Transforming Primary Mathematics
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Transforming Primary Mathematics

Understanding classroom tasks, tools and talk

Mike Askew

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eBook - ePub

Transforming Primary Mathematics

Understanding classroom tasks, tools and talk

Mike Askew

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About This Book

Fully updated to reflect the new curriculum, the revised edition of Transforming Primary Mathematics sets out key theories and cutting-edge research in the field to enable teachers to take a fresh look at how they teach mathematics.

The book encourages teachers to reflect on their own beliefs and values about mathematics, and asks them to question whether their current methods meet the needs of all learners, and the challenge of having high expectations for all. It provides clear, practical approaches to help implement fundamental change in classroom environments, and offers motivational teaching styles to ensure meaningful mathematics learning.

Chapters take an inspiring, sometimes controversial, and often unconventional look at the subject of mathematics, by:

  • endorsing the use of a 'new mathematics' – one based on problem solving, modelling, inquiry and reasoning, not on abstract rules, memorising, and regurgitation


  • arguing that there is more to maths teaching than 'death by a thousand worksheets'


  • challenging norms, such as the practice of sorting children into sets based on their perceived mathematical ability


  • asking whether mathematical ability is innate or a result of social practices


  • examining what a 'mastery' approach might entail


  • highlighting the role of variation in supporting learning


  • advocating an environment where teachers are encouraged to take risks.

Transforming Primary Mathematics is for all primary school teachers who want to make mathematics welcoming, engaging, inclusive and successful.

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Information

Publisher
Routledge
Year
2015
ISBN
9781317357551
Edition
2

1 Introduction

DOI: 10.4324/9781315667256-1
Some people I’ve encountered in various phases of my career seem more certain about everything than I am about anything. That kind of certainty. . . [displays] an attitude that seems to misunderstand the very nature of reality – its complexity and ambiguity – and thereby to provide a rather poor basis for working through decisions in a way that is likely to lead to better results.
(Rubin and Weisberg 2003)

Is there a ‘problem’ in primary mathematics education?

The wealth of books and resources available supporting the teaching of primary mathematics suggests there must be a problem with how it is currently taught and learnt, otherwise there would not be a big market for these products. And the media keeps suggesting that too many children leave English primary schools without sufficient knowledge of mathematics. While the first version of this book was being written, the United Kingdom had a Labour government that oversaw the development of an innovative national curriculum for primary schools. With the election of a coalition government, primarily Conservative, that curriculum was quickly replaced with one that, if the popular press were to be believed, heralded a return to the ‘basics’ in mathematics. Despite these swings in policy, the majority of mathematics lessons that I have been privileged to visit continue to be pleasant experiences. Teachers are supportive and children are engaged in mathematics that seems purposeful. If the children do not find the mathematical content terribly exciting, at least they do not appear to be stressed. Despite all the hoopla about it, I do not think there is a big problem in primary mathematics.
Yet I still believe and hope that learning mathematics in primary schools could be an exciting (in a calm sort of way) experience for more children and open up the possibility of mathematics being something that they want to pursue beyond what they are required to do. The research evidence, and my experiences, lead me to believe that making mathematics more engaging and at the same time more challenging will lead to higher standards. Contrary to popular opinion, most children rise to the challenge of ‘hard’ mathematics rather than shy away from it, as long as it is not simply getting harder for the sake of it. Children’s accepting challenging mathematics is, however, dependent upon a number of factors including a particular style of classroom ethos, close attention to the mathematical challenges presented, and support for children in their efforts. The purpose of this book is to explore each of these.
This introductory argument revolves around three questions:
  • What is good mathematics teaching?
  • What is mathematics teaching good for?
  • Who is mathematics teaching for?

What is good mathematics teaching?

