Teaching Mathematics at Secondary Level E-Book

Tony Gardiner

• Herausgeber: Open Book Publishers
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2016

Beschreibung

Teaching Mathematics is nothing less than a mathematical manifesto. Arising in response to a limited National Curriculum, and engaged with secondary schooling for those aged 11-14 (Key Stage 3) in particular, this handbook for teachers will help them broaden and enrich their students’ mathematical education. It avoids specifying how to teach, and focuses instead on the central principles and concepts that need to be borne in mind by all teachers and textbook authors—but which are little appreciated in the UK at present.This study is aimed at anyone who would like to think more deeply about the discipline of ‘elementary mathematics’, in England and Wales and anywhere else. By analysing and supplementing the current curriculum, Teaching Mathematics provides food for thought for all those involved in school mathematics, whether as aspiring teachers or as experienced professionals. It challenges us all to reflect upon what it is that makes secondary school mathematics educationally, culturally, and socially important.

Leseprobe

TEACHING MATHEMATICS ATSECONDARY LEVEL

Teaching Mathematics atSecondary Level

Tony Gardiner

http://www.openbookpublishers.com

© 2016 Tony Gardiner

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Tony Gardiner, Teaching Mathematics at Secondary Level. Cambridge, UK: Open Book Publishers, 2016. http://dx.doi.org/10.11647/OBP.0071

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This is the second volume of the OBP Series in Mathematics:

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Contents

About the author

Introduction and summary

I.Background: Why focus on Key Stage 3?

II.The general advice in the Key Stage 3 programme of study

1.Aims

2.Working mathematically

III.The listed subject content for Key Stage 3

1.Number (and ratio and proportion)

2.Algebra

3.Geometry and measures

4.Probability and Statistics

IV.A sample curriculum for all—written from a humane mathematical viewpoint

1.Very Brief version

2.Brief version

About the author

Tony Gardiner, former Reader in Mathematics and Mathematics Education at the University of Birmingham, was responsible for the foundation of the United Kingdom Mathematics Trust in 1996, one of the UK’s largest mathematics enrichment programs. Gardiner has contributed to many educational articles and internationally circulated educational pamphlets. As well as his involvement with mathematics education, Gardiner has also made contributions to the areas of infinite groups, finite groups, graph theory, and algebraic combinatorics. In the year 1994-1995, he received the Paul Erdös Award for his contributions to UK and international Mathematical Challenges and Olympiads. In 1997 Gardiner served as President of the Mathematical Association, and in 2011 was elected Education Secretary of the London Mathematical Society.

Introduction and summary

This extended essay started out as a modest attempt to offer some supporting structure for teachers struggling to implement a rather unhelpful National Curriculum. It then grew into a Mathematical manifesto that offers a broad view of secondary mathematics, which should interest both seasoned practitioners and those at the start of their teaching careers. This is not a DIY manual on how to teach. Instead we use the official requirements of the new National Curriculum in England as an opportunity:

•to clarify certain crucial features of elementary mathematics and how it is learned—features which all teachers need to consider before deciding ‘How to teach’.

In other words, teachers will find here a survey of some of the mathematical background which schools need to bear in mind when choosing their approach, when thinking about long-term objectives, and when reflecting on (and trying to understand and improve) observed outcomes.

We leave others to draft recipes for translating the official curriculum into a scheme of work with the minimum of thought or reflection. This study is aimed at anyone who would like to think more deeply about the discipline of “elementary mathematics”, so that whatever decisions they may take will be more soundly based. Feedback on earlier versions suggested that this analysis of secondary mathematics and its central principles should provide food for thought for anyone involved in school mathematics, whether as an aspiring teacher, or as an experienced professional—challenging us all to reflect upon what it is that makes secondary school mathematics educationally, culturally, and socially important.

The contents demand repeated reading, and should be weighed and digested slowly.

•The reader should begin with the very short Part I, which sets the scene.

•We suggest they should then work through Part II, which concentrates on the Aims etc. of the published curriculum, and on the general requirements in the section headed Working mathematically. But readers should not worry if some aspects remain unclear on a first reading.

