A Practical Guide to Transforming Primary Mathematics

Activities and tasks that really work

Mike Askew

• Publisher: Routledge
• Year: 2015
• Print ISBN: 9780415738453
• eBook ISBN: 9781317813255
• Edition: 1
• Topic: Education
• Subtopic: Early Childhood Education

A Practical Guide to Transforming Primary Mathematics offers inspiration and ideas for all training and practising teachers committed to making mathematics enjoyable, inclusive, engaging and successful. The companion to Mike Askew's bestselling book, Transforming Primary Mathematics, this practical guide focuses on showing you how to unlock the powerful potential of a small set of consistent principles and practices, known as the teaching tripod, to develop a coherent approach to teaching mathematics.

Organised around the major strands of the curriculum - number, calculations, shape and space, measures, and data handling – it offers an accessible introduction to the teaching tripod, a careful choice of tasks, supported by a range of tools that extend our natural abilities and held together by careful attention to classroom talk. A range of classroom tasks, each including key learning outcomes, clear links to the framework, links to relevant research, and suggestions for making the tasks easier or harder, are offered for every topic, helping you plan units of work for meaningful learning.

A Practical Guide to Transforming Primary Mathematics offers all teachers a vision, rationale and ideas for how teaching can support better learning of mathematics but also encourage learners to see themselves as being capable of learning mathematics, and wanting to learn it.

1 Tasks

Tasks lie at the centre of learning and teaching mathematics. Tasks bring mathematical activity into lessons, by which I mean mental activity, not just physical activity. Tasks have to meet the needs of specific pupils, so some writers encourage teachers to design their own tasks. As a former primary school teacher I know that time demands make this a high expectation, and having devoted my time teaching in higher education exclusively to mathematics education, I’ve also come to appreciate just how difficult task design is. Besides, there is such a plethora of tasks now available on the Internet I am not sure any more need creating. But not all tasks are created equal. I think it is a more reasonable expectation that teachers need to make good choices between tasks and a primary aim of this book is to support that decision making.

I do this in two ways. In this overview chapter I set out some ideas and principles about choosing tasks. Then, in the chapters devoted to specific curriculum topics I provide a number of elaborated examples. These examples are not meant to be an exhaustive list of all the tasks needed to develop understanding of and proficiency in the primary mathematics curriculum. My aim is to provide a sufficient number of examples to illustrate the general principles set out here, so that when subsequently faced with choosing tasks you, the reader, have some sense of what might make one choice better than another.

This chapter examines two general aspects of selecting tasks. It’s Sunday afternoon, Monday’s topic is subtraction and a search on the Internet has brought up a range of tasks: how to choose between them? The first part of this chapter examines what might influence choice making at such a moment in time.

Taking a longer–term view, introducing a mathematical topic to a class, then the type of task needs to change over time. Tasks offered to learners at the beginning of a unit of work on subtraction, when many of the ideas to be worked on may be relatively new, need to be different from the tasks offered towards the end of the unit when the ideas are beginning to bed down for learners. In the second part of this chapter I present a model for a cycle of tasks over time.

Choosing tasks with aims in mind

There is growing acceptance that learning mathematics is not a simple set of skills, but a complex interplay of a number of ‘proficiencies’, one model of which is set out in a major U.S. review of research into primary (elementary) mathematics – Adding It Up: Helping Chlidren Learn Mathematics (National Research Council, 2001). The authors of that report identify five proficiencies: conceptual understanding, procedural fluency, adaptive reasoning, strategic competence and productive dispositions.

Various versions of these proficiencies have found their way into National Curricula around the world and at the time of writing England’s new National Curriculum highlights three of them:

• fluency

• problem solving

• reasoning.

Taken together, these provide a balanced set of aims for teaching and learning that can lead not only to mathematical competence but also to understanding. When selecting a teaching and learning task we therefore need to be clear about which of the proficiencies of fluency, problem solving or reasoning the task is most likely to encourage learners to engage in and ‘good’ tasks will invoke more than one proficiency.

It’s a popularly held view that, in primary mathematics, fluency needs to be addressed first: that problem solving and reasoning have to follow on from some ‘basic’ proficiencies in core skills. However, research suggests otherwise. Not only are all three proficiencies inter–related, there is also increasing evidence that reasoning is the most important, so let us look at these in reverse order.

