My aims for the year
My teaching aim for the year is to create an equitable classroom culture where all students’ ideas are valued. I would like design a learning environment where students feel comfortable making conjectures and revising their answers in public, in the vein of Magdalene Lampert and Deborah Ball. I am guided by the principles of dialogic teaching and the concept of voice as advocated by Robin Alexander and Adam Lefstein. I would like students to work collaboratively to develop wider mathematical and social skills, with a view to tackling status problems, following the work of Elizabeth Cohen and Ilana Horn.
What connects these approaches? It is a view that effective learning requires students and teachers to talk and listen to each other.
They are also connected by the view that teaching is complex, and that by over-simplifying teaching we ignore the many interactions and relationships that contribute to a productive learning environment. As teachers we are faced with a series of complex dilemmas that demand in-the-moment decisions backed by a repertoire of approaches. In order to make good decisions we need to be able to anticipate what might happen, interpret what is happening, and have the skill and flexibility to carry out our decisions.
Opening, not closing
The biggest change in my teaching this year can be summarised by opening rather than closing. I want to create a culture of reciprocity and openness in my classroom, where student responses are viewed as opportunities: to open discussion, to add to the cumulative shared knowledge of the class, to agree and disagree, to make conjectures and explain, to consider alternative approaches and ideas that may not be the same as our own, to assign competence to a wider range of skills.
This week I have been putting this into practice. It has been difficult; my initial reaction as a teacher is to correct and guide. I have had to consciously correct myself, intentionally finding ways to open responses back to the individual students, groups of students, and the whole class, in order to help them make sense of their learning and gain a shared understanding.
For example, on receiving an incorrect, interesting or unexpected comment during whole-class discussion, instead of correcting or closing down the response, I might ask, “What do we think about this?” or perhaps, “Does anyone have anything to add to this?” I want to present a range of (possibly partial) solutions to problems, to open discussion around the relative mathematical merits of each approach.
During collaborative work, I aim to set open tasks that require students to work together AS a group not just IN a group, to make and explain conjectures, to agree and disagree with each other. I am experimenting with a range of approaches of varying degrees of ‘openness’, from working on teacher-generated problems (with a range of possible responses), through to more open inquiry approaches that require students to generate and solve their own problems.
When working with students one-to-one, instead of correcting errors, I might say: “That is interesting; why do you think this is the case?” or open the response to the whole class, asking: “What do we think of this?”
Although these are simple strategies, it takes a change of mindset and a conscious effort to ensure that students understandings are not closed down, either by you as the teacher, or by other students, but rather opened up for reflection and discussion.
Openness requires trust between all members of the classroom; students must be happy to present and revise their responses in public. In order for this to happen, we must aim to create an equitable environment where all students are perceived by each other to have equal status. Everything that happens in the classroom must be designed with this aim in mind.
The language we use is fundamental. Instead of answers being viewed as correct or incorrect, any (partial) response is viewed as contributing to a more complete understanding of the problem. Every response is valuable. I continually use the language of conjecture, explanation and revision, of agreement and disagreement, to encourage students to share responses and participate in discussions.
Alongside the use of language is the need to assign competence to multiple abilities. Each lesson I have taught this week has had clear mathematical (content) aims, wider mathematical aims (such as making conjectures and explaining why), and wider social aims (for example, listening to others' views). Students who exhibit these abilities are praised specifically and publicly; in this way every student has something to offer every lesson, not just those who have greater prior knowledge or who are quicker at calculations.
We have worked on establishing norms for collaborative work. I was initially sceptical of this idea, and started with an informal approach to group work. However, I was not satisfied with its success and through discussion with the students, we developed a set of ‘rules’ that we have all agreed follow. We currently have four norms:
· Work AS a group, not just IN a group.
· You are only stuck if the whole group is stuck.
· Take turns to speak and listen to each other.
· The aim of working as a group is to reach a group agreement.
I have become very aware that every decision I make as a teacher will contribute to the culture of the classroom.
For example: as a department, we decided to set a start-of-year diagnostic test. I decided to allow students to work in pairs (although no pair was allowed to converse with another pair), with each pair jointly responsible for their result on the test. Although this is not a particularly good model for group-work, it worked well (perhaps due to group accountability), and gave the message of cooperation not competition. It was also useful for students in remembering previously forgotten content.
In evaluating this test, simply marking questions right and wrong didn’t fit well with the culture I wanted to promote. I selected various questions that students found difficult, and asked them to work in larger groups to discuss various responses to these questions (some fully correct, some only partially), using the language of conjecture and revision, not correct/incorrect. In this way, we arrived at a shared understanding of effective approaches to these questions, whilst consolidating the use of appropriate language.
I have been working on developing a framework for moving from individual or small-group work towards whole-class dialogue. The framework that I am using is from Five Practices for Orchestrating Productive Mathematical Discussions by Margaret Smith and Mary Kay Stein (there is a summary here). The five practices are:
· Anticipating what students might do.
· Monitoring what students are doing during the lesson.
· Selecting students’ work that highlights key ideas.
· Sequencing students’ work in an order that advances understanding.
· Connecting students’ ideas in a coherent discussion.
This approach is nothing new, and is something that many teachers do informally. However, the key here is a keen focus on listening to students, using their understanding to build a productive mathematical discussion.
When students are working on a task, the natural instinct as a teacher is to get involved, ask questions, to give help when stuck. However, the monitoring phase requires teachers to step back and observe what is happening; this is crucial in allowing students time to make mistakes, to discuss approaches in their own voice.
