Developing pedagogical understanding
What aspects of teachers’ practice should we look at in order to meet the learning needs of high attaining pupils more effectively?
What is ‘purposeful talk’ and how can we develop it in our classrooms?
How can we improve questioning in mathematics lessons?
How do we develop children’s conceptual understanding in mathematics lessons?
Things to think about
“Pedagogy is the why, what and how of teaching. It is the knowledge and skills teachers need in order to make and justify the many decisions that each lesson requires. Pedagogy is the heart of the enterprise. It gives life to educational aims and values, lifts the curriculum from the printed page, mediates learning and knowing, engages, inspires and empowers learners – or sadly does not.”
Introducing the Cambridge Primary Review, (Cambridge Primary Review, 2009)
A distinction is often made between general pedagogical knowledge and pedagogical content knowledge, which could be in simple terms be described as the difference between our more general understanding of how to teach effectively and a more specific understanding of how to teach mathematics effectively. Excellent mathematics teachers draw on a strong range of general teaching techniques, but combine this with a well developed understanding of how to encourage the best mathematics learning.
Jennifer Piggott in her article Mathematics enrichment: what is it and who is it for? (Piggott, 2004) argues that enrichment is as much about developing a positive learning ethos in mathematics lessons as it is about the content of the curriculum. Teachers with strong pedagogical understanding enable children to experience the enjoyment that can come from mathematical enquiry. Such teachers are, for example, skilled at developing purposeful mathematical talk in their lessons, where children share and discuss their ideas in order to develop their understanding. Children are consistently encouraged to use higher order thinking in their mathematics lessons. This helps children develop the resilience needed to rise to the challenge of more demanding mathematics and the ability to talk about their mathematical reasoning.
Such approaches create possibilities for teachers to enrich the curriculum for high attaining children, rather than having to rely on curriculum acceleration.
This section of the microsite focuses mainly on the development of purposeful talk, questioning and reasoning.
What does research tell us?
Rich, purposeful talk by children is one of the ingredients of powerful learning. Robin Alexander in his booklet Towards Dialogic Teaching (Alexander, 2008) reviews research that indicates that too often talk in classrooms is low level and dominated by the teacher. In contrast, his thinking about dialogic teaching puts rich talk at the heart of learning, where, for example ‘children listen to each other, share ideas and consider alternative viewpoints; children articulate their ideas freely ... and help each other to reach common understandings; teachers and children build on their own and each other’s ideas and chain them into coherent lines of thinking and enquiry’ (p. 38)
Such ideas can be seen in the work of Neil Mercer and colleagues, who over a long period have researched the impact of purposeful classroom talk on learning. Discussing talk between children in classrooms, Mercer distinguishes between:
Disputational talk, characterised by disagreement and individualised decision making;
Cumulative talk, in which speakers build positively but uncritically on what others have said;
Exploratory talk, in which partners engage critically but constructively with each other’s ideas;
(Mercer and Littleton, 2007)
His research suggests that when children are encouraged explicitly to make greater use of exploratory talk in the classroom, their learning improves.
Applying these more general pedagogical approaches in mathematics teaching is not something that all teachers find easy. In particular, teachers who think of mathematics as being mainly about the development of secure computational skills are likely to find it difficult to see how mathematics lessons can be ‘dialogic’. Nick Pratt (2006) has investigated ways in which mathematics lessons can become truly interactive, where children actually need to talk about their mathematics. Questions such as ‘Is a square a parallelogram? Is a parallelogram a square?’ give rich opportunities for children to use their prior learning to develop their mathematical reasoning and their ability to explain their ideas to others.
In his article Effective Questioning and Responding in the Mathematics Classroom, John Mason explores different types of questions that teachers typically use – controlling, cloze technique, genuine enquiry, meta-questions and open and closed questions. He urges teachers to develop awareness of the extent to which they use different question types (Mason, 2010). In her research, Debra Myhill, using a different typology of questions, analysed video recordings of lessons and found that speculative and process questions (ie ones involving higher order thinking) were only about a quarter of all teacher questions. (Myhill et al, 2006).
