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Part I
Issues
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1 Engaging in mathematics
In the first two years of my teaching career in the UK, I experienced classes that would rarely listen to what I had to say. I felt I managed to ‘turn off’ students from mathematics who may have begun the year relatively keen and enthusiastic. Students seemed engaged primarily in an exploration of my boundaries rather than anything mathematical. The failings were of my own making. In my third year of teaching, I moved school and began a research collaboration with Laurinda Brown that was pivotal to my finding a way of being in the classroom that allowed something more interesting to take place. The ideas in this book have been developed almost entirely through that collaboration (see Brown and Coles 2008).
When I began teaching I had little sense of how to set up a way of working with a class so that the focus was on the mathematics. Reading accounts of other people’s successful classrooms did me no good, as I was not given access to information on what they did to set up the context in the first place such that students would listen to each other (or the teacher). In writing this book I have partly had in mind that ‘me’ who began teaching in the UK in 1994. I hope, over the course of the book, to offer some images, experiences and labels (symbols) to help think about what is involved in getting students engaging in school mathematics.
Freud referred to educating as one of the ‘impossible’ professions (along with healing and governing) ‘in which one can be sure beforehand of achieving unsatisfactory results’ (Freud 1976: XXII, 248). This reminds me of a story.
I taught a student Further Mathematics A-level some years ago. We were just starting the subject in my school and so I was allowed to run the course with one student, meeting for just a couple of hours each week. Due to the limited contact time, the student taught himself one module entirely. This was the module in which he received his highest marks.
Reflecting on the experience I can of course comfort myself with the notion that I must have equipped the student with the skills to teach himself. It is still striking, however, that he did best without my direct help. Maybe we should not be surprised. There is a view of biology that suggests the actions of living beings should be seen as a result of their own internal structure – the world around can trigger a response but never determine what that response will be (Maturana and Varela 1992). If I kick a stone, its path is determined by the energy, direction and manner of my strike. If I kick a dog, what happens next will mainly be a result of the dog’s character and metabolism, not the energy transferred from my boot (Bateson 2000: 229). Living beings are triggered into action by others, but how a living being responds is a result of its structure and history (Maturana and Poerksen 2004). Taking this view of biology to its logical end, suggests that ‘instruction, in the strict sense of the word, is radically impossible’ (Stewart 2010: 9). A teacher cannot determine change in her students. As teachers, we cannot make students act, all we can do is trigger and provide feedback on actions. How a student responds, and therefore what they learn, is a function of their own self. We cannot instruct/structure a student’s mind directly.
The impossibility of instruction resonates with a paradox about teaching and learning which has been recognised since antiquity. Plato (Meno 80d) had a view of learning as a recognition of the new, and asked how this is ever possible, since to recognise something I need to know already what I am looking for. More recently, in a presentation, Anna Sfard (2013) referred in her own language to essentially the same paradox: ‘To participate in a discourse on an object you need to have already constructed this object but the only way to construct an object is to participate in the discourse about it’ (quotation taken from the conference presentation slides).
There is something in these paradoxical statements that points to why teaching is so hard and why students sometimes do not engage in learning mathematics. When things go well, a student participates in the game, takes part in the discourse of mathematics and, in doing so, begins constructing mathematical objects that allow further participation – a virtuous cycle. When things are not going well, a student sees the discourse of mathematics as alien, does not participate and so does not construct the objects that others do, who are participating, and the discourse begins to seem further and further away as a possibility – a vicious cycle. What makes the difference is engagement.
Valerie Walkerdine (1990) also sees engagement in mathematics as about learning a new discourse. In a powerful critique of the contemporary practice in classrooms, she demonstrated how participating in the discourse of ‘reason’ is always harder for the ‘Other’ in a society: the women, the poor, the minority groups. Logic is seemingly always on the side of the oppressor. Walkerdine tracked the way that girls with high IQs aged four can come to be regarded as ‘stupid’ by the age of ten. Walkerdine (1990: 55) argued that it is
One of the roles of the teacher, then, is to find a way of hooking students into giving their attention to, and taking part in, the discourse of mathematics or, to borrow a Buddhist phrase, of tripping them into engaging. And the longer students have felt excluded from this discourse, the harder it may be to engage them. Walkerdine’s research implies it behoves us all, as teachers, to be particularly mindful of the groups and individuals in our classrooms who are excluded.
It is sometimes possible to catch yourself in a moment of exclusion, to remember what it must feel like to experience the events of a lesson as occurring at some distance from yourself. Another story comes to mind.
