## Introduction

This year I am researching my own practice through noticing and recording what I feel are significant teaching experiences, in my aim to move towards opening-not-closing and valuing student voice.

This post contains

*accounts-of*my experiences, not

*accounts-for*; I have made no attempt to evaluate or explain, in line with John Mason's Discipline of Noticing.

They are records of what feels significant to me; helpfully you will find that some of the events resonate with your experience.

## Account #1: Deeka

I had realised that I was running out of time. There were around 12 minutes of the lesson remaining on a Friday, and I wanted to start a new topic next week in line with the Scheme of Work. I had not made the progress I wanted in ensuring students appreciated the link between the the graph of a quadratic and the solution to quadratic equations.

I drew a range of examples of quadratic functions on the board, firing various questions about them around the classroom in order to activate the students, trying to involve them in the construction of the argument. I could see from their expressions that a few students were finding the argument hard to follow. I turned to one of these students, Deeka, and asked her: "

*So, how does the graph tell us the solution to the equation x^2 + 4x + 5 = 2?*"

I waited for her response. She was looking down at her book, avoiding eye contact. I could see that she was feeling uncomfortable. I wanted to give her time to answer. After a difficult silence of around 10 seconds , she said: "

*Sir, you're putting me on the spot!*"

I have been working hard this year on building trust with my students; this was the opposite of how I want my students to feel; I want them to feel supported and encouraged. However, I also felt a mixture of frustration and pressure to get to the end of the lesson before the time ran out. I pointed to the place on the graph where the solutions might be found, but by this time I could tell that Deeka was not feeling comfortable enough to venture an answer.

I did not want to effect her status by moving onto someone else, and, coupled with my desire to provide closure to the lesson, I decided to explain the solution on the board.

What are the implications of my actions? How does this account resonate with your experience?

## Account #2: Forgetting or never-knowing?

I watched one of my colleagues teach a lesson on sketching exponential graphs. He had asked students to complete a number of problems. However, the students were not progressing as he expected or would like; in particular, many of them had not remembered how to apply function transformations. He spent the last 15 minutes of the lesson going through the problems with the class at a reasonably quick pace in an attempt to reach his lesson objective.

When we talked about the lesson afterwards, he expressed his frustration, saying:

*"I never finish what I have planned! Why do they always forget? They don't remember the prerequisites... or perhaps they never understood them in the first place."*

Does this account resonate with your experience?

## Account #3: Students' pace of learning

In the book Culture and Pedagogy, Robin Alexander describes his experience of the lessons he observed across 5 countries:

Of all the differences in the way classroom time is handled, pace is the most striking. Some lessons press on relentlessly and even exhilaratingly while others seem to be suspended in time or crawl painfully towards their eventual conclusion.

In chapter 15 of the book, he talks about pace of teaching and student learning:

A fast pace in teaching is not necessarily a virtue. It may prematurely foreclose important lines of enquiry. It may disadvantage those children who need longer to achieve understanding or complete a task, and although it is not acceptable to speak of ‘slow learners’ we know and accept that children learn at different speeds.

What are the causes and effects of differences in students' pace of learning in your practice?

## Account #4: Performance

I explained how we can sketch a quadratic once we have found the intersections with the axes (by putting x=0 and y=0 into the equation of the curve). I then asked students to sketch the following family of curves, varying one coefficient and then the other, to help them witness the effects and perhaps allow them to make generalisations from particular examples:

Students performed well on this task. However, by monitoring their work, I was suspicious that some students were just following a process and had not gained a deep understanding of the connection between the equation and the graph.

I asked them to work in pairs to find the equation of this quadratic curve and state the co-ordinates of the y-intercept:

I asked them to work in pairs to find the equation of this quadratic curve and state the co-ordinates of the y-intercept:

Many of the students found this difficult, justifying my suspicions.

However, it did create a great deal of discussion. After a while, one student called Mahir demonstrated to the others how he solved this problem by "going backwards," starting with the roots and creating the factorised expression of the quadratic curve. The other students gave him a spontaneous round of applause.

How does this account resonate with your experience?

However, it did create a great deal of discussion. After a while, one student called Mahir demonstrated to the others how he solved this problem by "going backwards," starting with the roots and creating the factorised expression of the quadratic curve. The other students gave him a spontaneous round of applause.