•Ultimately all the sections are interlinked; but we expect the reader will then select sections in Part III (the listed Subject content) which are of most immediate interest—whether Number, or Algebra, or Geometry and measures, or Probability and Statistics—and extract whatever is found useful. Again, each section may bear repeated reading over a number of years, so do not be frustrated if at first some parts appear more immediately applicable than others.

•Part IV is a revised version of our “humane mathematics curriculum for all, written from a mathematical viewpoint”. This is offered as a “sample” rather than as an ideal “model”. It tries to avoid the hubris of some recent reforms and to show how more modest goals mesh together over time, and with each other. For example, we include stages intended to ensure that everyone should manage to learn their tables by the end of primary school, with reinforcement in lower secondary school (even if some pupils achieve fluency earlier); and though we emphasise the central role of fractions for everyone in secondary mathematics, we avoid their early introduction.

The reader is assumed to be an active reader. We repeatedly emphasise drawing, calculating, and making; but we have left these delights for the reader, who should always have pencil and paper to hand. In particular, problems and calculations included in the text should be tackled before reading on, and diagrams described in the text should be drawn.

The important messages are best understood in the context where they arise. However, we were advised to include a summary of some of the key messages at the outset. We therefore end this Introduction with a list of some of the most important messages that arise in the ensuing text, even though many of these messages cannot be easily summarised. Hence we also urge readers to construct their own list of key principles as they work through the main text.

•Key Stage 3 (lower secondary school, age 11–14) is a crucial transition stage, which needs concerted support (see Part I).

•We need to recognise that, if what is learned is to bear fruit in the medium term, whatever is taught needs to be analysed and taught within an organised didactical framework.

•What is taught also has to build on what is already known, so teachers need to exercise judgement about pupils’ readiness to progress.

•Mathematics can be daunting; but everyone can make progress with perseverance. So it is important to pace the initial material to allow this message to register.

•Whenever possible one should exploit opportunities for pupils to calculate, to draw, to measure and to make things for themselves.

•Whenever possible, one should establish and check pupils’ grasp of the inner structure of elementary mathematics through on-going class oral and mental work.

•Regularly extend routine oral and mental work to encourage an atmosphere in which thoughtful conjectures are expressed and tested, and where proof is increasingly valued.

•Actively develop pupils’ powers of remembering. Gradually extend the range and scope of important results and methods that pupils understand and know by heart. Help them to see that having to work things out from scratch each time seriously restricts the kind of problems one can tackle and solve.

•Each theme must be given sufficient time and variety for pupils to achieve the kind of robust fluency, and the shift of focus that is needed for subsequent progression.

•Special and recurring attention needs to be paid to strengthening key themes (such as place value, fractions, structural arithmetic, simplification, ratio and proportion) in a suitably robust form.

•An effective programme must allow pupils to appreciate links and connections, and to gradually become aware of the way in which simple ideas from different mathematical domains relate to each other.

•Always look for alternatives to ‘acceleration’. Aim for all pupils to achieve robust mastery in sufficient depth to maximise their preparation for subsequent progression. The easier a pupil finds a topic, or a group of topics, the more important it is for them to master that topic in serious depth before moving on.

•Use carefully designed sets of graded exercises that range from the very simple to the general, routinely exploring the more demanding ‘indirect’ variations, which are needed in many subsequent applications.

•Recognise the link between each direct operation or process (such as addition, or multiplying out brackets) and the corresponding inverse operation or process (such as subtraction, or factorising). Whilst fluency in the direct operation is essential, its main purpose is to serve as a foundation for solving the harder, and more important inverse problems. In particular, resist the temptation to break harder inverse problems into manageable (direct) steps.

•Routinely include simple word problems alongside technical exercises, so that pupils learn to identify and extract relevant information from short (two or three sentence) problems given in words.

•Regularly include short, non-routine problems (including two-step and multi-step problems), that cultivate pupils’ willingness to face the unexpected, and to think how to link known techniques into effective solution chains.

•Routinely re-visit old material and replace old methods by more flexible, forward-looking alternatives. Distinguish clearly between backward-looking methods (that may deliver answers, but which hinder progression) and forward-looking methods (that may at first seem unnecessarily difficult, but which hold the key to future progression).

