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School Effectiveness and School Improvement

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School Effectiveness and Teacher Effectivenessin Mathematics: Some Preliminary Findings fromthe Evaluation of the Mathematics EnhancementProgramme (Primary)

Daniel Muijs & David Reynolds

To cite this article: Daniel Muijs & David Reynolds (2000) School Effectiveness and TeacherEffectiveness in Mathematics: Some Preliminary Findings from the Evaluation of theMathematics Enhancement Programme (Primary), School Effectiveness and SchoolImprovement, 11:3, 273-303

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School Effectiveness and School Improvement 0924-3453/00/1103-0273$15.002000, Vol. 11, No. 3, pp. 273–303 © Swets & Zeitlinger

School Effectiveness and Teacher Effectiveness inMathematics: Some Preliminary Findings from theEvaluation of the Mathematics EnhancementProgramme (Primary)

Daniel Muijs and David ReynoldsUniversity of Exeter, School of Education, UK

ABSTRACT

In this study the effect of teacher behaviours and classroom organisation on pupils’progress in mathematics was studied in years 1, 3 and 5 of primary schools in the UKparticipating in a mathematics intervention programme. Data on a total of 78 teachers and2,128 pupils was collected. Teacher behaviours were measured using a classroom obser-vation instrument developed for the project, and pupils were tested in March and July of1998 using a curriculum-appropriate Numeracy test developed by the National Founda-tion for Educational Research. Background data on pupils was also collected at bothtesting occasions. Using multilevel modelling techniques it was found that teacher behav-iours were able to explain between 60% and 100% of pupils’ progress on the Numeracytests. Amount of time spent teaching the whole class was not related directly to pupils’progress, but structural equation models were tested in which time spent teaching thewhole class was found to be related to effective teaching behaviours and thus indirectly topupil progress. The implications of the study for British educational policies and foreducational research more generally are discussed.

INTRODUCTION

In the past 20 years, an ever-growing knowledge base on effective teach-ing has been built up. Starting with research in classrooms looking purelyat ‘quantity’ of behaviours (e.g., Stallings & Kaskowitz, 1974), researchhas now moved on to consider ‘quality’ of behaviours as well. A numberof scales have been developed to this end, and have further enhanced theteacher effectiveness knowledge base.

Address correspondence to: Daniel Muijs, University of Exeter, School of Education,Heavitree Road, Exeter, Devon, EX1 2LU, UK. Tel. +44 1392 264761. E-mail:[email protected]

Manuscript submitted: February 1, 2000Accepted for publication: June 7, 2000

274 DANIEL MUIJS AND DAVID REYNOLDS

This, mainly American, research has produced a large number of fac-tors that are associated with higher pupil achievement. Although thesefindings are usually presented in general terms, a lot of the research hasbeen done in connection with mathematics teaching, making these find-ings probably especially relevant to this subject (Reynolds & Muijs,1999a). One of the main factors related to mathematics achievement scoresin this body of research is opportunity to learn, usually measured as eithernumber of pages of the curriculum covered or percentage of test itemstaught. This variable is clearly related to factors such as length of theschool day and year, but also to effective classroom management whichmanages to maximise time-on-task, the amount of time pupils spend ac-tively engaging with the curriculum (Brophy & Good, 1986; Creemers &Reezigt, 1996; Hafner, 1993).

Another consistent finding is that effective teachers emphasise academ-ic instruction as their main classroom goal and have an academic orienta-tion. They therefore create a businesslike, task-oriented environment andspend classroom time on academic activities rather than on socialising,free time, etc. This factor would seem to operate at the school as well as atthe classroom level, and in a wide range of contexts and countries (Borich,1996; Brophy & Good, 1986; Cooney, 1994; Creemers, 1994; Griffin &Barnes, 1986; Reynolds, Sammons, Stoll, Barber, & Hillman, 1996; Schee-rens & Creemers, 1996).

Time on task is, as mentioned above, strongly influenced by classroommanagement. Effective teachers are able to organise and manage class-rooms as effective teaching environments in which academic activities runsmoothly, transitions are brief, and little time is spent getting organised ordealing with resistance (Brophy & Good, 1986). The latter is obviouslyrelated to good behaviour management, which entails clearly instructingpupils on proper behaviour procedures at the start of the year so that pupilsknow what is expected of them during lessons, closely monitoring theclassroom, reinforcing wanted behaviour, and discouraging undesired be-haviour (Borich, 1996; Brophy, 1986; Brophy & Good, 1986; Evertson &Anderson, et al., 1980; Lampert, 1988; Secada, 1992).

Moving to the actual teaching, research has found that pupils learnmore in classes where they spend most of their time being taught or super-vised by their teachers, rather than working on their own. In these classesteachers spend most of their time presenting information through lectureor demonstration. Teacher-led discussion as opposed to individual seat-work dominates. The teacher carries the content personally to the student,as opposed to relying on textbooks or maths schemes to do this. Informa-tion is mainly conveyed in brief presentations followed by recitation and

275SCHOOL AND TEACHER EFFECTIVENESS IN MATHEMATICS

application opportunities. In this type of instruction the teacher takes anactive role, rather than just ‘facilitating’ pupils’ learning. Use of examplesis important, and teachers should strive to make presentations lively andengaging to maximize gain. This type of instruction is usually referred toas direct instruction. (Borich, 1996; Brophy & Good, 1986; Galton, 1987;Lampert, 1988).

Achievement is maximised when the teacher not only presents materialactively, but does so in a structured way by beginning with an overviewand/or review of objectives. Teachers need to outline the content to becovered and signal transitions between lesson parts. Attention must bedrawn to the key points of the lesson, subparts of the lesson should besummarised as it proceeds, and the main ideas should be reviewed at theend of the lesson. In this way the information is not only better remem-bered by the students, but also more easily understood as a whole ratherthan as a series of isolated skills. In this respect, it has been found to beespecially important in mathematics to clearly link different parts of thelesson and of the curriculum. New knowledge needs to be linked to pupils’prior knowledge, and mathematical ideas must be linked and not taught inisolation (Borich, 1996; Brophy & Good, 1986; Lampert, 1988). Informa-tion must be presented with a high degree of clarity and enthusiasm. Forbasic skills instruction, the lesson needs to proceed at a brisk pace (Brophy& Good, 1986; Good, Grouws, et al., 1983; Griffin & Barnes, 1986; Sil-ver, 1987; Walberg, 1986).

