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Effective Teachers of Numeracy in Primary Schools: Teachers' Beliefs, Practices and Pupils' Learning.

Mike Askew, Margaret Brown, Valerie Rhodes, Dylan Wiliam, David Johnson
King's College, University of London

Paper presented at the British Educational Research Association Annual Conference
(September 11-14 1997: University of York)

Abstract

This paper reports on part of a study examining the links between teachers' practices, beliefs and knowledge and pupil learning outcomes in the development of numeracy with pupils aged five to eleven. From a sample of 90 teachers and 2000 pupils, we developed detailed case studies of 18 teachers. As part of these case studies we explored the teachers' beliefs about what it means to be numerate, how pupils become numerate and the roles of the teachers. From the data three sets of belief orientations were identified: connectionist, transmission and discovery. Results from pupil assessments suggest that there was a connection between teachers demonstrating strong orientation to one of these sets of beliefs and pupil numeracy gains.

1 Aims of the study

The aims of the study Effective Teachers of Numeracy, funded by the UK's Teacher Training Agency (TTA) were to:

  1. identify what it is that teachers of five to eleven year olds know, understand and do which enables them to teach numeracy effectively;

  2. suggest how the factors identified can be more widely applied.

The working definition of numeracy used by the project was a broad one:

Numeracy is the ability to process, communicate and interpret numerical information in a variety of contexts.

Evidence was gathered from a sample of 90 teachers and over 2000 pupils on what the teachers knew, understood and did and outcomes in terms of pupil learning.

Studies have pointed to the importance of establishing of a particular classroom culture (Cobb, 1986) , raising the issue of teachers' belief systems about mathematical knowledge, how it is perceived as generated and learnt, and the impact upon pupils' learning. It may be that beliefs about the nature of the subject are more influential than mathematical subject knowledge per se (Lerman, 1990; Thompson, 1984) .

Many studies, particularly in the USA, focus on effective classroom practice and routines (Berliner, 1986) but research demonstrates the difficulty that teacher experience in adopting new practices without an appreciation of and belief in the underlying principles (Alexander, 1992) . Further, teachers may have adopted the rhetoric of 'good' practice in teaching mathematics without changes to their actual practices (Desforges & Cockburn, 1987) . While teachers' classroom practices and subject knowledge were also foci of this research, this paper concentrates on the findings related to teachers' belief systems. (For full details of the research see Askew et al., 1997).

2 Identifying effective teachers of numeracy

Careful identification of teachers believed to be effective in teaching numeracy was crucial to this study. The idea that effective teachers are those who bring about identified learning outcomes was our starting point for the project. We decided that as far as possible the identification of effective teachers of numeracy would be based on rigorous evidence of increases in pupil attainment, not on presumptions of 'good practice'.

From an initial sample size of all the primary schools in three local education authorities (some 587 schools), together with Independent (private) schools, we selected eleven schools, providing a sample of 90 teachers. We selected the majority of these eleven schools on the basis of available evidence (national test scores, IQ data, reading test scores and baseline entry assessments) suggesting that the teaching of mathematics in these schools was already effective.

A specially designed test ('tiered' for different age ranges) of numeracy was administered to the classes of these 90 teachers, first towards the beginning of the autumn term 1995, and again at the end of the spring term 1996 (classes of five year olds were only assessed the second time). Average gains were calculated for each class, providing an indicator of 'teacher effectiveness' for the teachers in our sample.

In order to broadly classify the relative gains, the teachers were grouped into three categories of highly effective, effective, or moderately effective. This classification was made by putting the classes in rank order within year groups according to the average gains made (adjusted to take into account the fact that it was harder for pupils to make high gains if their initial test score was high). The cut-off points between high, medium and low gains were made on pragmatic grounds, so that classes in each year group fell into three roughly equal groups but avoiding any situation where classes with nearly equal adjusted gains were allocated to different groups. The groups were not based on any predetermined quantitative differences between the classes based on expectations of what a 'medium' gain should be.

