seen in terms of personal growth, while othersrelate to changes in teachers’ classroom practice.

The research suggested that teachers value timeaway from school to think about teaching andlearning mathematics. They find it useful to talk toone another other beyond their own setting.

It seems that teachers value the impetus providedby engagement with further study at a university.They seemed to enjoy reading research literatureand there was evidence that they were beginningto introduce ideas from the literature into theirclassroom practice.

Teachers were beginning to participate inprofessional activities outside of their schools, such as attending research conferences.

Julie-Ann Edwards is seniorlecturer in mathematicseducation at the University ofSouthampton. The teacherstaking part in this study wereRachel Jones, Liz Perry, JonathanEacott, Nicke Edge, Neil Gulliverand Paul Robinson.

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Improving primary ITTstudents’ subject knowledgeof mathematicsMike Geall and Sue Hornby

It is widely acknowledged that many primary school teachers lack confidence in theirmathematical knowledge. There is also a need to develop mathematics subject specialistsin primary schools. This project aimed to address both these concerns through adopting a coaching model where aspirant mathematics subject specialists coached generalisttrainee teachers. They built up a model of coaching in four cycles of action research.

Aims of the studyThe project aimed to give final year primary B.Edmathematics specialists the opportunity to coach non-specialist primary B.Ed students and to investigate theeffectiveness of various models of coaching.

Aims of the studyThere is general agreement that subject knowledge ofmathematics is a component of effective mathematicsteaching. Marjon (University College Plymouth) uses arange of strategies to engage and motivate students ininitial teacher training (ITT), to monitor and developtheir subject knowledge of mathematics alongside thestudy of the primary mathematics curriculum and itsassociated pedagogy. For example, the primarymathematics team suggested that a whole B.Ed. cohortused the Mathematics Teaching Self-evaluation Tools

on the NCETM portal. This project introduced coachingas another strategy to improve students’ subjectknowledge.

The publication of the Williams Review (2008)highlighted the need to develop mathematics subjectspecialists amongst primary teachers. Futuremathematics specialists may come from studentsopting to take a specialism module in primarymathematics. For these students, this project aimed tolay the foundations of a training process. It was thoughtthat providing them with the experience of coachingtheir peers would help develop them towards takingthe role of the mathematics subject specialists in their future schools.

The research element of the project was put in place todevelop approaches to coaching in cycles, with earliercycles informing the development of later cycles.

An example of a partially completed self evaluation on the NCETM portal.

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Four cycles of action researchThe project used an action research methodology,adopting a spiral model of planning, enacting andreflecting on outcomes before starting a second spiral.There were four action research cycles.

The project leader carried out coaching sessions withPGCE students during the autumn term of 2008, in order to investigate any problems that seemed to ariseduring coaching. The main finding to emerge was thatstudents are reluctant to spend up to one hour with amathematics tutor discussing their mathematics subjectknowledge. This suggested the importance of creating asupportive coaching environment in which students feltable to discuss their understanding of mathematics.

In the second cycle the B.Ed.3 students created atemplate for coaching sessions based on the ‘STRIDE’model: Strengths, Target, Real Situation, Ideas, Decisionand Evaluation. They designed a coaching pro-forma and a post coaching session evaluation sheet, designedto structure peer coaching and evaluations.

The documents were then used in experimentalcoaching sessions, with one student acting as the coach and another as a learner. A third person (anotherstudent, the tutor or NCETM project supporter) observedthe session. In the light of observations, from all involvedin these sessions, the documents were refined. This wasthe third action research cycle.

The coaching sessions took place on two separatecoaching days. The B.Ed3 students coached B.Ed2students (learners), using the documents developedtogether to guide their work. All sessions lastedapproximately an hour. Some coaches worked in pairs,some alone. The number of learners ranged from one to three. The data from these two coaching sessions, the final cycle of development, forms the main basis of the evaluations, conclusions and recommendations of the project.

FindingsBoth groups of students reported that they felt thecoaching sessions were beneficial to both their ownmathematical subject knowledge development, and theirunderstanding of how others tackle mathematical ideas.The B.Ed.2 students being coached reported feeling slightlyapprehensive beforehand, but that the session waswelcoming and supportive. They appreciated the attentionpaid to them as individual learners.

It was generally agreed that ‘coaching is a personalprocess’, and that each coaching session is different andtherefore requires a variety of approaches to be mosteffective. Each learner has different mathematics, andindeed learning, experiences and so one to one coachingallows for a much more specific, personalised approach. It seems that individual target setting at the beginning of the coaching process is very effective in helpingdetermine the focus of the session. Targets also provide a yardstick against which learning can be assessed.

Most learners reported that completing practical tasksduring the sessions helped to secure their knowledge. They said that they found that working through examplesand discussing methods during the sessions embeddedtheir knowledge.

There was a general consensus that an hour was a goodlength of time for a coaching session. The coachesachieved a fine balance between friendly chatting to put the learner at ease on the one hand, and a focus on mathematics on the other.

What could this study mean for you and your students?You might like to implement coaching in your school or college.

Think about teachers coaching their colleagues. Howcan the coaches help their colleagues feel ‘safe’?

Think about Year 9 students coaching Year 8 students.Would this be possible and viable? How could theyjointly set targets?

If you are involved with primary ITT, you couldconsider using or adapting the coaching modelsuggested here.

What has been learned?It is important to take the specific needs ofindividual learners as the starting point forcoaching sessions and to adopt deliberatestrategies to determine these needs.

Implementing coaching sessions has hadunexpected benefits in terms of developingrelationships between different cohorts ofstudents.

The action research model adopted seems tohave influenced the way the B.Ed 3 studentsdeveloped a sense of ownership of the project.The level of their engagement was higher thanexpected and may account for the success ofthe project.

Mike Geall is a senior lecturer in primary ITT at Marjon(University College Plymouth).Sue Hornby was a B.Ed 3student when this project took place.

Thanks to all the ITT students who participated in this study.