Although we have noted above that teachers need to spend a significantamount of time instructing the class, this does not mean that all seatwork isnegative. Individual seatwork or small group tasks are a vital componentof an effective lesson, as they allow pupils to review and practice whatthey have learnt during instruction. To be effective, however, tasks mustbe explained clearly to pupils, and the teacher must actively monitor theclass and go round the classroom to help pupils, rather than sitting at her/his desk waiting for pupils to come to her/him. The teacher needs to beapproachable to pupils during seatwork (Borich, 1996; Brophy & Good,1986).

This focus on the teacher actively presenting material should also notbe equated to a traditional lecturing and drill approach, in which the stu-dents remain passive during the lesson. Effective teachers ask a lot ofquestions and involve pupils in class discussion. In this way, students arekept involved in the lesson, while the teacher has the chance to monitorpupils’ progress and understanding. If pupils are found not to understand aconcept properly, it should be retaught by the teacher. Teachers mustprovide substantive feedback to students resulting either from pupil ques-


tions or from answers to teacher questions. Most questions should elicitcorrect or at least substantive answers. The cognitive level of questionsneeds to be varied depending on the skills to be mastered. The best strate-gy would appear to be to use a mix of low-level and higher level questions,increasing the latter as the level of the subject matter taught gets higher.There should also be a mix of product questions (calling for a singleresponse from students) and process questions (calling for explanationsfrom the students), although effective teachers have been found to askmore process questions than ineffective teachers (Askew & William, 1995;Brophy & Good, 1986; Evertson & Anderson, et al., 1980).

When students answer a question correctly, this should always be ac-knowledged. However, this does not mean that teachers should use praiseindiscriminately. Effective teachers are usually quite sparing in their useof praise, as, when overused, praise can become meaningless. More praiseis needed in low ability and low SES classrooms, as pupils tend to be lessself-confident. If students give an incorrect response, the teacher shouldindicate that the response is incorrect in the form of a simple negation,avoiding personal criticism of students which may deter them from partic-ipating in classroom activities. Teachers should try to rephrase questionsand provide prompts when pupils are unable to answer them. Studentsshould be encouraged to ask questions, which should be redirected to theclass before being answered by the teacher. Relevant student commentsshould be incorporated into the lesson (Borich, 1996; Brophy & Good,1986). It is therefore clear that effective teaching is not just active, butinteractive as well.

In addition, effective teachers have been found to use a varied teachingapproach to keep pupils engaged, and to vary both content and presenta-tion of lessons. Specifically with respect to mathematics, a lot of researchhas attested to the importance of using a variety of materials and manipu-latives, in order to be able to assist mental strategies and to more easilytransfer mathematical knowledge to other situations and contexts.

Classroom climate is the final factor that teacher effectiveness researchhas found to be significant. As well as businesslike, the classroom envi-ronment needs to be suitably relaxed and supportive for pupils. Teacherexpectations need to be high. The teacher should expect every pupil to beable to succeed. They need to emphasise the positive in each child. Forexample if a pupil is not particularly good at algebra, s/he may still begood at another area of mathematics such as data handling. These positiveexpectations need to be transmitted to the children (Brophy & Good, 1986).

In recent years, a lot of attention in mathematics research has focussedon issues connected to children’s construction of knowledge. The main


assumption of this constructivist school, which builds heavily on the workof Jean Piaget, is that children actively construct their own knowledgerather than passively receiving it. Children, according to this view, do notacquire learning from the teacher, but through seeking out meaning andmaking mental connections in an active manner (Anghileri, 1995; Askew,Rhodes, Brown, William, & Johnson, 1997; Nunes & Bryant, 1996). Useof correct mathematical language has also been considered an importantaspect of effective mathematics teaching by some commentators.

Most of the research undertaken has, as mentioned above, been in theUS. In Britain only three major studies of teacher effectiveness have takenplace so far. The first of these, Galton’s ORACLE project, found thatteachers labelled as ‘Class Enquirers’ generated the greatest gains in theareas of mathematics and language, but that this finding did not extend toreading. By contrast, the group of ‘Individual Monitoring’ teachers madeamongst the least progress. It is important to note that the more successful‘Class Enquirers’ group utilised four times as much time in whole classinteractive teaching as the ‘Individual Monitors’ (Croll, 1996; Galton &Croll, 1980).

The second important British teacher effectiveness study is the JuniorSchool Project (JSP) of Mortimore, Sammons, Stoll, Lewis, and Ecob(1988), based upon a 4-year cohort study of 50 primary schools, whichinvolved collection of a considerable volume of data on children and theirfamily backgrounds (‘intakes’), school and classroom ‘processes’ and ‘out-comes’ in academic (reading, mathematics) and affective (e.g., self con-ception, attendance, behaviour) areas.

This study reported 12 factors that were associated with effectivenessboth across outcome areas and within specific subjects such as mathemat-ics. Significant positive relationships were found with such factors asstructured sessions, use of higher-order questions and statements, frequentquestioning, restricting sessions to a single area of work, involvement ofpupils and the proportion of time utilised in communicating with the wholeclass. Negative relationships were found with teachers spending a highproportion of their time communicating with individual pupils (Mortimoreet al., 1988).

A recent, more qualitatively oriented study by a team from King’sCollege, London (Askew et al., 1997) looked at teachers’ beliefs, knowl-edge and attitudes rather than behaviours. The researchers identified threetypes of teachers described as discovery oriented (strongly contructivisti-cally oriented with the teacher seen as facilitator), transmission oriented(teachers using purely direct instruction with little interaction) and con-nectionist (balancing a focus on the teacher, as in the transmission orienta-


tion, with a focus on the learner as in the discovery orientation) and foundthe latter to be more effective than both other types. Results also pointed tothe importance of making connections between different parts of the cur-riculum.