3 Teacher case study data

Research on the links between knowledge, beliefs and practice suggested a mix of techniques to elicit teachers' knowledge and understanding backed up by classroom observation to examine actual practices. From the sample of 90 teachers we worked closely with 18 teachers who formed our case study teachers providing data over two terms on classroom practices together with data on teacher beliefs about, and knowledge of, mathematics, pupils and teaching. These teachers were identified in advance of the second round of pupil assessment, and chosen through discussion with head teachers and, where appropriate, with advice from the LEA inspectors and advisors. While the emphasis was on identifying effective teachers, the group of 18 were chosen so that their pupils were evenly distributed across ages 5 to 11 (year groups 1-6).

3.1 Classroom observations

In total, 54 lessons were observed, three for each of the case study teachers. Data gathered included a focus on:

3.2 Case study teacher interviews

Fifty-four interviews were conducted, three for each case study teacher:

The data were analysed using qualitative coding methods and the constant comparative method to build up models of belief systems (Lincoln & Guba, 1985; Miles & Huberman, 1984; Strauss & Corbin, 1990)

4 Orientations in teachers beliefs.

From the analysis of the case study data three models of sets of beliefs that emerged as important in understanding the approaches teachers took towards the teaching of numeracy:


             connectionist         transmission          discovery              

Beliefs      Being numerate        Being numerate        Being numerate         
about what   involves:             involves:             involves:              
it is to be  • using both          • the ability to      • finding the answer   
a numerate   efficient and         perform set           to a calculation by    
pupil        effective methods of  procedures or         any method;            
             calculation ;         routines;                                    

             • confidence and      • confidence and      • confidence and       
             ability in mental     ability in paper and  ability in practical   
             methods;              pencil methods;       methods;               

             • selecting a method  • selecting a method  • selecting a method   
             of calculation on     of calculation        of calculation         
             the basis of both     primarily on the      primarily on the       
             the operation and     basis of the          basis of the           
             the numbers           operation involved;   operation involved;    
             involved;                                                          

             • awareness of the    • confidence in       • confidence in        
             links between         separate aspects of   separate aspects of    
             aspects of the        the mathematics       the mathematics        
             mathematics           curriculum;           curriculum;            
             curriculum;                                                        

             • reasoning,          • able to 'decode'    • being able to use    
             justifying and,       context problems to   and apply mathematics  
             eventually, proving,  identify a            using practical        
             results about         particular routine    apparatus.             
             number.               or technique.                                

Beliefs      Becoming numerate is  Becoming numerate is  Becoming numerate is   
about        a social activity     a individual          a individual activity  
pupils and   based on              activity based on     based on actions on    
how they     interactions with     following             objects.               
learn to     others.               instructions.         • Pupils need to be    
become       • Pupils learn        • Pupils learn        'ready' before they    
numerate     through being         through being         can learn certain      
             challenged and        introduced to one     mathematical ideas.    
             struggling to         mathematical routine                         
             overcome              at a time and                                
             difficulties.         remembering it.                              

             • Most pupils are     • Pupils vary in      • Pupils vary in the   
             able to become        their ability to      rate at which their    
             numerate.             become numerate.      numeracy develops.     

             • Pupils have         • Pupils' strategies  • Pupils' own          
             calculating           for calculating are   strategies are the     
             strategies but the    of little importance  most important:        
             teacher has           - they need to learn  understanding is       
             responsibility for    standard procedures.  based on working       
             helping them refine                         things out yourself    
             their methods.                                                     

             • Misunderstandings   • Misunderstandings   • Misunderstandings    
             need to be            are the result of     are the result of      
             recognised, made      failure to 'grasp'    pupils not being       
             explicit and worked   what was being        'ready' to learn the   
             on.                   taught and need to    ideas.                 
                                   be remedied by                               
                                   reinforcement of the                         
                                   'correct' method.                            