Reference:Williams, P. (2008) Independent Review of Mathematics Teachingin Early Years Settings and Primary Schools. London: DCSF

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Changing the culture inmathematics lessonsSarah Little

Sarah and her colleagues felt that they needed to move away from the use of limitedfactual questions in the classroom and refocus their attention on questions that could be used to explore and develop pupils’ understanding and learning. Their first objectivewas to plan and conduct classroom dialogue in ways that would help pupils to learn.

They wrote a new programme of study and shared it via a virtual learning environment(VLE). This provided teachers with the resources and ideas they needed. It allowed themmore time to plan their lessons carefully to include effective questioning in their lessonplans; to encourage discussion between pupils.

Aims of the studyThe study aimed to:

• change the outlook on learning and the working methods of pupils in mathematics lessons;

• encourage teachers to adopt approaches based on the ideas of ‘Assessment for Learning’ – in order to generate a classroom culture of questioning and deep thinking in which pupils would learn from shared discussions with teachers and with one another;

• develop resources for the school VLE.

Dimensions of the studyThe project was carried out at Newport Girls’ HighSchool in Shropshire. The school is a selective, 11-18state school with 400 students on roll. Two Year 7 andtwo Year 8 teachers were involved in delivering thenew programme of study to all pupils in these yeargroups. The Deputy Headteacher and the LocalAuthority Advisor were also involved in monitoring and evaluating the impact of the project through lesson observations.

The students at the school were accustomed to ‘chalkand talk’ lessons. Whilst they were high achieving andvery successful in national summative tests, formativeanalysis of their work showed that they struggled toapply their knowledge in different contexts.

In general, the girls did not cope well with the transitionfrom GCSE to A level as they had not developedindependent study skills in lower school. Sarah feltconfident that if she replaced the traditionalmathematics textbook, full of graduated exercises, with a programme of work containing challengingproblems, it was possible to change pupils’expectations of mathematics lessons. She felt thatchallenging problems would provide the girls withdiscourse opportunities that would encourage them to examine their ideas and think about how well they knew the mathematics involved.

Activity

A DB C

Semi-circles are drawn on AB, AC, BD and CD

Where AB = BC CD

Show that the large circle is divided into 3 portions

of equal area.

An example from the Newport High School VLE

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Changing the classroom cultureTo change the classroom culture Sarah and her colleaguesneeded to move away from routine, limited, factualquestions and refocus attention on the quality anddifferent functions of classroom questions.

They started to write a new programme of study (on theirVLE) which included open questions and problem-solvingtasks that might encourage classroom discourse. Theywanted to start posing problems where the most obviousanswer was either wrong, or only partially correct, so thatthey could start challenging pupils to defend their ideas.

They wrote the first few modules of the VLE and taught themto Year 7 and Year 8 in the summer term of 2007. The VLEincluded extensive materials, and teachers worked with thesealmost exclusively, so that the textbook was hardly used.

They spent departmental meetings reviewing eachmodule thoroughly and deciding whether each part of theVLE needed improving/replacing. This was good for qualityassurance. It generated in-depth discussions about howeach member had used each problem and what theoutcomes had been. As a result they spent much of theirtime sharing good practice.

What happenedStudents involved in the project felt that their lessons were more challenging, interactive and fun. Many felt thatthis led to a better understanding of the subject.

“It has made a big difference because it is challengingand makes you think and work out problems.”

Sarah noticed that the most able students were makingthe greatest progress with the new style lessons and werereally excelling, owing to the higher level of challenge inlessons. The majority of the girls responded well to theincreased challenge and made very good progress. Theproblem solving approach appeared to lead to betterattainment for the students.

Teachers reported that the students had become morepositive about using problem solving approaches andnoted a change in the students’ attitudes.

“It has been interesting to see how the girls’ haveprogressed from the start of the year when it was clearthat they had a fear of problem solving and makingmistakes, to a stage now, where they are willing to share their ideas and reasons behind them”.

The teachers involved felt that they had developedprofessionally through teaching using a variety of learningstyles. They felt they had developed better questioningskills. Much of this, they reported, had come about throughdiscussion of their practice in departmental meetings. Theyreported that the materials on the VLE had helped themdevelop, by providing them with resources which seemedto allow them more time to plan lessons.

“I am finding that with the VLE I have all the resourcesthat I need for each learning objective and my planning time can be spent on developing myself as a teacher. I have more time to think about the deliveryof the materials, how I will use questioning to aide thegirls’ understanding and what materials would be good for group/pair/individual work.”

The findings of the lesson observations supported theseresults and concluded that the introduction of the newprogramme of study had a positive impact on the teachingand learning of mathematics within the school.

What could this study mean for you and your students?If you would like to provide more challenge for your students, you might like to use the ideasunderpinning ‘Assessment for Learning’ as a startingpoint. These ideas can help you move towards aclassroom culture where questions are more openand flexible, and students are asked to explain their thinking more.

You could create a VLE with a bank of qualityresources and activities. This will help your wholedepartment to teach problem solving skills to allyour students.

What has been learned?• The teachers in this department have changed

the learning culture in their KS3 lessons by introducing the new programme of study.

• Teachers no longer teach didactically and introduce discussion as a regular feature in mathematics lessons, where students offer their solutions and learn by listening to the solutions of others.

• Teachers and students discuss all answers, whether correct, partially correct or incorrect so that as students verbalise their solutions they can see their own mistakes and learn from them.

• Students are now able to tackle and solve a wide range of challenging problems in a varietyof contexts and explain their reasoning.

Sarah Little is Head ofMathematics at Newport Girls’High School. Other teacherstaking part in the study wereAdam Jones, Lynne Baker andJean Evanson.

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Improving attainment in the primary school usingproblem solving approachesViolet McLaren

Violet and her colleagues wanted to improve achievement in mathematics in theirschools. They noticed that pupils seemed to lack confidence in problem solving activities,tended to work alone and were not sufficiently challenged. They were concerned thatpupils were not enjoying mathematics as much as they might. They decided to changetheir ways of working to encourage group work, discussion amongst pupils, and more useof practical equipment. They began to use new approaches in the classroom, they sharedmathematical activities and raised the profile of mathematics in their schools.

