Effective teachers of numeracy

The study was set up to identify:

• key factors which enable teachers to put effective teaching of numeracy into practice in primary schools
• strategies that would enable those factors to be more widely applied in numeracy teaching.

By collecting and analysing data about teachers’ beliefs, knowledge, understanding and practice, the researchers’ intention was to identify what it was that made some teachers of numeracy more effective than others. (A definition of effective numeracy teachers is given in the next section).

Evidence in the form of pupils’ class gains in tests offered the researchers the opportunity not only to triangulate their data but also to assess the extent to which pupils in the schools developed mathematically over the period of the research and provided a measure of teacher effectiveness.

The report incorporates an extensive analysis of the factors that had an effect on the practice of the teachers in the study. This offers teachers a helpful framework, which they might use to reflect on their own practice and the beliefs, knowledge and understanding that underpins it.

Central to the study was the identification of teachers regarded as effective in teaching numeracy. Firstly, the researchers defined effective numeracy teachers as those who help pupils:

• to acquire and apply knowledge of numbers, number relations and number operations based on the integration of understanding, techniques, strategies and application skills
• to learn how to apply their knowledge and skills in a variety of contexts.

The researchers then considered the possible means of judging teacher effectiveness and suggested the following as potential sources of evidence:

• teacher behaviour
• pupil behaviour
• pupil learning outcomes.

They recognised the inherent difficulties of the first two approaches. In the case of teacher behaviour, there was very little information about what sort of teacher practice leads to effective pupil learning. Indeed the research project itself aimed at providing such evidence. The difficulty in measuring teacher effectiveness in terms of pupil behaviour, on the other hand, lay in the fact that learning is difficult to observe directly and there is a lack of evidence about the kinds of pupil response that are indicators of sustained learning.

The researchers defined ‘effective teachers of numeracy’ as those who bring about identified learning outcomes. Teacher effectiveness, of course, could only be related to pupil gains at the end of the research. The researchers applied a two-stage approach to sampling effective teachers, comprising:

• identifying a sample of ‘focus’ schools that appeared to be effective in teaching mathematics
• identifying a sample of teachers, believed by headteachers to be effective in the focus schools.

Data was collected by a number of methods including:

• questionnaires
• classroom observations
• structured and semi-structured interviews
• pupil test results.

What did the research discover about teachers’ beliefs? The researchers found three dominant sets of beliefs and approaches to numeracy teaching and learning amongst the teachers:

• connectionist
• transmission
• discovery.

Having identified these broad patterns the researchers used the resulting models to analyse other data about the pupils and teachers involved in the study. The resulting descriptions of the three approaches and their impact on learning are thus rooted in evidence but are also extended to create models. For clarity, we have described these models in their fully developed forms rather than in the partial forms that emerged from the analysis of the data.

The models describe broad bands or orientations, and cover a number of features of teacher behaviour. Of course, any individual teacher may show some characteristics from each of the three models over the course of his/her teaching, although s/he will usually show patterns of work that fall predominantly within one model.

The researchers suggest that the main ways in which the three orientations differ include beliefs about:

• what constitutes numeracy in pupils
• how pupils can learn to become numerate
• how best to teach pupils to become numerate.

The researchers observed differences among teachers in their approach to numeracy teaching methods. For example, teachers adopting a mainly transmission approach valued methods that were based on the use of standard procedures and routines. Teachers who adopted mainly discovery methods, advocated a practical approach to problem solving Those with a connectionist orientation encouraged methods that placed the highest priority on mental methods. They also regarded it as important that pupils were aware of different methods of calculation and were able to choose methods in relation to their effectiveness and efficiency in solving the problem. Those with a transmission approach emphasised paper and pencil methods.

The researchers observed differences in the way teachers treated pupils as learners. Teachers whose approach was broadly transmission or discovery emphasised either teaching or learning although they differed greatly in the emphasis they placed on one or the other. The former placed emphasis on teachers teaching pupils to follow instructions, while the latter regarded individual activity as central to pupils' learning of numeracy.