Beliefs      • Teaching and        • Teaching is seen    • Learning is seen as  
about how    learning are seen as  as taking priority    taking priority over   
best to      complementary.        over learning.        teaching.              
teach        • Numeracy teaching   • Numeracy teaching   • Numeracy teaching    
pupils to    is based on dialogue  is based on verbal    is based on practical  
become       between teacher and   explanations so that  activities so that     
numerate     pupils to explore     pupils understand     pupils discover        
             each others'          teachers' methods.    methods for            
             understandings.                             themselves.            

             • Learning about      • Learning about      • Learning about       
             mathematical          mathematical          mathematical concepts  
             concepts and the      concepts precedes     precedes the ability   
             ability to apply      the ability to apply  to apply these         
             these concepts are    these concepts        concepts               
             learned alongside                                                  
             each other.                                                        

             • Connections         • Mathematical ideas  • Mathematical ideas   
             joining mathematical  need to be            need to be introduced  
             ideas needs to be     introduced in         in discrete packages.  
             acknowledged in       discrete packages.                           
             teaching.                                                          

             • Application is      • Application is      • Application is best  
             best approached       best approached       approached through     
             through challenges    through 'word'        using practical        
             that need to be       problems: contexts    equipment              
             reasoned about.       for calculating                              
                                   routines                                     


Table 1: Key distinctions between connectionist, transmission and discovery orientations towards teaching numeracy.

These orientations are "ideal types". No one teacher is likely to fit exactly within the framework of beliefs of any one of the three orientations. Many will combine characteristics of two or more.

However, it was clear that those teachers with a strong connectionist orientation were more likely to have classes that made greater gains over the two terms than those classes of teachers with strong discovery or transmission orientations.

Analysis of the data revealed that some teachers were more predisposed to talk and behave in ways that fitted with one orientation over the others. In particular, Anne, Alan, Barbara, Carole, Claire, Faith (the teacher initial matches the school code, so Anne and Alan are from same school), all displayed characteristics indicating a high level of orientation towards the connectionist view. On the other hand, Beth and David both displayed strong discovery orientations, while Elizabeth and Cath were both clearly characterised as transmission orientated teachers.

Other case study teachers displayed less distinct allegiance to one or other of the three orientations. They held sets of beliefs that drew in part from one or more of the orientations. For example, one teacher had strong connectionist beliefs about the nature of being a numerate pupil but in practice displayed a transmission orientation towards beliefs about how best to teach pupils to become numerate.


                      Highly effective    Effective           Moderately          
                                                              effective           

Strongly              Anne,    Alan                                               
Connectionist         Barbara                                                     
                      Carole,   Faith                                             


Strongly                                                      Cath                
transmission                                                  Elizabeth           


Strongly discovery                                            Beth                
                                                              David               


No strong             Alice               Danielle, Eva       Brian               
orientation                               Dorothy, Fay        Erica               


Table 2. The relation between orientation and effectiveness

The connection between these three orientations and the classification of the teachers into having relatively high, medium or low mean class gain scores suggests that there may be a relationship between pupil learning outcomes and teacher orientations.

5 Links between orientation and practice

5.1 Orientation and the role and nature of mental strategies in pupils becoming numerate

All the teachers, whether leaning towards a connectionist, transmission or discovery orientation saw some aspects of mental mathematics as important. Knowing basic number bonds and multiplication facts provided a baseline of expectations within all three orientations.

However, the connectionist orientated teachers viewed mental mathematics as going beyond this recall of number facts. Mental mathematics did not involve simply knowing number bonds but having a conscious awareness of connections and relationships to develop mental agility.

This mental agility meant that for the connectionist teachers mental mathematics also involved the development of flexible mental strategies to handle efficiently number calculations. Working on mental strategies, they believed, laid foundations that extended the pupils' levels of competency. Developing confidence in flexible mental methods meant that pupils would be able to tackle calculations for which methods had not been taught.