A number of teacher effectiveness factors have also been studied incontinental Europe, most notably in the Netherlands. However, a reviewof Dutch research found disappointing results, with teaching factors suchas whole-class teaching, achievement orientation and time spent on home-work being positively related to pupil outcomes at the primary level inrespectively 3, 4 and 4 studies out of 29 (and negatively related in none),while differentiation and cooperation were negatively related to outcomesin 2 and 3 studies respectively, and positively related to outcomes in none(Scheerens & Creemers, 1996). However, where significant results areobtained they tend to support the conclusions of the American and Britishstudies, Westerhof (1992) for example finding that a large proportion ofvariance could be explained by the factors ‘interactive teaching’, ‘givinginstructions’, ‘criticising’ (a factor better described as correcting errors),‘soliciting responses’ and ‘conditioning’.

The limited number of British studies into what makes teachers differ-entially effective makes this study timely and important, as, in view of thisdearth of research and the mixed results of the research in the Netherlands,it is clearly necessary to explore whether the American research basereadily translates to a British context. Traditionally, teacher effectivenesshas not been at the heart of British educational research, which has oftenbeen sociologically inspired and lacking in a strong pedagogy traditionsuch as that existing in continental Europe, while the powerful schooleffectiveness tradition has tended to concentrate on the school level to theexclusion of classroom level factors (Reynolds et al., 1996). Recently,however, British interest in this research base has grown, largely as aresult of weak performance in the Third International Maths and Sciencestudy, and the publication of the ‘Worlds Apart’ report (Reynolds & Far-rell, 1996), which focussed on the effectiveness of the whole-class interac-tive teaching methods employed in educationally successful East Asiancountries such as Taiwan, which led to the government adopting a whole-class interactive strategy for maths teaching in primary schools, the ‘Na-tional Numeracy Strategy’ (Department for Education and Employment[DfEE], 1998). In view of the recent enthusiasm for this method, thequestion needs addressing of whether whole-class interactive teachingreally is effective in raising mathematical achievement.

We agree with Brophy’s (1986) contention that effective teaching islikely to be a conglomerate of behaviours. It is unlikely that one isolated


behaviour will make the difference. Rather, it is the combination of effec-tive teaching behaviours that will lead to better performance in pupils. Asfor the debate on whole class interactive teaching, whenever it is foundthat spending more time teaching the whole class together is effective, thequestion arises of why this should be the case. We propose here that theso-called ‘black box’ between teaching the whole class and gain scores isin fact filled by the effective teaching behaviours, whole-class teachingthus allowing teachers to teach (effectively). It is this hypothesis that willbe explored in this study. Furthermore, while researchers such as Brophy(1986) have hypothesised that it is the cumulative impact of effectiveteaching behaviours rather than any one or group of behaviours that makesthe difference, neither the interrelationship between behaviours or theircumulative impact have so far been studied using the sophisticated statis-tical techniques we now possess. In this study we will propose a model inwhich the separate teacher behaviours form a number of scales: classroommanagement, behaviour management, direct teaching, interactive teach-ing, individual practice, classroom climate, varied teaching, constructivistmethods and mathematical language. It is hypothesised that these scaleswill be influenced by the percentage of time spent in whole-class teaching,as this type of teaching is expected to create the conditions for effectiveteaching to occur (Reynolds & Muijs, 1999a, 1999b).

THE GATSBY MATHEMATICS ENHANCEMENTPROGRAMME PRIMARY

Notwithstanding the evidence discussed above, there has long been a ten-dency in Britain to prefer methods of teaching based on an individuallearning approach, in which the student is expected to learn at his/her ownpace during most of the lesson, usually going through maths schemes/workbooks, while the teacher sits behind his/her desk marking pupils’work. Recently, partly as a result of the poor performance of British pupilsin international comparative tests (e.g., TIMSS), and partly because of thepoor numeracy of British adults as evidenced by a number of national andinternational studies (e.g., The Basic Skills Agency, 1997) the effective-ness of these methods has come under ever closer scrutiny, which has ledto a number of reform programmes in mathematics education, emanatingfrom national government and local education authorities and charitabletrusts.

One such reform programme is the Gatsby Mathematics EnhancementProgramme Primary, funded by the Gatsby Charitable Foundation (a Sains-


bury Family Charitable Trust), which aims to improve mathematics teach-ing at the primary level, in part through helping teachers to attain moreeffective whole-class teaching strategies. An evaluation of this programmewas funded by the Gatsby Charitable Foundation, and carried out by theNewcastle Educational Effectiveness and Improvement Centre at the Uni-versity of Newcastle. The evaluation of the success or otherwise of theproject will be discussed in other publications. In this article, we will lookat what the data have shown us about effective whole-class interactiveteaching in primary schools. Looking at the effectiveness of whole-classinteractive teaching in this way is particularly timely, in view of the factthat the British government is advocating this teaching method through itsNational Numeracy Strategy (DfEE, 1998), which all schools are advisedto follow, although it has to be pointed out that the fact that all teachers inthe project have been trained to use whole-class interactive teaching meth-ods will limit the variance in teacher behaviours we can observe in thisstudy.

SAMPLE AND METHODS

Data on 16 primary schools in two local education authorities involved inthe Gatsby project, along with three control schools in another local edu-cation authority, are included in this analysis. All teachers in years 1, 3 and5 were observed during maths lessons by trained observers, making a totalof 24 teachers in year 1, 26 in year 3 and 28 in year 5. Inter-observerreliability had earlier been established by all observers observing the samefour lessons as .81 (sig. < .001) using Cohen’s Kappa. A total of 2,128pupils were involved. An observation schedule developed for the project,the Mathematics Enhancement Classroom Observation Record (Schaffer,Muijs, Kitson, & Reynolds, 1998), which was based on a number of exist-ing reliable instruments such as the SSOS, was used in the classroom.During lessons observers counted the number of pupils on/off task every 5minutes, and wrote down what was going on during the lesson in a detailedfashion. This included classroom organisation, which was coded as eitherwhole class interactive teaching (the teacher is teaching the whole class inan interactive way), individual seatwork (pupils are working on their own,for example doing an exercise on a worksheet or in a workbook), smallgroup work (pupils are working collaboratively on a task, in pairs or largergroups), and lecturing the whole class (the teacher is teaching the wholeclass in a non-interactive way, lecturing pupils and not engaging themthrough questioning or discussion). The amount of time in minutes spent


in each of these types of classroom organisation could then be calculatedfor each lesson. After each lesson the occurrence and quality of 65 teacherbehaviours was rated on a scale from 1-5, coded as follows: 1 = behaviourrarely observed, 2 = behaviour occasionally observed, 3 = behaviour oftenobserved, 4 = behaviour frequently observed, 5 = behaviour consistentlyobserved and na = not applicable. It is the rating of behaviours, classroomorganisation and time on task measures which will form the basis of thisarticle. The detailed lesson scripts are more qualitative in nature, and willbe qualitatively analysed at a later date.