The final version owes much to many friends and colleagues, whose comments on successive drafts kept alive the vision of trying to write something of value in difficult times: I hope they will accept my profound thanks without my running the risk of trying to name them all. The London Mathematical Society provided essential support for this project over an extended period. But the book would never have seen the light of day without the endless encouragement and Herculean efforts of Alexandre Borovik.

I. Background: Why focus on Key Stage 3?

When designing a mathematics scheme of work for Key Stage 3, the obvious move would be to try to adapt the official programme of study.1

However:

•the programme of study incorporates some startling omissions of essential content that simply cannot be skipped (to give just two examples: there is no reference to the subtleties of teaching the arithmetic of negative numbers, or of combining negatives and ‘minus signs’ in algebra; nor is there any explicit mention of isosceles triangles, or of deriving and using their properties in other settings);

•many of the officially listed themes require careful interpretation in other ways;

•in the official programme of study the connections between topics are rarely elaborated; and

•the grouping and sequencing of, and the progression through, topics is far from clear.

In short, the programme of study needs to be supplemented and ‘fleshed out’ (and sometimes corrected). Moreover, unlike the programmes for Key Stage 1 and Key Stage 2,

the programme of study for Key Stage 3 has no year-by-year structure and no accompanying Notes and guidance.

The fact that we need to think more carefully about mathematics teaching at Key Stage 3 has been a theme of the Ofsted triennial reports on mathematics:

Mathematics: Understanding the score (2008)2

and

Mathematics: Made to measure (2012).3

These reports have not been as widely read as they deserved. Their analysis is unusually forthright for official documents, and provides a sobering starting point for any school seeking to review its mathematics provision at Key Stage 3. The reports summarise observations from hundreds of inspections—but they do so in an unusually constructive spirit. For example, having classified half of secondary maths lessons, and more than half of the schemes of work, as being either ‘inadequate’ or ‘requiring improvement’, Ofsted went out of their way to provide down-to-earth advice.4

This down-to-earth Ofsted DIY guide begins with a four-page table contrasting

•the general features of “good mathematics teaching”

with

•those of “mathematics teaching deemed to require improvement”.

The Ofsted guide then presents a string of specific examples chosen to clarify the differences between ‘weak’ and ‘more effective’ mathematics teaching, and to challenge schools to reflect on, and to improve, their own teaching. Hence this collection of examples and advice should probably be taken seriously by any school seeking to revise its published scheme of work for Key Stage 3.

Key Stage 3 mathematics teaching is important because it marks a transition from the more informal approach in primary schools to the formal, more abstract mathematics of Key Stage 4 and beyond. Hence those teaching Key Stage 3 classes need a clear picture of how the constituent parts of secondary mathematics interlock, and how Key Stage 3 work can best support progression—first progression to Key Stage 4, and then to Key Stage 5 (at ages 16-18). In this regard the 2012 report Made to measure highlights the uncomfortable fact that (p. 4):

“More than 37,000 pupils who had attained Level 5 at primary school gained no better than grade C at GCSE in 2011. Our failure to stretch some of our most able pupils threatens the future supply of well-qualified mathematicians, scientists and engineers.”

This illustrates the extent to which current provision at Key Stage 2 and Key Stage 3 fails to lay the necessary foundations for subsequent stages, and raises the question of how to improve provision at Key Stage 3. The question is especially relevant given that so many schools feel unable to allocate their strongest mathematics teachers to Key Stage 3 classes. So there is clearly a need to provide more detailed guidance for those who teach at this level.

The quality of existing support and guidance at school level is summarised in the key findings of the 2008 report Understanding the score (p. 6):

“Schemes of work in secondary schools were frequently poor, and were inadequate to support recently qualified and non-specialist teachers.”

