Mathematics in the Primary School

A Sense of Progression

Sandy Pepperell, Christine Hopkins, Sue Gifford, Peter Tallant

• Publisher: Routledge
• Year: 2014
• ISBN:
  9781317796695
• Edition: 3
• Topic: Education
• Subtopic: Education General

Now in its third edition, Mathematics in the Primary School has been updated to reflectrecent mathematics curriculum documentationand revised standards for QTS.

Key areas include:

• The role of talk in learning maths
• Teacher questioning
• Development of children's reasoning
• Creative engagement with maths
• Assessment for learning and self assessment
• Suggested resources for teachers including ICT

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Providing a coherent set of principles for teaching primary mathematics across the main topics in the curriculum, the authors explore children's understanding of key areas of mathematics, at reception, infant and junior levels. Important principles and teaching approaches are identified, including the use of calculators and computers, and there is an emphasis on mental mathematics and problem solving supporting key issues raised by the Williams review (2008). Case studies are used throughout to illustrate how different teaching approaches are put into practice and how children respond to them, and there is advice on planning, organisation and assessment of mathematical learning in the classroom.

Emphasising the importance of teachers' own mathematical knowledge and offering clear guidance and practical advice, this book is essential reading for students, NQTs and practising teachers with a focus on primary mathematics.

SECTION 1

Using and applying mathematics

MATHEMATICAL THINKING

Whether or not we articulate them, the answers we have to questions such as, ‘What is mathematics really about?’ and ‘What is mathematics for?’, provide the basis for our approach, as teachers, to the teaching and learning of mathematics. Our answers to these and similar questions will depend on our own experiences as learners and these experiences will have been generated from the views of those who taught us. Put another way, our opinions are formed from what and how we were required to learn and the reasons for doing so that we were given (or not!). If the major emphasis was placed on the acquisition of facts and the need to memorise and to make use of rules unthinkingly, then this is what we will bring first to our teaching. This is what we will view mathematics to be about. We will think of it as simply school mathematics – something we merely pick up and put down and this solely in the context of the classroom. If this is the nature of our experience, there is also the danger that we have suffered the anxiety over mathematics which Williams (2008) reported is still a significant factor in our society. If the way in which we were taught, however, has challenged us to view mathematics as an area of learning which is stimulating, rewarding and something which touches us personally, if our focus is on making connections and enjoying mathematics’ potential for being a means of interpreting and understanding the world, then these will be the views we bring and to which our children will be exposed.

As a prelude to teaching mathematics, therefore, the first task for teachers is to reflect on and to analyse their experiences of learning it. Teachers must assess the quality and nature of their experiences and seek ways to ensure that the ‘messages’ about mathematics that are given to children are appropriate in that they stimulate engagement, enthusiasm and deep thought. Children should work with teachers who are committed to the idea that the acquisition of knowledge and skills is important. Equally, however, children must have the opportunity to learn with teachers who recognise that knowledge and skills are of limited value as end products. It is in the addressing of Using and Applying mathematics that children can begin to get to the heart of mathematics and find purpose and fulfillment in learning about it. What Using and Applying mathematics involves, and how it can be addressed in the classroom, is discussed in detail below.

1.1 WHY DO CHILDREN LEARN MATHEMATICS IN SCHOOL?

Responses to this question usually include:

• the need to solve practical everyday problems;

• to use mathematics in other areas of learning;

• to learn to reason logically;

• to gain some satisfaction and enjoyment from exploring this area of human knowledge.

These basic reasons lead naturally to an emphasis on mathematical thinking. It is of little use if a child can correctly complete pages and pages of calculations but does not know how to get started when faced with an unfamiliar problem or a task presented in a practical rather than a written form. It is harder for children to relate the mathematics they do in one situation to that required by another than is sometimes assumed. We need to encourage children to generalise and abstract from particular situations and experiences, so they can move from concrete to abstract and back again. Moving in this way, from concrete to abstract and vice versa, involves children in modelling, in representing things in different ways. This relates to the essential nature of mathematics as the discovery and application of numerical and spatial relationships. Generalising is therefore involved both in the process of abstraction and in the transfer of mathematical ideas from one context to another.

An approach to the teaching and learning of mathematics which develops the skills needed to tackle unfamiliar problems and the processes involved in doing mathematics needs to focus on generalising mathematical relationships. However, this central aspect of mathematics is perhaps the most difficult to develop. Because of this, it is the one concentrated on in this section and is a recurring theme throughout the book.