As mentioned earlier, these behaviours were hypothesised to form ninesubscales: classroom management, behaviour management, direct teach-ing, individual practice, interactive teaching, varied teaching, mathemati-cal language, classroom climate and constructivist methods (a number ofitems which were culled from constructively oriented theory and researchmentioned above were included to reflect this position. These includeencouraging pupils to use their own problem solving strategies and con-necting different areas of maths to each other and other curriculum areas).Internal consistency of the scale scores (measured using Cronbach’s Al-pha) was over .8 for all scales.

All pupils were tested using the National Foundation for EducationalResearch’s Numeracy tests, which were administered twice, in March andin July of 1998. These tests consist of two sections for each year, onewritten and one mental, and are designed to accord with the English Na-tional Curriculum in mathematics. The scores of the tests had a reliabilitybased on Cronbach’s Alpha of over .8 in all years in this study (see Table1). Data on free school meal eligibility, English comprehension, specialneeds status and gender were also collected from the school.

Table 1. Internal Consistency of Test Scores (Using Cronbach’s Alpha).

Year 1 Year 1 Year 1 Year 3 Year 3 Year 5 Year 5written A written B mental written mental written mental

March .93 .89 .90 .97 .96 .98 .96

July .96 .93 .94 .98 .96 .98 .98


RESULTS

Means and CorrelationsMeans and standard deviations of the test scores and teaching scales usedin these analyses are given in Table 2. Pupil gain scores were calculated bysubtracting scores on the March test from scores on the July test. In Table3 the correlations of the teaching scales, classroom organisation, and timeon task measures with each other and with pupil gain scores from theMarch to July tests are given. As can be seen in Table 3, seven teachingfactors, classroom management, behaviour management, direct instruc-tion, review and practice, interactive teaching, varied teaching and class-

Table 2. Means and Standard Deviations of Test Scores and Scales.

M (SD)

Year 1 written test A - March 7.55 (3.03)Year 1 written test B - March 3.51 (2.94)Year 1 mental test - March 4.37 (3.52)Year 3 written test - March 16.06 (8.15)Year 3 mental test - March 9.18 (5.28)Year 5 written test - March 21.77 (9.47)Year 5 mental test - March 12.96 (5.90)Year 1 written test A - July 9.88 (3.70)Year 1 written test B - July 5.34 (3.53)Year 1 mental test - July 7.32 (4.26)Year 3 written test - July 20.26 (9.46)Year 3 mental test - July 11.57 (5.36)Year 5 written test - July 24.53 (10.64)Year 5 mental test - July 15.04 (6.66)Classroom Management 15.39 (3.09)Behaviour Management 13.79 (2.85)Direct Teaching 28.85 (5.81)Individual Practice 13.85 (2.48)Interactive Teaching 47.92 (8.48)Constructivist Methods 10.58 (3.34)Mathematical Language 6.87 (1.50)Varied Teaching 9.45 (2.69)Classroom Climate 27.03 (4.41)Time on Task 90.55 (6.32)Percent Whole Class Interactive 51.63 (14.30)Percentage seatwork 31.45 (13.30)Percentage Small Group work 5.55 (11.22)Percentage Whole Class Lecture 3.85 (5.16)Percentage Transitions 12.04 (6.75)


room climate are consistently significantly positively correlated with pu-pil gain scores in all 3 years. Correlations were strongest in years 3 and 5on the written test. All these factors were significantly correlated with gainscores in all years. The use of correct mathematical language was signifi-cant only in year 1 and on the written test in year 5, and the use ofconstructivist methods was not significant in any of the tests except theyear 3 mental test, where the relationship was negative. Possibly, positiveeffects of these methods may take longer to translate into achievementgains than the 3-month period discussed here.

Time on task percentage and percentage of time spent on seatwork,teaching the whole class interactively, lecturing the whole class, smallgroup work and percentage of time spent on transitions were not stronglyrelated to pupil gain scores, although some weak positive correlations withgain scores and some weak negative relations with seatwork were found inyears 1 and 3, but not in year 5. Time on task was likewise only weakly

Table 3. Pearson Correlation Coefficients Teacher Behaviour Scales - Pupil Gain Scores.** = significant at the .01 level * = significant at the .05 Level.

Scales Year 1 Year 1 Year 1 Year 3 Year 3 Year 5 Year 5written written mental written mental written mental

test test test test test test testform A form B

Classroom Management .12** .21** .26** .34** .15** .34** .17**Behaviour Management .13* .19** .25** .40** .16** .32** .15**Direct Teaching .24** .22** .32** .32** .14** .36** .22**Individual Practice .18** .17** .26** .35** .15** .34** .21**Interactive Teaching .20** .24** .24** .38** .18** .39** .18**Constructivist Methods .09 .03 .07 .04 –.18** .03 –.09Mathematical Language .22** .19** .12* –.01 .09 .13** .01Varied Teaching .20** .24** .28** .37** .25** .34** .14**Classroom Climate .17** .23** .21** .28** .13** .36** .16**Time on Task .05 .10* .15** .21** .05 .02 .10*Percent Whole ClassInteractive .16** .11** .16** .26** .10* .03 .01Percentage seatwork –.12* –.13** –.13** –.20** –.07 –.06 –.03Percentage SmallGroup work .02 .00 .00 –.14** –.10* –.14** –.12**Percentage WholeClass Lecture –.02 –.05 –.06 –.07 –.22** .30** .07Percentage Transitions –.10* .04 –.06 –.04 –.08 –.13** –.02

284 DANIEL MUIJS AND DAVID REYNOLDST

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related to gain scores, although the direction of the relationship was posi-tive in all cases.

The teaching factors are strongly correlated with one another (all corre-lations over .55), with the exception of the constructivist methods andmathematical language scales, which are weakly to moderately (but stillpositively) related to the other scales. (See Table 4).