The ‘Executive Summary’ (p. 4) noted:

“Evidence suggests that strategies to improve test and examination performance, including ‘booster’ lessons, revision classes and extensive intervention, coupled with a heavy emphasis on ‘teaching to the test’, succeed in preparing pupils to gain the qualifications but are not equipping them well enough mathematically for their futures. It is of vital importance to shift from a narrow emphasis on disparate skills towards a focus on pupils’ mathematical understanding. Teachers need encouragement to invest in such approaches to teaching.” [emphasis added]

And the ‘Recommendations’ (p. 8) included:

“Schools should […]

•enhance schemes of work to include guidance on teaching approaches and activities that promote pupils’ understanding and build on their prior learning.”

Pages 19–25 of the 2008 report provide useful additional details: Figure 4 on p. 19, and Figure 5 on p. 24 summarise the observed weaknesses in secondary schools, and the surrounding paragraphs make clear suggestions as to what needs attention.

The 2012 report Made to measure echoes, and reinforces the concerns expressed in the 2008 report:

p. 9:

“Teaching was strongest in the Early Years Foundation Stage and upper Key Stage 2 and markedly weakest in Key Stage 3.” [emphasis added]

p. 18:

“Learning and progress […] were least effective in Key Stage 3, where only 38% of lessons were good or better and 12% were inadequate” [emphasis added]

p. 19:

“[…] Quick-fix approaches were particularly popular. Aggressive intervention programmes, regular practice of examination-style questions and extra provision, such as revision sessions and subscription to revision websites, allowed pupils to perform better in examinations than their progress in lessons alone might suggest.

These tactics account for the rise in attainment at GCSE; this is not matched by better teaching, learning and progress in lessons, or by pupils’ deeper understanding of mathematics. In almost every mathematics inspection, inspectors recommended improvements in teaching or curriculum planning, in most cases linked to improving pupils’ understanding of mathematics or their ability to use and apply mathematics.

[…] It remains a concern that secondary pupils seemed so readily to accept the view that learning mathematics is important but dull.” [emphasis added]

The analysis in this book may be seen as an attempt to help schools respond to one of the main ‘Recommendations’ in the 2012 report (p. 10):

“Schools should:

•tackle in-school inconsistency of teaching, making more of it good or outstanding, so that every pupil receives a good mathematics education

•increase the emphasis on problem solving across the mathematics curriculum

•develop the expertise of staff:

–in choosing teaching approaches and activities that foster pupils’ deeper understanding, including through the use of practical resources, visual images and information and communication technology

–in checking and probing pupils’ understanding during the lesson, and adapting teaching accordingly

–in understanding the progression in strands of mathematics over time, so that they know the key knowledge and skills that underpin each stage of learning

–ensuring policies and guidance are backed up by professional development for staff to aid consistency and effective implementation.”

The seriousness of the current situation summarised in these two reports, and the weaknesses in the published Key Stage 3 programme of study may explain why these notes and guidance grew into an ‘extended essay’, rather than being effectively distilled into a punchy DIY manual. Despite (or perhaps because of) this, we hope that all teachers (from those just beginning their careers, or those aiming to take responsibility as Head of Department, to the most experienced practitioners), and those who train teachers will find that what follows provides food for thought, and that schools will find what is presented here helpful in reviewing their current provision in lower secondary school.

1National curriculum in England: mathematics programmes of study, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study; https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/239058/SECONDARY_national_curriculum_-_Mathematics.pdf

2http://webarchive.nationalarchives.gov.uk/20141124154759/http://www.ofsted.gov.uk/resources/mathematics-understanding-score

3https://www.gov.uk/government/publications/mathematics-made-to-measure

4http://webarchive.nationalarchives.gov.uk/20141124154759/http://www.ofsted.gov.uk/resources/mathematics-understanding-score-improving-practice-mathematics-secondary

II. The general advice in the Key Stage 3 programme of study

Schools will naturally try to implement and adapt the published programme as it stands. It is therefore important to decide

•when it is safe simply to copy what is listed;

•when the given list of topics needs to be reordered or supplemented in some way; and

•when there are strong mathematical reasons to reinterpret an official requirement (and to clarify in one’s own mind why it needs to be reinterpreted).

Hence the remaining sections of this book are presented in the form of a line-by-line commentary (where comment seems needed) on the published programme. The present part, Part II, concentrates

•on the Aims etc. which appear on page 2 of the published programmes of study (Section 1 below), and

•on the broad expectations discussed in the section headed Working mathematically on pages 4 and 5 of the published programmes of study (Section 2 below).