5.2 Orientation and teacher expectations

The connectionist orientated teachers placed strong emphasis on challenging all pupils. They believed that pupils of all levels of attainment had to be challenged in mathematics. Being stretched was not something that was not restricted to the more capable pupils. They had high levels of expectations for all pupils irrespective of ability. Intelligence was not seen as static and all pupils were regarded as having the potential to succeed.

In contrast the transmission and discovery orientated teachers may provide challenge for the higher attaining pupils but structured the mathematics curriculum differently for lower attaining pupils.

5.3 Orientation and style of interaction

The connectionist teachers' lessons were generally characterised by a high degree of focused discussion between teacher and whole class, teacher and groups of pupils, teacher and individual pupils and between pupils themselves. The teachers displayed the skills necessary to manage effectively these discussions. The teachers kept pupils focused and on task by organising these discussions around problems to solve, or sharing methods of carrying out calculations.

In school A, one of the most effective schools, there was a consistent approach to interacting with pupils throughout the years. Right from age five pupils were expected to be able to explain their thinking processes. Because the pupils were explaining, rather than simply providing answers to questions that the teacher already knew the answer to, the lessons were characterised by dialogue. In this discussion both parties, teacher and pupils, were having to listen carefully to what was being said by others. The result was pupils who, by eleven, were confident and practised in sharing their thinking and challenging the assumptions of others.

5.4 Orientation and the role of mathematical application

For the discovery or transmission orientated teachers, application of knowledge involved pupils putting what they had previously learnt into context. Problems presented 'puzzles' where the pupils already have the required knowledge and the challenge is only to sort out which bit to use. Alternatively, problems were a means of demonstrating to pupils the value of what they are learning.

The connectionist orientated teachers also recognised the importance of being able to apply computational skills. But over and above this they did not see it as a necessary pre-requisite that pupils should have learnt a skill in advance of being able to apply it. Indeed, the challenge of an application could result in learning.

6 Discussion

The importance of these orientations lies in how practices, while appearing similar may have different purposes and outcomes depending upon differences in intentions behind these practices.

We would suggest that these orientations towards teaching mathematics need to be explicitly examined in order to understand why practices that have surface similarities may result in different learner outcomes. While the interplay between beliefs and practices is complex, these orientations provide some insight into the mathematical and pedagogical purposes behind particular classroom practices and may be as important as the practices themselves in determining effectiveness.

Other teachers may find it helpful to examine their belief systems and think about where they stand in relation to these three orientations. In a sense the connectionist approach is not a complete contrast to the other two but embodies the best of both them in its acknowledgement of the role of both the teacher and the pupils in lessons. Teachers may therefore need to address different issues according to their beliefs: the transmission orientated teacher may want to consider the attention given to pupil understandings, while the discovery orientated teacher may need to examine beliefs about the role of the teacher.

References

Alexander, R (1992). Policy and Practice in Primary Education. London: Routledge.

Askew, M; Brown, M; Rhodes, V; Wiliam, D & Johnson, D (1997). Effective Teachers of Numeracy: Report of a study carried out for the Teacher Training Agency. London: King's College, University of London.

Berliner, D C (1986). In pursuit of the expert pedagogue. Educational Researcher, 15(7), 5-13.

Cobb, P (1986). Contexts, goals, beliefs and learning mathematics. For the Learning of Mathematics, 6(2), 2-9.

Desforges, C & Cockburn, A (1987). Understanding the mathematics teacher. Lewes: Falmer Press.

Lerman, S (1990). Alternative perspectives of the nature of mathematics and their influence on the teaching of mathematics. British Educational Research Journal, 16(1), 53-61.

Lincoln, Y S & Guba, E G (1985). Naturalistic Inquiry. Newbury Park, California: SAGE Publications.

Miles, M B & Huberman, A M (1984). Qualitative Data Analysis. Beverley Hills: Sage.

Strauss, A & Corbin, J (1990). Basics of qualitative research: grounded theory procedures and techniques. Newbury Park, Cal.: Sage.

Thompson, A G (1984). The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105-127.

This document was added to the Education-line database 28 November 1997