The classroom organisation factors, percentage time spent on whole-class interactive teaching, percentage time spent on seatwork, percentagetime spent on group work, percentage time spent on whole class lecturingand percentage time spent on transitions were also related to the teacherbehaviour factors. The main positive relationship was with whole-classinteractive teaching, which was significantly correlated with the seven‘behaviourist’ effective teaching factors (termed effective teaching scale).Time spent on seatwork was (more weakly) negatively correlated with theteaching factors, as was time spent on small group work and time spent ontransitions. Lecturing the whole class was only weakly related to the effec-tive teaching factors, but was more strongly positively related to the use ofmathematical language and constructivist methods. Time on task was sig-nificantly positively related to all nine teaching factors, with correlationsranging from .32 to .59.

Multilevel AnalysesIn order to see whether the teaching factors are still related to pupil testscore gains once pupil background is controlled for, multilevel modelswere computed. This technique partitions the variance between the differ-ent levels at which it is measured (e.g., pupils and classrooms), thus bothattenuating the problem of underestimating standard errors in single-levelanalyses of clustered sample data, and allowing us to look at whether thereare actually significant differences in pupil gain scores at the classroomlevel before enabling us to look at the explanatory power of the variablesat the different levels.

In these multilevel analyses, we will look at pupil scores on the July1998 tests, controlling for performance on the March tests, which areentered as a predictor of July test scores in order to gauge the effect of thepredictor variables on pupil gains in mathematics. We will also introducethe main pupil background variables, free school meal eligibility (a dum-my variable: eligible or not), gender, English comprehension (English firstlanguage or not), and the pupil’s age in months at the time s/he took thetest. In a second phase, rather than entering the effective teaching scalesseparately, a composite effective teaching variable will be entered (calcu-lated by summing the scores on the scales for classroom management,


behaviour management, direct teaching, individual practice, interactiveteaching, varied teaching and classroom climate), this to help avoid multi-collinearity which could otherwise result from using the highly intercorre-lated teaching scales as predictors in the analyses. Time on task and per-centage whole class interactive teaching will also be introduced, as willconstructivist teaching and use of mathematical language. A dummy vari-able indicating whether or not the school was involved in the Gatsbyproject (interven) will also be added. Results can be found in Tables 5–7.

As can be seen in Table 5, the percentage of unexplained variance inpupil attainment in maths between March and July at the classroom levelranged between 7.2% (year 5 mental test) and 23.4% (year 1 mental test),and was significant in all analyses. This does not, of course, necessarilymean that this variance is all down to teacher factors, as classrooms differfrom each other on other dimensions as well.

In the second model the pupil background variables free school mealeligibility, pupil age at test, English comprehension and gender were add-ed.

These variables (see Table 6) explain less than 5% of the variance ingains on the tests from March to July. Free school meal eligibility was themost consistently significant variable, being significant in year 1 and onthe written test in year 5, and almost reaching significance on both tests in

Table 5. Multilevel Model 1: Intercept Only.

Year Year 1 Year 1 Year 3 Year 3 Year 5 Year 5written written mental written mental written mental

A B

FIXED:Intercept 4.03 3.09 4.68 5.43 4.88 6.25 4.86(Constant) (0.42) (0.27) (0.42) (0.58) (0.36) (0.93) (0.56)March 0.78 0.69 0.66 0.90 0.72 0.83 0.78Test Score (0.04) (0.04) (0.04) (0.02) (0.03) (0.03) (0.04)

RANDOMvariance 0.74 0.67 2.58 3.19 1.17 5.96 1.47level 2 (0.31) (0.29) (0.90) (1.14) (0.45) (2.23) (0.68)variance 6.37 6.44 8.45 20.63 10.21 37.93 18.88level 1 (0.41) (0.41) (0.54) (1.20) (0.59) (2.32) (1.16)

% Var. level 2 10.4% 9.5% 23.4% 13.4% 10.2% 13.5% 7.2%


year 3. Age in months was significant for the written test form b in year 1and the written test in year 3, with older pupils making more progress, andEnglish comprehension was significant for progress on the written testform a in year 1 and the written test in year 5, in both cases pupils withEnglish as their first language making more progress.

An interesting finding is the fact that these background variables ex-plained a larger proportion of level 2 variance than of level 1 variance.This points to the homogeneity of primary classrooms, which tend torecruit pupils from relatively homogeneous catchment areas with respectto socio-economic status. On the whole though, it has to be pointed outthat these variables did not explain pupils’ progress on the Numeracy testswell. This is not to say that these background variables are unimportant to

Table 6. Multilevel Model 2: Intercept and Pupil Background.


FIXED:Intercept –0.35 –1.85 0.25 –11.79 –0.31 –0.98 –2.80

(2.67) (2.71) (3.17) (5.56) (3.93) (9.32) (6.56)March test scores 0.73 0.66 0.63 0.88 0.70 0.81 0.76

(0.05) (0.05) (0.04) (0.03) (0.03) (0.03) (0.04)Age in Months 0.04 0.06 0.04 0.17 0.05 0.05 0.06

(0.03) (0.03) (0.04) (0.06) (0.04) (0.07) (0.05)Gender –0.42 –0.40 –0.03 –0.34 0.29 –0.17 0.14

(0.22) (0.22) (0.26) (0.37) (0.26) (0.53) (0.37)English 1.27 0.39 0.58 1.32 0.70 2.26 1.09comprehension (0.51) (0.51) (0.62) (0.78) (0.54) (0.98) (0.66)FSM eligibility 0.75 0.46 0.80 0.72 0.57 1.34 0.45

(0.25) (0.26) (0.29) (0.42) (0.30) (0.59) (0.41)

RANDOMLevel 2 intercept 0.64 0.61 2.39 3.16 1.11 4.33 1.14

(0.27) (0.27) (0.85) (1.13) (0.43) (1.57) (0.50)Level 1 intercept 6.16 6.34 8.32 20.16 10.10 37.55 18.84

(0.39) (0.40) (0.53) (1.71) (0.59) (2.30) (1.15)Percentagevariance level 2 9.4% 8.8% 22.3% 13.6% 9.9% 10.3% 5.7%

EXPLAINED VAR. Compared to prev. modelTotal 4.4% 2.2% 2.9% 2.1% 1.5% 4.6% 1.8%Level 2 13.5% 11.6% 7.4% 0.1% 5.1% 27.3% 22.4%Level 1 3.3% 1.6% 1.5% 2.3% 1.1% 1.0% 0.2%


pupils’ achievement in mathematics. Rather, it seems likely that the ef-fects of pupil background are already incorporated into their performanceon the March test, and therefore do not strongly affect their progress in theshort time-span studied here.