Over the period of the study, teachers have gradually introduced these new ways ofworking and have found that SATs scores have improved and pupils seem to enjoymathematics more.

Aims of the studyThe aims of this study were to:

• improve teacher knowledge in the teaching of mathematical strategies;

• develop children’s problem-solving skills;

• raise the profile of numeracy throughout the schools;

• develop children and teachers’ confidencein mathematics;

• improve numeracy areas within classrooms;

• show an increase in SATs results.

Dimensions of the studyThis project encouraged teachers to adopt problemsolving approaches with their classes. This includeddesigning a mathematics area in the class assigned to:layered targets in mathematics, a working wall whichwill teach the necessary strategies in order to reach thetargets, practical equipment suitable for the target, andproblems which require the strategies learned to beapplied in a range of situations. Teachers were asked to encourage children to work with discussion partnersin small, mixed ability groups. Working walls areclassroom displays for numeracy which displayoutcomes, modelled examples and success criteria.These walls enable children to know what they arelearning and how this learning process develops over a period and hence they support children’s learning.

The project involved Year 2 and Year 5 teachers in twofederated primary schools in Swindon, and teachers ofother year groups became involved over the course of the project. It took place between September 2007and May 2009.

Working walls are classroomdisplays for numeracy which displayoutcomes, modelled examples andsuccess criteria. These walls enablechildren to know what they arelearning and how this learningprocess develops over a period and hence they support children’s learning.

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An integrated approach toproblem solvingStaff at both schools received training on the use ofproblem solving approaches in mathematics teaching.They were introduced to the ‘Working Wall’ concept andhow to use it to promote problem solving. Teacherslearned about the following strategies:

• decide on a mathematical area of focus;

• display a range of objectives relating to the chosenarea of mathematics;

• develop numeracy areas with supportive apparatus/vocabulary;

• display and teach appropriate heuristics – including group discussion;

• display breakdown of strategies to be taught and allow these to be interactive;

• once strategies are learned, apply them to real problems;

• presentation of solutions to share thinking.

“The increases in SATs results are due to the fact that children know how to interpret and apply thestrategies they have learned.”

ResultsThese results relate to both schools involved in the project.

Year 2: In April 2008, data showed that 36% of children wereon target to reach a Level 2b/3 in Key Stage 1 SATs in 2009.No children were on target to reach a Level 3.

Further data was taken every six weeks and analysed.Results have continued to show accelerated progress. AtFebruary 2009, 84% of children were on target to reach aLevel 2b+, an increase of 48%. 21% were now on target toreach a Level 3 – an improvement of 21%.

Year 5: In April 2008, data showed that 96% of children were on target to reach a Level 4b+, with 42% on target to reach Level 5, in Key Stage 2 SATs in 2009.

Further data was taken every 6 weeks and analysed. Resultshave continued to show accelerated progress. At February 2009, 98% of children were on target to reach aLevel 4b+, an increase of 2%. 67% were now on target to reach a Level 5 – an improvement of 25%.

Observations of children’sbehaviourObservations of classrooms suggest that the children are more confident in mathematics. This is particularlynoticeable with girls who now put their hands up andcontribute more often. All the children seem to be more motivated and they report that they like the new ways of working.

“The children are really motivated and enjoy working in this way. SATs tests aren’t scary anymore”.

What could this study mean for you and your students?You might like to try using problem solvingapproaches in your classroom. Think about usingworking walls and physical apparatus to supportpupils as they work on problems.

Try getting together with other teachers to comparedifferent ways to support the introduction of problemsolving approaches.

If you want to raise the profile of mathematics in yourschool or college, working walls could help.

What has been learned?The main learning points are as follows:

• Children respond very well to being able to discuss problems and strategies together in a non-threatening way.

• Presenting ideas in groups leads to greater confidence in mathematics and an improvement in the use of mathematical vocabulary.

• Teachers report feeling far more confident in their teaching of mathematics.

• Children’s results improve when they are taught using this holistic approach.

• Raising the profile of mathematics in the classrooms increases motivation and performance.

Violet McLaren is a leadingteacher and coordinator fornumeracy in Swindon. Teachersfrom Moredon Primary andNursery School and RodbournePrimary School took part in the study.

This project has been shortlisted for a TES supplementaward in the ‘numeracy initiaitves’ section.

Year 2 pupils working together on a problem solving activity.

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Mathematics trips: why they are worth the hassleJane Newns

The ‘Excellence and Enjoyment’ strategy for primary schools suggests that childrenshould be given opportunities to learn in a range of contexts. Jane Newns and thecolleagues of her Gloucestershire network decided to explore the possibilities of usingeducational visits in the pursuit of enjoyment and excellence in mathematics. Theyrecognised, however, the ‘hassle factor’ of school visits and set out to create, trial andrefine resources to support other teachers planning for mathematics trips. Observationsof their own classes using these resources on mathematics trips convinced them thatmathematics trips were ‘worth it’.

Aims of the studyThe project aimed to:

• develop ideas for quality mathematics visits that could be used by schools to give children mathematical learning opportunities outside the classroom;

• develop and share resources that would be useful to other teachers of mathematics in planning similar visits;

• develop success criteria for children taking part in trips;

• trial the resources with their own classes.

Dimensions of the studyThe study is underpinned by the philosophy thatexperiential learning contextualises subject matter,enabling learners to connect with ‘real-life’ situations.This philosophy is taken from the government’s‘Excellence and Enjoyment’ agenda which suggests that‘it is important that children have a rich and excitingexperience at primary school, learning a wide range ofthings in a wide range of different ways’.

Subject leaders from five primary schools inGloucestershire had previously formed a network(Numer8Network) which broadly aimed to improve the quality of learning and teaching in their respectiveschool contexts. These teachers felt that usingmathematics as the vehicle for educational visits wouldsupport their philosophical approach. However, they all agreed that mathematics trips were time-consumingand required a great deal of planning; they thought that many teachers would think they were ‘more hassle than they were worth’. They aimed to produceresources to support teachers in taking their classes on mathematics based educational visits. One of theguiding principles of the project was that the visitsthemselves should be as low-cost as possible. They were therefore conscious that if they wanted teachers to access the resource package, then the locations for the visits would have to be accessible to the majority of schools.

