Mathematics Subject Leader In-Depth Study Module 2 (3 hours)
Help all staff involved in the teaching of mathematics to develop their pedagogy in order that it might deliver improved results
By studying this module you will consider ideas about leading a mathematics department that is improving its teaching and learning, or pedagogy and therefore improve the results that the pupils can achieve. In particular you will come to know more about:
- Taking a lead on pedagogy
- What pedagogies there are and how they help in learning mathematics
- Learning theories and their effect on pedagogy choice
- The impact of assessment on learning
- Involving other staff
- Managing Change
The sections can be studied in the order presented here or you can click on one of the sections below to take you to a section that particularly interests you:
- Taking a lead on pedagogy (30 mins)
- What pedagogies are there and how do they help learning? (40 mins)
- Learning theories and their effect on pedagogy choice (30 mins)
- The impact of assessment on learning (30 mins)
- Involving other staff (20 mins)
- Managing change (20 mins)
- Reviewing the module (10 mins).
Back to top
1. Taking a Lead on Pedagogy
Taking a lead on pedagogy means helping teachers understand that developing their practice is about providing the best learning environment that they possibly can for their pupils. Therefore it means reflecting on what everyone does now and what they might do even better, understanding what the best practice looks like and using the skills of coaching and mentoring to make improvements. Click the title of the section that you want to study:
A Growing Department
Teachers in a department that is helping its pupils learn knows about learning because it is learning and growing itself. Here you will reflect on what it means to have a growth mindset.
Developing a growth mindset in your department
Do you have a growth mindset?
Use the questionnaire in Folio 2.1.1
to see if you and your department have a fixed or growth mindset.
A growing department loves to learn
Meeting together one lunchtime each week to ‘do maths’
Solving problems together can extend mathematical knowledge and encourage sharing of pedagogical knowledge. See NRICH
for some problems to get you started.
Always have a ‘learning maths’ item that starts the meeting agenda
If this is going to be on the agenda then it must be at the start otherwise it will get shunted off by organisational or administrative issues.
Setting weekly maths puzzles for staff
Many maths departments are surprised how much other staff like to try their hand at maths. Sharing setting (and marking) a weekly maths puzzle can encourage discussion within the department and raise the status of mathematics in the school.
Challenging the department to set up a whole day maths challenge for a whole year group with protected time to do so
By working together to set a new and challenging maths day the department will learn from one another and provide the pupils with an exciting day’s learning.
These are just a few ideas to get you started. The main features of a growing department are that the teachers are willing to take risks, make mistakes, support one another and learn from each other.
Looking at Growth Leadership
Exemplifying good practice
Use the Case Study in Folio 2.1.2
to look at 'growth' leadership.
Read the case study. How would you proceed if you were leading this department? Make some notes.
Read page 2 'Ideas and Suggestions' to see how other people have attempted to engender a growth mindset in such a department.
Who does the exemplifying?
On the face of it this may seem to be a ‘no-brainer’ – whoever does the good practice exemplifies it!
Well not always.
- Often in a small department there is one teacher who is forward thinking and the others feel that that person’s practice is ‘not for them’. If that teacher is always the one who does the exemplifying, that could be counterproductive
- Observing practice that is ‘on the way’ to embedding an approach can allow those feeding back to say what is going well and identify next steps in continuing to improve. This helps the observers understand what they are trying to achieve for themselves.
- Planning a lesson together and observing one teacher following the joint plans can be powerful. All the planners meet together afterwards to dissect the lesson, what worked well and what wasn’t as good. It is not the teacher’s ‘performance’ that is critiqued but the pupils’ learning from the planned lessons. Japanese teachers plan in this way and some mathematics teachers in America and England are trying it out too. Find out more in the Mathemapeda entry Lesson Study.
