**Teaching is complicated**

In my previous post I asked the question:

*How do we decide how to teach?*The truth is that there isn't a simple answer. Anyone who thinks they know a simple answer, or a 'best' way to teach, is almost certainly missing the point.

Perhaps all we can say with certainty is that there is no single best way to teach, as described in this excerpt from the booklet Adding it up: Helping Children Learn Mathematics:

This booklet includes an

*instructional triangle*(below), which shows various relationships we might consider when designing instructional activities (i.e. anything that happens in a classroom):

When designing instructional activities, there are a number of relationships to consider, none of which happen independently of each other, and all of which happen in context.

A similar view is held by US teacher-researcher Magdalene Lampert. In her excellent book Teaching Problems and the Problems of Teaching, she describes teaching as:

... the practice of structuring activities of studying in relation to particular content and particular students.

Lampert’s message is that we should aim to consider the full complexity of classroom interactions when planning and teaching, something she calls ambitious teaching, rather than attempt to reduce teaching to set of routines and best practices; here is an interesting blog on this.

Whatever our view of teaching, we must think very carefully when planning what and how to teach. As Graham Nuthall describes in his book The Hidden Lives of Learners (p79):

…[instructional] activities need careful designing so that students cannot avoid interacting with relevant information.

All of this suggests that any discussion around a single ‘best’ method of teaching does not fully consider the complexity of teaching. This is a daunting thought.

However, we can work towards becoming more knowledgeable about the complexity of teaching, and find ways to strengthening the many relationships in our own classrooms.

**1. Student-content relationships**

Strengthening

*student-content relationships*is the main reason we teach – to help students learn our subject.

However, the student-content relationship is also the most problematic because it is not easily visible to us as teachers. We try to

*observe*learning as carefully as possible, and then use this information, along with the other relationships, in order to

*strengthen*the student-content relationship.

Skilful teachers aim to observe the changing student-content relationship through a range of formal and informal methods (assessments). We then use these assessments formatively to select appropriate instructional activities that will strengthen the student-content relationship further.

This process of observing and strengthening the student-content relationship is part of the answer to the question "

*How do we decide how to teach?"*

We decide how to teach based on how we think our students will interact with the content, on our

*knowledge of the student-content relationship*- but also based on the other relationships in the instructional triangle; our relationships with the students, their relationships with each other, and our relationship with the content.

This is the skill of teaching. We use our knowledge of each relationship to decide how and when to give students guidance. So the question is not whether lessons should be student-led or teacher-led, but rather:

*Which instructional activities and scaffolding should I provide to which students and when, in order to help them learn the content and skills I want them to learn?*

It is our role as a teacher to gain as much knowledge as we can about these relationships for all of our students, and use this to provide the right level of scaffolding in order to help them towards their aims.

**Case #1: A comment on my previous post**

I would like to give an illustration of how this process might work in relation to a suggestion I made in a previous post.

In this post I suggested that one way to balance 'discovery' and 'transmission' approaches could be through the use of ‘pre-teaching problems’, in which students should work on problems before the lessons, before listening to teacher exposition.

I received this comment in reply from David Wees (@davidwees):

It is interesting to me that your final conclusion suggests that one should use explicit instruction after giving students problems to work on. I propose a friendly amendment; one should give students problems to work on and then decide what instructional approach to use next with explicit instruction being one choice... At a meta-level, one should study the impact of both approaches on student learning (using a variety of different metrics) and use this information to help inform future decisions.

David's absolutely right! The decisions we make as teachers should be based on our knowledge of the student-content relationship. He embodies this idea with the comment that we should “

*study the impact of (both) approaches on student learning (using a variety of different metrics) and use this information to help inform future decisions*”.

**2. Teacher-content relationships**

Most would agree that effective teachers must have good mathematical

*subject*knowledge and, perhaps more importantly, good mathematical

*content*knowledge - knowledge of the curriculum, methods of teaching various concepts, our experience of student misconceptions, and so on.

Our knowledge of the content we teach is vital to strengthening student-content relationships. However, this strengthening is reciprocated: our knowledge of how students interact with content is also vital in strengthening our teacher-content relationship.

How can we know which methods are effective unless we measure their impact on student learning? How can we know what misconceptions to address if we don’t observe the student-content relationship carefully?

For this reason, the teacher-content relationship is always changing, and is always in need of development. Added to this is the need to engage with the ever-chaging body of educational research. We must always seek to strengthen our teacher-content relationship.

**Case #2: Learning about teaching Maths**

One way to develop teacher-content relationships is through good quality

*subject-specific*CPD that occurs

*in context*. However, in over 10 years of teaching I have received little CPD that has significantly strengthened my teacher-content relationship.

In my department next year, we are aiming to address this problem,

*to work together as a department to learn about teaching maths to our students*. There will be two strands to this:

Watch this space to see how this works over the course of next year!

