Teaching Arithmetic in Primary Schools

Audit and Test

Richard English

• Publisher: Learning Matters
• Year: 2012
• ISBN:
  9780857257260
• Edition: 1
• Topic: Education
• Subtopic: Elementary Education

There is a new and increasing emphasis on the importance of teaching arithmetical skills in primary schools. This text outlines what is meant by arithmetic and enables trainee teachers to build their own confidence in teaching arithmetic. It covers all aspects of arithmetic including recall of number facts and traditional pen and paper methods. Each arithmetical approach is illustrated with detailed examples and readers can access their own learning in each area. Guidance on when and where to use different approaches to enhance children?s learning is included and interactive activities highlight essential links between theory and practice.

Abouth the Transforming Primary QTS series

This series reflects the new creative way schools are begining to teach, taking a fresh approach to supporting trainees as they work towards primary QTS. Titles provide fully up to date resources focused on teaching a more integrated and inclusive curriculum, and texts to draw out meaningful and explicit cross curricular links.

1. Getting your head around arithmetic

Learning Outcomes

By the end of this chapter you will: have a clearer understanding of what is meant by the terms ‘arithmetic’, ‘numeracy’ and ‘mathematics’, both in a historical context and in relation to this book; be aware of the past, current and likely future curricular requirements in relation to arithmetic; understand the key issues and challenges facing primary schools in relation to the teaching of arithmetic.

What is arithmetic?

A good starting point for a book about arithmetic is a consideration of what we understand it to mean. Before discussing the historical interpretations and the interpretation used throughout the rest of this book, let us first see what your thoughts are about arithmetic.

Activity

What is your definition of arithmetic? Think about it for a few minutes and jot down a few notes. What experiences and recollections, possibly from your childhood, have informed your view of what constitutes arithmetic?

There is a strong possibility that your definition relates to calculation, perhaps with a particular emphasis on mental arithmetic and traditional pencil and paper procedures with the numbers set out one underneath the other. You may even recall having a weekly mental arithmetic test when you were a child at school, although I wonder if you were actually taught how to calculate mentally. In my own experience children were expected to perform well in mental arithmetic tests, but were never taught the necessary skills and techniques. A parallel situation would be to expect someone to do a driving test without having had any driving lessons!

Arithmetic, mathematics and numeracy

Arithmetic is a word usually associated with bygone days and this would seem to be supported by quotations provided in the introductory section of the Cockcroft Report (DES, 1982, page xii). One of these originates from Her Majesty’s Inspectors in 1876, who state that In arithmetic, I regret to say, worse results than ever before have been obtained. Another is an extract from a Board of Education Report of 1925, which states that Many have experienced some uneasiness about the condition of arithmetical knowledge and teaching at the present time. A third quotation is from a Mathematical Association report of 1954:

Experience shows that a large proportion of entrants [to trade courses] have forgotten how to deal with simple vulgar and decimal fractions, have very hazy ideas on some easy arithmetical processes, and retain no knowledge of algebra, graphs or geometry, if, in fact, they ever did possess any.

(DES, 1982, page xii)

If you would like to read more examples of the historical debate about mathematical education in the primary school, dip into a very interesting article by Alistair McIntosh (1981), details of which can be found in the Further Reading section at the end of this chapter.

The quotations presented above, perhaps in conjunction with your own reflections, lead to the conclusion that arithmetic relates to the mechanical processes of adding, subtracting, multiplying and dividing, either mentally or using pencil and paper. Importantly, this view of arithmetic is usually associated with a complete absence of any understanding of the underlying processes involved in its execution. Arithmetic was seen as a set of rules and procedures, which, if followed precisely, would yield correct answers, but often at the expense of much anxiety on the part of those executing them. Arithmetic in this sense is therefore a subset of mathematics, and indeed is not even a complete representation of what we usually refer to as number work.

The shift away from the widespread use of the word ‘arithmetic’ came about in the 1960s and 1970s with the advent of initiatives such as the Nuffield Mathematics Teaching Project from 1964 to 1971 and the publication of the Plowden Report in 1967. The widening of the mathematics curriculum and the move towards pupil-centred, as opposed to teacher-centred, education resulted in the expression ‘arithmetic’ no longer being an accurate representation of children’s experiences, and as a consequence it fell out of favour.

The expression ‘numeracy’ became popular in the 1990s with the introduction of the National Numeracy Project in September 1996. This pilot project, the forerunner of the National Numeracy Strategy, had a clear focus on the development of number skills and solving number problems, and so the first Numeracy Framework made no reference at all to areas of mathematics such as shape, space, measures or handling data. The definition of numeracy was modified further in The Final Report of the Numeracy Task Force, which presented it as being:

A proficiency that involves a confidence and competence with numbers and measures. It requires an understanding of the number system, a repertoire of computational skills and an inclination and ability to solve number problems in a variety of contexts. Numeracy also demands practical understanding of the ways in which information is gathered by counting and measuring, and is presented in graphs, diagrams, charts and tables.

(DfEE, 1998, page 11)

This much broader definition of numeracy, together with the subsequent publication of the Framework for Teaching Mathematics (DfEE, 1999), indicates a complete blurring of the boundaries between what is understood by the terms numeracy and mathematics; they had, in effect, come to mean the same thing, certainly in the primary phase, if not more widely.

With regard to the falling out of favour of the expression ‘arithmetic’, it should be noted that this word appears not even once in either the 1999 version or the 2006 version of the Framework for Teaching Mathematics. However, since the change of government in May 2010 and the demise of the National Strategies in March 2011, the word has acquired new found popularity, even if only among politicians. Shortly before the 2010 election, the shadow schools secretary, Michael Gove, stated in an interview to The Times newspaper that:

Most parents would rather their children had a traditional education, with children sitting in rows, learning the kings and queens of England, the great works of literature, proper mental arithmetic, algebra by the age of 11, modern foreign languages. That’s the best training for the mind and that’s how children will be able to compete.