Teachers with a connectionist orientation emphasised the complementary nature of teaching and learning and valued classroom activity, which involved pupils working together with other pupils and teachers to overcome difficulties and to reach shared understandings.

The researchers also observed that many teachers with a connectionist approach believed that mathematics should not be taught in a fragmented way and that where appropriate pupils should be introduced to some of the complexities of mathematics. Teachers with either of the other orientations were observed to be more cautious in the approaches they adopted. Numeracy content was more likely to be compartmentalised, and pupils judged to be ‘ready’ before they could go from one idea to another.

Another difference observed by the researchers related to feedback. Whilst teachers with different beliefs about numeracy teaching and learning may seem to act on feedback from pupils in similar ways, the particular beliefs of the teachers can lead to very different outcomes. Those with a transmission orientation were likely to treat pupils’ errors as a result of pupil carelessness or lack of attention and simply correct errors rather than make them explicit. Connectionist teachers who believed that the major factor in learning is that pupils engage and struggle with processes, used pupils’ errors as a means of engaging with them in order to further their understanding.

To read a case study in which teachers used pupils’ errors to create new strategies for learning, see the case study about the Sharp Lane Primary School Project Report.

A major finding from the research was that those teachers with a strongly connectionist orientation were more likely to have classes that made greater gains over the two terms than those classes of teachers with strongly discovery or transmission orientations.

Another finding was that the connectionist teachers who were highly effective had engaged in extended continuing development (CPD).

The researchers acknowledge that there is no unique description of the effective numeracy teacher. However, they do highlight approaches from their study which, for these 90 teachers and their pupils, appear to contribute to effective numeracy teaching and which include:

• stressing the connectedness of numeracy ideas rather than compartmentalising them
• using pupils’ descriptions of their own methods and reasoning as starting points for engaging with numeracy concepts
• an emphasis on enabling pupils to select strategies according to whether they were both effective and efficient (see, for example, the illustration below)
• emphasising the development of mental skills
• ensuring that all pupils are challenged
• encouraging purposeful discussion about choice of strategies
• using assessment to inform planning and teaching.

The researchers suggest that the connectionist approach to problem solving encouraged methods that were both efficient and effective. As an illustration they gave the following example:

”…while 2016 – 1999 can be effectively calculated using a paper and pencil algorithm it is more efficient to work it out mentally.”

Further examples of how effective numeracy teachers used these strategies are presented in the following sections.

All the teachers involved in the research encouraged pupils to have rapid recall of basic number facts. Knowing number bonds was a skill that teachers of all orientations – connectionist, transmission and discovery – saw as important. However, connectionist orientated teachers viewed mental arithmetic as more than this. They regarded number bonds as the starting point for the development of a conscious awareness of connections and relationships with which to underpin mental agility. As one teacher showing a strong connectionist orientation explained:

”I think you’ve got to know that they are inverse operations those two (addition and subtraction), and that those two (multiplication and division) are linked, because when you are solving problems mentally you are all the time making links between multiplication, division, addition and subtraction… I think mental agility depends on seeing relationships between numbers and being aware of links.”

Connectionist teachers also stressed the importance of estimation:

”If you’ve got a good ability to estimate and to know what a sensible answer is then you’re very quick to pick up if something doesn’t sound right. And if you can estimate you can get very quick on mental arithmetic and you get very quick on oral skills.”

For an illustration of approaches adopted by one school, which was not part of the original study, to improve mental arithmetic strategies used by children, see the case study about an investigation into pupils’ strategies for mental mathematics.

Lessons presented by teachers identified as highly effective generally involved a lot of task-related discussion between the teacher and pupils and among the pupils themselves. One teacher explained in the following terms:

”If I am honest with myself I probably spend more time talking with them than doing exercises and things like that…because I want them to be able not just to give an answer, I want them to be able to explain the process and what they are doing.”

