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Deep Progress in Mathematics - The Improving Attainment in Mathematics

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Anne Watson

Els De Geest

Stephanie Prestage

Deep Progress in Mathematics:The Improving Attainment in Mathematics Project

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Deep Progress in Mathematics:The Improving Attainment in Mathematics Project

This work was supported by the Esmee Fairbairn Foundation (grants 01-1415 & 02-1424) and the Universities of Oxford and Birmingham. Note that the viewsexpressed in this booklet arise from the research and analysis done by the research team, not necessarily the institutions supporting the work.

Anne Watson

Els De Geest

Stephanie Prestage

Technical aspects of the research were assisted bySolange Amato, Thabit Al-Murani and Emily Macmillan

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Contents

PageTitle PageTitle

Teaching low attaining students 1

The project 3

Deep Progress in Mathematics 4

The students 5

The teachers 7

Beliefs 9

What do the teachers do? 11

Establishing working habits 13

Generating concentration and participation 15

Interacting and responding verbally 18

Reacting to correct and incorrect answers 21

Giving time to think and learn 23

Working on memory 25

Visualising 27

Students’ writing 29

Students’ awareness of progress 32

Giving choice 33

Being explicit about connections anddifferences 35

Dealing with mathematical complexity 37

Typical task types 39

Structures of lessons 42

Extending tasks 43

Differences in practice for different classes? 45

What did the teachers need? 46

Were these methods successful? 47

Technical information about the research 52

Bibliography and resources 54

First published 2003by University of Oxford,Department of Educational Studies15, Norham Gardens,OxfordOX2 6PY

ISBN: 0 903535 68 8

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If you share this belief, this booklet willshow you how a group of variedteachers try to put the belief intopractice, in many different ways, with arange of results. If you do not sharethis belief, this booklet will have littleto offer you.

In spite of this wealth of research, itis customary for students in the

lowest achieving groups to be givenrepetitive, procedural, fragmented,disjointed, simplified mathematics.

This booklet describes how a group often teachers taught low attaininggroups in secondary school, and whatfeatures were seen to be important.They were neither told what to do, norhow to do it. Instead, they had ashared commitment to improving theattainment of their lowest attainingstudents by building on the belief that:

all students can think hard aboutmathematics, and thus do better atmathematics.

There are twoaspects to lowattainment inmathematics:

not knowingenoughmathematicsand not knowinghow to learnmathematics

Teaching low attaining studentsTeaching low attaining students

What do we know already?

In 1976 Richard Skemp described how a‘relational’ understanding of mathematics wasmuch more powerful, long-lasting and useful thanan ‘instrumental’ understanding.

In 1986 Brenda Denvir and Margaret Brownshowed that learning mathematics is not a linearprocess, and that being immersed in one aspect ofmathematics can frequently lead to unexpectedlearning.

In 1987 Afzal Ahmed published the results of aproject with teachers who worked with lowattaining students. They found that nearly all thestudents were able to use sophisticated thinkingskills and learnt better if they were given time tomake choices, to discuss, and to exploremathematics.

In 1997 researchers at King’s College Londonfound that teachers who made connections,

‘connectionist teachers’, were more successful thanthose who taught in a technical or fragmented way.

In 1997 Jo Boaler showed that students wholearnt mathematics through open-endedexploration and problem-solving did better intests, and other types of mathematics assessment,than similar students who were taught only toperform techniques and follow procedures.

In 1999 Carol Dweck showed that learners andteachers who believed that intelligence wasflexible, and the goal was to learn as much as theycan, were more successful than those who believedin finishing tasks and passing tests.

In 2003 a comparative study of seven nationsfound that, whatever the lesson, task or assessmentstyle, teachers in highest achieving nations tendedto focus on relationships, connections andcomplexities within mathematics, not reducingeverything to technical performance (Hiebert et al.).

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4

Deep progress in mathematics

In this project we intended that those who hadbeen placed in groups of low-attaining students

would make deep progress in mathematics.

Deep progress means that students

■ learn more mathematics,

■ get better at learning mathematics,

■ feel better about themselves as mathematicsstudents.

Sometimes the last of these follows from the othertwo; sometimes students have to feel better beforethey learn more; sometimes students have toredevelop good learning habits before they can moveon and learn more. Ideally, students will makeprogress in all three aspects.

The teachers in this Project believethat students need to start

secondary mathematics on a good basis.There is a public belief that thesestudents need to ‘catch-up’ by rushingthrough earlier concepts again veryquickly. This approach could, if usedunwisely, lead to a diet of repeatedtopics and repetitive tasks. The beliefin ‘catch-up’ is partly generated bygenuine concern about students’ abilityto handle more advanced concepts,partly by political targets, partly by theprovision to all schools of materialswith which to do this task, and partlyby the misinterpretation of thesematerials and their purpose by schoolmanagers and governors.

We know, from research, that groups ofsuch students are often

■ Given repetitive work of a kindwhich is very simplified.

The Project

■ Offered mathematics in a step-by-step way.

■ Focused on arithmetic in imaginary‘everyday’ contexts.

■ Expected to memorise unconnectedtopics and methods.

■ Expected to re-do, again and again,work they have done before becausethey may have forgotten methods.

Project teachers believed that therewere better ways of helping theirstudents become good mathematicallearners than these, which may onlylead to short-term, superficial, success.

Indeed, some of these students workharder than any others in school,because all work is so difficult for them.

The aim is to generate deep progressin mathematics.

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The belief guiding theImproving Attainmentin MathematicsProject (IAMP) is thatmore good can bedone by helpinglearners developthinking skills andunderstandingthroughout all theirmathematicslessons, than byteaching them onlyto perform andremember particularmethods and topics.The students areprepared to learnmore mathematicsand work well withmathematics in laterlife. All students arewilling and able towork hard.

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■ Limited teaching methods: Theymay have been expected to learnonly in one way, e.g. verbal, whenthey may respond better visual orphysical methods.

■ Low expectations: Early weakness inthe subject may have consignedthem to lower sets where thecurriculum did not expect muchfrom them.

■ Learned helplessness: This conditionis reported as a result of beingsupported and helped too much andnot having developed the self-confidence to do anything withoutsupport.

■ Reading and writing difficulties:These may lead to a reluctance to dowritten work, which may lead tounderestimating what they can domentally.

■ Language difficulties: Even smalllanguage problems, such as notdistinguishing between prepositions,can create obstacles in mathematicsparticularly if previous teaching hasbeen procedural.

■ Physical and physiological problems:They may have diagnosed orundiagnosed physical problems,such as deafness, poor sight,allergies which provoke excessiveenergy or tiredness, sleep problems.

■ Cognitive problems: some havegeneral cognitive problems whichaffect all their learning, such asshort-term memory deficiency, andnot been shown ways to overcomethese.

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”

I really love toteach bottom sets,because that’swhere all thepeople go wholearn in differentways from theteachers, and itis really excitingworking withthem.

“When students enter secondary schools, many who have not

achieved level 41, or have only justmanaged to achieve it, are already onthe way to feeling that they are failingat mathematics. They may then belabelled as ‘low attainers’ and placed inlow-achieving groups. Their pastachievements have been low relative toothers, yet there are many reasons whythey may not have reached the covetedlevel 4, and none of these reasons relateto innate lack of fixed ability:

■ Disrupted schooling which has ledto fragmentary mathematicsexperience, possibly missing somekey conceptual developments.

■ Cultural differences: Expectations ofbehaviour at home may be verydifferent to what is expectedat school, in particular explaining

The students

yourself, or challenging others, maybe unacceptable at home;it is also possible that the nature ofknowledge and the notion of truthmay differ between home andschool.

■ Social and emotional difficulties:They may have had problemslearning how to behave inclassrooms, or may have not found away to get on with particularteachers; they may lack the socialskills required to get positiveattention; they may belong to aparticular group which isexpected to have low achievement.

■ Lack of specialised teaching: Theymay have had teachers whosemathematics was not very strong,and who could not find ways to helpthem learn.

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1 Level 4 of the England and WalesNational curriculum for mathematicsis seen by some as the idealminimum level of attainment on entryto secondary education.

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extended tasks, commentingpositively in evaluation tasks.

■ Being more willing and able totackle routine, non-routine andunfamiliar tasks.

■ Working more mathematically,expecting to find coherence in tasks,expecting mathematics to makesense.

■ Doing better than expected, or thancomparison groups, on certain typesof question in national and in-house tests.

■ Showing improvements inbehaviour and attendance.