Exploring current practice
Before you set up a system of exemplifying good practice as leader of the department, you will need to find out what really goes on in your department. You could do this through:
- Observations, formal and informal
- Work trawls
- Discussions, interviews, chats
Finding good practice
On the Finding Good Practice sheet
- what you want to improve in your department,
- what you need to look at to find out about the current situation
- how you might go about observing and monitoring
There is an example on the form.
There are many different models of observation. For example:
- Evaluation observation – usually used as part of formal monitoring/school self-evaluation or a performance management system. Each school will have their own observation form for such observations and there are many examples available on the web. The purpose of these observations is to evaluate what is currently happening other steps will need to be put in place if teachers are to see them as opportunities for learning and improvement.
- Paired observation - this is also known as peer observation. A pair of teachers decides on their own focus for improvement in their practice. They discuss what new approaches they might try and how they might go about implementing the improvements. Each teacher then observes the other in order for the pair to learn more about the improvements they are seeking to make.
- Learning from observation - it is important that as many observations of teachers as possible are regarded as learning opportunities both by those observing and those observed.
Mentoring, coaching and action research
Mentoring, coaching and action research are so important that they have a much bigger section devoted to them in Module 5. In this section you are invited to start thinking about how you can use these ideas to develop pedagogy in your department.
Departments that use mentoring, coaching and Action Research as part of their continuing professional development usually have an ethos of exploration, risk taking and steady improvement within the team.
Mentoring offers support to colleagues from more experienced members of the team. A new member of staff may have a mentor appointed in order to help them become accustomed to the mores and systems of a new department. Someone who has developed skill in an area of teaching could mentor others as they develop their own practice.
The coaching relationship is more equal and is often called co-coaching to emphasise this. A department may establish coaching pairs within its team in order to develop a common area of practice or may encourage less formal pairings around an area that two teachers wish to work on.
Action research is an excellent way to innovate within the department and could involve just individuals or the whole department. The idea is to use the method to form, try out and improve an idea to improve the pupils' learning.
Back to top
2. What Pedagogies are there?
What is Pedagogy?
Pedagogy could be called:
design for learning
a theory and method for engaging with education
Dictionary: the science and art of teaching; the function and work of a teacher.
From the Latin paedagogus, a slave who escorted children to school and generally supervised them.
There are as many pedagogies as there are teachers and classes. When studying this section, you will be asked to think about the different pedagogies that you already use and how you can extend your own skills. Then you will think about extending the repertoire of pedagogy your department uses and improve the learning climate. Click the title of the section that you want to study:
What pedagogies are there?
Here is a list
of some common pedagogies that mathematics teachers use. Think about last week’s lessons and complete this table to show the number of times you used these pedagogies within your lessons.
Was there a pedagogy that you used more than any other?
Were there any that you did not use at all?
Was there a pedagogy that you frequently use that is not on the list?
Some pedagogies you use only rarely but variety is important – pick one that you use less than the others and think how you could use it soon.
Using Different Pedagogies
For example Pythagoras could be taught by:
- Discovering the relationship by
The teacher using a dynamic geometry package to draw and measure squares on a triangle and showing the pupils the resulting relationship. Pupils drawing and measuring their own squares on the sides of triangles and finding their own relationships
A series of problems that use Pythagoras’ Theorem in a variety of places around school. Each group has a different starting place and has to finish as quickly as they can whilst still getting the right answers
- Does it work with other shapes?
The pupils work in groups around a computer. They know Pythagoras’ Theorem works for squares, but does it work for triangles, semicircles, hexagons and so on? Why could it work for other shapes?
- Ordering the proof in groups
The teacher writes a proof for Pythagoras and photocopies it. It is then cut into stages. The pupils have to get all the stages in the right order.