How does this account resonate with your experience?

## Account #5: 90%

Last week, I watched one of my colleagues teaching a lesson where students were working individually on a range of problems of increasing difficulty.

She was working very hard; I recorded that she spent around 90% of the time moving round the class helping individuals or small groups of students, addressing their queries as they got stuck.

Does this resonate with your experience? What are the possible implications?

## Account #6: QFT

I have been experimenting with an inquiry approach and the use of prompts rather than questions. In an introductory lesson on quadratics, I wrote:

*"All quadratic equations have two solutions."*

I decided to use the Question Formulation Technique (QFT) to allow students to create their own questions about this prompt. Under the QFT, students create as many questions as they can without pausing to evaluate them. They then prioritise the questions; they wrote their prioritised questions on the board.

The students then chose which questions they wanted to answer and worked on them in small groups. I then sequenced the questions in an attempt to create a logical argument, and invited responses to each question. This process took around two lessons.

There were a number of interesting general questions such as

*"Why do all quadratics equal zero?"*which led to a discussion about how writing a quadratic in the standard form underlies each of the solution approaches: factorising, the formula and completing the square.

What are your views on QFT?

## Account #7: Deciding what to do

In the article Consolidation from MT243, Anne Watson talks about teaching students how to decide what to do:

There is not a great deal of point in learning to add if we only think to add when someone tells us to add.I think this every time I flip through the kind of ‘practice’ books that are sold in supermarkets.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.

Does this resonate with your experience? What are the possible implications for your teaching?

## Account #8: What do we think?

Here are some of my students' responses when asked to solve the equation

**x^2 - 2 = 0.**

I decided to put these responses on the board and asked:

*"What do we think?"*We then talked about the student thinking that led to these conjectures.

How does this resonate with your practice?

## Account #9: Types of pace

In

*Culture and Pedagogy*, Alexander suggests that

*pace is not unitary*; he distinguishes between 5 types of pace:

**Organisational pace:**The speed at which lesson preparations, introductions, transitions and conclusions are handled.**Task pace:**The speed at which learning tasks and their contingent activities are undertaken.**Interactive pace:**The pace of teacher–pupil and pupil–pupil exchanges and contingent factors such as maintaining focus and the handling of cues and turns.**Cognitive or semantic pace:**The speed at which conceptual ground is covered in classroom interaction or the ratio of new material to old and of task demand to task outcome.**Learning pace:**How fast pupils actually learn

Does this categorisation resonate with your experience? If not, how would you categorise 'pace'?

## Account #10: Habits

Every lesson I start with around 5 short starter questions. Students filter in and I complete the register. Some students complete these questions quickly and others only complete a couple of them. It is often the same students who finish first, and the same students who only complete a couple of questions. Once a few students have completed all the questions, I go over starter questions briefly and start the 'lesson-proper'.

Does this resonate with your practice? What are the possible implications?

## Account #11: Meta-cognition

In the paper Knowing When, Where and How to Remember: A Problem of Metacognition, Ann Brown highlights the complexity of metacognition for any problem-solving system:

Some idea of the complexity... can be gleaned by considering the operations attributed to the central processor, interpreter or executive...

Being capable of performing intelligent evaluation of its own operations is an essential characteristic of the central mechanism; some form of self-awareness, or explicit knowledge of it's own workings is critical for any efficient problem-solving system.

The basic requirements of such an executive demonstrate the complexity of the issue. It must include the ability to (a) predict the system's capacity limitations, (b) be aware of its repertoire of heuristic routines and their appropriate domain of utility, (c) identify and characterise the problem at hand, (d) plan and schedule appropriate problem-solving strategies, (e) monitor and supervise the effectiveness of those routines it calls into service, and (f) dynamically evaluate these operations in the face of success or failure so that termination of strategic activities can be strategically timed.

These forms of executive decision-making are perhaps the crux of efficient problem-solving because the use of an appropriate piece of knowledge, or routine to obtain that knowledge, at the right time and in the right place is the essence of intelligence.

Apart from the implications for my students, this paper made me think about the executive (or what John Mason calls the 'inner witness') within

*me*when I am teaching.

What does this account mean to you?