Tasks with a focus on reasoning

A recent major longitudinal research project looked at the mathematics of 8- and 9-year-olds and what understandings correlated with later attainment. The primary finding was that, ‘Mathematical reasoning, even more so than children’s knowledge of arithmetic, is important for children’s later achievement in mathematics’ (Nunes, Bryant, Sylva, & Barros, 2009, p. 1).

Correlation, of course, does not imply causation – it could be that pupils who have a talent or taste for mathematics reveal this through being able to reason, but over a wide range of studies it is becoming clear that reasoning in mathematics – asking why or how an answer is correct, not simply focusing on getting it right – is as important, if not more so, as arithmetical fluency.

Yet there is also evidence that the aim of developing mathematical reasoning is the least attended to in many mathematics lessons and that is not because primary pupils are not capable of mathematical reasoning – they most certainly are. I think this aim is neglected partly because reasoning is the most difficult proficiency to identify when selecting tasks. For example, the majority of mathematical tasks that come in a written form – pages in a textbook or worksheets – rarely address reasoning as it is hard to convey reasoning on the printed page. Taking music as an analogy, no piece of sheet music can present tone or a sense of rhythm – it is only in the playing of the music that such things emerge. A mathematics worksheet might provide a prompt to activity, but as with music, mathematical reasoning emerges in the playing, the doing, of mathematics.

Mathematical reasoning cannot be ‘done’ in isolation of mathematical content. You cannot simply go into a classroom and announce that the lesson is about reasoning and invite pupils to think! Pupils have to have something to reason about, say, subtraction or capacities of containers. Yet as soon as the content – the direct object of learning – comes into focus, attention to that content can take over and eclipse the reasoning. The lesson ends up focusing on whether or not the subtractions are correct, rather than whether it makes sense to use the same calculation method for 146 − 23 and 257 − 248. Or the focus is on how to record litres and millilitres rather than reasoning about the multiplicative relationship of a litre being 1 000 times as big as a millimetre. It is not a question of focusing on reasoning or on content, but on developing both in tandem. In the extended examples in the content chapters I provide advice on how to enact this. I also suggest that a key way of bringing more reasoning into mathematics lessons is by making problem solving an integral part of all lessons.

Tasks focusing on problem solving

There are two ways to think about problem solving in the mathematics classroom: teaching for problem solving and teaching through problem solving.

Teaching for problem solving is probably the most common approach – problem solving is treated as the application of ideas previously learned. Pupils are taught how to, say, add or subtract, and then given problems that require addition or subtraction to solve them. While, obviously, pupils do need to use the mathematics that they have learned, the issue with much teaching for problem solving is that what is offered to pupils are not genuine problems, in that the learner is initially stumped and has to think deeply about what to do, but that problems are often just exercises practising what has been taught but ‘wrapped up’ in an often spurious context.

The ‘flipping’ of this approach – teaching through problem solving – rests on two assumptions. First, that mathematics is, at heart, a problem solving discipline and learners can come, through problem solving, not only to understand mathematics but also to appreciate its power. Second, that pupils come to school already skilled as problem solvers. Given a suitable problem to solve, the mathematical learning comes about through looking at ways to refine their informal solutions, to express them in mathematical terms. To distinguish problems chosen to help learners engage with new mathematical ideas, I call these foundational inquiries, and go on to discuss them in detail later.

Tasks focusing on fluency

Fluency is primarily (but not exclusively) about skills and procedures. It is the mathematical equivalent of working on scales when learning to play a musical instrument and, like scales, mathematical fluencies are a means to an end, not ends in themselves. Practising pages of, say, long division just for the sake of becoming fluent in the algorithm is not mathematically valuable if the pupil cannot appreciate when a problem may need to involve long division.

There are two aspects to becoming fluent in mathematical skills such as being able to use a ruler accurately or carry out the algorithm for long multiplication. First, having a clear sense of what expertise in the skill looks like: many skills can be modelled by the teacher or other pupils through direct instruction that shows how to set up a ruler to measure a line segment or demonstrates how to set out the digits for a long multiplication of a three-digit number by a two-digit number. Second, pupils then need to practise!

Objects of learning: direct and indirect

The language here, taken from variation theory (see TPM), of ‘objects of learning’ (rather than learning intentions or objectives) carries the connotation that pupils are supposed to take something – an object of sorts – as a result of the tasks they engage in. When choosing tasks for lessons, the focus is most often on the direct object of learning – the explicit mathematical content, for example, subtraction or fractions or 3D shapes. The direct object of learning is what pupils would say in response to the question, ‘What did you do in mathematics today?’

