effective teaching and learning numeracy

NumeracyEFFECTIVE TEACHINGAND LEARNING

SUMMARY REPORT

Diana Coben, Margaret Brown

Valerie Rhodes, Jon Swain, Katerina Ananiadou

Peter Brown, Jackie Ashton, Deborah Holder, Sandra Lowe

Cathy Magee, Sue Nieduszynska and Veronica Storey

Numeracy Short LIVE WEDNESDAY 20/1/07 17:19

Published by the National Research and

Development Centre for Adult Literacy and

Numeracy

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5 Introduction

6 The effective practice studies

7 Main findings

9 Recommendations

10 Background to the study

13 The learners

19 The teachers

23 How practice and progress are linked

29 Conclusions

30 References

NumeracySUMMARY REPORT

RESEARCH TEAM

Principal investigators

Diana Coben, Margaret Brown

Researchers

Valerie Rhodes, Jon Swain, Katerina Ananiadou

Researcher and administrative support

Peter Brown

Teacher researchers

Jackie Ashton, Deborah Holder, Sandra Lowe

Cathy Magee, Sue Nieduszynska and Veronica Storey

SERIES EDITOR

John Vorhaus

EFFECTIVE TEACHING AND LEARNING


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■ The Skills for Life Strategy has led tounprecedented investment in adultliteracy, language and numeracy (LLN),major reforms of teacher education andtraining, and the introduction of corecurricula and national standards inteaching and learning. We have a uniqueopportunity to make a step change inimproving levels of adult skills. But untilrecently too little was known abouteffective teaching and learning practices,and reports from Ofsted and the AdultLearning Inspectorate repeatedly drewattention to the quality of teaching, andthe need for standards to improve.

It has been a strategic priority at theNational Research and DevelopmentCentre for Adult Literacy and Numeracy(NRDC) to investigate teaching andlearning practices in all the subject areasand settings in Skills for Life: to report onthe most promising and effectivepractices, and to provide teachers andtrainers, along with policy-makers andresearchers, with an unparalleledevidence base on which to build on theprogress already made.

Our findings and recommendations arereported here, and in the four companionreports covering reading, writing, ESOL

and ICT. The five studies, which havebeen co-ordinated by NRDC AssociateDirector John Vorhaus, provide materialfor improving the quality of teaching andlearning, and for informingdevelopments in initial teacher educationand continuing professional development(CPD). We are also preparing a range ofpractitioner guides and developmentmaterials, as a major new resource forteachers and teacher educators. Theywill explore and develop the examples ofgood and promising practicedocumented in these pages.

Until recently adult numeracy wasunder-researched and underdeveloped,and it was often not distinguished fromliteracy in policy documents andinspection reports. However, the profileof numeracy has been steadily rising,following confirmation by national andinternational surveys of low levels of skillamongst the adult population. This studyis the largest undertaken into adultnumeracy in the UK, and it represents asubstantial advance in our understandingof the practices that contribute tosuccessful teaching and learning.

Ursula Howard, Director, NRDC


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Introduction


■ The NRDC’s five Effective PracticeStudies explore teaching and learning inreading, writing, numeracy, English forSpeakers of Other Languages (ESOL)and information and computertechnologies (ICT), and they set out toanswer two questions: • how can teaching, learning and

assessing literacy, numeracy, ESOLand ICT be improved?

• which factors contribute to successfullearning?

Even before NRDC was set up it wasapparent from reviews of the field thatthere was little reliable research-basedevidence to answer these questions.Various NRDC reviews showed thatprogress in amassing such evidence,though welcome where it was occurring,was slow. Four preliminary studies onreading, writing, ESOL and ICT wereundertaken between 2002 and 2004.However, we recognised the urgent needto build on these in order greatly toincrease the research base for thepractice of teaching these subjects.

The inspiration for the design of the fiveprojects was a study in the United Statesof the teaching of literacy and Englishlanguage to adult learners for whomEnglish is an additional language(Condelli et al., 2003). This study was thefirst of its kind, and the lead author,

Larry Condelli, of the AmericanInstitutes for Research, has acted as anexpert adviser on all five NRDC projects.

The numeracy team’s research began inAugust 2003 and was completed inMarch 2006. We investigated approachesto the teaching of numeracy, aiming toidentify the extent of learners’ progress,and to establish correlations betweenthis progress and the strategies andpractices used by teachers. The studyinvolved 412 learners and 34 teachers in47 classes. Two-thirds of the classeswere in further education colleges; theaverage teaching session was just undertwo hours and average attendance inclass was eight learners. In all, 250learners were assessed on theirmathematical understanding and 243completed attitude surveys. Thisoccurred at two time points in order toassess progress. Classes were observedbetween one and four times during eachcourse. Background information wascollected on teachers and learners, andwe carried out interviews with 33teachers and 112 learners.

The ICT study differed from the others inthat its first phase was developmental,its sample size was smaller, and it had ashorter timescale, completing in March2005.

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The Effective Practice Studies



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Progress

We found evidence of significantprogress, with an average gain of 9 percent in test scores, although there wasa wide range of average gains betweendifferent classes.

Learners’ attitudes were more positiveat the end of the course, with thechanges tending to be greatest for olderpeople.

Once learners overcome initial anxietyabout the course and aboutmathematics, numeracy courses canhave a significant and positive effect ontheir identities. They can improveconfidence and self-esteem, and enablelearners to develop new aspirations andform new dispositions towards learning.

For some learners, to maintain theirlevel of skills, knowledge andunderstanding is a sign of personalprogress.

Time to learn

Evidence from the National Center forthe Study of Adult Learning and Literacy(NCSALL) in the US suggests thatlearners require between 150 and 200hours of study if they are to progress byone level within the Skills for Lifequalification framework. However,although average attendance by

learners between our first and secondassessments was only 39 hours, wefound that many had made significantprogress. Others needed longer toconsolidate their learning.