**3. Teacher-student relationships**

Whether you believe that students are in your lesson to just learn mathematics, or whether you hold a wider view of education, it is clear that teacher-student relationships are important.

But how exactly do you go about strengthening them? Well, like everything else in teaching, it's complicated. Every relationship depends on you and each student separately, but also with you and the students as a group.

To add to this complexity, the teacher-student relationship cannot be separated from the teacher-content and student-content relationships; students will be more willing to cooperate with a teacher who has the ability to help them learn.

But perhaps here lies the first step towards strengthening the teacher-student relationship –

*strengthening the teacher-content and student-content relationships*. Helping students learn the content will go a long way to building cooperation, trust and respect.

I tell students, and anyone else who asks, that students are in my classroom primarily to learn mathematics. I make it very clear to them that this my job. My classroom culture is that we are there to work and learn mathematics, and that nothing should stand in the way of this. There are objectives and routines - expectations - that I expect students to follow.

However, this is not the whole picture. The real reason for the routines and expectations is that what I really want is to create a classroom environment where everyone feels comfortable and happy, where everyone wants to be there, where everyone wants to learn and be successful. I want make sure that students know (implicitly or explicitly) that I am here to help them, whether that's academically or otherwise. I want them to help them because I care about my relationships with my students

*for their own sake*, not just for the sake of learning maths.

I know this is all a bit woolly, but I hope this explains in part how and why I go about trying to build strong teacher-student relationships.

**Case #3: Mathematical discussions**

We need to talk to, and listen to, our students, both on a personal level and for important pedagogical reasons.

Discussions are crucial to effective learning of mathematics; they bring thinking out into the public and allow us to build a classroom culture where all students’ ideas are valued. In this way, they strengthen teacher-student and student-content relationships. There are huge gains to be made from well-structured classroom discussion.

However, there are risks in making thinking public; poorly handled discussions may have negative impact on the learning of some students. In their excellent book Intentional Talk, Elham Kazemi (@ekazemi) and Allison Hintz (@ahintz124) describe the importance of handling discussions sensitively:

Discussions … open up the possibilities that students will share their partial and incorrect understandings. How we respond to errors and partially developed ideas sends important messages about taking risks.

Discussions give us the opportunity to model approaches to doing mathematics, and the language we use is crucial.

In the book Teaching Problems and the Problems of Teaching, Lampert talks about how she uses the word

*conjecture*as a replacement for the word

*answer*, in order to signal that partial (or wrong) answers are a part of the mathematical process, and are open for discussion and revision (p77-78).

In their paper Making the most of Mathematical Discussions, Megan Staples and Melissa Colonis use the term 'work in progress’ to describe students partial or incorrect solutions:

These ideas are built on by Kazemi and Hintz in their book, in which they provide structures for holding productive classroom discussions. In a structure they call 'Troubleshoot and Revise', they describe how as teachers we can (p112):

...use errors for advancing mathematical thinking [as well as] thinking carefully about how we can treat students as sense makers and find the logic in students' partial understandings as they facilitate mathematically productive and socially supportive discussions. We want students to know that thoughtful mathematicians voice their confusions; thinking collaboratively through errors can help everyone better understand mathematics.

Notice again the importance of the language; when we talk about errors with students, we are talking about

*revising*

*partial understanding.*

The template below for planning to use the Troubleshoot and Revise strategy can be found here (Appendix F). I am looking forward to using this in September to help structure the discussions around Exam Co-op (see Case #4 below).

**4. Student-student relationships**

Classroom discussion is closely tied in with student-student relationships. What students say, how they interact, has a huge impact on how students learn. In The Hidden Lives of Learners, Graham Nuthall describes the importance of peer relationships (p87):

What is not often understood is just how powerful the social world of peer relationships is in shaping how and what students learn. Exchanging relevant information occurs very frequently in most classrooms whether the teacher is aware of this happening or not. Much of the knowledge students acquire comes from their peers.

He suggests that teachers must take into account the power of peer relationships, and concludes that (p104-106):

In all the classes we studied, the teachers used a mixture of whole-class, small-group and individual activities. As might be expected, the frequency of peer interactions was significantly higher in small-group activities. But during individual activities and whole-class activities, there was still the constant presence of peer interactions.

This finding raises an important question about how teachers could and should effectively manage a classroom. If a significant part of what a student learns is through informal, often spontaneous peer interactions, what, if anything, should the teacher do about this?

What seems to emerge from the research is that effective group work requires students to acquire a set of attitudes and beliefs. In other words, there is a need to develop the classroom as a learning community.

There is a huge body of research evidence that confirms that working collaboratively helps students learn more effectively. Yet, here is my confession: In recent years, the amount of planned student-student interaction in my classroom has reduced to virtually zero.