Table 7. Multilevel Model 6. Intercept, Pupil Background and Classroom Variables.


FIXED:Intercept –0.36 –5.11 –3.06 –21.16 –1.26 0.49 –7.45

(3.27) (3.36) (5.30) (6.58) (5.14) (10.42) (7.91)March test score 0.71 0.64 0.62 0.89 0.71 0.82 0.76

(0.02) (0.04) (0.04) (0.02) (0.03) (0.03) (0.04)Gender –0.38 –0.36 0.00 –0.41 0.28 –0.34 0.09

(0.22) (0.22) (0.26) (0.36) (0.26) (0.52) (0.37)English 1.00 –0.01 0.39 1.36 0.73 1.97 0.88comprehension (0.50) (0.51) (0.62) (0.72) (0.53) (0.91) (0.68)FSM eligibility 0.71 0.42 0.81 1.00 0.69 –1.19 –0.39

(0.25) (0.26) (0.29) (0.40) (0.29) (0.57) (0.41)Age in months 0.06 0.05 0.04 0.15 0.05 0.05 0.06

(0.03) (0.03) (0.04) (0.01) (0.04) (0.07) (0.05)Intervention –0.05 –0.76 –2.10 0.08 0.16 –0.24 0.08

(0.70) (0.73) (1.37) (0.60) (0.53) (1.03) (0.08)Whole-class 0.01 0.015 0.04 0.01 –0.01 –0.00 0.02interactive (0.01) (0.010) (0.02) (0.02) (0.02) (0.02) (0.02)Time on task 0.08 0.00 –0.02 0.00 –0.04 0.16 0.01

(0.02) (0.23) (0.04) (0.04) (0.04) (0.06) (0.05)Constructivist 0.16 –0.03 0.09 0.04 –0.12 0.10 –0.08methods (0.06) (0.06) (0.11) (0.06) (0.06) (0.09) (0.08)Mathematical 0.09 0.28 –0.12 0.01 0.10 –0.20 –0.08language (0.14) (0.14) (0.26) (0.20) (0.18) (0.17) (0.15)Effective Teaching 0.02 0.03 0.05 0.06 0.03 0.08 0.04

(0.01) (0.01) (0.02) (0.01) (0.01) (0.01) (0.01)

RANDOMLevel 2 intercept 0.06 0.08 0.92 0.18 0.41 0.00 0.38

(0.10) (0.10) (0.39) (0.28) (0.24) (0.00) (0.34)Level 1 intercept 6.15 6.34 8.32 20.22 10.12 37.34 18.82

(0.39) (0.40) (0.53) (1.17) (0.59) (2.24) (1.15)% variance level 2 1.0% 1.2% 9.9% 0.9% 3.9% 0.0% 2.0%

EXPLAINED VAR. Compared to prev. model (total)Total 8.6% 7.6% 13.7% 12.5% 6.1% 10.9% 3.9%Level 2 90.6% 86.9% 61.5% 94.3% 63.1% 100% 66.7%Level 1 0.2% 0.0% 0.0% 0.0% 0.0% 0.5% 0.0%


In the third model the effective teaching scale and the other classroomteaching factors were introduced in order to ascertain whether they wereable to explain part of the remaining between-classroom variance oncepupil background had been taken into account.

It is clear from the results presented in Table 7 that the effective teach-ing scale explains a significant percentage of between-classroom variance,once pupil characteristics have been controlled for. This variable washighly significant in all years and on all tests, and the teaching variablestogether explained between 61.5% and 100% of the remaining between-classroom variance, causing between-classroom variation to become in-significant in most cases. Explained variance was highest in year 5. Of theother two variables, only the percentage of time spent teaching the wholeclass interactively as opposed to time spent on seatwork or group work,was (weakly) significant in the year 1 mental test analyses but not in thehigher years. Time on task was only significantly positively related toprogress in mathematics on the written test in year 5 and the written testform a in year 1. Constructivist teaching methods were significantly posi-tively related to gains in year 1 on the written test form a and negatively togains on the mental test in year 3. Use of mathematical language wassignificantly related to gains on the form b test in year 1.

These percentages make clear that the aggregate effect of effectiveteaching behaviours is highly practically significant, explaining as it doesin this sample the majority of the variance in between-classroom test gainsnot explained by pupil background factors. The significance of the differ-ence between effective teaching behaviours is illustrated in Table 8, fromwhich it becomes clear that, holding all other variables constant, beingtaught by the teacher scoring highest as opposed to the teacher scoringlowest on the effective teaching scale can increase a pupil’s scores on the

Table 8. Predicted Differences in Scores on the July Numeracy Tests for Pupils Taught bythe Weakest and Strongest Teachers, Holding All Other Variables Constant.


Difference in 2.05 2.05 4.46 5.70 2.85 8.57 3.88point score% Difference 10.7% 14.6% 24.7% 12.9% 12.4% 17.8% 12.9%Test range 0-19 0-14 0-18 0-44 0-23 0-48 0-30


test by between 10% and 25%. It would seem that the effective teachingscale, which accounted for by far the majority of the predictive power inthese analyses, is quite well able to distinguish effective from less effec-tive teachers at the primary level and for the elements studied here, al-though it must be remarked that, due to the short time elapsed between thetwo testing instances, the total variance in gain scores to be explained wasquite limited.

Structural Equation ModelsWhile these analyses did not find much evidence for a direct effect ofclassroom organisation (whole class interactive) or time on task on pupilgains, this does not necessarily invalidate the aforementioned hypothesis,as long as there is evidence of an indirect relationship as hypothesisedabove. Some tentative evidence for such a relationship may be garneredfrom the correlations between these variables and pupil gain scores, asevident in Table 3 above.