Teaching strategies

Teachers valued ‘flexibility’ as a keyfeature of effective practice. Thediversity of learners, contexts andsession lengths meant that no onepattern of lesson activity appeared to beoptimal.

A wide range of teaching approacheswas observed, although whole class andindividual work predominated.

Most teachers gave clear explanations,which were much valued by learners.They also broke work down into smallersteps and gave feedback to learnersabout their work.

Most teachers followed a set scheme ofwork, and few incorporated learners’personal interests. It was also lessusual for teachers to differentiate work,make connections to other areas ofmathematics, or ask higher-orderquestions to encourage higher-levelthinking or probe learners’misconceptions.

Although activities were often varied,

Main findings


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there was little use of practicalresources or ICT, little group orcollaborative work, and it was unusualto find learners collaborating with, andlearning from, each other.

Teaching and learning relationshipsOver 90 per cent of learners interviewedexpressed a high level of satisfactionwith their course and their teacher.Learners were usually highly engaged.They were often, but not always,challenged and stretched; they weregenerally given time to gainunderstanding, and the majority hadtheir individual needs met.

Learners recognised that therelationship between the teacher andeffective learning was critical. It wasimportant for teachers to develop goodrelationships with learners and to treat,and respect, them as adults. Classroomobservation indicated that teacherswere enthusiastic and generous ingiving praise, and there was a high levelof mutual respect.

Teachers’ qualificationsThe teachers were generallyexperienced and well-qualified, withmany having previously taughtmathematics in primary and/orsecondary schools. Teachers’ subjectknowledge was generally adequate.

Twenty-seven (79 per cent) of the 34teachers reported having a formalqualification in mathematics or arelated subject (e.g. science). Thirtyteachers (88 per cent) said they had ateaching qualification and six (18 percent) reported having a subject-specificLevel 4 qualification for teachingnumeracy to adults.

It is often assumed that individualsholding high qualifications inmathematics are able to teach basicconcepts at lower levels ofmathematics. We did not always findevidence of this. Some teachers reliedon methods they had been taught atschool.

Classroom observationindicated that teacherswere enthusiastic andgenerous in giving praise,and there was a high levelof mutual respect



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Development work and quality

improvement

In all teacher education programmesfor adult numeracy, there should be arequirement for teachers to have a firmunderstanding of basic concepts suchas place-value, multiplication anddivision.

Teachers need a firm grasp of subjectand pedagogical knowledge, and alsosubject-specific pedagogical knowledge.This enables them to be flexible in theirapproaches, and to cater to the diversityof learners and provision in adultnumeracy.

Policy

Adult numeracy education should beseen as part of mathematics education,and as a discrete subject in relation toadult literacy and other Skills for Lifeareas. This should be reflected in policydocuments and in the organisation andinspection of provision, so that, forexample, adult numeracy provision iseffectively co-ordinated with othermathematics provision offered bycolleges and other organisations.

Research

Further research and developmentshould be undertaken into learnerassessment in numeracy at Skills forLife levels with a view to developing an

appropriate assessment instrument forresearch purposes. More sensitivitywould be achieved if an instrumentwere designed to focus on a narrowrange of initial attainment: for example,Entry Levels 1 and 2 or Entry Level 3and Level 1.

A bank of secure, reliable and validquestions should be available to matchassessment questions to individualteaching programmes, and therefore toprovide a more genuine test of learningin relation to teaching.

More research is required to explorelearner and teacher identities.Learners’ identities affect attitudes,motivations, dispositions towardsmathematics and education in general,relations with peers and teachers, andfuture expectations and aspirations.Teacher identities also matter: we needto know how much personal investmentteachers make both as numeracyteachers and as people who areknowledgeable about numeracy andmathematics.

Recommendations


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■ This study is set within the context ofthe Government’s Skills for Life Strategyto improve adult literacy and numeracyin England, and took place against abackdrop of policy changes in adultnumeracy education, post-14mathematics education and training,initial teacher education and concernsabout skills levels among adults.

Skills for Life defined numeracy as theability ’to use mathematics at a levelnecessary to function at work and insociety in general’. The strategy’s targetis for 1.5 million adults to improve theirliteracy and numeracy skills by 2007. TheSkills for Life Survey commissioned bythe Department for Education and Skills(DfES) in 2003 suggested that nearly halfof all adults of working age in England(15 million) were at or below the levelexpected of an average 11-year-old innumeracy. At the same time, a disturbingpicture of adult numeracy educationbegan to emerge, with a shortage ofexperienced teachers and teachertrainers.

What counts as effective practice in adultnumeracy education in this context isboth complex and straightforward. It isstraightforward in so far as adultnumeracy provision is inspectedaccording to standards set out in theCommon Inspection Framework by the

Office for Standards in Education(OFSTED) and the Adult LearningInspectorate (ALI). However, complexityarises as the relationship betweeneffective teaching and successfullearning in adult numeracy has yet to beestablished. Our study represents a steptowards this goal.

The project was in two phases, in2003/04 and 2004/05. The target was torecruit a minimum of 250 learners,assess their attainment and attitudes attwo points during the year in which theywere in the study, interview bothlearners and teachers, observe thestrategies their teachers used, andcorrelate these strategies with changesin learners’ attainment and attitudes.

The research team consisted of theproject directors, professionalresearchers, and six teacher-researchers.

Our sample and methods

Adult numeracy tuition is diverse interms of the range of provision, settings,teachers and the different purposes oflearners. It is offered both as a discretesubject and ‘embedded’ in other subjectsand vocational areas. Reliable data onthe adult numeracy teaching workforceare unavailable, but it is likely that suchteachers vary in their experience ofteaching adults in different contexts,

Background to the study



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their knowledge of mathematics andnumeracy/mathematics pedagogy, andtheir teaching qualifications. We aimed toreflect the diversity of numeracyprovision, and the range of adultlearners, who include growing numbersof non-traditional learners and 16 to 19-year-olds.