## Account #12: Student generated examples

I want students to start generating their own examples. We needed to practise factorising quadratics, so I posed this question:

*"Think of an example of a quadratic expression that is easy/hard/impossible to factorise."*

Here are the student responses to this question:

In both classes, students enjoyed the challenge of creating and solving difficult problems. As each problem got solved, I wrote the solution on the board, and students turned their attention to factorising those that remained, and/or checking whether the expression in the last column were 'unfactorisable'.

On the second board, the last expression to be factorised was x^2 + 1/2.x - 18.

We spent around 10 minutes debating whether it

*could*be factorisable, and what it means to say a quadratic is factorisable.

What are your thoughts on this approach?

## Account #13: The subjectivity of pace

In their paper

*Beyond a unitary conception of pedagogic pace: quantitative measurement and ethnographic experience,*Adam Lefstein and Julia Snell add to Alexander's conception of different types of pace with the observation that pace can be both

*objectively measured*and

*subjectively experienced.*In their work analysing classroom discourse, they discovered an apparent paradox:

**

The classroom with the slowest pace - measured as discourse moves per hour - was the one we had experienced as most brisk and riveting.

They go on to describe one particular lesson, taught by 'Ms James':

At one level the pace is brisk and business-like: focused on getting through the task at hand in a direct manner. However, it is precisely this business-like manner that slows down our experience of the episode. The pattern of questions is repetitive and predictable.

Likewise, the pupils offer stock answers, referring in each instance to previously identified generic features, which have been rehearsed throughout the lesson. One result of the focus on generic features, combined with the brevity of pupil responses, is that we receive practically no information about the texts under discussion; there is no issue to engage our attention, no controversy, tension or puzzle to occupy our mind.

They account of the brevity of pupils' responses as follows:

Ms James’ rapid and snappy questioning exhibits a sense of urgency. However, one paradoxical effect of this urgency is that in her urgency to push the lesson forward Ms James ends up doing the bulk of the work.

In our discussions with her, Ms James frequently complained about her class’s lack of cooperation in whole class discussions, which she attributed to low ability and/or reticence to speak up in front of the group. In light of these comments... we suggest that the culture of Ms James’ classroom involves a positive feedback loop in which pupils’ hesitation to respond encourages Ms James to both lower the cognitive demands of her questions and also do the bulk of the work of answering and elaborating herself.

They conclude that the

*subjective experience*of pace is:

...rooted in the meaningful content of the conversation, including the extent to which this content is new and/or surprising to participants, if and how the conversation matters and how participants treat one another’s contributions.

At their extremes, objective and subjective pace may be inversely related: meaningful and important content requires us to slow down in order to attend and think; less consequential ideas require that we speed up, to get through the material as quickly as possible.

Does this resonate with your experience?

## Account #14: Katy

Intrigued by a subjective notion of pace, I asked one of my colleagues, Katy, to come and watch me teach, and record

*her*(subjective) feeling of 'pace' alongside the activities that were happening in the classroom.

She recorded that the parts of the lesson with the 'paciest feel' were those when students worked together in small groups.

Does this resonate with your experience?

## Account #15: In-the-moment pedagogy

Here is the summary of the paper The Importance of Teachers’ Mathematical Awareness for In-the-Moment Pedagogy by John Mason:

For us, then, teachers’ disciplinary knowledge of mathematics is more than a demonstrated competence in advanced mathematics, a specialized pedagogical content knowledge, and a set of practiced competencies that arise through years of practice. It is also a way of being with mathematics knowledge that enables a teacher to structure learning situations, interpret student actions mindfully, and respond flexibly, in ways that enable learners to extend understandings and expand the range of their interpretive possibilities through access to powerful connections and appropriate practice.

Learning to be responsive through sensitivity to students, rather than depending on reactions derived from habits, requires preparing to be present in the moment. We suggest that any apparent paradox in preparing to be spontaneous is oxymoronic only from a stance of articulated knowing about and an addiction to established procedures as the basis of expertise. From a stance of constantly working to expand and enrich what is noticed, becoming ever more aware of both what one is attending to and in what manner—in short, developing one’s own being, the phrase is intensely meaningful. But it points to becoming a teacher as a lifelong adventure, not a short-term activity.

How does this account resonate with your experience?

## Summary

What are the common themes among these accounts?

How do they resonate with your experience?