Planning for tripsTo begin, the teachers met to plan a strategy for the 10-month project. They decided that they wanted toproduce a resource that could be used for every visit, byevery phase/year group. They then wanted to trial theseresources with their own classes. They agreed that theguidance for each visit would have a similar format:

• A location to take the children.

• A mind-map of mathematical ideas and activities that link learning on the visit to Number and the Number System, Data Handling and Shape, Space and Measure. Using and Applying mathematics is integrated into each aspect.

• A detailed sheet that set the context of the visit, also containing:

- Learning intentions for the visit;

- Learning outcomes;

- Probing questions;

- Links to the Primary Framework for mathematics;

- Success criteria – where appropriate;

- ICT links;

- Display ideas for back in the classroom.

After deciding on the framework for each visit, the teachers chose the following locations:

• A woodland walk. This could be a local walk around a park if necessary.

• A visit to a place of worship. It was felt this was more accessible than a visit to just a church.

• A sports venue. This could be as grand as a local rugby/football club, or as small as a recreation ground/local sports and fitness centre.

• A visit to a shop. This could involve either a visit to the local supermarket, or corner shop in the village.

• A maths trail around the school grounds. It was felt that this would enable schools to take account of the mathematics in their own school environment.

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Developing the ResourcesThe teachers developed resources after making theirown visits to chosen locations. They explored themathematics they could find in the location, andtogether developed activity mind-maps, ensuringthat there would be depth of learning for each yeargroup. They added two boxes of guidance notes foruse by teachers; one was a ‘points to consider’ boxwhich detailed short advice, and the other was a linkto other locations of similar interest. It was felt thatthe list of activities on the mind map would betransferable if teachers wanted to adapt them to adifferent location.

The group turned their attention to planningguidance for teachers. They found this challenging, asthey wanted to ensure that all activities were linkeddirectly to learning objectives and learning intentionstaken from the Primary Framework for Mathematics,but in a usable format that could be taken andreferred to quickly on the visit.

On trialing the mindmaps and planning sheets in amock-up of a visit, the teachers discovered a flaw inthe notions of success criteria they had used; theyneeded to make a distinction between the steps thata child would need to demonstrate in order toachieve the learning outcome, and the learningoutcome itself. The success criteria would thereforealso act as an assessment opportunity for the teacherto see if a child could solve a mathematical challengeon the visit, by using a set of steps s/he had beentaught during classroom-based learning.

Developing success criteria took the group a longtime to get right, and they exchanged ideasfrequently in face-to-face meetings and using email.They eventually worked out guidance they feltshowed clear steps to success, which took intoaccount progression between age-ranges.

The next step was to trial the visits with classes ofchildren from the network schools, and see howsuccessful they were.

To complete the project, the teachers produced apage for each visit, showing where links could bemade between the suggested activities to beundertaken on the visit, and the blocks and units of learning from the Primary Framework forMathematics.

What could this study mean for you and your students?You might be thinking that you would like to dosomething to contribute to the ‘excellence andenjoyment’ strategy in your school.

You could organise a mathematics-based trip foryour class - this study shows that trips can be veryworthwhile and enjoyable. Some of the resourcesproduced by the teachers in this project (see www.ncetm.org.uk/enquiry/7971) could help withyour planning.

If you work in a secondary school, you could adaptthe ideas presented in this project.

What has been learned?

• The teachers discovered that applying mathematical learning in the context of outdoor activities motivated the children considerably. The children were fully engagedin the learning, and teachers could quickly see opportunities for assessing ‘Using and Applying’ mathematics. Children’s talk was enhanced as they reasoned with their friends about the mathematics they were undertaking.Clear links to classroom-based learning were evident.

• The teachers realised that the mathematicsvisits could be placed either at the beginningor the end of a block of learning. Either way, the activities the children were doing were integrated with prior learning.

• An area for future work might be to take theproject further into secondary education.

Jane Newns has worked for theGlosmaths team and is anAdvanced Skills Teacher inmathematics. She teaches atHarewood Junior School,Gloucester. Other teachers takingpart were David Crunkhurn(Kingsholm C E Primary School,Gloucester), Lynsey Willis(Fieldcourt C E Infant School,Gloucester), Marie Sheehan(Castle Hill Primary School,Gloucester), Jessica Wade (CaltonInfant School, Gloucester).

Tim Foster and Pete Griffinhave provided support for the project.

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Learning throughmathematical modellingGeoff Wake

In England, with the introduction of the new Key Stage 3 curriculum and functionalmathematics at Key Stage 4, there is a move towards making mathematics more relevant to pupils. Many teachers are interested in developing their skills and pedagogicapproaches in order to equip them better to meet these challenges. One possibleapproach is mathematical modelling; an approach rooted in real life problems, which is seen to provide pupils with opportunities to learn mathematics in contexts that arerelevant to their experiences.

In this project five teachers and two Local Authority advisors worked together with theUniversity of Manchester to develop case studies of examples of these ways of working.They published these case studies on the Internet.

Aims of the studyThe project aimed to:

• introduce teachers to mathematical modelling as an approach to teaching mathematics and support them in trying out this approach in their classrooms;

• develop a project web site to disseminate case studies and resource materials

Mathematical modellingIn England, teachers are currently working towardsimplementing new programmes of study at Key Stage 3 and functional mathematics at Key Stage 4. Real world problems are seen to be an important part of these new ways of working withmathematics.

Mathematical modelling approaches start byidentifying a problem in the real world. Before this can be worked on mathematically, it may need to be simplified, perhaps ignoring some of the morecomplex features of the situation. Simplificationallows us to set up a mathematical model of thesituation, which gives us a mathematical problem to solve. Using appropriate mathematical skills and techniques and working accurately leads to amathematical solution. This needs to be interpretedin terms of the real world situation. How valid thesolution is depends on how the model was set up in the first place, and crucially therefore, whatassumptions were made. It is possible to traverse the cycle again making the model reflect the realsituation more closely - refining the model.