1. Aims

1.1.[Aims p. 2]

Mathematics is an interconnected subject in which pupils need to be able to move fluently between [different] representations and mathematical ideas.

Elementary mathematics derives its power from the way a simple idea sometimes has other interpretations, and from the way simple ideas from different domains can be combined to deliver more than one might expect. The published programme of study does not always make it easy to identify these connections and interactions. Hence it is important to consider how to sequence and to link the listed material in a way that clarifies and develops the interdependencies between topics and ideas.

For example, if we consider the most familiar idea of all—namely ‘place value’—schools may recognise the need to reinforce:

•how the place value notation for integers works, and how it extends to decimals;

•that it does so in a way that links

–the more familiar positive powers of 10 (tens, hundreds, thousands),

–with negative powers of 10 (for places to the right of the decimal point);

•the fact that powers of 10 multiply together in a way that foreshadows the index laws for general powers;

•that the written algorithms of column arithmetic, which were developed in primary school for integers, extend naturally to decimals—giving plenty of opportunity to reinforce both the procedures themselves and why they work, and hence to strengthen pupils’ sense of ‘place value’.

Schools will benefit from identifying such recurring themes and important connections for themselves, and from organising the required Key Stage 3 content so that pupils come to appreciate these themes and connections. Some of these are very basic. The next ten bullet points indicate a few selected examples to illustrate the need

–to consider each of the requirements listed in the programme of study,

–to decide what links need to be explicitly mentioned, and

–where possible to include these in any scheme of work.

•The way work with pure numbers (that is, numbers like 1, 23, , or −67.8, stripped of any units), and the arithmetic of integers and decimals, links to simple applications—where purely numerical calculations allow one to solve problems involving measures, and to make sense of, and solve, all sorts of ‘word problems’.

•The way multiplication and division of decimals and fractions hold the key to routinely solving almost any problem involving rates, or percentages, or ratios, or proportion.

•The way blind calculation gives way to simplification and “structural arithmetic”, which links naturally to effective calculation in algebra.

•The way “I’m thinking of a number …” problems should at first be tackled without algebra (as ‘inverse mental arithmetic’), but can later be formulated as a simple equation in one unknown, then routinely solved.

(i)ax + b > 0 (or ax + b⩾ 0) with a > 0, having solution x > (or x⩾)—i.e. a ‘half-line’; or alternatively to

(ii)ax + b < 0 (or ax + b⩽ 0) with a > 0.

•The fact that two simultaneous linear equations can be solved exactly, and that the solution is the point of intersection of the two lines corresponding to the linear equations (provided the two lines meet).

•The way short and long division (combined with a little algebra) shows that fractions correspond precisely to terminating or recurring decimals.

•The way the basic property of parallel lines forces the sum of the angles in a triangle to be equal to the sum of the angles at a point on a straight line.

•The way the congruence criterion and the parallel criterion allow us to justify the standard ruler and compass constructions, and to prove the basic facts about areas (of parallelograms and triangles), which lead to a proof that in any right angled triangle the square on the hypotenuse is miraculously equal to the sum of the squares on the other two sides, which then links with coordinate geometry by allowing us to calculate exactly the distance between any two given points in 2D or in 3D.

1.2.[Aims p. 2]

Pupils should build on Key Stage 2

This is excellent advice—provided it is suitably interpreted. Key Stage 3 has to start out from pupils’ experience at Key Stage 2. But this prior experience also needs to be revisited and developed in fresh ways if it is to be used as a reliable foundation for further work. In commenting on this principle, we consider one example in modest detail (1.2.1), then digress to make three important general points (1.2.2–1.2.4), indicate some further examples more briefly (1.2.5), and end with a gentle warning about the likely impact of the Key Stage 2 programme of study on Key Stage 3 (1.2.6).

1.2.1Mental calculation work should not end with Key Stage 2. It should continue in Year 7, but should increasingly use what pupils know in a way that exploits structure, rather than calculating blindly.