Think of a topic that you are about to teach, graphs, algebra or whatever
Write down four distinctly different pedagogies that you could use to help the pupils learn the objective. Try to think of:
- an active pedagogy like the race above
- a pedagogy that uses ICT
- a pedagogy where working in groups is important as in the different shapes or proof approaches above
- a whole class but exciting pedagogy where you model a different way of working
Improving the department's repertoire
Using collaborative learning
There is a great deal of information available ot help develop your department's pedagogical skills. For example:
Swan, M. Improving Learning in Mathematics, DfES Standards Unit (2005)
Questioning for thinking
- Thinking time built in
- Role of silence
- Open questions
- Treat all answers as interesting
- "Mistakes are where we start learning"
- Pupils asking the questions
KS3 Publications DfES Pedagogy and Practice Unit 7 Questioning and Dialogue
KS3 Publications The Standards Site Pedagogy and Practice Unit 7 Questioning
Lee, C. (2006) Language for Learning - assessment for learning in practice. OUP
Wragg, E.C. and Brown, G. (2001) Questioning in the Secondary School, Routledge Falmer.
Building a conjecturing atmosphere
Encouraging the pupils to reason and justify
Mathemapedia: Questioning Mathematically
Mason, J. Burton, L. & Stacey, K. Thinking Mathematically, Prentice Hall.
Using discussion as a learning tool
Swan, M. Improving Learning in Mathematics, DfES Standards Unit (2005) - Developing Questioning and Managing Discussion.
Do some pedagogies suit some areas of maths but not others?
Think of an area of maths that you might teach using:
Do some pedagogies suit some topics better than others? There is no right answer to this, often it is better to think of using a variety of pedagogies so that the class experience thinking and learning in many different ways
- discussion as a learning tool;
- whole class questioning;
- pupil questioning and/or conjecturing;
- collaborative learning.
Creating the classroom climate
Changing the classroom climate
- Takes time for the
- teacher to change
- pupils to change
- Doesn't go right all of the time!
- Don't do it on yor own, involve as many others as you can!
- Use coaching and mentoring to support change
- Make the change an action research project.
What counts as a good classroom climate?
Now decide on how best to make changes in the classroom climate
On Folio 2.2.1
note down the way that your ideal pupil will act in the classroom.
- Will they answer questions thoughtfully?
- Say exactly where they get stuck?
Now think what actions by the teacher would enable the pupil to act in this way in the classroom.
What changes are needed to improve the classroom climate?
Organising the physical environment to support different pedagogies
Decide on one change in the classroom climate you would like to make.
This may be from your ‘wish list’ if you have already studied Module 5, or could be one pedagogical change that you would like to make.
- What needs to change?
- How do you want to make changes?
- What will count as a successful change?
Who will you make changes with?
- Choose one class rather than many
- Choose one or two classes that are happy to work with you and you are happy to work with
- Work with one or more other members of the department so that you can share ideas
How will you evaluate the changes?
- Use action research methodology?
- Use formal/informal observations?
- Discuss the outcomes in the department?
Now make that change!
A physical environment that supports different pedagogies must be flexible. The desks and chairs may have to be moved and resources must be available, either all the time or space will be needed to set them out. Display boards can be used as learning boards by displaying learning prompts constructed by the pupils, displays that the pupils add to as they learn more about a topic, posters that tie in with the work in hand and vocabulary that the pupils may need. It is also important to think how ICT will fit into the learning environment. When using ICT to learn mathematics pupils often need to make notes of their ideas; will there be space to do this next to the computers or will the pupils need clip boards for notes?
Arranging the furniture
Use Folio 2.2.2
to complete this task.
Look at the four different ways that furniture is arranged in the classroom
- may indicate that the teacher has a transmission model of learning? (See Module 1 section 3 for transmission learning)
- could allow pupils to be involved in the learning process?
- would allow the pupils to see and hear one another easily?
- may allow the pupils to discuss ideas easily
- as a small group?
- as a whole class?
- encourage the pupils to collaborate easily?
- would be too difficult to arrange in your classroom?
- would you only use occasionally?
- could be used most of the time in your classroom?
Of course there is no right or wrong answer to these questions, only what is right for you.