The research was undertaken in learningcontexts throughout England, includingadult numeracy, Return to Employment,Foundation ICT, family numeracy, GCSE,workplace-based groups, JobcentrePlus, a prison and a ’vocational taster’numeracy course for young people withlearning difficulties, in both day andevening classes. Providers included FEcolleges, a neighbourhood college, acommunity group, the Army, a prison, alocal education authority (LEA) and aprivate training provider.

Research sites were sought throughadvertising but when this produced onlyone result, this was supplemented bysites found through professionalcontacts. As a result, sites wereclustered near to the researchers innorth Lancashire, Gloucestershire andLondon, with additional sites in Kent,Cambridgeshire and the South-west.

Our sample was thus neither randomnor fully comprehensive andrepresentative. However, settingsbroadly reflected the range availablenationally and the proportion of learnersin them. We hope our sample also maybe reasonably representative of theteaching workforce, although since all

teachers were in a sense volunteers,there may be some bias towards thosewho are more effective.

A total of 412 learners participated in thestudy, and we observed 34 teachers and47 classes; 17 in Phase 1 (2003/04) and30 in Phase 2 (2004/05). Thirty-one ofthese classes were in FE colleges, four inadult/neighbourhood colleges, two infamily numeracy, four in workplaces, twoJobcentre Plus, one Army trainingcourse, two in prisons and one privatetraining provider. Class sizes rangedfrom one to 23 learners, with an averagesize of eight. A minority of the classesobserved (28 per cent) had a learningsupport assistant or volunteer. Mostwere daytime classes and lastedbetween one and three hours.

Phase 1 was used to develop ourresearch instruments, which we trialledextensively. In both phases 1 and 2, weassessed learners at the beginning, Time1 (T1), and near the end, Time 2 (T2), oftheir learning programmes. We observedteaching sessions, surveyed learners’attitudes to numeracy and interviewednearly every teacher and a sample oflearners. We also gathered backgroundinformation on all learners and teachers.

A total of 250 learners took theassessment at both T1 and T2, and 243completed the attitude survey at bothtimes. As well as providing quantitativedata, the project team interviewed 112learners and 33 teachers to gain insightsinto effective practice.


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A typical

assessment

item

A ■■ £1.18 B ■■ £0.82 C ■■ £0.80 D ■■ £0.59

If I buy 2 jars of jam, how much change would I get from £2.00?

59p 59p

What is distinctive about this study?

Two issues proved particularlyproblematic in this study: encompassingthe diversity of provision and teaching,and measuring learners’ progress.

Adult numeracy education takes variousforms, occurs in various contexts, andhas a wide range of teachers andlearners. It is extremely heterogeneousby comparison, not only with numeracyand mathematics education in schools,but with other Skills for Life areas. Thereis also a wider issue of the difficulty ofsimulating in classrooms the situations inwhich mathematics occurs elsewhere.Hence an assessment might not give aclear indication of an individual’sstrengths and weaknesses whenconfronted with mathematics outside theclassroom. These factors make it difficultto produce generic research instrumentsable to encompass the full range oflearners, teachers and forms of provision,or to draw conclusions that can begeneralised across the whole sector.

Measuring learners’ progressWe reviewed several standardised tests

before deciding on a modified version ofthat used in the Skills for Life Survey. Theassessment covered a range ofcurriculum areas and difficulty levels,from Entry Level 1 to Level 2. The needfor our assessment instrument to bepracticable – and not take up too muchclass time – was also problematic, andwe therefore chose a written test thatcould usually be completed in about 30minutes and be administered in onesitting. The 20 items were all multiple-choice and incorporated photographsand diagrams, with only simple text. Anexample of an Entry Level 2 question isshown (above).

Researchers were able to read or explainthe meaning of questions for ESOLlearners or those with languagedifficulties. Calculators were available butwere seldom used. However, inretrospect, the assessment instrumentwas insufficiently sensitive with respectto the range of learners in the study. Itparticularly lacked validity with learnersat or below Entry Level 1, those withlearning difficulties, and learners whosereading or command of English was poor.



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■ The learners were fairly equallygender-balanced. They werepredominantly in the younger agegroups, with 40 per cent between theages of 16 and 19. In a questionnaire,more than 40 per cent of learnersreported their ethnic group as whiteBritish, with the second largest groupbeing Bangladeshi. Almost three out offive learners in the sample said Englishwas their first language. Forty per centof the sample were in full-timeeducation, and approximately 15 percent were employed full-time. Theaverage age at which learners leftschool was 16, and almost 40 per centalready held at least one maths ornumeracy qualification. Around one in10 reported being permanently sick ordisabled. Nearly a quarter of thesample reported at least one factorthat adversely affected their ability tolearn. Dyslexia was most frequentlymentioned, with around 7 per centciting this.

In some classes the range of abilitywas relatively small; in other classesthe wide range of ability was seen as aproblem. Even in classes working at asimilar level, an individual might bestrong in one curriculum area butrelatively weak in another. This tallieswith the Government’s description ofadult learners as having ‘spiky profiles’.

Other classes had distinctivepopulations, such as those for peoplewith learning difficulties and ESOLlearners.

In some classes teachers often found itdifficult to motivate learners. Thisproblem was particularly acute whennumeracy was part of a vocationalcourse. Jobseekers’ classes also had adistinct population, as some learnersattended for eight weeks and others upto six months, which made planningdifficult.

Differences between learning

numeracy as a child and as an adult

Many of those interviewed spoke ofanxiety about returning to learning tostudy numeracy, and most of thesewere women. However, not all learnershad worries and this was particularlytrue of the 16 to 19-year-olds, as manywere, in effect, continuing at school.