This is what the majority of books and articles and research papers set out to address. In writing this book I am adding to the mountain of advice (although, as I hope will become clear, I offer my advice as conditional – it might work – rather than as dogmatic ‘it should work’). I started teaching in the late 1970s and moved into research in the early 1990s: most of my career has been directed towards trying to answer this question. I don’t think I’ll ever have the definitive answer to it. That is because the nature of teaching is, and always will be, an adaptive challenge, rather than a technical problem (Heifetz et al. 2009).
Heifetz et al. (2009) argue that we can find solutions to technical problems through our current expertise. Plenty surround us in our daily lives. Not so long ago I had to travel from Sheffield to London and I needed to be in London by a certain time. The Internet provided me with the tools to plan my journey, book my train ticket, and pay online so that I could relax in the knowledge that none of these decisions or actions had to be done at the last minute. Next time I have a similar ‘technical problem’ I’ll be familiar with the steps to go through to solve it.
In contrast, adaptive challenges require solutions that have yet to be found; solutions that are different from current practices. Teaching mathematics for the uncertain times we live in is an adaptive challenge. It is imaginable that future generations will meet this adaptive challenge by deciding to stop teaching mathematics altogether. I cannot imagine why or how that could come about, but my lack of imagination is no reason for assuming that mathematics education needs to continue in its current form. No doubt there must have been moments in time when schoolmasters could not imagine any education could be complete without a sound grounding in Latin or Greek, but where are they now? And at least one brave education district did see what would happen if mathematics were not mandated to be taught (at least as separate lessons). The results were surprising (see box).
Stop teaching mathematics
Surprising as it may seem, one school district in the USA (Manchester, New Hampshire) did experiment with taking mathematics out of the primary (elementary) curriculum. Teachers’ concerns over the burdens of a too heavy curriculum led the district education officers to persuade half of the schools in the district to drop formal mathematics lessons from the timetable. They did not remove mathematics completely from the curriculum: the teachers were encouraged to pick up and teach mathematical ideas as they arose in other subjects. So, for example, measuring could be taught in the context of science lessons, or averages and percentages in the context of a topic such as looking at sport and its statistics. There was a state test towards the end of the final year of elementary school (when children were around eleven years old), so the officers agreed that teachers could reinstate mathematics lessons in that final year to ensure that the learners were familiar with the content and conventions of the test. To check out the impact of the initiative, they also gave all the learners a separate test of mathematical problem solving.
When the results of the state test came in, it was not possible to distinguish the learners who had been in the mathematics-lessons-free classrooms from those who had been taught it formally throughout their elementary school years. Perhaps more strikingly, on the problem-solving assessment the children who had been taught in the mathematics-in-context schools significantly out-performed the others (Benezet 1935a, b, c).
All this took place in the 1930s; in these days of high-stakes tests and accountability, I doubt anyone anywhere would be brave enough to replicate this experiment.
Policy initiatives in England, and elsewhere, tend to treat the teaching and learning of mathematics in primary schools as a technical problem: we already know from current practices the solution to engaging children and raising standards. If teachers can be better ‘trained’ to ‘deliver’ mathematics using existing techniques then all will be well. The current national curriculum partly addresses this by a renewed emphasis on formal calculations and arguing for increased time practising. While teaching arithmetic to Victorian clerks might have been a technical problem, teaching mathematics now is too complex to be reducible to a prescriptive set of techniques. We need to work with a view of mathematics teaching as an adaptive challenge. That means trying out new ways to teach and in particular allowing pedagogies to emerge rather than imposing them.
This does not negate the fact that, literally, tomorrow, children across the globe are going to go into mathematics lessons. We cannot use pedagogies that have not yet been invented to teach them – we can only work with what we have got. We have to work with our current knowledge, but we need to treat that knowledge as ‘conditional’. Things work to the best of our current knowledge within the current conditions of teaching. We must be cautious of claims for descriptions of current ‘good mathematics teaching’ being what is needed in the near, or far, future.
Treating teaching as an adaptive challenge does not, however, suggest an ‘anything goes’ approach or that teachers simply have to make it up as they go along. We have to work with the best of our current knowledge without that getting set in stone. As teachers we need to be prepared to take some manageable risks. I am sharing in this book what I consider to be some of the ‘best’ of the current knowledge (that I am aware of) from the research and some stories of the manageable risks that, working together, teachers and I have tried.

What is mathematics teaching good for?