An illustration of encouraging children to explain their methods was provided by this exchange in a Year 2 class:

Teacher asks the children if anyone can add 9 to 36 without using their fingers. She asks a child to explain how they knew the answer was 45. “Well I knew that 36 add 10 is 46 and I took off one.” The teacher reiterates the method and another child says that it can be done by taking off one and then adding 10.

To find out about strategies used by teachers in whole class situations to foster classroom interactions, see the case study about teachers’ interpretations of effective whole-class interactive teaching in secondary mathematics classrooms.

A Year 1 teacher provided another example of how teachers can foster classroom interactions between children. The children in her class modelled numbers to 100 by putting cubes in two hoops to represent tens and ones respectively, recording and then reading the numbers on a hundred square. The teacher asked the children how the process might be extended to larger numbers. One child suggested adding a third hoop and a ‘lively discussion’ took place about where it should be placed and the order in which the digits should be recorded.

Teachers who encouraged classroom interactions were also keen to help children with their reasoning. To read how one school developed strategies aimed specifically at improving children’s explanation skills, see the case study about developing skills in mathematical explanation.

Teachers of all three orientations appreciated the importance of pupils being able to apply their computational skills to real-life problems. For many transmission teachers, however, application of knowledge only involved putting what they had already learned into context:

”I will give them just some basic sums set out ready for them similar to the ones we’ve done today and then problems where they have got to actually extract the information and use what we are using today.”

Connectionist orientated teachers, like their colleagues, also emphasised the ability to apply computational skills. But unlike transmission and discoveryorientated teachers, they did not regard it as essential that pupils should have learned a skill before they tried to apply it. One strongly connectionist orientated teacher explained that she believed that pupils benefited from situations in which they did not always have the skills available:

”It is not always a good thing to always be able to do something, because there is no challenge there then. If you are not always able to do it that is when you start thinking in a different approach and broadening your base.”

Another connectionist teacher echoed this view, explaining that he did not hesitate to provide pupils with challenges that they might not succeed at. With one class, he had set up a challenge in which children had to compare two pie charts. Both pie charts showed preferences of populations for different pastimes but one referred to a population of 80 while the other referred to one of 100. Some pupils did not realise that although 40 people in each sample had stated the same preference (for computers) this did not represent the same proportions of each sample. This required them to relate numbers of people both to the proportions shown in the pie chart and to percentages. The teacher noted:

”…although they are very comfortable with the idea of a pie chart, they haven’t really got the idea that a pie chart is actually a precise thing…”

The teacher went on to explain that even when pupils got something wrong it provided the teacher with insight into their thinking.

For an example showing how one school investigated mental arithmetic strategies, including one that built on children’s errors to create improved strategies, see the case study about Sharp Lane Primary School Project Report.

Connectionist teachers also generally believed that activities should challenge all children not just the most able. As one teacher put it:

”…but I have the same expectations for the children, I always think about it as not so much what the children are doing as what they have the potential to do.”

When transmission teachers listened to pupils, the researchers suggest, they
were listening for how well the pupils’ explanations matched their own rather than engaging in a dialogue. Similarly, such teachers appeared to use pupil performance marks more to check that what had been taught had been learned rather than that the children understood.

Connectionist teachers, on the other hand, used continuous and varied means of assessment to build up detailed profiles at both class and individual student level, as the following comments illustrate:

”So it’s all sort of assessment and focus teaching all the time. I assess every day – what activities have gone on and where each group goes next.”

”Every piece of work I do, I just keep a sheet like that…and I write my own notes on there if a child has a specific problem. My planning, my search to find the most suitable method of teaching a child… And that just comes from my own experience and my observations and my constant assessment that I use.”

”I use continuous assessment from talking to the children, and from listening to what is going on and looking at the work they are doing.”

To find out more about formative assessment in the classroom, see the previous Research of the month study Inside the black box by Paul Black and Dylan Wiliam.