I went into our garden to pick some runner beans from a bean plant in our vegetable patch. Bean plants are tall with large leaves and the runner beans are of the same colour, growing from the stem of the plant. At first I could not make out any runner beans. After some while of looking and wondering if there were any beans on the plant at all, I saw a runner bean, perhaps by chance, and picked it. Then quite quickly I noticed another runner bean, then another. Having started to see the beans, I realised they were everywhere on the plant!
As soon as I start seeing ‘beans’, I cannot not see them; it is then hard to engage with someone who was in the place I was just a moment before (and perhaps easy to interpret them as lacking ‘intelligence’ or ‘ability’). In essence, then, learning changes how we view the world, usually without us even realising.
Marton and Booth (1997) describe how, in the course of life, we normalise our world. As we learn something new and therefore experience a small part of the world in a different way (e.g., looking at a bean plant), that new way of seeing becomes normal and we are rarely sensitive to the shift that has taken place. Freudenthal (1978: 185) describes working with some children and asking them which number was least likely to occur when you roll a dice. The children agreed that ‘6’ was hardest to get (perhaps because you need to roll a ‘6’ to start in some games and it can seem as though you can never roll it). Freudenthal goes on to get students creating nets of dice and writing on the numbers. With no discussion and seemingly no discontinuity for the children, at the end of engaging in making their own dice, no child thinks ‘6’ will be harder to get than the other numbers. We do not always notice how we change.
One of the difficulties for the teacher, therefore, is to be sufficiently open to how students may be seeing situations differently from us (and particularly those students who are ‘different’ from us). How can we think about what activities to give students when they may be seeing the world in ways that we are no longer able to access and when we know that we cannot directly influence how they see the world? And linked to this question, what will engage students in wanting to change their way of seeing, in wanting to learn, in the first place?
Engagment requires vulnerability. Both teacher and student need to allow themselves to become vulnerable to the other, if there is to be a space where engagement is possible and where experiences can lead to the creation of new ways of thinking and new symbols.
An image of a classroom
To help think about some of the questions raised so far and to set up the themes of the rest of this book, I want to present an image of a classroom where students are involved in creative and independent work in mathematics. The problem with doing this is that inevitably, as a reader, you can only interpret the events as they appear on paper in terms of your own classroom experiences. Nevertheless, I begin by setting up the context by giving two transcripts from lessons. As you read the transcripts you will inevitably be confronted by reactions that may be judgemental (positive or negative). A recurring theme in this book is the usefulness of being able to put aside evaluations and focus on the detail of experience, to allow the possibility of seeing something in a different light from an initial reaction.
After the transcripts and a discussion of issues raised by them, I draw out issues linked to engaging students in school mathematics.
Context
The date is 2007 and the classroom is in a state comprehensive secondary school in the UK, on the edge of a city. The school serves a mainly white catchment area, including some wards with high levels of deprivation. One of the major issues faced by the school, as interpreted by many staff, was low educational aspiration among some students and some parents. I was the teacher in the classroom, head of the mathematics department at the time, and this was the start of my eleventh year of teaching at the school. The class was a year 7, meaning the students were aged 11 or 12. They had started secondary school at the start of September, so the transcripts are from some of their first mathematics lessons in their new school.
The problem they are working on is called ‘1089’. The next section is taken verbatim from the documentation of the mathematics department that would have been used by all year 7 teachers at the time, to guide their planning (i.e., the department ‘scheme of work’).
The beginning of year 7 – ‘1089’
We begin by articulating the purpose of year 7 for the students as being about becoming a mathematician and thinking mathematically, i.e.: thinking for yourself and so not asking the teacher if things are right, noticing what you are doing, e.g., patterns but then asking why patterns work, writing down everything you notice, being organised, doing things in your head.
We want to establish a purpose for the year that is removed from the content level of what we do in class and is an easily stated label that can gather complexity and meaning for each individual as the year progresses. We believe the purpose of ‘becoming a mathematician’ supports students in becoming aware of what they do when working in a mathematics lesson, by allowing them (and us) to question and reflect on whether something they (and we) do is mathematical or not.
Lesson extract
I issued the following instructions, at the same time going through an example on the board:
Several comments were made by students that they also got 1089 and the challenge I gave to the class was: ‘Can you find a number that does not end up as 1089?’
There are several reasons behind the choice of this activity as the first one with our year 7 groups. It is an activity we are familiar with, but more important for our purposes is the fact that it is self-generative. By that we mean that, having ...