At first sight the answer to this may seem glaringly obvious and circular. Whatever can mathematics teaching be good for other than teaching mathematics? A bit like asking, ‘What is a sieve good for?’, or ‘What is in this can of beans?’ The answer is in the question.
Any teaching, including the particular case of mathematics, actually teaches far more than the ‘content’: children are learning much more than just mathematics in mathematics lessons. They are learning a lot about themselves and about their peers and about relationships. A second aim of this book is to look at some of the other things that get learnt alongside mathematics and to argue for giving these equal, if not more, importance as the content. These are not ‘either/or’ aims of mathematics teaching, but complementary. What children learn about themselves and others directly impacts on the learning of mathematics at the same time as their experiences in mathematics lessons influence this other learning. This is a challenge brought sharply into focus by the move, in England, to a curriculum framed around yearly expectations of what pupils should learn. The ten-levels curriculum model that was previously in place allowed for pupils to progress, to a certain extent, through the curriculum at a pace that met individual needs. Now individual needs need to be met while at the same time cohorts of learners need to be moved through the curriculum at largely the same pace. It is even more imperative to ensure that learners do not come to identify themselves as mathematical failures if they experience difficulty in keeping up.

Who is mathematics teaching for?

Again we can take the obvious answer: the children in the class. Cynics may say that in the current climate of national testing mathematics teaching is for those who compile the league tables – the children are only a means to the end of making sure schools meet targets.
I want to add that today’s teaching is, in part, for the adult that the child will become. Given the rate of change that we are going through – generally accepted by commentators as the most rapid ever in our history – how best to prepare learners for an unknown future is a major challenge. If, as the writer L.P. Hartley said, the past is a foreign country then the future is an alien one. They not only do things differently there, they do different things. This makes the challenge for the primary mathematics teacher of today increasingly difficult. We cannot predict the future, and yet we need to prepare children for this uncertainty.
There is much talk of ‘equipping’ pupils for the future that they will be ‘entering’ when they leave school. But the future is not a place that exists, simply waiting for pupils to enter it. The pupils of today are the creators of future societies, not visitors to them. Does mathematics education have a role to play in helping prepare children for creating their futures? I think it does.
For example, recent practices established norms about different abilities in mathematics and enacted these through practices such as sorting pupils into high, medium and low groups and labelling individuals – level 4, gifted and talented and so forth. Society at large also labels people, for example in terms of their ‘race’. Genetics studies now demonstrate that there is no biological basis to the notion of race, that this is a construct that is socially created through interaction and relationships and social practices; there is no reality to the idea of race outside the social. Research now similarly challenges the long-held view that mathematical ability is innate and fixed (Shenk 2010). If the reality of mathematical ability is ungrounded and differences in attainment largely come about from social practices, then we need to look at the impact of these factors on learning mathematics and to question them. Looking beyond the classroom, the practice of grouping and labelling children in mathematics may contribute to legitimizing grouping and labelling more broadly and contribute to sustaining societal issues such as race. We have to accept that there may be a relationship between divided mathematics classes and divided communities.
The move to thinking of teaching and learning in ‘mastery’ terms challenges practices like ability grouping and setting between classes. At the time of writing the first version of this book, mastery was not a buzzword, but many of the approaches and practices that I advocate here – particularly thinking about how classes learn as much as how individual pupils learn – are pertinent to what a mastery approach might involve.
Given the impossibility of predicting the future the solution is not as easy as looking back and returning to ‘the basics’ as some would advocate. Nor can we justify dull mathematics teaching on the ‘trust me, you’ll need this later’ argument. Learning and teaching have to be worthwhile experiences in the here and now, while keeping an eye on the future horizon. Looking forward while attending to the present is a big challenge to today’s mathematics classrooms.

The joy of mathematics

Imagine two semi-circles, one much bigger than the other and joined at one end of each of their diagonals (Figure 1.1).
Suppose they are connected by a third semi-circle meeting them at the open ends of their diameters. If the angle between the two original semi-circles is large enough, then it is clear by looking (by inspection as mathematicians like to call it – not to be confused with school inspection) that the area of the third, new, semi-circle must be much greater than the area of the two original semi-circles put together (Figure 1.2).
Figure 1.1 Two semi-circles joined at a point
Figure 1.2 A large third semi-circle joining the original two
Figure 1.3 A sma...

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