Many learners contrasted theirexperiences of learning maths atschool with their current experience ofnumeracy education, highlighting thesmaller classes and the individualattention they now received. Many alsocited the relaxed atmosphere, theirfeelings of security, the lack ofpressure from teachers and peers, thesense of making progress, and the

The learners


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generally stimulating level of work.

A key theme that emerged from thelearner interviews was that, where theteaching is good, learners begin tounderstand more aboutnumeracy/maths, and withunderstanding comes greaterconfidence.

Learner motivations

Research has established thatlearners’ motivations for joiningnumeracy classes are many, intricateand often overlap. Most of our 412learners reported ‘getting aqualification’ as the main reason fordoing a numeracy course, with ‘gettinga better job’ being the second mostpopular response. The number ofadults over 20 who said they wanted tostudy numeracy to either provesomething to themselves or becomemore confident was more than twicethat of the 16 to19 age group.

In common with other researchfindings our data confirm that wantingto prove to themselves that they cansucceed in a high status subject is alsoa powerful reason. Although givingeducational support to their childrenwas only the fifth most popularresponse, it should be rememberedthat over 40 per cent of the samplewere 16 to 19-year-olds, and so wouldnot have children of school age.

Policy-makers often assume that amajor reason for people to attendnumeracy courses is to help them

function more effectively in the outsideworld. Our research, however,suggests that this was perceived bylearners as being a comparativelyminor reason.

What people felt about numeracy

The 77 learners who spoke about theirfeelings towards numeracy were morelikely to say that they liked numeracy(44 per cent) than disliked it (21 percent).

Findings from the attitude survey wereeven more positive: from a total sampleof 243, a large majority reported thatthey enjoyed numeracy learning (78 percent), and only 22 per cent stated thatthey did not enjoy it.

During the interviews, more than onein four learners said they were feelingmore confident about maths now thatthey were on the course. This was alsoreflected in data from the attitudesurvey.

Succeeding in what many learnersperceive as being a high status subjectalso led to higher levels of self-esteem.Some saw it as like being able to joinan elite club, and as one learner put it,‘it makes me feel like an educatedperson’.

The potential of learning numeracy inbeing able to change learners’identities is illustrated in thequotations below.

One female learner said:



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I feel equal. When I’m at work now I don’tfeel that I’m a second-rate person. Idon’t feel that I have to prove myselfanymore.

Her male classmate agreed:

To be able to do, like when you seemaths, and to be able to do it, makes meso proud, I am going somewhere. And Iwant to do more.

Views on the course

Learners were overwhelmingly positiveabout the course, with more than 90per cent expressing a high level ofsatisfaction. Learners seemed to likemost the relaxed atmosphere and theway they were treated as an adult; theindividual help and attention; thefriendliness of the other learners;working with and helping others; theteacher and the way numeracy wastaught; feelings of progress; and theirimproved confidence and self-esteem.

What makes a good numeracy teacher

We asked the learners what theythought makes a good teacher. In orderof frequency, a good teacher wasdescribed as someone who:

• Has good communication skills;explains things clearly using severaldifferent ways, including breakingdown concepts into small steps.

• Has good relations with learners:respects learners; does not makethem feel stupid; is approachable andlistens carefully to their needs.

• Makes maths interesting by being

imaginative and makes sure there isplenty of variety in each session.

• Does not lecture and talk too much. • Gives individual help. • Does not rush through the work. • Has a firm grasp of their subject.

Learners also wanted a teacher whowas cheerful, had a sense of humour,was relaxed and easy-going and madethem feel welcome. Above all, theywanted someone who was patient.

Some of the teachers we interviewedhad all of these qualities and were notfazed by what they encountered in theclassroom, as the following storyconfirms:

Should have seen him (adult learner)

last week actually. He comes up to me

and needs his shoelace doing up and

thinks I can do his shoelace up. He put

his foot up, I did his shoelace up and

said: “There you go” and he bent over to

kiss me, it must be what he does to his

Mum [laughs]…Personal space! And

there, you know, tying one person’s

shoelace when I’ve got someone else in

my other ear asking me about quadratic

equations because the maths exam is

on.

Female numeracy teacher

What is progress?

Learners’ progress in each class, asmeasured by the average gain in theirscores on the assessment instrumentbetween T1 and T2, is used to judge theeffectiveness of teaching and learning


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in this and the other NRDC EffectivePractice Studies. However, it isimportant that our findings are notover-interpreted. For instance, forsome people, to simply maintain theirlevel of skills in numeracy rather thanfalling back is itself a sign of personalprogress. We also know that learnersmay regress if they are not regularlyusing their skills. Moreover, as we havealready stated, we were aware that ourinstrument for measuring learners’progress was not as valid and reliableas we would have liked.

Gains in attainment

Of the 412 learners in both phases ofthe study, 250 completed anassessment at both time points,towards the beginning and end of theirnumeracy course. In this section we arepresenting data only on learnersassessed at both time points.

This group of 250 was compared withthe 162 assessed only at T1 in terms ofbackground characteristics such asgender, age and qualifications toestablish that the groups did not differsignificantly. The same test was usedboth times, usually with a gap of sevenor eight months, although the nature ofthe majority of the provision meant thatthe average number of teaching hoursthat learners received between pre- andpost-assessment for each class wasonly 39.

We found an average 9 per cent gainbetween the two time points across alllearners in Phases 1 and 2. The mean

gains are statistically significant, and ina test with 20 items it is equivalent to anaverage learner being able to answercorrectly two additional questions in thefinal test.

However, there was no correlationbetween number of hours of tuition andthe gain in scores. Although this seemscounter-intuitive and inconsistent withother research, this may reflectparticular circumstances. For example,there were a number of short courses inour sample, such as one highly intensivecourse run by the Army over five days.