An example of a modelling task

How many aircraft would an airline need to fly to thesedestinations from Manchester?

Image courtesy of Ryanair

“I particularly enjoy mathematicalmodelling because the situations areunstructured and the problems thatarise have alternative solutions. The tasks by their nature aredifferentiated and therefore areaccessible to almost all pupils”.

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What we didThis project involved five teachers, two Local Authorityadvisors and a tutor at the University of Manchester.

At the first project group meeting, participating teacherswere introduced to ideas about mathematical modelling,with a particular focus on classroom pedagogy. Thewhole group worked through a range of modellingactivities that they planned to use with their pupils.

This orientation session proved highly motivating and an agenda was developed that would allow each teacherto develop their own case study of professionaldevelopment, with a focus on their classrooms andtheir departments.

At subsequent meetings teachers shared the case studieswith the group, often by PowerPoint presentations that‘captured’ the essence of the their experiences anddevelopment. Participants discussed the cases,particularly focusing on:

• issues relating to changes in classrooms, such as how best to orchestrate group work;

• the need to adapt tasks so that the degree of ‘open-ness’ is appropriate for their classes;

• how to help pupils understand that, in modelling tasks, there is often no ‘right answer’.

Observations

• There is a need to support CPD initiatives that have their starting point in local communities of teachers working with colleagues in the same geographical areas. These initiatives are not necessarily led by a national agenda.

• Producing narratives of collaborative professionaldevelopment can act as a focal point that motivates teachers and supports and stimulates reflection on professional practice and its development.

• Mathematical modelling activities provide pupils with mathematical experiences that are different from the norm. They can develop both content and process skills in mathematics and promote mathematics as functional.

• For many teachers new pedagogic practices are required when they undertake modelling activities with their classes and they need support in developing these skills. Collaborative reflection on practice with like-minded colleagues seems to be very effective.

The project has had a major impact on the personnelinvolved. It has also had an immediate and wider impact,particularly with colleagues in schools and, mostimportantly, on pupils in classrooms.

What could this study mean for you and your students?If you are beginning to teach functional mathematics,think about using a modelling approach. This fits wellwith the emphasis within functional mathematics on‘real world’ problems. Pupils seem to enjoy theapproach. It allows them to explore meaningfulsituations and get a sense of the power ofmathematics.

Have a look at the project websitewww.education.manchester.ac.uk/research/centres/lta/LTAResearch/lema/. This is a usefulresource where you can find modelling tasks that otherteachers have used. You can also read about how theyfelt about introducing modelling into their teaching.This could help you plan how to make similar changes.

The work of the project suggests that the focus on a‘product’ (in this case a website) can have multiplepurposes that are important in supporting collaborativeprofessional development.

Mathematical modelling tasks may provide idealapproaches to the new Key Stage 3 programme ofstudy in mathematics, allowing the assesment of pupilperformance in using and applying mathematics andfunctional skills.

What has been learned?Three groups have learned from the CPD:

• Individual teachers have learned a range of new pedagogic approaches that have (re-) vitalized their teaching;

• School departments are beginning to learnfrom individual teachers and develop newapproaches to teaching and learning;

• Local Authority consultants saw how important it is for teachers to feel ‘ownership’ of their own professional development.

Geoff Wake is lecturer inmathematics education at theUniversity of Manchester. Theparticipants were Will Wilson(Salford LA), Jo Kennedy(Stockport LA), Rhian Davies(Marple Hall School), BenMatthews (Salford City Academy),Jon Horlock (Moorside HighSchool), Peter Pawlik (WalkdenHigh School), Kay Whiting(Reddish Vale TechnologyCollege).

The project web site organises and publishes arange of data that includes:

• case studies of teachers;

• case studies of classroom practice;

• case studies of Local Authorities and how they can support CPD of this type in mathematics;

• video sequences of lessons;

• mathematical modelling tasks;

• issues arising;

• links to other related developments.

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Implementing collaborativeplanning in the mathematicsdepartmentSteve Watson

Collaborative planning involves teachers working together to plan a sequence of lessons, try out the lessons and then evaluate their effectiveness. Steve and his colleagues at TheLindsey School were encouraged to adopt collaborative planning approaches by their Local Authority consultant. They formed two collaborative planning groups within theirschool and developed ways of working together.

Steve sees the adoption of collaborative planning approaches as a journey for the department,which is characterised by a change in culture towards more open ways of working.

Aims of the studyThe aims of this study were to:

• introduce collaborative planning as a working practice amongst the teachers of mathematics at The Lindsey School;

• identify and respond to barriers to collaboration;

• document the process and identify key themes to help other schools who want to move in this direction.

Dimensions of the studyThe major stimulus for Steve’s department to introducecollaborative planning was advice given by the NationalStrategies. This advice encourages mathematics subjectleaders to promote collaborative working to plansequences of lessons featuring ‘rich tasks’ and toevaluate schemes of work. Steve was supported by theLocal Authority advisor, who emphasised the importance

First stages in developing a unit plan

of evaluating, modifying and embedding collaborativeplanning in mathematics departments.

The study introduced this type of working at The LindseySchool in North East Lincolnshire, which is an 11- 16school with about 1000 students on roll. There isconsiderable pressure on a school like Lindsey toimprove, both in the short term by improving GCSEresults and in the long term through improving thequality of teaching and learning. Professionaldevelopment projects like the introduction ofcollaborative planning are seen to have the potential toinfluence both short-term and long-term change.

Initially the five full-time mathematics teachers attendeda day-long meeting run by the Local Authority wherethey developed a ‘unit plan’ for Year 7. Two collaborativeplanning groups were formed, one focusing on KeyStage 3 and the other on Key Stage 4. These groups metfortnightly during the school day, on alternate weeks.Where required, the teachers had their lessons covered.

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Collaborative planningThere was originally some scepticism about collaborativeplanning because the approach contrasted with themodel Steve’s department was used to, where teacherstended to work in isolation. If any collaboration did takeplace, this tended to be informal. However, Steve becameconvinced that a collaborative planning approach wasworth pursuing, particularly after being introduced to theidea of Japanese lesson study, in which members of adepartment plan lessons, teach and observe the lessonsand evaluate them.