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This booklet is aboutsimilarities anddifferences in theways the teachersworked.

The teachers were recruited in a variety of ways. Some joined as a

result of letters to all the schools in acounty, others through ITT partnershipconnections, or by personal contact.All we asked was that a teacher shouldbe able to commit herself to teachinglow-attaining students by focusing onthe development of mathematicalthinking alongside covering content.Ten teachers joined: 8 experiencedmathematics teachers and 2 in their firstthree years of teaching. 2 experiencedteachers asked to join about a term afterthe start of the Project; 2 left theProject early due to changes of job, andwere replaced by one junior colleagueand one experienced teacher. They alltaught at least one new-to-them classcontaining a significant number oflower attaining students in lowersecondary school - the focus class forthe Project.

The teachers

2 Sometimes we use actual namesin this booklet, sometimespseudonoyms depending on thecontext and agreements we havewith the teachers.

They knew it would be hard topersuade the students to respond tomathematics in a positive way. Therewere a few students who could not bepersuaded to engage with mathematicseven after two terms of effort, but themajority of students eventually startedto do better. ‘Doing better’ meansdifferent things to different teachers,but the following criteria were shared:

■ Being more active in lessons, forexample by participating indiscussion, asking and answeringquestions, volunteering for tasks,offering their own methods.

■ Being more willing to share ideaswith others: teachers, peers, wholeclass.

■ Showing more interest inmathematics, for example by doingmore homework, working on

Here are all thenames2 of ourProject teachers:David Askew

Rebecca Freeman-Moody

Andrea Pitt

Siobhan Thomas

Anthony Broadley

Sara Howes

Andrea Rigby

Claire Fletcher

Linda Juul

Kevin Slater

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Learners’ identityLearners can see their goal as to learn, or to finishtasks, or to fit in. Some cannot see how to fit in soappear to choose not to fit in. Some see silence andinactivity as safe ways to fit in. In this Project thegoal of teachers was to help learners have the goal tolearn. Intellectual engagement would be its ownreward. There would be no need to constructartificial ‘real-life’ contexts as motivational devices,although authentic contexts could, of course, beused mathematically where obviously appropriateand relevant for the students.

Taking account of realityWhile improvement of mathematical thinking andself-esteem are appropriate goals for educationists,students also need to be achieving in waysrecognised by the outside world. So finalexamination results are important, but students willbe better prepared for these if they understand somemathematics, and feel confident to tackle unfamiliarproblems, while being sure they have some funda-mental tools, such as arithmetic and calculator use.

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they are progressing in a high-status subject can bevaluable, particularly if they failed to make progressin their previous schools. But more than this, apositive experience of mathematics can empowerthem mentally because their own thoughts are beingvalued. Self-esteem can be developed through:

■ Believing that students desire to learn.

■ Responding to, using and generating students’own questions.

■ Fostering awareness of learning, such as focusingthe use of practice exercises on self-assessment.

■ Offering challenge and support instead ofsimplifying the work.

■ Enabling students to step out of their comfortzones and take risks, such as creating their ownhypotheses (for some of these students, no part ofa mathematics lesson is a comfort zone).

■ Developing ‘togetherness’ in classrooms whenworking on mathematics.

The main drive for the teachers was the shared belief that all students can learn mathematics,

so that the efforts made to help them do so weregoing to be worthwhile. Even these least-achievingstudents are entitled to the full breadth ofmathematics, including abstract reasoning. Theteachers all have similar sets of goals and beliefs intheir teaching of students who, on entry tosecondary school, have achieved less than targetlevels in mathematics.

Development of reasoning and thinkingAll students are entitled to learn mathematics in wayswhich develop thinking and confidence in problem-solving. Mathematics offers, as well as numeracyskills, logical reasoning, discussion and argumentabout abstract ideas, and an arena for careful analysis,categorisation, generalisation. It also offersexperience in solving unfamiliar problems, perhapsby looking for familiar structures and bringingknowledge and thought to bear on them.

Beliefs

Right to access mathematicsWhile many adolescents appear to have obstacles toengagement in mathematics, all students have theright to, and are capable of, full engagement with thesubject. Outstanding features of mathematics whichmake it interesting, and which make learning easier,are the inter-connections between different topicsand representations, and the relationships betweenand within mathematical structures.

Rights and responsibilities as citizensAll students are entitled to have access to themathematics necessary to function in society, beyondminimal functioning. They need to be able to solvemathematical problems, and solve problemsmathematically, with an awareness of number, spaceand probability, if they are to be good employees andcitizens.

Maths as a source of self-esteemFor a variety of reasons, success in mathematics canbe a source of self-esteem for students. Some maydo better in mathematics than in other subjects,particularly if literacy is also weak. The feeling that

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”“

”

In one observed lesson, the first of the day, Mandystarted with

I haven’t got a maths book, Miss,

I ripped it up and put it in the bin.

and proceeded to take twenty minutes to calm down,demanding attention frequently. She said thatalready that morning she had fought with her mum,left the house without washing, been spat at, and hada headache.

These really are students who have the potential tobe difficult, alienated and disruptive in any school.But the teachers all persisted with their highexpectations and what we are reporting is typical oftheir aims, purposes, goals, lessons and, in general,what was achieved.

In the pages which follow we tell you about howindividual teachers acted to achieve similar ends.

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The importance of the role of the teacher was veryevident in these classrooms: they were active andpurposeful. At any given moment in the lesson, anobserver (and presumably the students) could easilysee what the teacher was doing and wouldunderstand, either because it had been made explicit,or because it was an obvious response to the needs oflearners, why it was being done. Thus the goals andpurposes of teaching and tasks (which were usuallyabout learning and very rarely about upcoming tests)were known to students.

We do not pretend that these lessons were alwaysgood, always smooth, always rosy. They were not.For example, in many lessons Simon would deliver aconstant loud monologue of this kind:

It is boring; I can’t do this;

I can but I don’t want to;

I want to go home;

she knows we can do this

already.

“

In this section we report on aspects of practice which the Project teachers and researchers saw as

significant in their work. For example, teachers mayhave changed some feature of the way they taught asa result of thinking through their practice with others,or may have become aware that something about theway they taught was different to how colleagues orother observed teachers worked.

You may not be surprised by much of what we write,because it is largely a description of possible details ofgood teaching. Why devote this booklet to it then?Because the expectations these teachers had werehard for these students to meet, for whatever reason.We assume that the resistance existed because theseexpectations were unusual for them; they were out ofthe habit (or had never established the habit) ofworking effortfully and successfully in mathematicslessons.We have collected the characteristic features oftheir teaching under these headings, which arearranged in a continuum. This continuum describeshow teachers create an atmosphere in which studentsgradually become active learners of mathematics, thenthe nature of that activity, and then the tasks andstructures within which this took place.

What do the teachers do?

■ Establishing working habits

■ Generating concentration and participation

■ Interacting and responding verbally

■ Reacting to correct and incorrect answers

■ Giving time to think and learn

■ Working on memory

■ Visualising

■ Students’ writing

■ Students’ awareness of progress

■ Giving choice

■ Being explicit about connections anddifferences

■ Dealing with mathematical complexity

■ Typical task types

■ Structures of lessons

■ Extending tasks

■ Differences in practice for different classes?

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move on to harder things, and notice whensomeone did so.

■ To establish the need for a growing storehouse ofknowledge and techniques, students were givenspecial books in which they would write theirchosen examples as a reminder.

■ One teacher required sensible, thoughtful,participation in public mental arithmetic work,using the rest of the class as collaborators, andmanaged to develop a no-risk atmosphere inwhich this was achieved.

Behaviour can be altered, but it takes time,persistence and imaginative methods. Old habitshave to be replaced by new ones over time. A‘training’ approach(with clear expectations andrewards) might be effective. Anything whichdisrupts old expectations (including expectations ofthe teacher’s behaviour) is worth trying. Time hasto be given in lessons to establishing new habits,and time must be given over several weeks for themto become ‘habits’.

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■ To establish discussion, students were seated in aboardroom/ horseshoe style so they could seeeach other to talk to. Although they could stillsometimes use this as an opportunity to be silly,the teacher used this style often and studentsresponded well more and more frequently.

■ To establish real discussion, so that the teacher isnot the centre of all interactions, one teacherstopped sitting at the ‘front’ of the class and sat atthe ‘back’.

■ To establish continuity of work, one teacher keepsevery student’s work in their own plastic bag, sothat at the start of a lesson the student hassomething to open and start doing, containingtasks and artefacts which trigger memory fromthe last lesson.