The researchers analysed effective numeracy teaching in the context of a model that builds on the work of, among others, Bennett, Summers and  Askew, 1994; Shulman, 1987. In the model, teacher practice is the most important factor affecting learning outcomes and teacher practice itself is influenced by teachers’ beliefs and pedagogic content knowledge (see below).

It is a complex model that also incorporates the effects of pupils’ responses on teachers’ practices, beliefs and pedagogic content knowledge. The researchers also suggested that teachers’ perceptions of pupils’ knowledge, understanding and classroom behaviour would feed back to and influence teachers’ beliefs, knowledge and practice.

Teachers' Beliefs

The researchers regard teachers’ beliefs as a crucial element influencing teachers’ practice. They suggest that these beliefs are based on ideas about:

• what it is to be a numerate pupil
• how best to teach numeracy
• how pupils learn to be numerate.

The researchers stress the importance of the interactivity of the model described above and provide examples to illustrate this. For example, one teacher used the idea of getting pupils to explain their methods to each other as a strategy to control a lively class. The teacher then discovered that this approach was an effective way of helping pupils understand numeracy better. In this case, changes in the teacher’s practice had an effect on the teacher’s beliefs.

Another teacher explained how her beliefs about pupils’ abilities had been challenged and altered through CPD, which involved her in activities with pupils.

The researchers found that the importance of a teacher’s own subject knowledge was considerably more complex than they had assumed it would be. They found that there was little to distinguish highly effective teachers from the effective and moderately effective teachers in the sample in terms of:

• formal knowledge of mathematical concepts
• qualifications and experience.

What was important, the researchers stress, was the use of suitable teaching approaches to make the ideas accessible to pupils. They suggested that effective teaching required ‘pedagogic content knowledge’, which incorporated the following three elements:

• understanding of the numeracy knowledge appropriate to what is being taught
• knowledge of how pupils learn numeracy
• understanding of different teaching approaches for presenting information to pupils.

Evidence from the research also indicated that pupil gains in the tests were greatest for teachers who had a good understanding of the conceptual links between the areas of numeracy in the primary curriculum.

The effect of type and duration of CPD

Data about teachers’ experiences of CPD were collected by questionnaire from the full sample of 90 teachers and from interviews with the 33 case study teachers. Of the 88 teachers who responded to the questionnaire, 26% (23 teachers) had undertaken some form of extended CPD.

The researchers compared the mean gains of pupils whose teachers had engaged in mathematics CPD with those whose teachers had received other types of extended CPD. Results suggested that the classes of those teachers who had experience of extended mathematics CPD showed higher gains than those of other teachers including those who had undertaken some form of (non-mathematics) CPD. This finding highlights the importance of pedagogic content knowledge, ie, knowledge of teaching approaches within a specifically mathematics context.

Results also suggested that the length of time spent on CPD had an important bearing on the effectiveness of the CPD in furthering teachers’ skills. Analysis of the data showed that only those teachers who had engaged in CPD of at least 15 days duration were highly effective. The class scores of teachers having had three or fewer days of CPD in the previous year were indistinguishable from teachers reporting no days.

Data from the case study teachers indicated that certain aspects of mathematics-focused CPD helped teachers become more effective in putting over concepts to their pupils. Four of the five highly effective connectionist orientated teachers all identified an emphasis on the importance of working with pupils’ meanings and understandings as significant elements of their CPD. This had led them to appreciate more fully the role of mental strategies in helping themselves and their pupils build up the mental imagery that would enable them to work through numeracy problems more quickly and efficiently.

One teacher, whose Year 6 class had very much higher gains on the Y5/Y6 test than any other class, reported that her approach to numeracy teaching had changed considerably as the result of the CPD she had undertaken:

”(prior to the CPD)… I would have taught them a set way of doing your long division and the algorithms that I learned at school without bothering about what it means,…but (in a recent topic) they all did it in a way they could understand.”

The effect of teacher collaboration in the schools

When the focus schools were ranked according to performance in pupil tests for those pupils at the beginning of Year 2 and then again in line with the results of the tests for their Year 6 pupils at the end of the research, the results showed that two schools in particular seemed to have improved their ranking considerably.