In interpreting results such as these weneed constantly to bear in mind thedifference between correlation andcausation. For example, longer learningtime would be expected to causegreater progress, but there may beother underlying associations.

Table 1 shows the mean gains made byeach class; these are shown in rankorder within each phase. We found alarge spread in the mean class gains,with the largest at more than 30 percent and the lowest at -13. It should benoted that negative mean gains do notnecessarily indicate that learners knewless at the end of the course than at thestart. These indicate only that the meanscores on a small sample of items werelower.

Attainment gains and background

characteristics

We investigated whether any learnercharacteristics were related to the

3


Phase 1 classes

Rank Mean Number

order gain % of learners

assessed

1 27.04 9

2 13.89 6

3 13.34 5

4 12.5 6

5 9.52 7

6 5.56 6

7 4.17 4

8 3.67 5

9 3.18 11

10 2.17 10

11 1.25 4

12 1.15 2

13 -0.33 2

14 -0.67 5

15 -2.77 3

Phase 2 classes

Rank Mean Number

order gain % of learners

assessed

1 32.50 5

2 31.11 6

3 23.33 5

4 20.00 2

5 17.86 7

6 16.54 13

7 15.56 3

8 14.81 9

9 14.17 2

10 12.50 12

11 10.56 3

12 10.33 5

13 9.58 4

14 9.44 3

15 9.05 7

16 8.89 6

17 7.08 4

18 6.67 3

19 6.33 5

20 6.30 9

21 6.21 11

22 4.44 3

23 3.97 13

24 3.75 4

25 2.33 5

26 1.11 3

27 -1.11 6

28 -4.44 3

29 -12.78 3

30 -13.33 1

Table 1 Mean gain of classes in Phases 1 and 2 in rank order


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amount of progress learners made. Welooked at gender, age group, firstlanguage, ethnic group, having attendedanother numeracy class since school,reporting a factor affecting learning, andqualifications held. The only statisticallysignificant difference found was thatlearners who said they lacked a formalqualification in maths made greaterprogress.

We also explored differences inprogress in terms of the reasonslearners had given as their mainmotivation for doing the course. Theonly significant difference found wasbetween learners who stated that theywanted to become more confident asopposed to those who did not, with theformer group tending to make moregains than the latter.

Finally, there were no significantcorrelations between any of the teachercharacteristics measured and theprogress of the learners in their class.This may seem counter-intuitive, but itmirrors other findings about primaryteachers and progress in numeracy.Similarly, as with earlier studies, wealso found that young and/orinexperienced teachers were notnecessarily less effective than olderand/or more experienced teachers,possibly because of their more recenttraining experience.

How attitudes changed

A total of 415 learners completed the‘attitude to numeracy’ questionnaire atT1 and 254 at T2, with 243 completing it

at both time-points over the two years ofthe project.

From the 17 statements, learners had totick one of four options from ‘stronglydisagree’ to ‘strongly agree’. Thequestionnaire was designed to includestatements relating to usefulness,enjoyment and difficulty of learningnumeracy.

The correlations between attainmentand attitude scores were weak and non-significant, as was the correlation ofgains in assessment and attitudescores. These seem surprising findings;one might expect that those with higherscores would be more positive aboutmathematics and that learnersdemonstrating a positive change inattitudes would make most progress. Itmay have been that the attitude surveywas not capable of detecting theincreased enthusiasm for mathematicsthat we noted when interviewing somelearners.

Analysis of attitude sub-scalesThe 17 items of the attitudequestionnaire were divided into threegroups, according to the aspects ofattitudes towards numeracy theyintended to measure, namely perceivedusefulness (seven statements),enjoyment (five statements) anddifficulty (five statements). There weresmall changes in all three dimensions,all in the expected direction; that is,learners found numeracy more useful,more enjoyable and less difficult at theend of the course.



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■ Twenty-five of the teachers werewomen and nine were men. Eleventaught classes in Phase 1 only, 18 inPhase 2 only, and five taught classes inboth phases. The mean number ofyears of teaching experience innumeracy or maths was just over 13,while the vast majority had taught atLevels 1 and 2, GCSE and learners overthe age of 19. More than 66 per centhad taught in secondary schools and 24per cent in primaries. Twenty-sevenhad a formal qualification inmathematics or a related subject, suchas science. Thirty teachers reportedhaving a teaching qualification. Six saidthey had the new Level 4 qualificationfor teaching numeracy to adults.

Key elements of an effective lesson:

the teachers’ view

Teachers described their classes, andresearchers drew out the features ofwhat they regarded as an effectivesession.

• Being flexible and able to use avariety of approaches toaccommodate learners’ needs;learners work at different speeds,and activities sometimes take moreor less time to work through thananticipated.

• Enabling learners to makeconnections to other areas of

mathematics. • Good planning, including anticipating

learners’ responses. • Starting from where the learners are,

providing a variety of activities, and avariety of ways of doing things thatincorporate learners’ own methods.

• Extending learners beyond theircomfort zone.

• Getting learners to interact, andviewing learning as a social activity.

• Encouraging learners to make theirthinking explicit to the teacher and toother learners; allowing them toarticulate what they understand.

The final point was important becauseit helped learners not only to practise a‘technique’, but also to assimilate anunderlying concept. One teacher, whosetwo Jobseekers’ classes were in the topthree in terms of gains at Phase 2, said:‘You have always got to hear learnersspeak.’