It can be difficult to record the vast amount of information that might be happening; the skill is to focus on what might constitute important points for discussion. I have been experimenting with a monitoring map; another approach might be to make a table/list of things you are looking for (as part of the anticipating phase) and fill this in as the lesson progresses.
Selecting and sequencing students’ work is the key part of the process, but also the most difficult. In reality, you are continually monitoring, selecting and sequencing at the same time, all the time. You are looking for common approaches or perhaps something unusual or interesting that will advance mathematical thinking towards your aims.
A dilemma that you will face is to what extent you should follow approaches that may not be as you expected, that may not directly lead to your mathematical aims, but will allow different student voices to be heard. This is a decision that must be made relative to the students and your aims. You are thinking about the order and the method in which you will connect these ideas together in order to create a coherent discussion about the mathematics that is important.
All of this takes practice and skill, especially with open tasks where students are exploring a range of conjectures; there are many dilemmas to be considered. Whilst the approach is demanding and will take time to master, the benefits become clear as soon as you try it!
The case of Katy
One of my colleagues, Katy, has similar aims for her classroom culture this year, and was interested in using the five practices approach described above in order to move from collaborative work towards a whole-class discussion. We arranged that I would watch her teach and see how it worked; this is part of a model in my department where we have agreed to watch each other teach once a fortnight.
We started by anticipating what the students would do by working through the task together.
During the lesson, we both monitored the students for around 15 minutes while they started the activity. At one point around half-way through this period, Katy and I had a discussion (a sort of in-the-moment CPD) and agreed that there were simple things that would encourage students to work more collaboratively, such as better use of the physical environment (getting students to sit closer together, working on one piece of paper).
We also discussed ways to encourage less vocal students to get involved, perhaps by asking them what they thought about what the rest of the group was doing, or suggesting that we would come back in a few minutes to ask them what the group had discovered. I modelled the approach to Katy and she tried it for herself - it worked really well.
We then continued to monitor the students; it was clear that these simple ideas were having an immediate impact. Then I sat back and observed as Katy progressed through the selection, sequencing and connecting phases of the lesson.
In discussion after the lesson, Katy said that she felt the interventions were hugely beneficial. I was worried that she might find it off-putting, or perhaps might undermine her confidence in some way, but this was not the case. Of course, this might not be the best approach with another teacher, and I will have to be careful when using it.
We also discussed some other dilemmas that Katy faced. I was interested to know how and why she selected and sequenced in the way she did, among other things. One thing that became apparent as part of this conversation was that Katy was very clear about her social aims for the lesson, but wasn’t fully clear on her mathematical aims.
She has since taught a similar lesson and has found having clear mathematical aims a crucial element in helping her decide how to select and sequence. Not only this, but it helped her assign competence to multiple abilities. This will form the focus when we go through this process again in a couple of weeks time.
Teaching as teacher learning
Following this conversation with Katy, I asked some teachers and teacher-educators on Twitter, whose views I greatly respect, what they thought about the questions I asked Katy, and how they provided feedback to teachers. Here are their thoughts:
Bryan Meyer (@doingmath) had some very interesting comments. He views the role of observer as a data collector, with the teacher as researcher. He uses these coaching maps for pre- and post-observation conversations, from the book Cognitive Coaching. I particularly like the idea of impressions and analysing causal factors.
Michael Pershan (@mpershan) talked about the sensitivity of teachers (I know, I’ve been one for 10 years!!) and how all lesson feedback might be more usefully framed as interpretation of events, with any feedback being used as part of planning for the next lesson.
David Wees (@davidwees) gave some great examples of how he only looks at a single focus that is related to the goal of the teacher. He gave an example where he recorded every question (of 170) that a teacher asked. Another example was when he tracked the teachers movements for a whole lesson; it looked something like this:
Mike Ollerton (@MichaelOllerton) explained how we likes to write a short account about an interesting or unexpected event, and discuss what might have caused it.
Elham Kazemi (@ekazemi) did not contribute to this discussion, but has talked about teacher time-outs here, which are relevant in light of the discussion Katy and I had during her lesson. I love her vision of teaching as teachers learning (as well as students).
These are all amazing suggestions and I am going to consider how and when I might implement them. For my part, I am also interested in the use of audio to record and analyze dialogue in the vein of Adam Lefstein (@alefstein). I am also interested in evaluating what is happening in the classroom using of these high-leverage practices from teachingworks, this rubric the Boston Teacher Residency, and the 5x8 card from SERP, perhaps focussing on just one of the competencies for a lesson as agreed through discussion with the teacher.
All of this in one week! So much has happened already, but there is so much more to work on.
What has been really interesting for me is the difficulty I have had in changing my approach. I have set my own aims; I am hugely motivated to achieve them, and I have done a great deal of reading around these practices - but it is still very difficult to maintain these approaches. Perhaps this is why little changes in teaching, and why CPD is often ineffective - it is hard to change habits!
The danger is that it would be easy to slip back into my old habits, to rely on less-ambitious teaching methods that do not take account of the full complexity of what is happening in my classroom. It is crucial that I consciously ensure that everything that happens in my classroom does not damage the culture that I am working hard to create. Of course I will make mistakes, and practices will need to be reviewed and adapted to suit the needs of my students, but I must try to stay true to the ambitious classroom goals that I have set. This will take a great deal of conscious thought and intentional planning, a lot of practice and a lot of reflection.
Stay posted for week 2 (if I have the time!). If you have any comments on any of this, please post below!