■ To establish the recognition that the kind ofanswers which, though correct, students had beengiving for years (and been praised for) were nolonger appropriate, one teacher would ask for:‘my favourite answers?’ and praise with ‘that is one ofmy favourite answers’ to indicate that it was time to

Establishing working habits

Teachers used a variety of ways to develophelpful learning habits. Some worked explicitly

on contradicting ambient assumptions aboutbehaviour. Statements like ‘they can’t concentrate’were taken to mean ‘they don’t concentrate’; ‘they can’tlisten’ as ‘they don’t listen’. Teachers would then try tocreate lessons in which ‘they do concentrate’ and ‘theydo listen’. All of these took time, creating new habitstakes new routines which need constant repetition tobecome habitual. The teachers did not give up; theykept plugging away at creating new habits. Here aresome examples of what individual teachers did:

■ To establish listening, aural questions wereused at the start of lessons as a settling-down task.

■ To establish effort, students were made to look ata problem on the board in silence, told that theaim was not for them to solve it, but to thinkabout it, and then they were asked for ideas.Over time, as this became routine, more andmore students actually did think about the givenproblem.

■ To establish atmosphere, effort, concentrationand memory, students in one class were given thesame sheet of multiplication facts to complete atthe start of every lesson and worked through it insilence for 15 minutes. The aim was to completemore and more each time, to compete withprevious personal best.

■ To establish listening and concentration, answersare read out once only, very quietly.

■ To establish the expectation that everyone will bethinking, students know that anyone in the classmight be asked to answer, after some thinkingtime.

■ To establish the expectation that everyone will betalking about the maths, pairs of students discussmaths and all pairs have to report back on whatthey have found.

■ To establish homework organisation habits,students had to first establish the habit of takingpaper to and fro from school successfully! Whenthis was established, they had to write somethingthey had learnt on the paper.

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■ All teachers use ‘hands-down think’, where noone is expected to raise hands to answer. Theclass all sit and think and anyone might be askedto answer. There is extended time to thinkabout their answer.

■ Students are asked to give answers to workedquestions, rather than the teacher or textbookgiving them.

■ Students create questions for each other.

■ Small whiteboards are used so that everyone canshow an answer or idea or example.

■ Teachers explicitly use what is interesting instudents’ work to direct the next part of a lesson:‘Can we have a look at something I have just noticedin Sam’s work?’

■ One teacher gave time for students to cue in tothe mathematics by starting with a silentthinking task which related to what had beendone in a previous lesson.

■ One teacher uses elaborate stories with abeginning, middle and end to exemplify

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■ Recognising that some students have difficultydoing homework for a variety of reasons: the taskmay have been posed in an ambiguous or vaguemanner, it may really be too hard, there may beproblems at home and so on. Flexibility abouthomework expectations was the norm on theProject; some ran homework clubs, or usedschool detention sessions as a maths help clinic.

■ Memory was explicitly worked on by someteachers; sometimes they would say ‘you need toremember this so listen hard’, others would askstudents ‘how could you remember this?’ or ‘howdid you remember this?’

■ Each student may have to produce something tocontribute to the class; these can be compared,classified, or ranked according to difficulty.

■ Students are asked by all teachers to explain ideas,concepts and definitions in their own words.

■ Some teachers deliberately ask questions withmultiple answers which everyone can answer toensure all can participate, and to generate enoughanswers for the class to consider.

Generating concentration and participation

Here are some ways in which Project teachersexplicitly generate a studious atmosphere of

concentration and participation. Of course this takestime, but in this Project no one gave up trying toachieve these. We noticed that participation doesnot always imply concentration. One can beinvolved physically, such as taking part in a physicalrepresentation of a graph or of a data set, withoutconcentrating on meaning. However, to concentrateone has to participate to some extent.

Teachers worked on concentration and participationby:■ Structuring whole class tasks so everyone has

to take part.

■ Asking students to check and make commentsabout each other’s work.

■ Making ‘listen to each other’ important byasking for students to say it in their ownwords, or to take each other’s ideas intoaccount in their answers, or to do some workbased on another student’s idea. This could bedone implicitly; for example, one teacher asked:

‘Can you say that again to Mark because you aresaying it very well?’

■ Getting students to listen to, or read, questions indetail before they answer.

■ Giving students the responsibility to monitortheir own efforts. Again, this can be doneimplicitly, emphasising high expectations inapparently casual interactions.

Teacher: Liane, are you working well?

Liane: No

Teacher: Are you doing something about it?

Liane: Yes, miss, I will make it good work.

■ Asking students to imagine something and givingtime for an image to develop.

■ Using visual and kinaesthetic teaching methods aswell as verbal and practical methods.

■ Using rewards such as chocolate used to startparticipation, until a sense of intrinsic interest hasa chance to develop (so that chocolate becomesirrelevant).

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Student-student interactions■ Students listen to replies from other students to

questions asked by teacher.

■ Students are invited to write on board, or onOHP, so their ideas can be discussed.

■ Students generate examples and methods, whichare shared publicly and students make commentsabout them voluntarily.

■ Students working together manage to coercereluctant peers into the activity, creating peer-pressure to work. One student said to another,who had just interrupted the teacher: ‘just bequiet, then you’ll be alright, won’t you?’

■ Students in one class are free to move around anddiscuss their work with others; one is observed totell another she cannot do it and they worktogether for a while.

■ At times, all utterances are directed towards theteacher; at other times, students talk directly toeach other, even in whole class discussion.

Interacting and responding verbally

One of the most exciting things to emerge fromthis Project is the variety of ways in which

interaction in classrooms between student andstudent, or between teacher and student, promotelearning. It is particularly noticeable that moststudents learnt (or relearnt) to talk to each other inuseful ways when that was the clear expectation ofthe teacher. This is in contrast to the common habitwith such groups of separating students andexpecting them not to interact. Teachers would varythe room and seating arrangements, for exampleputting tables in a horseshoe, or placing themselvesnear the back of the room, in order to ‘upset’ normalpatterns of interaction and promote more student-student talk. We are not pretending that allinteractions were about mathematics, but instead weidentify the kinds of interactions which were aboutlearning and give examples of these. To over-controlinteractions means that these would also be lost aswell as the distracting kind, but this meant thatimprovement did not happen overnight, nor wasimprovement sustained in every lesson.

mathematics in context; thus she hopes herstudents will build a complex picture in theirminds of the story, the situation and themathematics which they can relate to personallyand emotionally.

There is a connection between engagement andlearning; students cannot learn unless they areengaged, and engagement is a combination of social,emotional, intellectual and task characteristics.Teachers had to work on all these facets to ensureengagement. All teachers believed that learners’concentration and participation could be developed.

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■ Some students need help from the teacher indeveloping organisation strategies in order to learn.

The quality and patterns of interaction can bechanged so that students participate in the creationof mathematics in the classroom. Gradual shiftshave been made so that students are moreresponsible for their own learning, and what theydo is valued by the teacher and other students.

■ Students ask for, and are given, individual help:teacher may demonstrate how to do something,or give instructions, but is more likely to probewhat the student thinks already, or to ascertainwhat they understand.

■ Teacher offers counter-examples to students’conjectures to promote further thought.

■ Teachers see their role as helping with learning,not mainly as markers of work or keepers of thepeace.

■ Students mark each other’s work and discussdubious answers.

■ Tasks which clearly require discussion aresometimes given, e.g, those where more than oneapproach is possible, or where ways to get startedare not immediately obvious.

Student-teacher interactions■ Teachers have standard questions which they use

frequently enough that students expect them andcan ask them for themselves, for example ‘what isthe same and what is different?’, and ‘is it always,sometimes or never true that ....?’

■ Teacher asks questions, which might be open,closed, or structured in ways which graduallyexpose a topic, or an idea, or a generality ofmathematics.

■ Extended waiting time is used, before and afteranswers are given.

■ Students give answers and methods or reasons.

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■ Students might be invited to comment on eachother’s answers - the teacher may not indicatewhich are correct or incorrect, or which can befurther developed.

■ Teacher expects students to use technicallanguage and asks for such uses to be explicit andrepeated.

■ Discussion sessions can consist of remarks allchannelled through the teacher but are morelikely to involve interchange between many,orchestrated by teacher.

■ Teacher writes everything students say on board,for later discussion and criticism by whole class.