One concern of the department was that many of theteachers were not mathematics specialists. The LocalAuthority advisor suggested that it was important toproduce a document that any teacher could follow and he and Steve discussed how this could help non-specialists. The emphasis on producing documentationsometimes seemed to get in the way of what Steve was trying to do.

“My agenda was to transform the way in whichteachers approach teaching and learning and not so much a documented output”.

About the journeyIntroducing collaborative planning is not straightforward.Steve saw it as a journey rather than as an event. Steve feelsthat the department has travelled part of the journey buthas not got to the destination yet.

Collaborative planning represents a significant culturalchange and imposing cultural change can result inresistance. For example, some teachers placed a highpriority on producing unit plan documents while otherswere more interested in trying out new approaches. Theyfound that tensions emerged, particularly in the Key Stage 3group because they were not used to collaborating.

All members of the department are now optimistic aboutthe idea of collaborative planning. They managed to plansome units of work collaboratively, but in the main theywere sharing practice and discussing teaching and learningin general. Perhaps this suggests there are stages in thejourney to collaborative planning or lesson study.

Observations aboutimplementing collaborativeplanningIt is important to take into account the interests andconcerns of all those involved in collaborative planning inorder to develop a shared vision.

Changing the culture and the ways of working of adepartment needs to be managed skilfully and sensitively.

Steve suggests that the guidance given by the NationalStrategies should include support for school leaders andsubject leaders. This support should acknowledge thetensions that might arise between top down leadershipmodels and more collaborative practices. It should includesupport for school leadership and subject leaders inmanaging this.

What could this study mean for you and your students?If you are thinking of implementing collaborativeplanning as a departmental practice, take intoaccount the tensions that might arise. It is importantto be sensitive to the relationships betweenindividuals within the department in order tounderstand what might get in the way of effectivecollaborative working.

Other departments could learn from this initiative byrecognising that change can not be imposed fromabove; it needs the individuals involved to becommitted to the enterprise and this can only bepossible if the needs and interests of individualteachers are taken into account.

Teachers could learn from this project by recognisingthat changed ways of working are possible, but thatthis might require them to take risks and to beprepared to be open with their colleagues.

What has been learned?Steve described two major changes. One was that, despite some tensions, the teachers in thedepartment were working together to plan unitsof work and they were trying out new ways of teaching.

The other was in his own understanding of thedynamics of the department and the importanceof taking these into account. He emphasised thevalue of behaving like a researcher, which for himinvolved standing back from what was happening,reflecting on what was happening and takingsteps to resolve problems. The latter he wasbeginning to achieve by interviewing the teachersand management and directly confronting theproblems he had identified.

Steve Watson is Key Stage 3Lead Learner at Lindsey School.All members of themathematics departmentcontributed to the study.

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Using ‘masterclasses’ toexplore teaching approaches,with a focus on teachingtrigonometrySteve Wren

The mathematics department at Comberton Village College has a good record ofachievement with students, and the mathematics teachers are considered to be effective.However, they began to be dissatisfied with the ‘recipe’ approach to teaching and learningtrigonometry and were looking for ways in which to encourage more active learning in thisarea of mathematics. Led by Steve, they established a model of CPD which encouragedthem to think deeply about their current practice and to try out new ways of teaching.Members of the team reported that they thought the experience had been valuable andthat it had encouraged them to think more deeply about pedagogy.

Aims of the studyThe aims of the study were to:

• run formal within-school departmental CPD meetings to share practice;

• introduce all members of the mathematics team to ‘guided discovery’ or ‘active’ approaches to teaching trigonometry;

• try these approaches in their own classroom whilst being observed by others;

• to investigate the effectiveness of this particular formalised model of sharing good practice.

Dimensions of the studyThe teaching of some core topics, such as trigonometry,can sometimes turn into a series of recipes for studentsto learn and ‘regurgitate’ in examinations often withoutan understanding of what they are doing or why.Despite teachers’ dissatisfaction with this approach,experience has shown that they are unlikely to changetheir way of teaching unless they have seen an approachused ‘live’ in a lesson. Discussions and shared lessonplans may catch their attention and ‘sound’ or ‘look’good, but frequently do not provide sufficient evidenceor support to convince teachers to try a new approach.However, there is evidence that when teachers jointlyobserve a lesson delivered by a colleague, followed bygroup discussions, they are more likely to try newapproaches and resources with their own classes. Theproject was put together within this context.

The project took place at Comberton Village College, an 11-16 school situated about five miles west ofCambridge in a village setting. Examination results inmathematics are consistently high with about 85% of students gaining A*-C grades, with 40% gaining A*/Agrades. The mathematics department consists of someexperienced teachers who have been at the college forsome years, along with several staff who have joined theschool more recently, during a period of growth instudent numbers.

The teachers in the mathematics team are all regardedas highly effective teachers and they adopt a range ofteaching approaches. They are treated as professionalswho should choose how they wish to teach the topicscontained within the scheme of work to their classes.The Head of Department aims to ensure that CPDopportunities exist to introduce staff to new ideas,which they might choose to add to their ‘toolkit’ ofteaching approaches. In addition, staff are always willingto share good practice but, owing to the nature of theirbusy school lives, opportunities for informal sharing ofpractice occur less frequently than everyone would like.This project sought to provide CPD for teachers througha more formal sharing of practice.

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Planning, teaching and evaluatingMembers of the department met in small groups to discusshow they currently teach trigonometry. Staff took alongexamples of materials they use and pupil exercise books tohelp with their discussion. There was a deliberate decisionnot to direct the discussion but to allow a free range of ideasto emerge. It was felt that this would empower the teachersbecause they would be able to collect whatever ideas andinformation they wanted to ‘widen the toolbox’ of activitiesand teaching approaches that they could use with theirclasses. Many of these discussions were recorded.

“I am looking forward to seeing different ways ofteaching this topic – I have been teaching it the sameway for years.”