In one of the two schools, key features of practice among the numeracy teachers in the school were:

• strong leadership by the connectionist orientated teachers
• discussion of teaching methods and activities at a more detailed level than seemed to be the case in other schools
• provision of time for key teachers to work collaboratively with other teachers in the classroom.

A flavour of the kind of collaborative activity that teachers in this school found helpful is provided by one of the teachers who led the activities:

”I work closely with each (year group) team so I talk about the work I do and they talk about the work they do. Then we try to pass things around and we have a lot of discussion about the problems the children have, how we can solve them, and I will look for things to support them.”

Identifying the ‘focus’ schools

The focus schools were a group of schools already known to be performing well above expectations in relation to numeracy. The schools came from three local education authorities in London and the southeast of England. The local education authorities were able to provide the research team with extensive school-level data about pupils’ performance in numeracy.

Each LEA assisted the researchers in identifying effective focus schools, using a number of data sources which included IQ scores, reading test scores, baseline assessments and national test results. They also included two independent schools. The final sample of six focus schools comprised four state primary schools and two independent preparatory and prepreparatory schools.

The researchers included schools that reflected different socio-economic intakes in different environments – inner city, suburban and rural.

Identifying the teacher sample

The six focus schools and five ‘validation’ schools (see below) provided 66 and 24 teachers respectively for inclusion in the study. From the six focus schools, 18 teachers – three from each school – were selected by the researchers for case studies in order to provide data about classroom practices and teachers’ beliefs about, and knowledge of, mathematics, pupils and teaching. The three teachers from each school were identified as those most likely to be highly effective. Selection was based on the headteachers’ recommendations and, where appropriate, advice from LEA inspectors and advisors. The selection was made so that teachers were evenly distributed across Years 1 to 6.

The research design raised the important question of validity of findings. Because the teachers in the focus schools were selected on the basis that they were highly effective in teaching numeracy the researchers needed to establish a way to ensure that the findings really did relate to effective teaching and that the same characteristics could be observed in teachers outside the sample.

To ensure as far as possible the validity of their findings about teacher effectiveness, the researchers selected five ‘validation schools’ in order to provide a variety of effective, average and less effective teachers of numeracy.

Like the focus schools the validation schools were selected based on:

• evidence of performance in the teaching of mathematics, based on national test and other results
• being representative of a range of schools in terms of size, socioeconomic backgrounds of pupils and environments (inner city, suburban and rural).

The researchers also used a range of instruments for collecting different kinds of data to dig under the surface of teachers’ practice. For example, one method was the ‘personal construct’ interview (see next section), which enabled the researchers to supplement other data they had collected.

Data about teachers’ beliefs and pedagogic content knowledge was collected using a number of methods including:

• a questionnaire administered to all 90 teachers in the focus and the validation schools
• observation of 84 lessons, three for each of the 18 case study teachers in the focus schools and two for each of the 15 teachers in the validation schools
• interviews with the six headteachers of the focus schools
• fifty-four interviews with case study teachers in the focus schools, three for each of the 18 teachers
• an interview with each of the 15 validation teachers
• pupil tests, designed to measure gains in numeracy, at the beginning and end of the six month period of the research.

The questionnaire provided background data, which included the organisation and planning for mathematics teaching and the training and continuing professional development of teachers. It also provided the researchers with data about teachers’ perceptions of teaching styles and of their beliefs about teaching, learning and assessing mathematics.

Lesson observation enabled the researchers to examine a number of aspects of teacher practice including organisational and management strategies, teaching styles and pupil responses. Particular foci of attention were the type of instruction provided by the teacher and the use of more sophisticated strategies for calculation as opposed to reliance on counting based methods.

The three interviews consisted of:

• a background interview, which provided evidence to supplement the questionnaire on training and experience, in addition to information about beliefs, knowledge and practices in teaching numeracy
• a ‘concept mapping’ interview, which explored teachers’ mathematical understanding in relation to numeracy teaching
• a ‘personal construct’ interview, which explored teachers’ beliefs and knowledge about the pupils they taught (more details are given below).