■ Teacher invites students to share in exposition, orin worked examples, not just by ‘gap-filling’ butby offering suggestions and reasons, or sayinghow one line follows from the one before.

■ Students ask questions about what they do notunderstand, or about other possibilities.

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■ Shift the responsibility for ‘knowing what to do andhow to do it’ to the pupil.

■ Use their responses to pace the lesson: ‘Do we allagree with that?’ when it is time to move on; ‘Tellme more’ when reflection is needed.

Answers, whether right or wrong, are materialfor consideration and for learning, not end pointsexcept where speed, accuracy and memory are thefocus of the task. Answers, therefore, are oftengiven the status of conjecture to be tested out anddiscussed.

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and was chosen to answer by the teacher, butsaid: ‘I have forgotten’, the teacher just said:‘Alright, you’ve forgotten’ without making anissue of it.

■ It is very rare for Project teachers to accept thefirst correct answer given without furtheropportunity for learners to think about it. Forexample: ‘you answered that much too quickly;what other ways are there of working out a quarter?’

In general teachers:

■ Provide a range of cues for the maths.

■ Are interested in what students know and whatthey do not know.

Tim: I don’t know nothing, Miss.

Teacher: Let’s go back to what you do know then.

■ Develop expectations of interest, e.g. by saying:‘Whatever you know I am interested in knowing’.

■ Expect all to be involved and to have somethoughts about the questions asked.

One particular type of interaction which isworth separate attention is the teacher’s

reaction to answers. A teacher can choose to react toall answers, right answers, wrong answers, or to notreact at all. These decisions create different dynamicsin classrooms, and sometimes not reacting at allpromotes further thought more effectively thancomments from the teacher.

Listening to pupils’ responses to questions is vital for‘reading’ the lesson and deciding where to go next,to move on, to recapitulate, and to provide morerehearsal space. The less articulate the students, themore agile the teacher needs to be.

Students in the Project classes learn that incorrectanswers are valuable in providing learningopportunities. It is frequently the case that studentswho feel free to answer in an unthreateningatmosphere will self-correct as soon as they hearthemselves saying something. The Project teachersare good at developing reflection on incorrectanswers, and also on encouraging students to thinkfurther.

Reacting to correct and incorrect answers

■ A student has written an incorrect answer on theboard and the teacher says: ‘We are going to leavethis on the board as, even though it is wrong, it mightgive you some ideas about what to do’.

■ Some teachers say nothing about correct answersand continue to ask others until several studentshave also given correct answers.

■ Some teachers ask for explanations of bothcorrect and incorrect answers (thus givingstudents a chance to self-check and self-correct).

■ When students cannot answer, teachers avoidsimplifying the questions until it becomes triviallyanswerable. One teacher asks ‘how else could I askthis?’; another coaxes, saying things like ‘youremember when we did such-and-such and Simon saidso-and-so?’.

■ Teachers are sensitive about the inability toanswer questions, and recognise the social needfor some to look as if they are participating, or tolook as if they are not. Stewart had his hand up

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In general, teachers thought about theappropriateness of timing from the dominant pointof view of learning, not from a dominant point ofview of coverage. Rote-learning was avoided,although some aspects of mental arithmetic benefitfrom chanting and memory. There was a balancebetween time pressure used to encourage fluencyand effort, and space used for thinking, reasoning,considering, reflecting.

Covering the whole curriculum was important,but so was understanding what had been learnt.Working towards understanding is ultimatelymore long-lasting than mere acquisition ofprocedures.

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3 ‘Own pace’ can be slowed down by disengagement, but here the focus is onengagement so that working slowly does not equate with ‘wasting time’.

■ Students who gave wrong answers to mentalcalculation questions were told by one teacher‘you answered that much too quickly’, thus indicatingthe value of self-checking against speed.

■ Class discussions were often managed so thatthere was no apparent time pressure; for example,there was time to consider statements carefully,time to find out what everyone knew.

■ Sometimes there was implicit pressure becauseother students would be waiting for someone toanswer or finish something; other times there wastime pressure used constructively, such as inchanting to a rhythm in order to help studentsattain fluency aspects of mathematics.

Rather than rushing through topics, the teachers gave extended time for learning, and

often discussed this with students explicitly. Theybelieved that it takes time to reach a learning goal, ormake a new connection, although the learning whichis going on en route may be dense and busy.Discussion takes time, and students would benefitfrom being aware of the use of time to explore, mull,think again, to ponder and so on.

There were various ways this expressed itself inpractice:

■ Some teachers extended a single topic over severalweeks, in order to use many differentrepresentations, to ensure progression in thetopic, and to include repetition, use andexploration.

Deep progress was ensured by such extended tasks,and deep progress also included the development ofwork habits which allowed long tasks to besuccessful. Teachers gradually shifted learners’perceptions of how to pace their working.

Giving time to think and learn

■ One teacher put no time limits on any task, andexplained this to students: ‘sometimes you needmore time to think’.

■ Ideas might be shared before anyone felt they hadfinished, so that everyone might be helped byhearing how others were working.

■ Students can only learn at their own pace, not atan externally imposed rhythm3. This is in contrastto completing a procedural task within a giventime, which may not result in learning directlybut may provide raw material for later reflection.

■ Exercises are provided which need not befinished, but which offer choice and challenge aswell as practice. If the expectation is that studentswill learn as much as they can from an exercise,rather than merely ‘do’ as much as they can, thentime taken to finish is not an issue. One teacherwould say: ‘Take your time, go through verycarefully ... there is no hurry’.

■ Students can ask for more time to work on anexploration.

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■ Using writing to develop memory for chains ofreasoning.

■ Using mental mathematics to help developmemory and thinking skills because they have tosort out the mathematics in their heads while‘holding on’ to numbers and relationships.

■ As we have said before, memory was explicitlyworked on; sometimes teachers would say ‘youneed to remember this so listen hard’, others wouldask students ‘how could you remember this?’ or ‘howdid you remember this?’

Memory for what is important needs explicit work;this includes frequent return to main ideas, so thatmemory is necessary and important. Students needto know what is worth remembering. Explicit workis needed on words, techniques, facts and images tohelp conceptual recall. For most students, memorycan develop, particularly if there is some self-esteemto be gained by doing so - so rememberingmathematics needs to have value in the classroomand to be used as a marker of personal progress.

4 This isonly usedselectivelyfor fluency,not as ausual formof teachingtechniques.

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■ Implicit and explicit work on calculating andholding information: ‘you have to sort this out inyour heads; it’ll help you remember it’.

■ Getting students to return to a physical position,or location in the classroom, which related tosome previous activity.

■ Asking them to devise their own methods ofrecalling, and of testing their recall.

■ Asking them how they remembered something.

■ Repeating something which would be useful toremember enough times for it to become familiar4.

■ Combining words and movements, such as whenwriting algebraic expressions (drawing the fractiondividing line while saying ‘divided by’) or whenlearning multiplication facts (sequence of actionsaccompany tables), or when drawing angles (indicateturn with hands as they say ‘angle’).

■ Having objects, equipment, which is usedfrequently during work on a topic so thathandling the equipment brings back the memoryof what it was used for.

One of the roles of the teacher is to helpstudents know what is worth remembering in

mathematics. Students need to know what it isvaluable to remember, what will be useful in future ifit is fluent, and how to remember these. Overloadingstruggling students with meaningless mnemonics wasnot seen as helpful, although having some of thesearound can help students feel equipped formathematics. The Project teachers focused insteadon memory with meaning, sometimes trying to linkwords with visual or physical memory. Memory wasworked on at several levels: remembering from lessonto lesson provides continuity and a sense of short-term progress; working deeply on mathematics canaid longer-term memory, and memory formathematics can aid deep progress, and contribute toself-esteem when students are aware of the extent ofwhat they know. Hence teachers use a variety ofways to develop memory and here are some examples:

■ Reviewing areas of confusion through discussionin pairs of distinctions between, for example, areaand perimeter, or mean, median and mode.

Working on memory

■ Reviewing at start of lessons what was done inprevious lessons. One teacher does this by givinga related task for silent consideration to startthe lesson.

■ Ensuring new topics are mathematically linked toprevious topics, not fragmenting the subject.

■ Reminding students about central aspects of topicfrequently, such as technical terms, definitionsetc., as if they are tools for the task. For example:‘Maybe someone can run through what ‘mean’ meansfor us.’

■ Pointing out what they might need to recall,saying ‘listen carefully to this’.