To test whether the theoretical model we proposed above, in whichwhole-class teaching affects effective teaching behaviours and time ontask, which in turn affect pupil gain scores, it was decided to use structuralequation modelling. This technique, which measures the fit of pre-speci-fied directional relationships between the variables to the covariance ma-trix used, allows us to model directional relationships between variables,while also taking into account measurement error in the data. The LISRELprogramme was used to calculate the models.

Structural equation models were tested for each of 3 years. In order toallow the sample sizes to be such that the Chi-Square test would give us anaccurate estimate of goodness of fit, the sample was split in two for each ofthe 3 years. In this way it was also possible to obtain an impression of thereliability of the model.

In this model, in which only the classroom factors were included, Marchtest scores were posited to strongly affect July test scores, while the teach-ing factors in turn were hypothesised to also have a significant effect onthe July test scores, albeit clearly weaker than that of March test scores.Classroom organisation (whole-class interactive teaching) was hypothe-sised to affect effective teaching behaviours, and thus exert an indirectinfluence on pupil gain scores (July test scores controlled for March testscores). This because it was hypothesised that spending more time teach-ing the whole class would allow teachers to display more effective teach-ing behaviours than allowing pupils to work on their own for a larger partof the lesson. In view of the correlation between the constructivist scaleand the mathematical language scale, it was decided to let both load on a


latent non-behavioural teaching variable. This was hypothesised to affectJuly test scores. Whole-class teaching was expected to lead to higher timeon task rates, as it was hypothesised that spending more time teaching thewhole class as opposed to allowing pupils to work on their own for most ofthe lesson would allow the teacher to monitor pupils’ behaviour moreeasily, and would be less likely to lead to pupils getting distracted from thelesson. Time on task was in turn hypothesised to help create the conditionsfor effective teaching to occur. While we would assume that the relation-ship between effective teaching and time on task would in fact be recipro-cal, the modelling of reciprocal effects in LISREL is highly problematic atpresent. Models were tested separately for the 3 years studied. The errorsof the teaching factors were allowed to correlate with each other, as, due tothe way they were measured (by one classroom observer during the sameobservation), they were expected to covary to some extent. These modelsare depicted in Figures 1 - 6. A problem with this model is the fact thatthey do not take into account the multilevel structure of the data, as class-room level data is in this case disaggregated to the individual level. Thisclearly is cause for some caution when interpreting the results, as standarderrors, for example, are liable to be underestimated.

As can be seen in Table 9, the model fits in years 3 and 5, and the similarfit indices for the two subsamples suggests this model is quite stable in theseyears. Chi-square remained significant for the year 1 models, however, sug-gesting that the hypothesis is less well supported in this year. This confirmsthe weaker relationship between teaching factors and progress in mathemat-ics in year 1 found in the analyses mentioned above. Most paths suggestedin the theoretical model also reached significance, the only one failing to doso in a number of models being that from time on task to non-behaviouralteaching, which was not significant in year 5, and the path from non-behav-ioural teaching to test scores which was not significant in any of the models.It also has to be remarked that the loadings of the non-behavioural scales onthat factor were weak in a number of models.

Some other differences were found between the years in the strength ofthe significant paths in the models: the effect of whole-class teaching onteaching behaviours and time on task was strongest in year 1, and theeffect of effective teaching on test scores was strongest in year 3 andweakest in year 1. The path from whole-class interactive teaching to timeon task was not significant in year 1. Overall these differences suggest thatyear 1 differs somewhat from the older years.

However, despite these differences the overall similarity of the modelsis striking. As would be expected, especially in light of the short periodthat has elapsed between testing, test scores are quite stable over time.


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However, over and above the effect of test stability, differences in pupils’progress clearly do occur, and they would seem to be affected by teachereffectiveness, the path from which to July test results is significant in allcases. Effective teaching behaviours are in turn influenced by both time ontask and classroom organisation, and classroom organisation (the percent-age of time spent on whole class interactive teaching) in turn stronglyinfluences time on task. The only paths that failed to reach significance ina number of cases were the ones involving non-behaviourist methods.Both the influence of non-behaviourist methods on achievement and theinfluence of time on task on non-behaviourist methods were not stronglysupported by these models.

DISCUSSION AND CONCLUSION

This study clearly points to the importance of the effective teaching be-haviours as outlined in the teacher effectiveness research to successfulmathematics teaching in the UK. It has become clear, that while individualbehaviours may only explain a very small percentage of variance in pupilgains over time, taken together they are significant. This study also lendssupport to the view that these behaviours do indeed occur together ineffective teachers, thus forming a cluster of effective teaching behaviours.

The relationship of pupil gains to classroom organisation would seemto be an indirect one. It would seem that, rather than directly affectingpupil progress, whole-class teaching creates the conditions for effectiveteaching to occur, a result that could explain the fact that the amount oftime spent teaching the whole class, while found to be effective in a numberof American studies, was only found to be significant in 3 out of 29

Table 9. Structural Equation Model Fit Indices. Italics indicate non-significance at the .01level.

Chi Square (df) RMSEA GFI NFI CFI

Year 1, odd cases 174.49 (77) 0.06 0.94 0.96 0.98Year 1, even cases 193.35 (77) 0.07 0.93 0.96 0.97Year 3, odd cases 69.49 (56) 0.03 0.97 0.99 1.00Year 3, even cases 72.61 (56) 0.04 0.97 0.98 0.99Year 5, odd cases 79.64 (56) 0.04 0.97 0.98 0.99Year 5, even cases 77.66 (56) 0.04 0.97 0.98 0.99


primary level studies in the Netherlands reviewed by Scheerens andCreemers (1996). More whole-class teaching allows teachers to be effec-tive in a way that individualised approaches do not. That this findingemerged even from a study in which, through its nature, the vast majorityof teachers used a whole-class interactive approach is significant. Howev-er, it would be dangerous to generalise this finding ad infinitum. It is clearthat while spending a large proportion of the lesson teaching the wholeclass is beneficial, individual or group practice (as evidenced by the sig-nificance of this teacher scale) is clearly necessary to enhance pupil learn-ing. To spend 100% of all lessons teaching the whole class together wouldprobably be harmful. In this sense, the relationship between classroomorganisation and gains is probably a curvilinear one. The mistake someAmerican educators have made, in extrapolating findings on the impor-tance of academic learning time to such an extent that play times havebeen virtually abolished, should not be repeated here.