The ‘personal construct’ interviews were important in that they helped the researchers triangulate other data they had collected. They used a series of questions about the mathematical learning of all the pupils in a teacher’s class to explore what the teachers regarded as similar and different about the learning of pupils in named groups of three. In this way, the researchers were able to elicit evidence about teachers’ beliefs about numeracy in the context of their knowledge about their pupils and about learning processes.

The pupil tests were based on a diagnostic test that had previously been designed and used at King’s College. The tests aimed to assess pupils’ mental dexterity with numbers and their ability to apply it. The researchers used an aural approach to testing. This had the principal advantages of enabling them to control the timing for certain questions and was helpful for younger children and weaker readers. The bulk of the analysis was based on the performance of pupils in Years 2 to 6 who were tested twice - at the beginning and at the end of the research - rather than on that of Year 1 pupils who were only tested at the end of the research.

All the classes made gains in the second test. The gains in mean class scores were used to classify the teachers of Years 2 to 6 who participated in the study into the following groups:

• highly effective (23 focus school teachers, 3 validation school teachers)
• effective (15 focus school teachers, 6 validation school teachers)
• moderately effective (17 focus school teachers, 10 validation school teachers).

Only six of the 18 case study teachers, selected on the basis that their headteacher judged them to be highly effective, came into the highly effective category. One reason for this was that nearly half the teachers judged to be highly effective based on pupil test results came from the same school - from which only three teachers had been selected for case study analysis at the outset.

Reasons for other highly effective teachers not being selected for the case study analysis may lie, according to the researchers, in the following:

• some highly effective teachers had been unwilling to participate in the case study work because of other pressures
• the headteacher did not recommend those teachers who were subsequently identified as highly effective
• headteachers did not always recognise the most effective teachers in their schools and knowingly or unknowingly may have proposed teachers for other reasons such as their skills in class control.

Teachers wanting to improve teaching and learning of numeracy may wish to consider the following implications of the findings of this research review:

• the researchers found that teachers with a ‘connectionist’ orientation were more likely to have classes that made greater gains than teachers with a ‘transmission’ or ‘discovery’ orientation.  Would you find it useful to review your own beliefs and approaches to teaching numeracy and consider to what extent they match those of the ‘connectionist’ teachers the research describes?
• ‘connectionist’ teachers used formative assessment as a means of identifying what their pupils had learned, where they were struggling and what needed to be covered next.  Could you make more use of formative assessment to identify what your pupils need to learn? (You can find out more about formative assessment in two other ‘Research of the Month’ summaries: ‘Raising standards through classroom assessment’ and ‘Assessment for learning: putting it into practice’
• effective teachers of numeracy developed their pupils’ mathematical thinking. For example, pupils articulated their reasoning verbally or in writing and discussed and evaluated their methods.  Could you make more use of this type of approach with your pupils? 
• schools in which pupils made good gains in numeracy were found to be those which encouraged open discussion and opportunities for staff to work together.  Would you find it helpful to work collaboratively with your colleagues and share your approaches to teaching?  For example would you find it interesting and helpful to explore with your colleagues the connections between the mathematical concepts that you teach?

Leaders may wish to consider the following implications:

• the researchers found that effective teachers of numeracy had access to courses where they learned about successful teaching approaches.  Could you make more opportunities available for your staff to participate in professional development activities?  (Practitioners may find it helpful to read another of our RoM summaries about the impact of continuing professional development (CPD) upon teaching and learning
• the case studies we have provided came from schools where teachers were working with researchers from higher education institutions on specific projects.  Have you considered encouraging teachers in your school to work with a higher education institution to carry out a school-based research project, such as investigating strategies that children use in arithmetic, especially faulty arithmetical procedures that lead to errors, or effective ways of helping pupils overcome difficulties they experience with maths?

Your feedback

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