■ Using pooled memory of class to create conceptmap, or spider diagram on board. For example,one teacher wrote ‘Facts’ on the board and asked:‘Who can tell me anything about polygons?’ She thenwrote everything which was said underneath it, tobe tidied up later.

■ Whole class discussions used to recreate techniques.

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Students can use a range of senses to engage with,understand and remember mathematics. Teacherscan offer a wide range of approaches, includingasking students to imagine visual, tactile andphysical experiences, both recalled and totallyimaginary.

The use of visualisation may be plannedthoughtfully and deliberately, comparing what islikely to be learned using different forms.

■ Use of physical actions to represent mathematics(e.g. walking to and fro along the numberline for negativenumbers; making graphs by standing in appropriate placeson a floor-grid etc.).

■ Asking students to re-enact previous visualisationsand actions.

All Project teachers emphasised visualisation for several reasons: many of their students were

better at visualising than verbalising and images maybe easier to recall for some students. The spoken andwritten word are sometimes insufficient for pupils tomake meaning, so alternative forms of representation,or indeed many forms of representation extend thepossibilities for learning. Sometimes visualisation wasdone mentally, sometimes through drawings anddiagrams. Visualisation appeared as a feature ofpractice in these ways:

■ Explicit appeal to the imagination in spatial and inalgebraic tasks (e.g. using an empty bag to representan unknown quantity).

■ Students’ own images are compared to anotherimage (given verbally or graphically).

■ Working in the imagination, then checking someother way. For example, Kevin asked students todraw reflections and rotations by imagining whatmight happen, then drawing it. He then gavethem tracing paper to check their work. This

emphasised the transformation process. If theyhad used tracing paper first, they may haveconcentrated on the detail of producing the image.

■ Making the contents of a ‘toolkit’ of visual andtactile objects with which to pursue several tasks.This also gives concrete ownership; handlingthe ‘kit’ gives instant recall of past tasks.

■ Appeal to numberline or other spatialrepresentations of number while doingcalculations, (e.g. a quarter is half and half again;1/3 x 3/4 visualised as one third of three parts ofsomething).

■ Focus on mental calculations.

■ Spider diagrams and concept maps are used toconvey links in mathematics, or as tool forawareness of learning, such as through self-assessment.

■ Use of equipment. For example, students mademultilink towers to create physical data sets whichwould have particular properties: means, modes,ranges etc.

Visualising

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5 But all teachers were very wary of allowing students to ‘colour in’ where it was notobviously connected to thinking about mathematics.

Teachers can consider the role of writing and weighthe obstacles against the benefits, providing support,frames and scribing where this allows students tofocus more on mathematics.

The effort of deciding what to write aboutmathematics can scaffold and support thedevelopment of conceptual understanding andreasoning. It can also support students’ certaintyabout concepts, and their memory for chains ofreasoning.

Teachers were very aware that writing maybe a problem for their students, but that

writing is a valuable means of communication inmathematics. They thought carefully about theiruse of writing and their expectations of students.They provided models, and scaffolding, forwriting, but also minimised its use where it mightbe a barrier to learning mathematics. Here aresome examples:

■ Use of writing frames so that students fill inblank spaces rather than have to writearguments in full. Some students were asked tofill in this sentence: ‘when I looked back over myanswers I saw that these were similar . . . , andthese were different . . . ’.

■ Students are encouraged to make their ownnotes about what they notice, or what theywould like to investigate next.

■ Teachers write students’ ideas on the board intheir own words (or students write their ownideas on the board). These can be hypotheses

about concepts and tasks, calculations, student-generated definitions, etc. Similarities anddifferences are discussed.

■ Students draw sketches or diagrams of spatialwork they have done tactile objects such ascut-outs, tiles, etc.

■ Minimal use is made of copying teachers’words, or producing written work solely tosatisfy external audiences such as parents andinspectors.

■ Mental work is emphasised where possible.

■ Use of colour where this aids thinking5,shows classification, or contributes to self-esteem through ownership. One student said:‘I like colouring-in, it helps me think’.

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Students’ writing

The relationship between writing, thinking,learning and remembering is neither obvious nortrivial, especially for students who may havedifficulties with general literacy.

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■ Asking students to write their ideas on stickylabels and stick on a poster and then to refer tothem in later lessons as they explore; this helpsthem to recognise their progress.

■ Students are asked to write down their owndefinitions, methods, reminders when they thinkthey have reached an understanding ofsomething.

The teacher does not have to be the audience forassessment. Students can get an implicit sense oftheir progress, and hence develop self-esteem, if tasksare structured to include marking progress of somekind at various stages.

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6 At the time of the Project, teachers were beginning to adopt some of the methodssuggested in Black et al (2002) but this list includes additional ideas

thought it was. I don’t have to say how I feel about itout loud, that’s embarrassing. This works because theteacher can go over the work again and again if shewants to.

■ Some games-type approaches to mathematicshave inbuilt mechanisms for monitoringprogress, such as having to remain sitting orstanding until some task has been completed.

■ Being asked to make up their own hardexamples, or to ‘make up one you can do now,but you could not have done a week ago’, cangive a sense of progress.

■ Some students are given a pre-test at the start ofa topic so that what they already know can bevalued and used, then students are in a betterposition to identify their own new learning.

■ Starting a topic by being given a relatively hardtask and students finding that, after a few lessons,they can do it effectively.

Students’ awareness of progress

Students in lower sets often find it hard toknow when they are making progress in

mathematics. Self-awareness is very important totheir self-esteem, and that there are a variety of waysin which this can be established in mathematicslessons, from the casual to the more formal.Teachers expected students to engage in a wide rangeof assessment activities, including self- assessmenttasks, traditional testing, and so on6. Self-assessmentusually involves reporting to the teacher, whileteacher-assessment, however informal, involvesreporting to the student.

In this section we focus on something more subtle.The emphasis in this section is on the learnerssensing that they are making progress throughouttheir work because the way the task is structuredoffers clear markers of progress. Here we list waysin which students in Project classes became awareof their own progress:

■ Students’ casual discussion about their progressduring tasks is a source of self-monitoring

relative to others. Even tiny interactive incidentscan make a difference:

Mhairi (calling across the room): June, we’ve done it!

June: so have I!

■ Sharing ideas, especially during tasks, can give asense of progress relative to others and the task.

■ Use of a range of ways of recording work onpaper, formally and informally, in words ordiagrams, can give a sense of how much has beendone, although not all the mathematics done canbe usefully recorded.

■ The teacher giving time to listen to students’ideas seriously, or recording all ideas on theboard, can give a sense of achievement.

■ Teachers put strategies in place which show thatstudents are being heard, as well as giving theteacher information.

Jilly said: I like it when we have to draw a trafficlight next to our work to show how hard or easy we

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■ Students can choose whether to spend furthertime on a task.

Teachers expected students to take responsibilityand provided situations in which choices had to bemade to promote this. They expect as much self-direction from students with lower achievement asfrom any other group.

through exploring possible perimeters of shapesmade by abutting four congruent squares,students can pose their own questions to explorefurther questions about squares and perimeters.

■ Offering their own interpretation of amathematical statement. When asked to ‘findthe sum of three consecutive numbers’, studentsrestated the problem in their own words. Allinterpretations were written on the board andcompared until agreement was reached aboutwhat it meant.

■ Teachers might plan lessons so that students cando ‘this’ or ‘that’; they do not have the choice todo ‘this’ or ‘nothing’.

Giving Choice

Students were given choice in many waysduring the Project, and some teachers were

aware they were giving more choice as the Projectprogressed. This allowed students to becomeemotionally attached to their work and develop theirown ideas.

It’s boring to be told what to do. It’s nice to have timeto choose ... kind of relaxes me. When you are toldwhat to do you don’t want to do the work ... when youare relaxed you want to sit down and do the work.

Students often had to account for the choices theymade, and this encouraged reasoning, explanationand justification and ensured that choices were madewith mathematical understanding.

Opportunities for choice included:

■ Choosing their own methods of working

■ Choosing their own methods of calculation.

■ Choosing a starting example for a mathematicaltask, such as: ‘draw your own quadrilateral, anyfour-sided shape you like; now join the mid-points of its

sides’; or ‘imagine you are cleaning windows of a sky-scraper, choose which floor you are on, now go up twofloors, down five etc. etc.’

■ Choosing their own examples to practise tech-niques. Having worked on percentages studentswere asked to find as many different percentagesof a given amount as they felt able to do.

■ Choosing how many examples to do in order tofeel confident. One teacher says: ‘do as many asyou need to in order to feel you know how to dothem’.