Researchers observed each teacher onan average of 2.4 occasions over thetwo phases. They completed a narrativesheet at the time of the session, andmade reflective observations that werecompleted retrospectively. The analysisof sessions was divided into sevenaspects:

• structure/organisation

The teachers


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Figure 1 Mean ratings for each teaching characteristic in order of frequency

0 0.5 1 1.5 2 2.5 3

Use of worksheets

Learner engagement

Evidence of mutal respect

Teacher gives clear explanations

Learner given sufficient time to gain understanding

Learners feel free to express themselves

Symbolic pedagogy

Teacher enthusiasm

Teacher gives feedback

Teacher demonstrates subject knowledge

Teacher monitors learning

Teacher gives praise and encouragement

Resources used to enhance and support learning

Teacher works through examples

Individual needs addressed

Learners challenged and stretched

Procedural pedagogy

Teacher engages in direct teaching

Teacher shares overall goals with learners

Teacher breaks work down

Variety of learning activities provided

Differentiation of tasks/activities

Whole class

Individual

Teacher is flexible

Contextual pedagogy

Conceptual pedagogy

Visual pedagogy

Learners working/learning with and from each other

Links to other areas of maths and real contexts

Learner discussions

Whiteboard

Teacher promotes discussion

Asks higher order questions

Practical apparatus

Calculators

Teacher sets up collaborative learning

Pairs

Opportunities for learners to raise own issues

Strategic pedagogy

Learners’ interests incorporated

Manipulative pedagogy

Groups

Plenary

Computers

Oral/mental starter

Games

Textbooks



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• teachers’ role• teaching process• learners and learning• teacher-learner relations• materials• mathematical pedagogy (Phase 2

only).

A numbering system from 0–3 was usedto indicate the emphasis the researchergave each characteristic of the lesson. 0indicated that it was not observed; 1,that it was observed to a very limitedextent; 2, that it was observed to someextent, and 3, that it was observed to ahigh degree. The reflection sheettherefore allowed us to describe thecharacteristics of the lessons as awhole.

Results for each class are based on theaverage ratings. Figure 1 is a summaryof the 48 categories that were used toanalyse teachers’ pedagogicalapproaches, and it presents the average

degree to which each was observedacross all the sessions observed.

Figure 1 shows that the most commonforms of organisation were whole-classand learners working individually. Therewas less group or collaborative work,and it was less typical to find learnersworking with, and learning from, eachother.

Most teachers followed a set scheme ofwork and rarely incorporated learners’personal interests. The main approachwas for teachers to show learnersprocedures, breaking concepts downinto smaller parts and demonstratingexamples. Worksheets were widelyused, with little practical apparatus,games or ICT.

Teachers generally had adequatesubject knowledge, gave clearexplanations and provided a variety oflearning activities. It was less usual for

Teachers generally hadadequate subject

knowledge, gave clearexplanations and

provided a variety oflearning activities


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teachers to differentiate work, makeconnections to other areas ofmathematics, or ask higher-orderquestions to encourage higher-levelthinking or to probe learners’misconceptions.

The majority of the teachers askedlearners to follow procedures usingsymbols (mainly numbers). There wasless emphasis on conceptualunderstanding or relating topics such asfractions to other areas of mathematics.Although about half of the sessionsshowed teachers relating mathematicaltopics to the world outside theclassroom, and employing visualtechniques to aid understanding, veryfew teachers asked learners to solveproblems or used concrete materials.

Mutual respect between teachers andlearners was high, and learners felt freeto express themselves. Teachers wereinvariably enthusiastic and gavelearners much praise andencouragement. They also usuallymonitored learning and gave feedback.

On the whole, learners were generallyhighly engaged; they were oftenchallenged and stretched; they weregiven time to gain understanding, andthe majority had their individual needsmet.

Teaching typologies and other factors

Based on the classroom observations,teachers were classified according totheir teaching approach. Three wereidentified: the connectionist and

transmission styles (Askew et al., 1997),and the constructivist/scaffolder style,after Bruner and Vygotsky. No one useda discovery approach – where theteacher believes that learners shoulddiscover the intended outcome of thelesson, guided by them and materials.

• The connectionist teacher frequentlymakes connections to other areas ofmathematics, including movingbetween symbolic, visual and verbalrepresentations.

• The transmission teacher is principallyconcerned with mastery of skills.Mathematics is seen as a series ofdiscrete packages to be taught insmall steps emphasising proceduresrather than conceptualunderstanding.

• Using the constructivist/scaffolderstyle, the teacher works alongsidelearners, co-constructing conceptsand asking questions.

Most teachers combine the differentapproaches to varying extents. Eachcould be perceived as appropriate todifferent purposes in teachingmathematics.



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■ We computed correlations betweenthe average scores for each of theclassroom characteristics observedand the class gains. Most of thecorrelations were relatively low and notsignificant. Since the gains were largerin Phase 2, and more observationswere made per class, we have usedonly the Phase 2 data with 29 classesfor the next section.

One significant positive correlation wasfound. This was between learners’progress and the extent of proceduralteaching. Procedural teaching involvesshowing discrete procedures forlearners to follow in order to carry outa computation or technique.

Another significant but low negativecorrelation was found between learnerprogress and the amount of individualwork. In other words, classes in whichthis characteristic was observed lesswere more likely to have made positivegains.

Almost every class contained individualwork at least to a limited extent, andthe relationship was not strong.Indeed, one of the classes that madethe best progress contained asignificant amount of individual work.However, classes in which individualwork was observed to a large extent

generally seemed to make a little lessprogress.

Teaching approaches and gains in

progress and attitude

Our study shows no clear relationshipbetween teaching typology(transmission, connectionist andconstructivist methods) and classprogress.

We also found no significantcorrelations either between any ofthese three factors, or betweenteaching typologies and changes inlearners’ attitudes.

This may seem surprising. However,such findings are not unusual inmathematics education literature,where it is first the learner variablesand second the curriculum that seem tobe the essential factors. The effect ofthe manner in which learners aretaught is either not detected or is verysmall.

It thus seems likely that in our studytoo, factors that cannot easily bedetermined in a large-scale survey mayhave more influence on learners’learning and changes in attitudes thanany difference in teacher behaviour.These include learners’ strength ofmotivation, self-discipline, aspirations,

How practice and progress are linked


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abilities and dispositions towardsnumeracy, socio-cultural backgroundand previous experiences both insideand outside the classroom.