•Pupils need to learn to be on the look-out for ways of extracting 10s and 100s in additions such as

or in multiplications such as

or

•Common factors among a list of added terms should be seen as an opportunity to ‘group’ using the distributive law, as in

rather than to calculate the left hand side blindly. In general, common factors among terms which are to be added or subtracted, multiplied or divided, should be seen as an opportunity to simplify and to cancel.

•Lots of simple work involving fractions should include (a) switching to common denominators (by scaling up both numerator and denominator) in order to simplify the arithmetic, and (b) moving in the opposite direction when using cancellation to simplify fractions.

Written calculation with integers also needs to be strengthened and extended to decimals—but we shall have more to say on this in Section 1.2.5 below.

1.2.2In Part I we saw clear evidence (in the two Ofsted reports) of the unfortunate consequences when a Key Stage seeks to maximise performance on immediately impending assessments, and forgets

that our primary responsibility is always to prepare pupils for the Key Stages that follow.

Pressure to “achieve” in the short-term often encourages pupils to become dependent on (and teachers to allow) ‘backward-looking’ methods that deliver answers in easy cases, but which sooner or later become an obstacle to progress. Hence any internal scheme of work needs to make a clear distinction between

(a)backward-looking methods that get answers in the short-term, but which trap pupils in old ways of working (as with finger counting, or idiosyncratic calculation methods, or reducing multiplication to repeated addition, or modelling questions about fractions in terms of pizzas—all of which may have transitional value, but which are known to block later progress if they become too strongly embedded), and

(b)forward-looking methods, that may seem unnecessary if the perceived goal is merely to get answers to simple problems at a given stage, but which are important because of the way they reflect the inner structure of elementary mathematics, and are often essential for progress at the next stage.

It is not easy for a mere listing of curriculum content to capture this crucial distinction. An effective primary school is one whose pupils are taught in such a way that allows them to flourish at Key Stages 3 and 4. Similarly, effective teaching at Key Stage 3 prepares the ground for, and leads to solid achievement at Key Stage 4 and beyond. Insofar as the revised programme of study incorporates this idea, it tends to do so in ways that are not immediately apparent, so we shall occasionally comment on how Key Stage 3 material impacts on mathematics at Key Stage 4 and beyond.

1.2.3The previous subsection drew attention to the distinction between backward-looking and forward-looking methods. Another important distinction is that between

•a direct operation (such as addition, or multiplication, or evaluating powers, or multiplying out brackets), and

•the associated inverse operation (such as subtraction, or division, or identifying roots, or factorising).

The distinction may be easier to appreciate if we consider a strictly artificial example—namely the “24 game”. Four numbers are given, and each is to be used once. These four numbers may be combined using any three operations chosen from the four rules (with brackets as required), with the goal being to “make 24”.

If one is given the starting numbers “3, 3, 4, 4”, then one scarcely notices the distinction between

When faced with the inverse challenge to “make 24 using 3, 3, 4, 4”, it is almost as easy to dream up a combination that works as it is to evaluate the expression once it has been invented. But

•evaluating the answer of a given sum is a direct, or mechanical, process, whereas

•juggling possibilities to come up with a calculation which produces the required answer of “24” is an inverse operation, which is far from mechanical (even if in this case it is rather easy).

The distinction between direct and inverse operations becomes slightly clearer if the given numbers are “3, 3, 5, 5”. Here the inverse task of coming up with a sum that delivers the required answer of “24” is significantly harder. The relevant tools are the direct processes of arithmetic—except that it is not clear which to use, so one has to scan what one knows, and select approaches which seem to be the most promising. It is precisely this willingness to juggle intelligently with numbers, and to think flexibly with simple ideas that is needed in many everyday applications. But once one is told what calculation to carry out, then the direct calculation “(5 × 5) – (3 ÷ 3)” is entirely routine.

This distinction between the direct operation (which is straightforward, and which requires only that one should implement a given calculation to check that the answer is equal to “24”), and the inverse operation (which is much harder, and which here requires us to invent a sum that has the required answer “24”), becomes markedly more clear if one is given the starting numbers 3, 3, 6, 6, and is left to find a way to “make 24” (or if one is given the starting numbers 3, 3, 7, 7; or 3, 3, 8, 8).