■ Choosing how hard to ‘push’ an idea, or extend avariable, to see what is possible.

■ Choosing their own variation of a task withinsome mathematical constraints. Students createdtheir own shapes for working withtransformations with vertices on the ‘dots’ ofdotty squared paper.

■ Deciding what they want to investigate about agiven situation. After discussing results obtained

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■ Students are asked to make their own distinctionsbetween mathematical objects which seemsuperficially similar, such as: means, medians andmodes; bar charts and histograms; expressionsand equations; equalities and inequalities.

As well as giving students the opportunity to noticechanges, links and similarities in mathematics,teachers can make these moments explicit. Studentslearn more about what is expected, what would helpthem understand more, and how to look moresearchingly at mathematics in future.

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■ When discussing the symmetries of a cube,students are reminded about their recentexperience with dice and the related probabilitiesof getting scores, thus linking both to the shapeof the cube and showing the value of such links.

■ When working with fractions, one teacher madeconnections with areas of rectangles which werenot trivial, i.e. which encouraged more thoughtabout both.

■ When students found themselves stuck, theteacher asked: ‘how did you get out of a similarsituation before?’ thus indicating they could makelinks based on similarities in the situations, andworking on their resourcefulness by modellinghow to deal with being stuck.

Being explicit about connections and differences

Teachers draw students’ attention to featuresof mathematics which are of a more general

nature than the current topic. The teachers do notexpect students to know what is worth noticing, sothey incorporate comments about good mathematicalhabits as they arise. They are sometimes very explicitabout features, ways of thinking, and relationshipswhich a stronger student might notice automatically.Some of the following examples are about learning tomake connections in their work and ways of working;some are about learning to discern subtle differencesand features of mathematics. Some of them may seemobvious, but being explicit about some of theconnections and distinctions we take for granted canbe very useful for students who have previously notbeen making the connections their teachers expected.These processes can be applied within or betweenmathematical tasks, topics and experiences.

Within tasks:■ Students check validity of statements for

themselves, so that they learn how statements andconjectures relate to examples they invent.

■ Using a particular layout of a calculation on theboard, the teacher changes the numbers beingused and says: ‘the information I am giving youhas changed’, thus indicating that students oughtto pay attention to differences.

■ Students are asked: what is changing and what isstaying the same?

■ When working with the perimeter of shapes madeup of four abutting squares, the teacher askedthem to consider the effect of each arrangement onthe perimeter, thus getting them to connectvariables, and also inducing a pre-conjecture stateof understanding that the variables might be related.

■ Asking students to articulate different methodsfor finding one answer, a common practice, theteacher then gets them to evaluate and discuss thepower of each method.

Between tasks:■ When multiplying positive and negative integers,

explicit connections were made to students’experience with natural numbers.

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Students can learn rigorous mathematics if therelevant kinds of rigour are discussed explicitly.Students can learn complex mathematics if thereis discussion and other forms of verbalisation tohelp them sort out the complexity.

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■ Students were encouraged to give detailedanswers about what they saw in diagrams oralgebraic expressions.

Dealing with mathematical complexity

Project teachers did not simplify mathematicsfor students, nor did they fudge difficult issues.

Instead they saw their job as helping students learnmathematics, with all its complexities. As well asoffering ways to deal with complexity inmathematics, each of the tactics which follow is alsoan example of learning to think in appropriate waysfor mathematics:

■ When working algebraically, students wereencouraged to use a wide range of letters, not justthe usual ones, to signify unknowns.

■ All teachers discussed similarities and differencesbetween mathematical objects and examples, thusmodelling the processes of classifying whichwould help students make sense of variety inmathematics.

■ Typical points of confusion, such as area andperimeter, were tackled head on, with explorationof shapes which increase area while reducingperimeter, and vice versa.

■ Suitable levels of accuracy were discussed andstudents given choice about which to use andwhen.

Students had been working on ‘squeezing anumber between 64 and 88, between 0.007 and0.009, between 7.09 and 7.11’.

Teacher: A lot of you would like to have youranswers checked.

Students, shouting: Me, me, me

Teacher: Shall I read out the answers?

[pause]

How many answers are there?

Students: Millions

Teacher: So can I read them out?

■ Proper terminology was generally used, withstudents given time to learn how to use it.

■ Students were asked to consider whetherstatements were always, sometimes or never true.

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discussion about properties of regular polygons,but could equally well be used to remindstudents that fractions have to be equal parts.

D. Students each have a pack containing 10 cut-outcongruent squares which they made in a previouslesson. All students have explored how to getmaximum and minimum possible perimeterswhen four squares are arranged edge to edge invarious configurations. Results are discussed.They are then asked to create their ownquestions as directions for future explorations.These ideas are shared so they can abandon theirown question and choose another if they like.

Comments: The reasoning processes they haveexperienced with the 4 squares, supported bywhole class discussion, can be re-used in caseswith more or fewer squares, thus they will get asense of progress as the reasoning will be easiersecond time around. They can monitor theirown progress by knowing if they have answeredtheir next question or not. The frequent use of

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C. A new topic is introduced part-way through alesson: polygons. Students discuss what theyknow about names and properties of polygonsand all these ‘facts’ are written on the board.They then draw two separate circles usingcompasses. The homework is to think of ways tocut their circles (pies) into various numbers ofequal pieces and to think about ‘how can I besure my circle is cut into equal pieces?’

Comments: The pooled knowledge provides thestarting point for new work; through this theteacher assesses what they know and also provideseveryone with a place to start thinking. Thehomework is open-ended and will generate theraw material for the next lesson. The task isposed as ‘being sure’; thus it is about reasoning,rather than just doing what is asked. The circle-drawing creates a link between topics, and thepotential link is not obvious from this lesson.Rather than make this explicit, the teacher retainsan element of surprise. In the next lesson, thesectored circles will be used to generate

The aims of all these tasks are complex. Themathematics is not simplified; there is no sense

of ‘finish’ in the tasks, since they are stated in wayswhich require extended thought; there are supportingprops in place (chocolate, materials, discussion); theyall involve reasoning of some kind; they all containpersonal challenge. If these learners react differentlyto others, it is that it takes them more time and theymay need more support because they need to developresilience, resourcefulness, language competence ormemory.

A. Teacher writes numbers on the board: 21, 4, 1, 5,5, 3 and the aim is to get 21 by using all thenumbers. This is to be worked on at home, but isvoluntary. Taking part earns a small piece ofchocolate, finding a lot of ways to do it earns alarger piece of chocolate. There are some ideasgenerated in class, mentally, before they move onto other work.

Comments: Chocolate is used as a playful reward.The underlying message is that home work shouldbe done, but that students who have no useful

work habits, and no self-esteem, need coaxingback in through creating a no-risk atmosphere.Thus help is generated in the lesson so all canunderstand what to do. The aim is not to findone method, but to find as many as you can; thuscontinued, extended exploration is beingencouraged.

B. After reminding students about different kinds ofaverage, they are given multilink to build towers.The aim is to build a family of towers whichtogether have given averages, such as a mean of 4,a median of 3, a range of 5 and so on. Familieswhich work are recorded on a worksheet.

Comments: Students are reminded that it isimportant to recall different averages. The taskencourages exploration of how averages are madeup, and awareness of the variation possible. Itfocuses on structure of averages, rather thanmethods of calculation. The worksheet framesthe writing so that recording is not problematic,nor is time taken away from thinking by writing,drawing up tables etc.

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Typical task types

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ones out’ from a collection of shapes, accordingto some constraints. After this there is discussionabout what was found in previous lessons, and theycreate their own questions to continue exploring.At the end, students share findings and use OHP toshow what they have found.

Lesson C: Students start with a number challenge;students volunteer to write their suggestions onthe board which have been worked out mentally.About half way through the lesson the new topicis started: data-handling. They brainstormeverything they recall about it. They have to eachinvent five questions to ask people and the wholeclass discusses potential difficulties with collectingdata to answer the questions. Almost the wholelesson is whole class discussion.

Lesson D: Students are given a worksheet and materialsto cut out and manipulate to find reflectionswithout tracing paper or mirrors. They make uptwo questions to be answered by their partner.Their work is checked by constructing the ‘same’reflections using tracing paper. There is no wholeclass discussion.

