## The state of Mathematics education

Teaching as telling appears to have survived early Modern Europe more or less intact.

This quotation by David Cohen is backed by a vast amount of evidence that the majority of classroom interactions follow an IRE (Initiation-Response-Evaluation) cycle between teacher and students. In an IRE cycle, teachers elicit known responses, students reply with the correct or incorrect response, and teachers evaluate that response. This model is so commonplace that Gage suggests that it

*“embodies something profoundly fundamental in the nature of teaching.”*

However, much of the research and theory about how we learn suggests that 'teaching as telling' may not be the most effective teaching method, as it may not lead to learners actively engaging with what is being told. The first principle of teaching in the book How People Learn is as follows:

The model of the child as an empty vessel to be filled with knowledge provided by the teacher must be replaced. Instead, the teacher must actively inquire into students’ thinking, creating classroom tasks and conditions under which student thinking can be revealed.

There have been many attempts to change the way teachers teach in an attempt to incorporate the theory and research of how we learn, and to move away from student passivity engendered by ill-structured teacher talk.

Perhaps the biggest examples of this in the UK are the National Strategies, which included an attempt to move teaching towards ‘interactive whole class teaching’. However, the report Interactive Whole Class Teaching in the National Literacy and Numeracy Strategies discovered they had little effect on disrupting the IRE cycle:

Far from encouraging and extending pupil contributions to promote higher levels of interaction and cognitive engagement, most of the questions asked were of a low cognitive level designed to funnel pupils' response towards a required answer. Open questions made up 10% of the questioning exchanges and 15% of the sample did not ask any such questions. Probing by the teacher, where the teacher stayed with the same child to ask further questions to encourage sustained and extended dialogue occurred, occurred in just over 11 per cent of the questioning exchanges. Uptake questions occurred in only 4 per cent of the teaching exchanges and 43% of the teachers did not use any such moves. Only rarely were teachers' questions used to assist pupils to more complete or elaborated ideas. Most of the pupils' exchanges were very short, with answers lasting on average 5 seconds, and were limited to three words or fewer for 70 per cent of the time. It was also very rare for pupils to initiate the questioning.

There are many reasons for the resistance to change. Firstly, the term ‘interactive whole class teaching’ is not particularly helpful, as all teaching is interactive in some sense. But, more importantly, teachers were expected to teach using complex strategies with little depth of knowledge about the theory behind the strategies, or training on how to go about implementing them.

Contradictory guidance from the UK government and Ofsted has not helped the situation; the original National Numeracy Strategy stated that lessons should have “

*a sense of urgency, driven by the need to progress and succeed*”, and Ofsted expected to see “

*rapid and sustained progress*” in 20-minute lesson observations. Although the talk of rapid progress has been replaced with “

*progress over time*”, the legacy remains; judgements on teaching and student progress are still made on observations of (parts of) lessons.

The need to demonstrate visible progress goes hand-in-hand with the IRE approach, rather than an approach that might involve prolonged periods of discussion and deliberation, where progress may be less certain. With the trusted IRE approach, teachers can minimize risk and uncertainty, keeping students on-task with sets of short questions, carefully sequenced to evidence progress towards clear content-based assessment objectives.

Of course, there are times when it may be beneficial to minimize uncertainty, or to teach at a pace with a number of short questions. However, sometimes it might be beneficial to slow the pace and introduce more uncertainty, in order to elicit a range of views and approaches, to reflect on appropriate methods, and aid understanding. Herein lays one of many of the dilemmas of teaching.

Introducing uncertainty and discussion comes with inherent risk; you are challenging students to think more deeply about, and perhaps what it means to

*do*, Mathematics. As Ilana Horn (@tchmathculture) explains in her excellent book Strength in Numbers: Collaborative Learning in Secondary Mathematics:

Many students have learned that getting work done quickly is an effective way to ‘do school’. They are often reluctant to slow down to work with a peer. They may feel that giving help interferes with their own learning, but on the contrary it often enhances it by presenting them an opportunity to make sense of their own thinking.

However, a range of approaches is required if students are to develop the wide range of mathematical skills outlined in the current National Curriculum. Students must be able to “

*recall and apply knowledge rapidly and accurately*”, but also develop skills such as “

*reasoning mathematically by following a line of enquiry, conjecturing relationships and generalisations and developing an argument, justification or proof using mathematical language.*”

It is unlikely that a single approach, IRE or otherwise, will be sufficient to teach the wide range of mathematical skills described here; it would appear essential that teachers should develop a

*repertoire*of approaches.