Three aspects involved in mathematical thinking – reasoning mathematically, mathematical communication and solving problems – underpin much of the discussion in this book.

Reasoning mathematically: reasoning mathematically involves children in:

• searching for patterns and relationships within their mathematical work;

• moving from particular examples to general statements about mathematical ideas and vice versa

• ;thinking logically in a mathematical situation.

Mathematical communication: mathematical reasoning also depends on mathematical communication; this involves encouraging children to articulate their thinking and to move towards using more abstract language, symbols and representations. This means children need opportunities for:

• talking about the mathematics:

• listening, reflecting and responding;

• explaining why a method works;

• posing their own mathematical questions;

• using mathematical language/terminology;

• using a range of representational forms (modelling).

Solving problems: this involves children in exploring relationships within mathematics and also using mathematics with understanding in a variety of contexts. This means children need opportunities for seeing mathematical similarities in apparently different situations and for mathematical decision making. This includes:

• making their own decisions about appropriate mathematical ideas and operations;

• drawing upon a variety of strategies;

• choosing tools and materials (for example number line, calculator, cubes, squared paper);

• making links between different aspects of mathematics and between mathematics and other subjects.

It is important to say here that, in trying to find solutions in mathematics, children will also have the important experience of discovering what does not work. They will try things which are not obviously successful. Children may find this discouraging, initially. They may feel that they have ‘done it wrong’. Here, the role of the teacher is to address the vitally important affective factors which influence children when they are learning. Part of thinking mathematically is a willingness to pursue a line of enquiry, to explore, to be open to possibilities. Anxiety concerning our ability to succeed restricts our ability to think, to consider possibilities, to ‘have a go’. Negative attitudes can lead to unwillingness even to engage with a task or merely to a desire to find a solution quickly; perhaps in order to demonstrate that we can ‘do it’ or simply to allow us to move on to something else. If the children’s attempts to solve a problem do not yield success then negative attitudes may be created or reinforced. Sensitive and appropriately-timed interaction with the children is vital, therefore. How a teacher reacts to children pursuing an idea which does not generate an immediate solution is crucial. Indeed, her reaction to children’s efforts will not only influence their experience with the specific situation but also, potentially, have a profound effect on their general view of mathematics and what it involves. It is the role of the teacher to explore, with the children, how an apparently unsuccessful attempt to solve the problem can contribute to an eventual solution. This can be brought about through careful discussion of the children’s ideas in order to draw out what has indeed been achieved. In short, the teacher might say, ‘Okay, that hasn’t worked, try something else’ or, alternatively, she could respond with, ‘Now that really helps. If it can’t be that … then what do you think it could be?’. If the teacher models this response consistently then the children will learn to incorporate it into their approach and to recognise that their efforts are of real potential value, even if they do not produce an immediate solution. In other words, if we know what does not work, then we can be on the way to discovering what does. Another aspect of what the children can describe as ‘failure’, is ‘getting stuck’. Once again, the teacher’s response is crucial in sustaining children’s involvement and maintaining positive attitudes and her role is to support and at the same time challenge the children. One strategy for achieving this is by prompting the children to try to think of something they have done before which might help. Another is to ask them to talk through what they have done so far in order to help them to generate an overview of the problem. There are other strategies available and many of them rest on the high-level, professional skill of questioning for more of which see below.

When planning children’s work, these three themes of reasoning mathematically, mathematical communication and solving problems should act as a checklist for judging the value of all of the mathematical activities in which the children engage.

Having set up opportunities for the children, the teacher’s role is to develop and sustain the children’s thinking. Case studies are used throughout the book to give some indication of how the teaching principles identified might influence classroom practice. The case studies are in different schools with children of different ages, but it will become apparent that similar questions are used to extend the children’s thinking.

Key questions

The thoughtful use of certain key questions by the teacher can encourage mathematical reasoning by the children. In practice it is not the questions alone which support the problem-solving but all aspects of the verbal and non-verbal interaction between the teacher and the children. By showing interest, puzzled looks and body language the teacher sustains the children’s thinking, while the tone of voice in which the questions are posed shows the teacher’s involvement with the problem. If the teacher poses these questions in many different situations, then the children have the opportunity to internalise the questions and develop as independent thinkers.

The questions serve to encourage different ways of reasoning. It should be noted that most of them are ‘open’, in other...