Why? In truth, I had become disillusioned with group work. It is harder to control what students do, and harder to assess what they learn. By asking students to work individually, I can ensure each student is on task and can assess what every student has learned more easily.

But I have never been happy with this. I know that students learn from each other, and I believe that there is a social aspect to learning mathematics. Do I want my students to view mathematics as something that is

*only*done individually?

As with all things, there should be a balance. There is a time for individual work, but it is impossible to argue with the research; removing the chance for students to work collaboratively removes important learning opportunities. It is time for me to redress the balance and learn more about how I can incorporate collaborative work into my lessons more effectively – but how?

In the book The New Circles of Learning (Johnson, Johnson, Johnson-Holubec), the authors use the term

*cooperative*(rather than collaborative) learning to describe effective small-group work (p35):

Many educators who believe they are using cooperative learning are, in fact, missing its essence. There is a crucial difference between simply putting students in groups to learn and structuring cooperation among students. Cooperation is not having students sit side-by-side at the same table to talk with each other as they do their individual assignments. Cooperation is not assigning a group report that one student does and the others put their name on. Cooperation is not having students do a task individually with instructions that those who finish first are to help the slower students. Cooperation is much more than this.To be cooperative, a group must have clear positive interdependence, and members must promote each other’s learning and success face-to-face, hold each other personally and individually accountable to do their fair share of the work, appropriately use interpersonal and small-group skills, and process how effectively they work together. These five essential components make small-group learning truly cooperative.

So it turns out there is more to structuring successful collaborative work that getting students to work in groups. A key component is

*positive interdependence,*which can only

*exist when students perceive that each group member’s efforts are required to group success. There are various ways of achieving this, but a simple approach is to design activities that*

*ensure*students

*must*work together to achieve a goal.

**Case #4: Exam Co-op**

Here is a simple idea that I am going use next year in an attempt to structure cooperative work and achieve positive interdependence in my lessons, whilst getting students to practice solving exam questions.

Consider this exam question, taken from the C1 exam, Jan 2013:

There are three parts to the question, suggesting it could be tackled by a group of three. In order for this to be a suitable question for this activity, first we need to check whether each part connects to subsequent parts.

To answer part (b) we must find the equation of line 2, although not necessarily in the form ax + by + c = 0. However, this form is arguably most useful as we can set x = 0 or y = 0 as required. This is something I would like the students to consider, perhaps when reflecting on the question as a whole. To answer part (c), we must answer part (b), but not necessarily part (a).

We break the question into three parts, making sure to include the relevant information for each part. Then we provide each group with the three parts (on different pieces of paper), with instructions on how to approach the task.

As a teacher, you will need to make a decision about how you want the group to approach the task. Do you as the teacher want to allocate parts to students, or do you want to let the students decide? Clearly part (c) is most challenging, so you may want to give this to the ‘most able’ student in the group, but of course in doing this you are making the others painfully aware of your view of their ability.

Here are the instructions. Note that I have included a follow-up question to give students the opportunity to reflect on the question as a whole, and have asked them to be prepared to discuss their solutions to the rest of the class:

And here are the three parts to the question:

This is a simple structure to start getting students working cooperatively and talking to each other about mathematics. However, it will not be enough to just give the activity to the students and let them get on with it. As a teacher I will need to research, test, and adapt various methods to ensure students are working truly cooperatively.

When discussing the solutions with the class, it will then be useful to use the structures such as those from Intentional Talk, such as Compare and Connect, What's Best and Why? and Troubleshoot and Revise to allow students to learn from any errors.

**Conclusion**

Teaching is complicated – there are many relationships to consider, all of which are dependent on each other. So-called ‘ambitious teaching’ is not about trying to 'fix' everything at once; it is about recognising and considering the complexity of classroom relationships when designing instructional activities.

With this in mind, it might be helpful to ask yourself questions like these when planning a lesson:

*How can I strengthen the student-content relationship?*

What are our aims? How should I design activities to reach these aims? How and when will I scaffold their learning? What are the advantages and disadvantages of different approaches? How will I find out whether the approaches are effective? Who can help me?

*How can I strengthen the teacher-content relationship?*

What are my aims? Am I happy with how my students are learning? If not, how can I improve or change my approach? Where can I learn more about this technique or that topic? How can I do it better next time? Who can help me?

*How can I strengthen teacher-student and student-student relationships?*

What are our aims, mathematically and socially? How can we create a learning community? What is our view of mathematics, and learning In general? What norms do we need in place for this to happen? Who can help us?

*Now what?*

Once you have answered these questions, it is time for action!

This post contains my thoughts on the various relationships, and gives a few simple practical approaches that I am going to make in order to improve my teaching to allow for these various relationships.

**What are your thoughts? What are you going to put in place to strengthen your relationships?**