“As an NQT I am looking forward to picking up some ideas on teaching method: this is typically a difficult module to teach.”

Teachers took it in turns to teach a ‘masterclass’ which wasobserved by other teachers. These lessons included activitiesor approaches that were likely to stimulate discussionamongst the teachers and were not thought of as‘exemplary practice’. Parts of the lessons were videoed forfuture use and it was suggested that snippets of the lessonsand subsequent discussions be hyperlinked into the schemeof work. In this way, new staff joining the department wouldbe able to benefit from the work that was undertaken.

“Even a filming of the masterclass is potentially helpful – it would be great to have an accessibleresource for different aspects of the curriculum anddifferent audiences…”

The ‘lead teacher’ and other teachers involved in eachlesson then met to discuss the similarities and differencesin approach between the masterclass lesson and currentpractice. Staff were encouraged to explore the rationalebehind the lesson and challenge/support their ownapproach to teaching the topic of trigonometry. Parts ofthis discussion were recorded.

“This encouraged me to think differently about myapproach and teaching method – I would like to try amethod which uses pupil discovery.”

“It was great to have the space and time for deeperthinking about pedagogy – the discussions (especiallythe one after the class) were really thought provoking.”

OutcomesThe key outcome of the project is that the department hasidentified a workable model of CPD that encourages detaileddiscussion and debate of teaching methodology. Thisapproach seems to have been effective and popular with theteam; several members of the team explicitly requested, inadvance of being asked, whether the department couldcontinue with similar training activities once the project wascomplete. Others commented positively on the CPD.

“Excellent way of doing CPD – on different levels, for teachers of different experience.”

“Every aspect of this project was helpful and valuable –including the seminars before and after the Masterclass.They helped me to reflect upon pedagogy – both inrelation to this topic and more widely.”

What could this study mean for you and your students?If you want to rethink the way you teach a topic, consider:

• talking to other members of your department about it – sometimes sharing ideas can help you move on;

• working as a department to tryout new ways of teaching – if you are all doing the same thing, you can compare success and failures;

• planning lessons together and using video to record one of these lessons – this will allow you to review not only the teaching but also the student learning.

What has been learned?The teachers in the mathematics departmentbelieve that constructive debates about currentand new practice are crucial to moving their ownindividual teaching forwards.

The teachers taking part in this study suggest thatthe shared understanding of effective selfimprovement will have a positive impact onthe mathematical learning of their students.

The department now has a model of CPD in placeto support sustained reflection and developingpractice within an experienced and established team.

Steve Wren is Head of Mathematics at CombertonVillage College. All members of the mathematics departmenttook part in the study.

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Graphical calculators have been used in mathematics classrooms for over twenty years.These devices are becoming more powerful and some can be used for more than makingcalculations and drawing graphs; for example it is now possible to run softwareprogrammes designed for use in the teaching and learning of specific mathematical topics.It is also possible to share the displays on students’ devices on the interactive whiteboard.

David and Pam investigated what happened when class sets of these devices wereintroduced in a secondary school.

Aims of the studyThe study aimed to investigate:

• the impact of personally-owned networkedhandheld technology in the mathematics classroom;

• how the device was used in learning mathematics;

• the ways in which the networking facility was used;

• how personal ownership of the graphical calculator (GC), and the chance to use it out of school, affected learning.

Dimensions of the studySmall software is sharply focused on specific topics inmathematics and is well established as a useful resourceto support learning in mathematics. Hand heldtechnologies like powerful graphical calculators (GCs) are often used mainly for graph plotting, but they areable to run versions of small software as well. Teachersare adopting ICT in the form of projection technologyand ‘interactive whiteboards’, but fewer are expandingaccess to the technology to the students. Where thisdoes happen, it frequently involves booking a special‘computer lab’, reducing the frequency and ad hoc use of ICT as part of the mathematics classroom.

The study was a two year enquiry which began in June2007 with the delivery of the GCs and was completed inMarch 2009. The study focused on the introduction of twoclass sets of TI84 GCs, together with a range of softwareand supporting accessories, into the teaching and learningpractices of a secondary school mathematics department.The school involved was an 11-18 mixed comprehensivespecialist mathematics and computing school ofapproximately 1600 students. Use of the GCs wasconcentrated in two top set Year 8 mathematics classes. In one class students were each given a GC to take homeduring the first year, while in the other class the GCs wereused in school only. The two mathematics teacherscentrally involved in the project were interested in thepotential of the GCs, but had no previous experience ofusing such technology in their teaching.

Small software formathematics on handheldtechnologyDavid Wright and Pam Woolner

The graphical calculator used in this project.

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David Wright is a tutor inmathematical education, at theResearch Centre for Learning andTeaching at Newcastle University.

Pam Woolner is a researchassociate at the Research Centrefor Learning and Teaching atNewcastle University.

Introducing the new technologyIt was decided to introduce the GCs through smallsoftware. This approach was chosen because adoption of new technology by teachers and learners tends to bemost successful where it does not involve a big initialcommitment in learning about its functionality, wherethe application fits in well to teachers’ existing practiceand where there is an immediate gain in ‘value added’ to the learning of the students.

The teachers were given GCs and some initial training(mainly focused on how to load and run small softwareon the GC) in the summer term 2007, then used them inclass from September 2007. The Navigator networkingdevice was delivered to them during this first term. Theyreceived training on it in February 2008 and began touse it in their lessons from that date. The teachers keptdiaries recording their reactions to the innovation.

Throughout the project researchers have worked with the teachers and students to investigate theirexperiences of using the handheld technology formathematics teaching and learning. Data was gatheredthrough informal interviews and meetings, classroomobservation, teacher diaries and pupil questionnaires.

What could this study mean, for you and your students?If you are thinking of introducing new handheldtechnology (such as graphical calculators) into yourschool, consider carefully the model of ownershipyou will adopt.

Think about what graphical calculators can be usedfor. ‘Small software’ can often run on these devicesand can provide useful ways of teaching orreinforcing specific mathematical topics.

Consider using networking to share your students’work on a whole class display. Think about usingdepartmental CPD time to present activities youdevelop for graphical calculators to other membersof the department.