The work of Mike Askew and colleagues (Askew et al, 1997) suggests that another key aspect of pedagogical content knowledge is the way teachers help children see connections between different aspects of their mathematics learning. This is something that can develop from purposeful talk between the teacher and the children. Teachers who consistently do this help children develop more sophisticated conceptual understanding, and this leads to better learning.
Implications for the use of the Year 6 Level 6 tests
Investigation of Key Stage 2 Level 6 Tests (Coldwell et al, 2013) discusses the risk that preparing pupils for the L6 test might lead to a narrowing of pedagogical approaches. Their analysis suggests that the more successful schools ‘had a vision for teaching and learning to which engagement in the L6 tests related’ (p. 97) while the less successful schools tended to have a focus only on measurable outcomes, with all their energy put into test preparation. The report concludes:
‘This indicates the need for policy to aim to drive home the vital importance of pedagogy and learning to counteract the tendency for some schools to respond to pressures by focussing on test preparation ...’ (p. 98)
Thus the use of L6 tests needs to be seen as part of the school’s wider policy on teaching and learning based on a full range of pedagogical approaches.
Working with colleagues
Changing teachers’ pedagogical practice is difficult. If you are asking colleagues to change, for example, the way they ask questions in mathematics lessons, then you are asking them to change habits built up over a long period of time, habits they may well feel work well. Furthermore, it is not easy to reflect on your teaching while you are in the middle of a busy lesson, and this adds to the challenge of changing pedagogical approaches.
Therefore if your evaluation is that the development of pedagogical practice is an important part of the development of provision for higher attaining children, accept that this is likely to be a long term project. It is likely that you will need to achieve several things:
colleagues come to see that other pedagogical approaches may have value;
colleagues develop understanding of how to apply different pedagogical approaches in lessons on a regular basis and they feel supported and affirmed in the steps they are taking;
colleagues feel that children’s learning is improved as a result of using different pedagogical approaches.
Without these, it is likely that any change is superficial and short-lived. If you keep a focus on these broader outcomes you will be able to decide on appropriate CPD strategies.
Taking the example of developing dialogic teaching approaches, here are some possible approaches you could use.
Helping colleagues see the value of dialogic teaching
show colleagues the Mathemapedia entry Volleyball not ping pong. Use this as a prompt for a discussion about the types of question we ask in the classroom, and whether we close down discussion by immediately telling children whether they are right or wrong, or whether we open up discussion by asking other children to respond to each other.
sse clips from Teachers TV videos which show teachers questioning in mathematics lessons. Discuss the way the teachers are questioning in the light of the ‘Volleyball not Ping Pong’ idea. These videos have useful sections to discuss, as well as having examples of rich tasks that are simple and easy to use:
use a rich task with colleagues. For example you could ask them to think about whether these statements are true or false: ‘All numbers have an even number of factors’, ‘No quadrilaterals have exactly three lines of symmetry’. Give them a few minutes to talk about this in pairs and then develop a group discussion with you in the ‘teacher’ role. Try to focus on getting your colleagues to respond to each other, rather than telling them if they are right or wrong. Record the discussion and then play it back and discuss how effective your own questioning was and what impact the questions you asked had on the discussion – did your questioning open up the discussion or close it down?
show colleagues the video clip of Tom Rainbow teaching and talking about his teaching. Tom is a secondary teacher, but the clip is very relevant to primary teachers.
use the NRICH article What’s all the talking about? by Bernard Bagnall as a prompt for a staff meeting discussion about the place of talk in mathematics lessons.
Helping colleagues develop understanding of how to develop dialogic teaching in lessons on a regular basis and help them feel supported and affirmed in the steps they are taking
the tasks that teachers give children need to be ones that encourage discussion. One approach is to use tasks where there are different possible answers. For example, discuss with colleagues the difference between asking children to identify how many lines of symmetry a rectangle has got and asking children to draw a quadrilateral that has two lines of symmetry, or the difference between giving children a worksheet with 2 digit sums and asking them to come up with five sums that all have the answer 67. Get them to develop ideas like this for covering core content with more open tasks
encourage colleagues to develop their understanding of rich tasks by using the Rich Tasks section of this microsite
show colleagues the Thinking Mathematically section of NRICH. This has links to activities which focus on specific aspects of mathematical thinking
develop a Lesson Study approach with colleagues where, for example, two teachers jointly plan a lesson which will incorporate mathematical discussion. One teaches the lesson and the other observes. If they feel confident enough they could also record the lesson. Ask them to bring their thoughts about the lesson back to a staff meeting. There is more about this in the Creating Sustainable Change section of the microsite.