Structures of lessons

Here we give some examples of lessonstructures to show the variety used by Project

teachers. There was no uniform lesson style, althoughall contained some discussion and sharing, not always asa whole group, some mental work and some reflectionon previous work. Contrary to a common belief thatlow attaining students and ‘difficult’ classes needfrequent changes of activity, these teachers worked ondeveloping sustained activity and opportunities forthinking about what they are doing.

Lesson A: Aural questions used as a settling device,students work in silence on their own; followedby discussion of answers and methods in whichthey mark their own work. Then there is an activitywhich starts with the teacher demonstrating andorchestrating discussion about possible ways totackle tasks. Students work on a worksheet whichincludes being asked to make up their ownquestions to tackle. The ‘own questions’ tasks act asa reflective device; there is no final whole class time.

Lesson B: Students continue work started in aprevious lesson, writing down their findings onsmall whiteboards to share; they are finding ‘odd

the word ‘perimeter’ is important, and varyingthe number of squares allows variation in area.The availability of more squares encourages thischoice. The personal packs create ‘ownership’,familiarity and memory.

E. Students are presented with pyramids, in whichthey are introduced to a structure which has therule ‘add the contents of two adjacent cells to getthe content of the cell below’.

In this class the task is varied so that:

7 See Journey into Maths by Alan Bell, David Rooke and Alan Wigley. Published byShell Centre (1978 &1979) www.mathshell.com

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(a) they have to fill in missing cells from lower downthe pyramid, given the upper cells.

(b) some cells have algebraic entries.

They are then given a work sheet to do, the finalchallenge being to make up some of their own.

Comments: This familiar task7 has not beensimplified for lower attaining students. They get acomplex version which requires symbolicmanipulation, reasoning, ‘working backwards’from answer to question, and also have to reflecton what they have done by creating their ownquestions. Thus they have been treated asthinking beings who can deal with complexity.

22 7 13

29 20

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opportunity to work on reasoning and higherlevels of abstraction.

Teacher (having written Wayne’s idea on the board):Can I add something or change something to make thatstatement true?

Teacher (sorting a pile of statements aboutquadrilaterals): Are there any statements which do notwork together?

Teacher (looking at several decimal equivalents ofunit fractions): What is the same and what is differentabout these?

Sustaining work on one topic over a period oftime promotes deep progress, awareness of progressand hence self-esteem and a sense of being alearner of mathematics. Concentration andparticipation enable tasks to be extended, throughlearners being actively engaged in thought. Therelationship between these aspects is complex andnon-linear; one does not guarantee another.There is no such thing as ‘a guaranteedextendable task’. The extension is created by theclass, the task, the teacher and the learning.

through accumulated experiences and givesstudents a sense of the range of possibilities in atopic. Some students had completed some workon ratios of two quantities successfully. They werethen offered similar work (demonstrations,practical tasks, exercises) on ratios of threequantities. This helped them become more fluentwith two, and also gain more understanding aboutratio, and then some asked if they could do four!

■ ‘Learn something new’ and ‘learn as much as youcan’ are goals for lessons.

■ The opportunity for students to make a personaltoolkit for a topic, so that opening the toolkit atthe start of each lesson triggers memory for whatwas done before, also provides raw material fornew questions. Some students compiled their ownlist of useful percentages and how to work themout; looking at the list suggested more whichcould be found, so students developed their own‘harder’ questions.

■ All situations in which there are variousmathematical objects or examples produced, orstatements such as facts or conjectures, provided

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All teachers found themselves working to sustainstudents’ interest in a topic over time, not

varying the task and topic frequently, but encouragingdeeper thought in a variety of ways. This often workedagainst normal practice of providing students whofound it hard to concentrate with frequent changes andnew tasks, or changing topic frequently to avoidboredom. Some of the ways this was achieved were:

■ When a student has ‘finished’, the teacher showshow further questions can be posed about thesame problem situation.

■ Teachers ask ‘why?’ and ‘what if?’ to encouragereasoning and justification, or further explorationof changes of different variables. This also modelsthe kinds of deep questioning which students cando for themselves in future.

■ When students are asked to create their ownexamples, they are encouraged to make up reallyhard ones which might present new challenges towork on. Some students who had been making uptheir own scalars for enlargement activities werethen asked to explore what would happen if theyused negative scalars.

Extending tasks

■ All hypotheses are followed with ‘why?’

■ Further changes can be made to a problemsituation, allowing students to apply what theyhave learnt so far. Having found out that whentwo lines cross you get two pairs of equal angles,which are supplementary, students are then givena third line and asked to find out about the anglesyou get if two of the lines are parallel.

■ Reflection on worked examples which studentshave completed, asking them to say which wereeasy and which were hard, and why, makes thewhole exercise the focus of study rather than thetechnique on its own.

■ Giving something familiar, in an unfamiliar form,helps make connections as well as doing the task.Some students are given a 100 square, but in adifferent lay-out. It is in a ‘snake’ shape, with 100in the top left corner and 0 in the bottom righthand corner. The teacher’s aim is to use students’sense of comfort from using the familiar 100-square to think afresh about number relationships.

■ Offering repetition for familiarity, but withsignificant variations each time, achieves fluency

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◆ expose what is already known,

◆ use this knowledge,

◆ use a range of representations,

◆ provide opportunities for mental andmathematical challenge,

◆ include markers for progress

■ Interacting with students in ways which sustaineffort, which sustain challenge, and yet whichbuild self-esteem.

■ Creating atmosphere and interactions whichimmerse students and teachers in mathematics.

They needed to be brave because some of themwere working in ways which contradicted normal

practices and expectations. They also needed to bebrave because the way they ‘covered’ content withtheir students did not conform to an easily-observedlist of topics. Working with others over time, as youcan do in a research project or professionaldevelopment course, helps.

They needed to be clever in :

■ Finding and creating time to spend on whatmatters.

■ Finding and creating tasks which:

What did the teachers need?

Some of the teachers used the same methods with all the classes they taught from the start.

Others found that methods they developed during theProject with students who had previously lowestattainment worked well with other classes too. At theend, we wondered what was different about teachingthe lowest attaining adolescents.

Most teachers claimed they did not teach differently.They had the same expectations in terms of taskcomplexity, effort and behaviour in all their classes.Some teachers used the same tasks with several classes,including higher sets.

Outside observers would say that they did teachdifferently, because they incorporated into their practicedeliberate strategies to change habits. This took time.It took students time to realise that they need to adoptnew habits, and that they will benefit from them.Teachers and students took time to find out what wasalready known, and to establish understanding beforemoving on.

Project teachers saw short-termism operating in generalin the education system, typified by:

■ Curriculum coverage needing to be completedbefore the next test;

Differences in practice for different classes?

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■ Students expected to display good work habitsimmediately and punished if they do not;

■ Moving rapidly from one task to another ifconcentration wanes;

■ Focus on finishing work, either in class or athome.

Teachers and learners in the Project were working to adifferent timescale. Short-termism had beenabandoned and long-termism was put in its place.

■ Longer was spent on each topic than wasrecommended nationally, but content coveragewas still important. Coverage without under-standing and memory was pointless - time was spenton understanding and memory.

■ Longer was spent on establishing good workhabits and undoing previously developed habits. Ifthis took a lot of lesson time, so be it.

■ Longer was spent on thinking and on particulartasks, to establish participation, reasoning,understanding and connectedness.

■ Focus on learning as much as possible, rather thanfinishing tasks.

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requiring application of techniquesand knowledge in an unusual way, ormulti-stage reasoning, students in theProject groups did significantly betterthan those in the comparison groups.

Progress in tackling extendedexplorations and reasoning tasksStudents in the Project groups of thesecond year of the Project were given amathematical situation to explore at thestart of the year9. Information wasgathered about the way this lesson went,the teachers’ view of it, and thestudents’ work. Work was graded usingtypical GCSE criteria for such work10.This task was repeated at the end of theyear to see if students were more willingto engage in extended mathematicalwork, and showed more interest andinitiative11. In all but a few individualstudents it was found that:

Comparative dataFive teachers provided data whichcompared their Year 7 test results to thosefrom comparison classes. The classeswere chosen from within their ownschools, and were not always parallelclasses in the sense that they had similardistributions of level scores at entry tosecondary school. Because of the numberof possible variables, and the small groupsizes, we collated all the data from Projectstudents rather than deal with individualschools8.

1. In general test performance, Projectstudents did roughly the same ascomparison classes who had followed amore procedural, coverage-based,approach to the curriculum, using officialguidelines. (In some schools they didbetter).