However, the recent increase in the amount of content in the already over-stretched GCSE and A-level curricula has resulted in teachers under increased time pressure to cover a wider range of content. And there is little evidence that the message has got through to the exam boards, where the majority of questions are still procedural, with little real problem solving or reasoning ability required.

Add to this the ever-increasing accountability of teachers for their students’ results, and we have little incentive for teachers to change what they have always done. An increase in content, formulaic exams and increased accountability means that few teachers are willing to take the time, effort and perceived risk of implementing different approaches.

## Should we teach wider mathematical and social skills?

*good enough*to get students through the current exams whilst minimizing risk for all. Which leads us to the crux of the matter: To what extent is the teaching of wider mathematical skills, or wider social skills, necessary, or important? And if we do think they are important, how and when should we teach them?

Some might not think wider

*mathematical*skills to be important at all, as students can succeed in the current UK exams by learning a set of procedures. I would argue that this might well be the case, especially at GCSE. However, you would hope that students with a wider range of mathematical skills would have a higher chance of exams success, particularly if exams ever catch up with the rhetoric of the National Strategies.

Some might not think wider

*social*skills are part of teaching Mathematics. However, I would argue that all teachers have a responsibility to address social issues in their classroom. Considering students spend the majority of their waking lives in a school environment, teachers must hold some of the responsibility for human improvement. This is not as easy to do in a Mathematics classroom as say a History classroom, where social issues form part of the content, but I think we can all structure our learning environment to promote social equality.

How should we address wider social issues in a Maths classroom? Firstly, let me make it clear that I do not think we should teach social skills out of context. That said, some teachers think that Maths teachers do not have to worry about developing social skills, that this happens implicitly as part of good teaching. I would agree that students gain some social skills implicitly by doing mathematics in a classroom of mutual respect, but I think we can go a little further than this: I think that social aims should form an

*explicit*part of Mathematics planning and teaching.

Horn describes this as a shift from viewing the teaching of Mathematics as the “e

*ffective presentation of ideas”*to the

*“design of effective learning environments”*. That is to say, we should move away from a coverage-based approach towards a consideration of the wider complexities of classroom interactions. Not only should we design our environment to meet mathematical aims, we should also design our environment to meet clear social aims.

In doing this, teachers must consider the impact their teaching has on student learning; how students learn influences cognition, as well as inter-related issues of motivation, affect and sense-of-self. This is a view of teaching that is wider and more ambitious in scope mathematically and socially.

## Knowing how to decide what to do

So, how can we teach wider mathematical skills? We must have a basis of procedural fluency; students must remember how to perform certain procedures, and perhaps it is here that IRE approaches have their place.

However, in order to teach wider mathematical skills, teachers must aim to expand their repertoire to include other approaches that allow students to learn how to decide

*when*to apply these procedures. In MT243, Anne Watson describes the importance of students being able to decide what to do:

There is not a great deal of point learning to add if we only think to add when someone tells us to add… It seems to me that the process of deciding what to do requires practice in deciding what to do. This is a different process to the practice required to become fluent in the doing. How do I decide what to do? In a mathematical situation I have years of experience on which to draw, so I can use similarity and memory to guide me. Perhaps that is not helpful in thinking about children’s learning.

Here is a shift in emphasis from doing, to

*deciding what to do*. In order to teach how to decide what to do, we must encourage students to justify why a certain approach might be more effective, to reason and make sense of what they are learning. This is described in this video by @davidwees.

Students and teachers must be prepared to listen to, and learn from, each other. We must create a culture and design activities that fosters

**mathematical dialogue.**

This is not an easy task. Horn describes a shift in how teachers might plan, from “

*How should I teach this?*” to “

*How will I get different kids with different strengths to talk about this with each other?*” We are using students as a resource to help each other learn, rather than the teacher being the only source of information. This takes a lot of research, skill and practice on behalf of the teacher.

We need to design an environment where cultural and

*intellectual*diversity are viewed as strengths, rather than obstacles to be overcome. It is just this diversity that leads to productive mathematical dialogue, where all students’ ideas are valued. Being good at Mathematics is no longer about who can get the most right answers; it is also about who can contribute to productive mathematical discussions, or who can approach a problem in a way that leads to a better understanding. We must change what we, and our students, understand by mathematical smartness:

Unless we address underlying conceptions of smartness, we risk reverting to the commonly held belief that group benefits struggling students because smart students help them. As long as we have a simplistic view of some students as smart, and others as struggling, we will have status problems in our classrooms. No doubt learners benefit from seeing more expert performance and should have opportunities to do so. But if we value only certainkindsof expertise, the same students will always play the role of experts. The question then becomes: What kinds of mathematical competence have a place in your classroom activities?