invite colleagues to watch you teach and ask them to observe your use of dialogic approaches and what impact they are having on the children. This can be particularly effective if you teach their class, not your own. Feedback to other staff in a staff meeting.
ask colleagues to read pages 30 to 41 of Improving learning in mathematics: challenges and strategies by Malcolm Swan (Swan, 2005). This gives many examples of ways teachers can encourage more active learning. Ask them to reflect on their own practice – to what extent do they use any of these approaches? Are there ones that they could use with their own classes?
Helping colleagues feel that high attaining children’s learning is improved as a result of using dialogic teaching approaches.
It is very hard to demonstrate conclusively that a particular pedagogical approach leads to better long term learning. Other approaches that can be used:
when observing lessons focus in particular on any evidence of dialogic approaches having an impact on the higher attaining children and their engagement and enthusiasm;
ask the children for their views on different teaching approaches. Have a regular (eg termly) series of meetings with small groups of high attaining children and use a standard set of questions as prompts. Feed back to colleagues. Over time this will build up a rich picture of the children’s attitudes to their mathematics learning.
Resources to use
The NCETM website has many resources which help develop approaches to mathematics teaching that encourage purposeful talk. For example:
Session 4 of The Resource Pack for Improving Learning in Mathematics at Primary Level has material on purposeful talk you could use in a staff meeting.
The NCETM Departmental Workshops are written for secondary teachers, but many could easily be adapted for use with primary teachers. You could use the section on Questioning to explore this topic with colleagues.
Questions, questions, questions has a number of practical suggestions for developing questions that encourage discussion.
Primary Proof? Is an NRICH article which discusses the how young children can explore mathematical proof
Themes: Speaking and Listening has links to several research papers of a more general nature on the importance of speaking and listening.
Find out more about Neil Mercer’s work by watching this video clip on YouTube
Improving learning in mathematics: challenges and strategies by Malcolm Swan (Swan, 2005) has a wealth of practical ideas for active learning and includes useful sections on small group and whole group discussions and on asking questions to make learners think.
How can we make connections in mathematics? is a rich resource which you could use to develop colleagues’ understanding of what this key idea means in mathematics learning.
For a broader look at aspects of mathematics specific pedagogy use the Self-evaluation Tools. You could use this as part of your own audit of the quality of mathematics teaching, or colleagues could use it as a self-audit. It can help identify development priorities and CPD needs. Many schools have found that it is particularly valuable using this with groups of colleagues since this generates discussion.
The Embedding in Practice section of the Self-evaluation Tools has further material that you could use to support development of pedagogical understanding.
Alexander, R. (2008) Towards Dialogic Teaching. Towards Dialogic Teaching. York: Dialogos.
Askew, M., Brown, M., Rhodes, V., Johnson, D. and Wiliam, D. (1997) Effective Teachers of Numeracy: Report of a study carried out for the Teacher Training Agency. London: School of Education, King’s College.
Cambridge Primary Review (2009) Introducing the Cambridge Primary Review. London: Routledge.
Coldwell, M., Willis, B. & McCaig, C. (2013) Investigation of Key Stage 2 Level 6 Tests. Centre for Education and Inclusion Research & Sheffield Hallam University.
Mason, J. (2010) Effective Questioning and Responding in the Mathematics Classroom [online].
Mercer, N. & Littleton, K. (2007) Dialogue and the development of children’s thinking. London: Routledge.
Myhill, D., Jones, S. & Hopper, J. (2006) Talking, listening, learning. Maidenhead: Open University Press.
Piggott, J. (2004) Mathematics enrichment: what is it and who is it for? British Educational Research Association Annual Conference, University of Manchester, 16-18 September 2004.
Pratt, N. (2006) Interactive Maths Teaching in the Primary School. London: Paul Chapman.
Swan, M. (2005) Improving Learning in Mathematics: Challenges and Strategies. Department for Education and Skills Standards Unit.