(Thomson and Sylvester, 2010)

and more recently, government schools minister, Nick Gibb, has stated that:

Our reforms will give all pupils a solid grounding in reading and arithmetic, with the right catch-up support if they start to fall behind.

(Paton, 2011)

Current and future statutory requirements in relation to mathematics and arithmetic will be considered later in this chapter and indeed throughout this book, although for the time being I will leave you to ponder on what is meant by ‘proper mental arithmetic’.

One final observation with regard to the resurgence in the use of the word ‘arithmetic’: the word is not used at all in Ofsted’s 2008 and 2009 reports about mathematics, but occurs frequently in the 2011 report, although this is not surprising given that:

This survey was conducted following a ministerial request for Ofsted to provide evidence on effective practice in the teaching of early arithmetic.

(Ofsted, 2011, page 5)

This resurgence in the use of the word ‘arithmetic’ is the reason why it features in the title of this book, but in doing so, one of the aims is to encourage teachers to move away from the narrow interpretations typically associated with the word.

The scope of arithmetic as presented in this book

Unlike the definitions of arithmetic discussed above, this book will adopt a broader interpretation. Yes, it will focus on calculations involving the four arithmetical operations, but there will also be a strong emphasis on arithmetical understanding, as well as clear progression in the development of arithmetical techniques. This will begin with the recall of number facts in Chapter 2, followed by a detailed examination of mental arithmetic in Chapter 3, where a guiding principle will be that existing facts can be utilised flexibly in many different ways to mentally juggle with numbers. This flexible approach to arithmetic, which will depend on the numbers involved as well as personal choice, could not be further removed from the notion of blindly following memorised rules and procedures, as is the case with the traditional view of arithmetic. The same is true of the development of pencil and paper arithmetic presented in Chapters 4 and 5. Here, the rules that are the traditional algorithms, which depend on memory rather than understanding, are viewed as possible endpoints in children’s arithmetical progression, not the starting points. The beginnings of pencil and paper arithmetic are therefore examined first in Chapter 4, building on the flexible mental methods presented earlier. Even when traditional pencil and paper arithmetic is introduced in Chapter 5 there continues to be an emphasis on understanding how these compact, efficient procedures actually work, so as to help you to move away from the notion of blindly following rules. Chapter 6 considers arithmetic involving fractions, decimals, percentages and ratios, and it makes use of the full range of arithmetical techniques discussed in Chapters 2 to 5. Finally, in Chapter 7, the vitally important role of technology is discussed, with a particular emphasis on calculators and spreadsheets.

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Research Focus: Relational and instrumental understanding

Richard Skemp’s seminal article, first published in 1977, presents two contrasting views of mathematical understanding: relational (knowing both what to do and why) and instrumental (rules without reasons), although it could be argued that the latter does not represent understanding at all. Examples of instrumental understanding include:

• the process of borrowing when using the traditional written method for subtraction;
• turning the fraction upside down and multiplying, when dividing by a fraction;
• taking a number across to the other side and changing the sign when solving equations;
• remembering that a minus and a minus is a plus when dealing with negative numbers.

Skemp argues that teachers and children will differ in their goals with regard to mathematical understanding. This can cause particular difficulties if the teacher is striving for relational understanding but the child is aiming for instrumental understanding (‘Just tell me the rule!’). Likewise, there will be a similar conflict if the mathematical understanding goals are reversed, that is, the teacher just wants to teach rules, but the child is striving for relational understanding.

Skemp also discusses the advantages and disadvantages of each type of understanding. For example, instrumental mathematics is easier to grasp, it can be taught quickly and the rewards can be reaped almost immediately. In other words, you can learn the method and get a page of correct answers in no time at all. It is for these reasons, combined with the pressure of exams and getting through the syllabus, that many teachers choose instrumental understanding as the goal for their children.

Regarding the limitations of instrumental understanding Skemp describes a scenario in which two people are visiting an unfamiliar town. One has separate detailed sets of instructions to get to and from various locations in the town. The other has explored the town, familiarised himself with the roads and built up a mental map of where everything is. The first person is all right as long as he follows the instructions precisely, but if at any time he takes a wrong turn he will be lost and will remain lost until he retraces his steps and starts again. In contrast, the second person’s mental map provides him with an infinite number of possibilities which will allow him to get from any starting point to any finishing point and, as Skemp states:

If he does take a wrong turn, he will still know where he is, and thereby be able to correct his mistake without getting lost; even perhaps to learn from it.

(1997, page 22)

If you are unable to obtain a copy of Skemp’s article, instead read Chapter 1 of O’Sullivan et al. (2005), which presents and discusses a large extract from the original.

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Curricular requirements for arithmetic

Before presenting the government’s current priorities for arithmetic, it is worth examining the statutory requirements as they have developed over the last 25 years.

Recall of number facts

In terms of statutory National Curriculum requirements since 1989 there has been a consistent expectation with regard to children’s recall of number facts. All versions of the programmes of study have indicated that children, b...

Table of contents

• Cover Page
• Halftitle
• Title
• Copyright
• Contents
• About the author
• Introduction
• 1 Getting your head around arithmetic
• 2 The rapid recall of number facts
• 3 Mental arithmetic
• 4 The development of pencil and paper arithmetic
• 5 Traditional pencil and paper arithmetic
• 6 Arithmetic with fractions, decimals, percentages and ratios
• 7 Arithmetic using technology
• Appendix: Model answers to the self-assessment questions
• Index