Best practice in classes where

learners made most progress

We went on to examine the highest-performing classes in greater depth tofind out whether any particular featuresdistinguished the teaching of theseclasses from the others. We selectedthe five classes that made the mostprogress, all achieving mean gains ofmore than 15 per cent. We thencompared these five with the fullsample in several ways. We also lookedfor differences in the teachers’background characteristics.

There was a considerable variety ofteaching typologies even within thesefive classes. Two teachers were judgedto use a connectionist/constructivistapproach, one aconnectionist/transmission approachand two a transmission approach.

While the teachers in the two best-performing classes (both of whichachieved more than 30 per centattainment gains) predominantly used abalance of constructivist andconnectionist approaches, other classestaught using similar approachesperformed much less well.

We found that the teachers of the fivegroups that made the highest gainsgenerally taught mathematicalprocedures more than the average

teacher, and very much more than theteachers of the five groups that gainedleast. This is consistent with proceduralteaching having the highest correlationwith gains for the whole sample.

Similarly, teachers of the five groupsthat made the highest gains made lessuse of practical activities than theaverage teacher and very much less usethan teachers of the five groups whomade the least gains.

It is important to guard againstconverting distinctions into causations.In this case it seems likely that teachersdecide to use practical equipment forlearners who have difficulties inlearning.

Teachers would be unlikely to try toteach formal procedures to learnerswho had experienced problems inremembering these in the past. It wouldbe more likely that they would do so forfaster learners. We do not thereforebelieve procedural teaching necessarilycauses greater learning and practicalactivities cause less learning.

Other aspects of what would normallybe regarded as effective practice wereused more among the lowest-performing classes, e.g. collaborativework, teaching of strategies as well asprocedures, teachers emphasisingmaking connections and hypotheses.

Again, we do not believe that these arecounter-productive activities. Indeed,we found classes where experienced



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researchers thought that the teachingwas generally very good but learnerprogress was weak. Equally, weobserved some teaching of lower thanaverage quality but where progress wasstrong.

Finally, we also checked to see if thecharacteristics of the teachers andlearners might have an effect. While theteachers’ background appeared to haveno discernible effects, it was noticeablethat the five highest-attaining classesall contained adults over 19 and tendedto be dominated by older learnerswithout any major language difficulties.No other clear associations were found.

Good practice in action

On pages 26-28 we present a detaileddescription of the beginning of oneteacher’s numeracy class as anexample of effective practice. It was anevening class at a London FE college,and the teacher had taught numeracyfor 21 years.

This class achieved an average gain ofmore than 30 per cent, with manylearners making exceptional progress.In addition, learners’ enthusiasmtowards numeracy was noticeable bothfrom the attitude surveys and the classobservations. This was achieved byusing a predominantly connectionist andconstructivist approach whichemphasised conceptual understandingrather than routine procedures.

The teacher created a non-threateningatmosphere and learners’

misconceptions were used as examplesto discuss with the whole group.Learners were encouraged to discussproblems and concepts both betweenthemselves and with the teacher,building a strong collaborative culture.Numeracy learning was viewed as asocial activity where understanding wasformed through discussion.

A variety of group, individual and whole-class teaching was used. However, evenwhen learning was organised on anindividual basis the learners were stillencouraged to discuss problems andhelp each other, developing a greaterunderstanding. The class was taught inan open style, which allowed higher-order, diagnostic questioning thatuncovered learners’ thinking.

A range of materials and teachingresources was used, from worksheetsto games and activities, includingwhole-class role-play. Calculators werefreely available. The teacher usedproblem-solving activities and was ableto change direction to respond tolearners’ needs.

The comments that appear in italics areretrospective and providecharacteristics of what we believeconstitute ‘effective’ practice.


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Time Content/focus of the session

7.00 Topic: Percentages

Becky (BH) holds up an individual mini-whiteboard (A4 whitelaminated card) with ‘%’ hand-drawn on it. She asks the learners totell her what it is and what it means. In response to one learner sayingit looks like a division sign she draws one [÷] on the main (fixed)whiteboard and initiates a discussion about the relationship betweenpercentages, fractions and decimals. She asks learners to call out different percentages that they had comeacross, and she writes them up on the main whiteboard.

BH writes up: ‘10% means divide by 10’. She makes no furthercomment

The teacher asks open questions; does not give answers; initiatesdiscussion, looks at relationships and connections and assesses learners’prior knowledge. The teaching is interactive and the teacher reinforcesunderstanding.

7.15 BH gives learners small cards with statements on two lines (e.g., Ihave 76. Who has 10 per cent of £6,500?) Learners have to read outtheir questions and answer if they have the right answer, otherwisekeep quiet. BH: ‘If your neighbour is quiet they may be asleep, so youcan look at your neighbour’s card.’ At the end Becky confirms to theclass that they were all able to calculate 10 per cent of the amount.

BH (having drawn on small whiteboard): ‘10 per cent of 30? So what’s5 per cent? So what’s 30 per cent? If I wanted 90 per cent of 500?’Greg says, ‘Take off 10 per cent’. BH asks for a number and Greg says‘300’: ‘50 per cent of 300? What’s 75 per cent of 300? Half is 50 percent, then halve that and add it to the 150. Notice we’re talking abouta half and a quarter.’ Learners call out the answers; Becky writes onlarge whiteboard. BH: ‘Can you see a pattern? What’s 55 per cent of300? You can do it however you like.’ Learners hold up theirwhiteboard cards as they do it. They ask each other what they’ve got.Becky helps one man (Moji). She asks (re 55 per cent of 300) ‘Whatwould be an easy percentage?’ Moji: ‘50 per cent’. BH: ‘Sandra, tellMoji what to do’ (she does). ‘One way is to use what you know here


and here’ (shows examples on main whiteboard).