What has been learned?Early adoption of technical innovations by‘mainstream’ teachers depends on:

• a relatively undemanding commitment initiallyin learning about its functionality;

• the application fitting in well to teachers’existing practice;

• a perceived immediate gain in ‘value added’ tothe learning of the students;

• readily available technical support to sort outany ‘hitches’

• including an ‘outside’ influence to sustain thepromotion innovative activities.

Planning, both ‘large scale’ schemes of workand ‘small scale’ lesson planning is crucial for the sustained use of the technology. This hasimplications for the allocation of time forprofessional development.

If teachers have a sense of ownership of theinnovation and students have personal ownershipof the technology, the innovation is more likely to be sustained..

OutcomesThe students reported that they found mathematicslessons both more enjoyable and interesting when theyused their GCs. The teachers said that that the GCs weregenerally well received. Some technical problems werereported, but in the main, the adoption of the technologywas seen to be unproblematic.

The GCs were used for specific mathematical topics, suchas angles, bearings and algebra. They were also used as a‘tool’, mainly for drawing graphs. It seems that the use ofthe GCs encouraged a move towards more conjecturingand experimenting, and importantly these activities tookplace in the ordinary mathematics classroom rather thanin a ‘computer lab’.

The Navigator facility was used to collate contributionsfrom students in a shared displayed result on the IWB andalso to display each class-member’s work as a series ofscreen-shots. When the Navigator was used in this wayfor ‘interactive whole class teaching’, all the studentsseemed to be engaged.

“… learners were able to look at and share the graphs they had managed to produce throughinputting equations, in the context of results fromother students. The display of screen shots allowedthem to see how others were rising to challenges such as the teacher’s request to produce a horizontalline. It also facilitated discoveries through discussion,such as about the quadratic curve that one studenthad accidentally produced.’”

There were some differences between the two classes interms of student perceptions about the GCs. The learnerswith personal ownership of the GCs tended to think thatthey learned better with the GCs and were less likely toperceive the devices as making no difference to learning.

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The NCETM fundedprojects schemeTeachers and researchers all over England are becoming engaged in enquiry intoinnovative initiatives of CPD for teachers of mathematics. The focus of the initiatives rangesin scope from the teaching and learning of specific topic areas of mathematics, such asalgebra, to establishing teachers’ communities of practice for mutual support and sharingideas and experiences.

This Teacher Enquiry Bulletin has reported on eleven of over 80 projects which have beenawarded grants. Snapshots of some of the other projects funded by the NCETM are givenbelow, and full reports, authored by the grant holders, are available on the portal:www.ncetm.org.uk/enquiry/funded-projects

PrimaryLynda Maple and her colleagues from seven primaryschools in North Islington formed a learning network to explore how to use talk in mathematics lessons more effectively.

Chris Williams established a project to investigate theeffect of setting on the achievement, attainment andself esteem of children in Years 4, 5 and 6.

Lucy Sayce from Reading was interested in developingstrategies for integrating cognitive conflict as a coreelement of teaching. By identifying strategies to locate and plan for cognitive conflict in any topic, teachers willdevelop pedagogic tools to widen their practice ofcognitive acceleration beyond published materials.

Cathryn Hardy worked with Year 2 and Year 3 teachers inseven schools – two infant and junior school pairs; threeprimary schools. This project identified effective modelsof CPD in the context of researching specific factors thatmay slow children’s progress in mathematics as theymove from Key Stage 1 to Key Stage 2.

Working with prospective teachers and NQTsJo Miller at St Teresa’s Primary School looked at howraising awareness and understanding of Dyslexia,Dyscalculia and Autistic Spectrum Condition (ASC)impacts on the teaching experience of mathematicsPGCE students. The project developed training materialswith the help of Special Educational Needs andSecondary Mathematics Education experts.

Alison Fletcher from the University of the West ofEngland set up a project to support the professionaldevelopment of a group of newly qualifiedmathematics teachers during their first two years ofteaching. She established a forum for the group wherethey could share, discuss and reflect on their earlymathematics teaching experiences.

Pat Stevenson from Mid Cheshire College worked withtrainee teachers, mostly working in FE. The majority ofthese trainees do not have principal responsibility fordelivering numeracy qualifications. The project aimed to develop the skills and knowledge of the participantsin devising ways of embedding numeracy in their own curriculum areas.

SecondarySimon Hart at Kirkley Community High School conducteda year long project looking into personalising learning for pupils in Years 9, 10 and 11. The project was split intotwo areas, coaching loops and learning guides.

Debbie Parks from St Helen’s local authority worked with six teachers from the ten secondary schools in St Helens. The project was developed to give theopportunity for teachers who wanted to improve theirpedagogy to meet together, to discuss developing theirpractice and to investigate why teachers do not seem to plan for activities that encourage pupils to think.

Adam Unwin-Berrey from North Axholme Schoolconducted a study to develop lessons and resources that motivate teachers, teaching assistants and pupilsthrough practical applications of mathematics.

Post 16Bernie Horan-Healiss worked with a team of colleaguesfrom across the college; support staff, learning supportstaff and teachers. These colleagues did not regardthemselves as numeracy specialists. The intention was toimprove both personal numeracy skills and confidencelevels for participants when using numeracy.

Pat Morton developed a Key Stage 5 network in an areaof London where teacher supply can be challenging and where there is a need to develop Key Stage 5mathematical skills and pedagogy in the face of risingdemand for both A level Mathematics and FurtherMathematics.

The NCETM Funded Projects Scheme is dividedinto two areas:

• Teacher Enquiry Funded Projects (TEFP), whichare suitable for teachers who are keen to undertake action research focused on a particular area of their teaching;

• Mathematics Knowledge Networks (MKN), for teachers who may want to work with other colleagues within a more structured approach.

Email Grace Lloyd (info@ncetm.org.uk)for information on how to apply for funding.

This Teacher Enquiry Bulletin was written, edited and coordinated by Marie Joubert and Rosamund Sutherland, at

the University of Bristol, on behalf of the NCETM.