- In general if the pupils are to collaborate and discuss ideas together they will need to be able to see and hear one another easily and often they do not need much space for written work so arrangement 2 might work well.
- The U shape (arrangement 4) could work well if you want to use whole class questioning and discussion as the pupils can see, hear and respond to the teacher and one another easily.
Back to top
3. Learning theories and their effect on Pedagogy choice
When thinking about what pedagogy to use it is important to consider how pupils learn and what will help them to learn best. Some classes will need to learn to behave before they can be helped to learn the concepts of mathematics; some classes will be ready to use discussion to move their understanding forward. So we need to consider what constitutes learning and then what the main theorists have to say about pedagogy in the classroom. Click the title of the section that you want to study:
What is learning?
Before thinking about how other people have descreibed learning, explore your own beliefs about how people learn.
What is learning?
What difference does learning theory make to pedagogy?
- Make some notes about your ideas on what constitutes learning
- Read Folio 2.3.1 'What is Learning?'
- How does the paper change, consolidate or extend your ideas on what learning is?
The major lines of learning thoery can be categorised under the following headings:
- Social construction
- Situated cognition
Learning Theories and Pedagogy
If you have not completed Module 1 then read Folio 2.3.2 which summarises each theoretical position on learning. If you have completed Module 1 then find Folio 1.3.2 and the information you noted on learning theories.
Each learning theory has something to say about the way that pupils learn best and hence pedagogy in the classroom.
Use Folio 2.3.3 (which builds on Folio 1.3.3 from Module 1) to note how each theory implies that classrooms should be organised
What difference will this knowledge make to your department?
Back to top
4. Assessment for Learning
“Assessment for learning is any assessment for which the first priority in its design is to serve the purpose of promoting pupils’ learning. It thus differs from assessment designed to primarily serve the purposes of accountability, or of ranking or of certifying competence.
An assessment activity can help learning if it provides information to be used as feedback, by teachers, and by their students in assessing themselves, to modify the teaching and learning activities in which they are engaged.
Such assessment becomes “formative assessment” when the evidence is actually used to adapt the teaching work to meet learning needs.”
Working Inside the Black Box
Black, Harrison, Lee, Marshall & Wiliam
In this section we consider what is assessment for learning and how assessment for learning can be used in your department. Click the title of the section that you want to study:
What do you think of as AfL?
AfL means giving the pupils a voice so that you (and the pupils themselves) know how well they are learning and how they could be helped to learn more or better.
Therefore the pupils must be given time to think and helped to articulate their knowledge.
What do you think of as AfL?
Write down five activities that you think of as Assessment for Learning, that is assessment that makes a difference to what and how pupils learn.
Assessment for Learning includes many activities. Your list might have included some of these ideas.
Using Assessment for Learning in your department
Using Assessment for Learning
Listen to the Vox extracts:
- Five key strategies for AfL
- More thougths on AfL
- Even more thoughts on AfL
or read Folio 2.4.1
Assessment for Learning in Maths.
- the key ideas that need to be developed in your department to fully use the power of AfL to improve learning
- how can different groups be helped to learn more effectively?
Back to top
5. Involving others in developing pedagogy
You don’t have to do everything, in fact it is often more productive to include other people in developing teaching and learning. You can distribute responsibility within your department and make use of the skills and expertise within the department or look outside the department, or do both! Click the title of the section that you want to study:
Using teachers in your department to develop pedagogy
Pedagogical ideas and skills are distributed across the department and teachers develop new ones because they take part in CPD or because their interest is sparked. Using the skills that are there is important.
What are the teachers in your department good at?
You will have found out what particular skills the teachers in your department have, by:
- observing them – maybe as a part of professional monitoring observations (e.g. performance management) and of less formal monitoring
- asking them
- use a questionnaire (see Folio 2.5.1 for an example),
- using part of every staff meeting for teachers to explain something that ‘went well’
- asking staff to suggest someone who is ‘good at questioning’ etc.