BH points out there are many different ways of doing percentages. Insome situations one method is good, in others, another method mightbe better. ‘171-2 per cent. If you think you know what to do, write it downon your board. 10 per cent; 5 per cent; 21-2 per cent. What have theydone here? Can you work out 171-2 per cent of 300?’ (shows it written onmini whiteboard with figures above each other). Learners work outeach element and then add them together. BH asks why they’ve addedthem. Learners explain. BH: ‘That’s VAT. It’s not too bad. Now try itwith my nice number (400). Just to see how comfortable you are withit, I’ll give you an even nicer number (800).’ Sandra gives the rightanswer. BH: ‘Did you do that in your head? That’s impressive. So 171-2per cent doesn’t hold any threats for you. How about 63 per cent? Howwill I break that down?’ (Learners call out different ways of breakingdown 63 per cent). BH: ‘Distinguish between ones you can do in yourhead and more tricky ones – you’d use a calculator for those.’

BH: ‘Let’s try 63 per cent of £800.’ She goes around the room (usingthe space in the middle) helping learners as appropriate, e.g., notlining numbers up. BH: ‘There’s a terribly dangerous thing happeningto everyone in the room and it’s all my fault! Karen, let me show whatyou did.’ She writes 400 wrongly aligned with the other numbers to beadded. BH: ‘Be careful that you always find percentages of the samenumber (800). Always refer back to the number you’re finding thepercentage of.’ BH: ‘Will 63 per cent be more than half or less thanhalf? Always think about doing a check. There are different ways ofchecking. We can learn some of those as we go along.’ BH: (writing onwhiteboard): ‘When you see 25 per cent what does it mean? A quarter;75 per cent, three-quarters; 331-3, a third.’

The teacher uses interactive games and asks questions. She builds on,and uses, learners’ strategies, points out that there are many differentstrategies that can be used, highlights that some may be better thanothers, and shows learners which ones to use. The teacher is, again,getting learners to look for patterns. The learners work collaboratively;some assume a teaching role and explain strategies to each other. Theteacher breaks maths down and works through examples. She points outthat there are different ways of solving problems. The teacher assesses


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different ways of working and asks learners to justify what they’ve done.She breaks maths down using learners’ own methods, and encouragesmental calculation. She gives praise and there is appropriate use oftechnology. The teacher monitors learning and identifies learners’misconceptions. She emphasises need for checking and reinforcesconcepts learned with whole group.

The narrative above covers only the first hour of the session and provides a partialview. Nevertheless, it shows how complex teaching is, how many decisions teachershave to make, and how hard they often have to work. We believe that this extractexemplifies some of the key features of effective practice. These resonate with theapproaches promoted by the DfES Standards Unit Improving Learning in Mathematicsproject that have been piloted with adult learners through the NRDC Maths4LifeThinking Through Mathematics initiative (www.maths4life.org.uk).

The narrative… shows howcomplex teaching is, howmany decisions teachershave to make, and howhard they often have towork


■ Taking all classes together, significantprogress was made over the length ofthe numeracy courses. There were,however, few significant links betweenprogress made and different classroomapproaches, and little associationbetween teachers’ characteristics likequalifications and experience or thenumber of teaching hours. There wasalso little association between the size ofgains and types of learner. There weresmall but positive overall changes inattitude.

We have found that effective approachesare difficult to determine solely fromquantitative data. The multiplicity offactors contributing to learning meanthat any effects that good practice mighthave are often compromised by otherconsiderations that contribute to, orconstrain, learner progress. In the end,our correlation calculations give littleindication of what constitutes aneffective approach in adult numeracyeducation.

This seems to suggest that factorswhich cannot easily be determined in alarge-scale survey may have moreinfluence on their learning than anyspecific easily-observed difference inteacher behaviour. These are: learners’strength of motivation, their self-discipline, their aspirations, their

abilities and dispositions towardsnumeracy, their socio-culturalbackground and previous experiencesboth inside and outside the classroom.

In one sense our findings are in linewith research literature that suggeststhat there is often only a partialrelationship between interactions inpedagogic settings and learning. Wetherefore caution against any attempt topromote a single method or approachthat can be applied across all settings.

Nevertheless, the qualitative strand ofour research bears out the view thateffective practice requires goodteacher-learner relationships andteachers being flexible in their responseto learners’ needs.

It also involves careful planning, well-grounded subject-specific pedagogy andmaking connections to and betweenother areas of mathematics.

Conclusions


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Askew, M., M. Brown, et al. (1997).Effective Teachers of Numeracy. London,King’s College London.

Condelli, L., Wrigley. H., Yoon. K.,Seburn, M. and Cronen, S. (2003). What

Works Study for Adult ESL Literacy

Students. Washington, DC: USDepartment of Education.

DfES (2003). The Skills for Life Survey: A

national needs and impact survey of

literacy, numeracy and ICT skills. London,Department for Education and Skills.

References


NRDC


University of London

20 Bedford Way

London WC1H 0AL

Telephone: +44 (0)20 7612 6476

Fax: +44 (0)20 7612 6671

email: [email protected]

website: www.nrdc.org.uk

NRDC is a consortium of partners led by the Institute of Education,University of London with:• Lancaster University• The University of Nottingham• The University of Sheffield• East London Pathfinder• Liverpool Lifelong Learning

Partnership

• Basic Skills Agency • Learning and Skills

Network • LLU+, London South

Bank University • National Institute of

Adult Continuing Education• King’s College London• University of Leeds

Funded by theDepartment forEducation and Skills aspart of Skills for Life: the national strategy forimproving adult literacy and numeracyskills.

NumeracyEFFECTIVE TEACHING AND LEARNING

www.nrdc.org.uk


effective teaching and learning numeracy

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