As the researcher Ference Marton points out, however, every lesson (and not just mathematics lessons) engages learners in indirect objects of learning (Marton, Runesson, & Tsui, 2004). Reasoning, problem solving and fluency are examples of indirect objects of learning. Pupils have to reason about something, a lesson cannot be directly about reasoning – there has to be some direct content to reason about. Similarly, problem solving has to involve problems that address some particular content and learners become fluent in some particular mathematics.

When choosing tasks we need to bear in mind different possible indirect objects of learning – reasoning, problem solving and fluency – that might be provoked. Whenever possible, we need to choose tasks likely to provoke indirect learning that helps pupils learn more about mathematics and the nature of mathematical activity, such as the importance of looking for patterns or seeking generalisations as well as helping them master some particular content.

Tasks with multiple indirect objects of learning

Although a task might focus on only one or two direct objects of learning, it is possible for tasks to serve several different indirect objects. Let us look at this through the example of a game, Say ten.

Say ten

A game for two players. The player to go first can start either by counting ‘one’ or counting ‘one, two’. Partners then take it in turns to count on from the last number, always counting on one or two. The player to say ‘ten’ is the winner. For example, play might go:

P1: One

P2: Two, three

P1: Four

P2: Five

P1: Six

P2: Seven

P1: Eight, nine

P2: Ten. I win.

The direct object of learning is for learners to play a counting game, but Say ten can address all three aims of fluency, problem solving and reasoning. At a basic level the game provides pupils with the opportunity to practise fluently counting on in ones or twos. With that object of learning in mind the task can be adapted to provide further practice in similar counting fluencies. The game might, for example, be changed to start at 30 and the winner is the person to say ‘50’. Or it might start at fifty and players count on in ones, twos, or threes. Or they might play the game counting backwards and win if they say ‘one’.

Another indirect object of learning of the game is being able to solve the problem of what a winning strategy might be. Playing Say ten, pupils soon come to realise that saying ‘seven’ is a winning move – whether their partner then says ‘eight’ or ‘nine’ they are certain to be able to say ‘ten’. Using this task in school, I find it interesting that many players stop at that point: having solved the problem of what number before ten ensures you win, learners stop there and do not look for further winning moves. Despite realising that saying ‘seven’ is a winning move, they do not go on to look at whether there is a strategy that ensures getting to say seven. The mathematician would expect there to be other winning positions, and learners closing down after finding seven is, I suspect, the result of most mathematics teaching that presents pupils with situations with only one answer to find.

A little reflection reveals that saying ‘four’ ensures that you can say ‘seven’ and saying ‘one’ then ensures you get to say ‘four’. The game effectively becomes one of ‘go first’. Encouraging pupils to go deeper into the problem of finding winning strategies thus introduces the indirect object of mathematical reasoning.

A big part of mathematical reasoning is looking for generalisations and a mathematician would expect that if saying ‘seven’ is a winning move, then this must generalise to there being other winning moves in variations of the game. The question then is, is whether there being a set of winning numbers is unique to this particular game, or whether all versions of the game have winning strategies. Thus, reasoning and generalising can be pushed further by altering the game to, say, being able to count on one, two or three and winning by being the first person to say ‘fifteen’. Pupils can be challenged to come up with a new rule that works for this variation (in this instance, counting back in fours from 15 produces the ‘unbeatable’ numbers). They can then be encouraged to conjecture (the mathematical term for coming up with a ‘theory’ that is not yet proved) as to whether there is a general rule that can be applied to any target number and any choice of steps to count on. Once they have explored different versions of the game, pupils can often articulate a generalisation along the lines of ‘count back from the target number in jumps one more than the highest number you can add’. That can be further explored to see whether it always holds true – what if you can count on two, three or four? Thinking can be provoked to go deeper: what if the g...

Table of contents

• Cover Page
• A Practical Guide to Transforming Primary Mathematics
• Title
• Copyright
• Contents
• Acknowledgements
• Introduction
• 1 Tasks
• 2 Tools
• 3 Talk
• 4 The Number System
• 5 Additive Reasoning
• 6 Multiplicative Reasoning
• 7 Geometry
• 8 Measures
• 9 Data Handling and Statistics
• References
• Index