2. In questions which focused on mathematical thinking, such as

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8 One reason for this was toavoid having to take account ofthe wide range of factorsdescribing the different kinds ofcomparison group: size,distribution of entry grades,experience of teacher and so on.

9 The task used was taken fromOllerton (2002)

10 The actual criteria used werethose developed by theAssociation of Teachers ofMathematics for 100%coursework GCSE in 1988-1994.They are more detailed thanthose used by major examinationboards, but fulfil the samestandards.

11 We used the same taskbecause responses to open-ended, unstructured, tasks arealways highly specific, and wewanted to compare before andafter responses as closely aspossible. The task providedpotential for variation in the way itwas explored, and there was littlelikelihood that students wouldrecall previous answers duringthe post-test. However, if theydid, then the marking criteria tookinto account repetitions.

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How did the teachers know their students were‘doing better’? They said:

Working on understanding has meant pupils notonly remember the activities but also methods andare transferring knowledge to exam-based work.Time spent earlier on understanding is worthwhile.Pupils I still have contact with are transferringwhat they learnt the previous year into this year’swork.

I see pupils gaining in confidence in maths andbeing willing to tackle more complex problems.

My pupils enjoy the lessons on the whole and areencouraged to think mathematically. They areengaged and motivated. They seem to have a deeperunderstanding of the concepts and think aboutapplying these in different structures. They areprepared to try. The test results are better than thoseof parallel groups.

They are developing greater understanding thanjust through practising a few examples beforemoving to another topic. They ask their ownquestions; there is more discussion.

Students are tackling more extended tasks and givemore thought to activities. Most students show apositive approach to their work. They seem to havemore self-esteem. The discussions allow the students toexpress ideas orally rather than always having to writeanswers, which can be a barrier for some of them.

But are teachers’ reports of ‘feeling better’ enough?We also looked at test results, progress in tackling anopen-ended task, and observation data.

Test and task resultsEach teacher had a different view about the value oftests, and the advantages and disadvantages of them,so we could not set up tests before and after theProject as there would have been no agreementsabout their style or even existence! Instead we useda range of kinds of test and task data as they becameavailable. This means that we cannot makestatements about results for all teachers and allstudents. However, here is a summary of the resultswe were able to collate without upsetting the normalpractices of the teachers:

Were these methods successful?

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Unprompted

Choosing appropriate techniques

Contribute examples

Describing connections with prior knowledge

Finding similarities or differences beyondsuperficial appearance

Generalising structure from diagrams or examples

Identifying what can be changed

Making something more difficult

Making extra kinds of comparison

Generating own enquiry

Predicting problems

Giving reasons

Asking to spend more time extending a task

Creating own methods and shortcuts

Using prior knowledge

Initiating a mathematical idea

Changing their mind with new experiences

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Prompted

Choosing appropriate techniques

Contribute examples

Describing connections with prior knowledge

Finding similarities or differences beyond superficialappearance

Generalising structure from diagrams or examples

Identifying what can be changed

Making something more difficult

Making comparisons

Posing own questions

Predicting problems

Giving reasons

Working on extended tasks over time

Sharing own methods

Using prior knowledge

Dealing with unfamiliar problems

Unprompted

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Students showed considerable improvement in theirwillingness to work on the task, and produced moreideas, more lines of enquiry and were prepared totake more risks. Achievement was significantlyhigher at the end of the year.

This means that the students who might normally beexpected to become more disaffected and reluctantto work were instead becoming more willing to workand more engaged. They showed significantimprovement in the use of thinking skills in this task.

In addition, teachers reported more enthusiasmamong students, and one class who had started theyear noisily rejecting the task asked for it to extendover several lessons when it re-appeared at the end ofthe year.

Observation data about mathematical thinkingAlthough we had not managed to agree a definitionof mathematical thinking, we all agreed that thefollowing ways of thinking and working wereimportant in mathematics, and that the studentsshould be given opportunities to use them. Oneexciting result of the project was that students beganto use these ways of thinking voluntarily elsewhere intheir work, not just when teachers asked them to.We are not claiming that all students in all classes didall these things, but that among these low attainingstudents we saw several manifestations ofsophisticated mathematical thinking, both promptedand unprompted:

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Technical information about the research

The participants in this Project met at leasttwice a term to discuss what we felt was

important, what our beliefs and intentions were, howwe could construct tasks and teaching situationswhich enabled students to achieve in mathematics,and to share good ideas and practices which worked.Some of the teachers began to co-plan lessons andothers found that the support of the groupencouraged them to stick with changes they weremaking long enough to see results, whereas on theirown they may have given up.

The research team did not tell teachers what to do,nor provide instructions about good tasks or goodways to organise teaching. Instead, teachers andresearchers all put ideas into the discussions. Whatteachers did in classrooms was, therefore, their ownideas or adaptations and adoptions of ideas fromothers. The researchers’ job was to provide support,ideas from experience and research literature,encouragement, and to collate data about practice.All teachers were provided with copies of recent

books which contained appropriate classroom ideasfrom which to choose. Data were constantly reviewedand issues for discussion selected from it, so thatsimilarities and differences between practices could beidentified, aired, shared and discussed. The followingkinds of data were collected and analysed:

■ Records of meetings held one day a term for twoyears.

■ Records of meetings in twilight time.

■ Records of visits to schools and lesson observationby research officer, Els De Geest.

■ Records of visits to schools by other academics,Anne Watson and Steph Prestage.

■ Teachers’ written notes about lesson plans andevaluations.

■ Videos of lessons.

■ Tape-recording and written notes of discussionssent to all Project members.

Students do at least as well as those in comparisongroups on standard tests. Students do better thancomparison groups on questions which involvemathematical thinking. Students’ willingnessand ability to take risks, and to engage withcomplex, open, extended tasks improvessignificantly.

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These results repeat what has been found in severalprevious studies in which students are taughtmathematics through a focus on thinking andunderstanding rather than focusing solely oncoverage of content. However, previous studies havenot focused specifically on the lowest achievingstudents, and they tend to focus on teachers who areusing certain ‘methods’ or ‘materials’. Thus thisProject shows that teachers who are free to innovatefor themselves are able to improve the attainment,engagement and mathematical thinking of low-achieving students. Their students were notdisadvantaged in the short-term, their test resultscompared well to other groups, and there weresignificant gains in participation, self-esteem, andtheir ability and willingness to engage with extended,unfamiliar and complex tasks.

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Ahmed, A. (Low Attainers in Mathematics Project)(1987). Better Mathematics. London: HMSO.

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Black, P., Harrison, C., Lee, C. Marshall, B.,Wiliam, D. (2002) Working inside the black box:assessment for learning in the classroom: London:King’s College.

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De Geest, E., Watson, A. and Prestage, S. (2003)‘Thinking in ordinary lessons: what happened whennine teachers believed their failing students could thinkmathematically’ in Proceedings of the 27thConference of the International Group for the

Psychology in Mathematics Education Conference,Vol.2, 301-308.

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Bibliography and resources

■ Written work relating to ‘before’ and ‘after’ tasks

■ SATs, Progress and Optional tests whereappropriate, including some from comparisongroups.

■ Occasional written statements by teachers aboutbeliefs and responses to particular issues.

■ Some students’ normal written work, includingself-evaluation and tests where these were done.

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© 2003 Anne Watson, Els De Geest, Stephanie PrestageAll rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known orhereafter invented, including photocopying and recording, or in any information storage or retrieval system, without the permission in writing from thepublishers.

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Skemp, R.R. (1976) Relational Understanding andInstrumental Understanding. MathematicsTeaching, No 77, 20-26.

Watson, A. and Mason, J. (1998). Questions andPrompts for Mathematical Thinking, Derby:Association of Teachers of Mathematics.

Watson, A., De Geest, E. and Prestage, S. (2002)Improving Attainment in Mathematics Project part 2:Developing Ways of Being Mathematical. Paperpresented at annual conference of the BritishEducational Research Association, University ofExeter, 2002.

Ollerton, M. (2002) Learning and teachingmathematics without a textbook. Derby: Associationof Teachers of Mathematics.

Ollerton, M. and Watson, A. (2001) InclusiveMathematics 11-18. London: Continuum.

Prestage, S. and Perks, P. (2001) Adapting andextending secondary mathematics activities: new tasksfor old. London: David Fulton.

Prestage, S., Watson, A. and De Geest, E. (2002)Improving Attainment in Mathematics Project part 1:Defining the Context. Paper presented at annualconference of the British Educational ResearchAssociation, University of Exeter, 2002.

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