Plan how you will ensure you know what skills the teachers in your department have (see section 1
of this module):
How will you use teachers' skills to develop pedagogy?
- in formal coaching or mentoring partnerships (see Module 5)?
- in informal partnerships?
- as an expectation following CPD?
- as part of an Action Research project (see Module 5)?
- in demonstration lessons?
- in short demos at department meetings?
- using video or observation rooms?
Make a plan to use all the teachers' skills in your department to improve pedagogy.
Using people outside the department to develop pedagogy
Who else can you recruit to develop pedagogy in your department?
There are many people outside your department who are willing and ready to help you develop teaching and learning in your department. These will vary from working with another department to develop an aspect such as questioning together using co-coaching to inviting in a specialist to talk about behaviour management to the whole staff and then working on the suggestions together.
Use the radial diagram in Folio 2.5.1
to consider who you might recruit to help you staff develop the pedagogies they use.
Make notes on who to you might use and who you might contact.
Back to top
6. Managing change
Managing change in a department is about creating a climate in the department that encourages reflection and innovation and systems that monitor the effectiveness of both current and innovative practice. Click the title of the section that you want to study:
Beware innovation overload!
Creating a climate that fosters reflection and innovation
A growing department
- reflects on what they are doing and asks can we do it better?
- decides on where they need to develop their practice and plan to work on it together
- only innovates where existing practice is not working
- learns separately and talks about it together
- learns together and talks about it together and separately
- uses coaching, mentoring and action research to embed good practice
- shares their practice with others, other teachers, other departments, other schools.
The Big Question
Consider all the sections in this module
- which section is most important for your department?
- how will you foster a climate of reflection and innovation that leads to growth in your department?
Monitoring results - how do we know when something is working?
Monitoring results can be a matter of scrutinising data, this area is considered in Module 6. However any of the actions that have been discussed in previous sections can be used to discover if current practice or an innovation is working effectively, for example:
- lesson observations,
- departmental meetings,
- informal discussions,
- the pupils’ voice
All these actions feed into your knowledge of the department and what is working within it and what is not. However it is well worth setting certain times when you reflect carefully on the evidence.
Take time to reflect on what you know
Write a paragraph or two on how well your department is doing. What has changed in the last six months? What has been successful, what hasn’t? What are the strengths of your department? What still needs to be improved?
- evaluate – it is easy to describe what the department is like and what it does, but this paragraph should be evaluative and make judgements about outcomes for pupils and the quality of provision
- be precise – base judgements on evidence and not what might happen
- explain impact – link judgements about the quality of provision to impact
- be transparent and specific – make the paragraph readable to your Head teacher or to governors or other stakeholders
- reflect stakeholders’ views - include what you know about the views of pupils and their parents and carers
Update this regularly as part of your self-evaluation process
Use Folio 2.6.1 to write a fuller self-evaluation based on the Ofsted criteria if you wish to.
Back to top
7. Reviewing the module
In completing this module, you will have considered:
1. Taking a lead on pedagogy
2. What pedagogies are there and how do they help in learning mathematics?
- creating a growing, learning department
- exemplifying good practice
- using mentoring, coaching and action research
3. Learning Theories and their effect on pedagogy choice
- what pedagogies are there
- using different pedagogies
- creating the classroom climate
4. The impact of assessment on learning
- what is learning?
- what difference do learning theories make to pedagogy?
5. Involving other staff
- what do you think of AfL?
- using Assessment for Learning in your department
6. Managing change
- using teachers to develop pedagogy in your department
- using people outside your department to develop pedagogy
- creating a climate that fosters reflection and innovation
- monitoring results
Each of these sections is intended to add to your knowledge of how to help all staff involved in the teaching of mathematics develop their pedagogy in order that it might deliver improved results.
Review the ideas that you have gained from each of the sections and reflect on which of the sections will make a short term difference to your department and which will add to your long term plans.