The centrality of the teacher in pupils’ learning processes is clear, how-ever. Any approach designed to let pupils learn on their own, with theteacher acting merely as a ‘facilitator’ is likely to fall short of the cognitivedemands of primary age children.

As is clear from the items included in the scales, it would be wrong todescribe this whole-class interactive teaching style as a ‘chalk and talk’drill-and-practice approach. The importance of engaging with students ata cognitively higher as well as lower level is clear from the inclusion ofitems such as asking open questions, allowing multiple answers etc. Assuch, this teaching style by no means precludes attention to higher-levellearning goals. An explanation for the value of these behaviours and relat-ed classroom observation factors can be found in the need for pupils at thisage for a good deal of explicit cognitive structuring (Stillings et al., 1995).

While this study thus clearly supports whole-class interactive teaching,a number of limitations of this study need to be pointed out.

The tests used in this study (while reflecting the English National Cur-riculum) and the short-term nature of this study, mean that we have stud-ied typical basic skills achievement gains only. It is not clear from thisstudy whether these effective teaching behaviours are also, or as strongly,related to longer-term and more cognitive outcomes, such as independentlearning goals or metacognitive development. It is possible, and indeedlikely, that other teaching methods are needed alongside the whole-classinteractive approach found to be effective in this study to attain thesegoals. Joyce and Weil (1996), for example, point to the possible utility ofmethods such as synectics or thinking skills approaches for achieving thistype of goal, while Slavin’s (1996) research points to the importance of


cooperative small group work, an approach not often used in Englisheducation, as evidenced by the fact that less than 5% of lesson time wasspent on average on group work (which was usually not organised to thestandards suggested by Slavin) in the lessons observed in this study.

These comments do not, however, invalidate the results of this study,unless one believes that basic numeracy skills are entirely unimportant inchildren’s mathematics learning. It should be clear that both basic skillsand long-term cognitive goals have their place in any learning situation,and an either/or approach to this problem is unwise in the extreme.

A further limitation is that while proper hierarchical modelling tech-niques were used for the multilevel regression model, this was not the casefor the structural equation model, for which the teacher level variableswere disaggregated to the individual level. This was due to the fact thatwhen using a two-level structural equation modelling approach, one isbasically using a multiple group approach, which means that for bothgroups/levels the assumptions of the approach have to be met. This in-cludes sample size, which in situations of perfect multivariate normalityof the data set may be as low as 50, but under more prevalent conditions isusually closer to 200. Our sample clearly does not meet this requirement atthe classroom level.

The short time-span between the two testing occasions is a furtherreason for caution with respect to these findings, and also explains why thetotal amount of variance in pupil gains is relatively small. This variance isfurther limited by the fact that schools in the Gatsby Mathematics En-hancement Project Primary were encouraged to engage in whole-classinteractive teaching styles, thus limiting the variance in classroom organi-sation as well as in teaching styles employed. The fact that all the teachersin the project were trained in whole-class teaching methods in the Gatsbyproject is presumably also responsible for the high overall time on tasklevels in this study.

Also, no data on teacher beliefs and attitudes were collected in this initialstage of the research. In view of the findings of amongst others Askew et al.(1997), this is a clear limitation, which will be addressed in the next phaseof the project in which we will be looking at teacher beliefs about and atti-tudes towards mathematics, teachers and mathematics teaching and at theirsubject knowledge as well as continuing to look at behaviours.

A possible worry about this method of teaching is the finding in thisstudy that boys made significantly more gains than girls did on manyoccasions. While it is not clear from this study whether this is a result ofthe teaching style, there is a danger that boys might dominate interactionwith the teacher in an interactive classroom. This is clearly an issue that


needs addressing, and teachers should be pointed to the need to involvegirls in any interaction in the classroom.

This research has clear practical implications for teacher training inmathematics. If it is possible to identify clusters of behaviour that appearto be significantly related to pupil gains, then it would clearly be benefi-cial to devote a significant amount of energy to training teachers in the useof these methods, especially in the early phases of teacher training. Thiswould also tie in well with the British government’s literacy and numeracystrategies, which support whole-class teaching methods. As Brophy (1986)has pointed out, not all the results from this and similar studies are imme-diately easily transferable to the classroom, but, as the research is based innaturalistic classroom settings, it is certainly more so than a lot of thepopular psychological research that is often used to justify impracticalindividualised and child-centred techniques. Knowing how children con-struct knowledge may be useful for teachers, but the practical utility ofbasing teaching strategies on such individual psychological theories mustbe questioned.

In future, longer-term research is needed to see how the relationshipsdevelop, and whether there is stability over the longer term, both withrespect to measures of teacher effectiveness and to relationships betweenfactors and achievement.

It is also imperative in future to do more fine-grained research, and toattempt to find ‘thresholds’ for the behaviours identified here as effective.Thus, while the results of this study would suggest that high levels ofwhole-class interactive teaching are beneficial, it would, as mentionedabove, be useful to find out more about where overuse of a particulareffective technique becomes harmful.

Future research also needs to take into account relationships betweenteacher behaviours and teacher beliefs and attitudes, which research (e.g.,Askew et al., 1997) has found to be related to teacher efficacy. As onecould also expect that teacher behaviour is related to teacher beliefs, it isessential to study these relationships further.

Also, while this (and a lot of the American teacher effectiveness re-search, for example Good, Grouws, et al., 1983) has concentrated on math-ematics, it is obviously of equally strong interest to study what works inother subjects, such as English, science, history etc. Extension of researchto other age groups would likewise seem appropriate. Finally, looking atthe effectiveness of teachers in fostering different outcomes (e.g., highercognitive level processes) might also be fruitful.

Overall, then, it would seem that there is plenty of work to do forresearchers who want to extend the British and international teacher effec-


tiveness research base, and thus contribute to a truly research-based pro-fession of teaching in the UK and beyond.

ACKNOWLEDGEMENT

This study was made possible thanks to funding by the Gatsby Charitable Foundation, oneof the Sainsbury Family Charitable Trusts. The authors also wish to thank all participatingschools and teachers.

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