This has important implications for student status. Horn defines status as the “

*perception of students’ academic capability and social desirability*.” Status and mathematical competence go hand in hand; how a student is perceived by others (and themselves) has a huge impact on how they learn.

We as teachers can alter the status of our students, as described by Elizabeth Cohen in her paper Producing Equal-Status Interaction in the Heterogeneous Classroom:

Because of differences inperceivedacademic ability, the high-status student will then expect to be more competent and will be expected to be more competent by others. The net effect is a self-fulfilling prophecy whereby those are seen as having more ability relative to the group … tend to dominate. Academic status characteristics are the most powerful of the status characteristics in the classroom.

Teachers can alter the status processes in a heterogeneous classroom by altering the expectations for competence that students hold for themselves as well as expectations they hold for one another.

Through assigning competence to a wider range of mathematical abilities, we can work towards a classroom culture where all students are perceived as competent mathematicians. In this way we can teach students wider mathematical skills

*and*promote academic (and social) equality, thus reducing the alienation from mathematics felt by those who are simply not as quick at calculations as others.

This passage from Magdalene Lampert's book Teaching Problems and the Problems of Teaching highlights the features, importance and the risks of this approach to teaching:

I am teaching everyone to participate in a classroom culture where students are publically willing to reason their way from confusion to making mathematical sense, are publically willing to talk about what they are thinking, and respect what others say even if it does not seem to make sense. This element of my practice is so fundamental to teaching students to engage in mathematical work that it leads me into dangerous social territory…

Implementing the type of teaching effectively is an ambitious task. David Cohen describes the rewards and challenges of this type of teaching in his paper Teaching Practice: Plus ca change... :

Such work can be fascinating, and students could learn a great deal about mathematical reasoning from it. But in order to do so they would have to tolerate considerable uncertainty: about the nature of arithmetical problems; about the procedures for solving these problems; about what the answers are, and what an answer is; and about how implausible answers can be detected, and plausible answers defended. If done well, this would lead on to questions about the nature of arithmetic, and what it means to know it. That would be all to the good: If done carefully, such work can be immensely illuminating. But it requires that students find ways to embrace uncertainty, that they adopt trying out--that is, hypothesis framing and testing--as a way of life in learning. To do so, teachers and students must devise instructional strategies that enable them to manage and capitalize on the higher levels of uncertainty and the more demanding thought required to manage it. Such strategies are available, but they make unusual demands on teachers and students.

Adventurous pedagogy invites students to define and attack fundamental problems, to be intellectual explorers, to share their ideas, arguments, and intuitions with classmates and teachers. While one can learn much from such work, much of it is learned from one's mistakes. Students must be ambitious--which is a polite way to say that they must take intellectual risks--and ordinarily they must take them in classrooms in which a large and possibly competitive audience watches and listens. Instruction of this sort requires that teachers find ways to engage students more fully in learning, but it also requires that they find ways to reduce or otherwise manage the possible personal risks of such greater engagement. It is no mean trick to intensify engagement at the same time as easing its risks.

## Towards dialogic teaching

**dialogic teaching**

*.*In his excellent book Towards Dialogic Teaching, Robin Alexander describes it not as one method of teaching, but rather:

…a repertoire of teaching skills and techniques, all underpinned by clear educational principles.

In this book, Alexander gives a brief summary of the features of dialogic teaching:

*Collective*: teachers and students address learning tasks together, whether as a group or as a class.*Reciprocal*: teachers and students listen to each other, share ideas and consider alternative viewpoints.*Supportive*: students articulate their ideas freely, without fear of embarrassment over 'wrong' answers, and they help each other to reach common understandings.*Cumulative*: teachers and students build on their own and each other's ideas and chain them into coherent lines of thinking and enquiry.*Purposeful*: teachers plan and steer classroom talk with specific educational goals in view.

However, there is much more to dialogic teaching than this; I would strongly recommend anyone wishing to understand the principles behind dialogic teaching in full to read the book.

Another excellent book on implementing dialogic teaching in practice is Better than Best Practice by Adam Lefstein and Julia Snell. They describe dialogic teaching as that which:

...examines a range of different aspects of classroom communication and interaction. These include communicative forms, interpersonal relations, the exchange and development of ideas, power, pupil and teacher identities …

Dialogic teaching considers the full complexity of classroom interactions. It requires each member of the classroom to learn how to talk and listen to each other. Lefstein and Snell go on to describe dialogic pedagogy as being:

…most helpful as a set of dilemmas to consider, concepts to think with, commitments to pursue and balance, and practices to add to our repertoires. It is less helpful as a narrow best practice solution to each and every classroom situation.

In my next post I will give some practical approaches for implementing dialogic teaching through collaborative work.