To sum up: the reasons why this distinction is important are that

•almost every mathematical technique one learns comes initially in a direct, or mechanical, form, but leads naturally to inverse problems (as addition leads naturally to subtraction);

•inverse problems are usually much more demanding than their direct cousins;

•mastery of the inverse form depends on a prior robust mastery of the direct form;

•but in the long run, it is the inverse operation which is generally more important.

Those who complain that pupils, or school leavers, cannot “use” what they are supposed to know, often fail to notice that what pupils have been taught (and what has been assessed) has usually focused on direct procedures, whereas what is required is the ability to think more flexibly when faced with some kind of inverse problem. Inverse problems often come in different forms, or variations something that has been a focal point of the recent teacher exchanges with Shanghai, where the idea of “exercises with variation” has emerged as a recurring didactical theme

Given this, one might expect formal assessments to include a strong focus on ensuring mastery of the many inverse operations and the ability to solve the standard inverse problems in elementary school mathematics. In reality, inverse processes have been neglected, or (worse) have been distorted by providing ready-made intermediate stepping stones that reduce every inverse problem to a sequence of direct steps. Why is this?

Direct operations are relatively easy to teach, and to assess. The associated inverse operations may be more important, but they are harder to assess. Inverse problems are more demanding, and cannot be reduced to deterministic methods. So they give rise to low scores, and they do so in a way that is hard to predict. This makes them distinctly awkward for those who devise test items within a target-driven and test-driven culture, where the assessors may be contractually obliged to return predictable results, and to avoid low scores. Hence, if such problems are set at all, they are usually adapted in some way to make pupil performance more predictable (for example, by breaking down the unpredictable inverse problem into a more manageable sequence of steps—each of which is essentially a routine direct task).

Teachers need to recognise the importance of such problems for pupils’ subsequent progress, and then devote sufficient time to them for pupils to achieve a degree of mastery. But it would obviously help if assessments regularly required, and rewarded, such mastery!

1.2.4The bald listing of content in the official programme of study is rather dry and formal—focusing on “what” rather than “how”. In one sense, this emphasis is healthy. But it ignores the essential interplay between content and didactics.

Procedural fluency is rightly stressed. But this emphasis is too often repeated in isolation—as though a robust grasp of place value (for example) will emerge spontaneously as a result of banging on about fluency in specified procedures. It won’t. So something more is needed. If it is to serve as a useful guide, a content list or programme of study needs to be constructed in a way that indicates, and supports, a clear underlying “didactical architecture”. In contrast, the given programme of study routinely misses the opportunity to convey key central principles (such as the contrast between backward-looking and forward-looking methods, or between direct and inverse operations), and important details (such as the key didactical stages which can lead from:

(a)talking about “half a pint” or “half an hour” in Year 2, to competence with the arithmetic of fractions in Year 9; or

(b)from meeting negative quantities for the first time in Key Stage 2, to calculating freely with negative numbers, and simplifying expressions which combine subtraction and minus signs in algebra at Key Stage 3/4).

The extension of long multiplication and division to decimals may need to be slightly delayed. When they are addressed, pupils need first to know how the decimal point behaves under multiplication and division by powers of 10, so that they can understand how this allows multiplication and division of decimals to be transformed into integer multiplication and division.

Short and long division develop the inverse of multiplication, in that they require pupils to use what they know about multiplication in a flexible way. When asked to divide 17 onto 918, the initial inverse question:

“How many times does 17 go into 91? And what is the remainder?”

requires greater mental agility than the two direct questions:

“What is 17 × 5?”, and “What is 91 − 85?”.

Short and long division also require pupils to string together a chain of steps, each of which is accessible, but where the whole chain has to be implemented 100% reliably for the process as a whole to succeed. And the power of the process becomes apparent when one discovers how it extends naturally to allow division of decimals. Later the division process helps to establish the remarkable connection between fractions and decimals.

Some pupils will benefit from the challenge of tackling (or extending their prior facility with) serious long division. This topic is listed in Key Stage 2 for all pupils. It is unclear what effect this may have; but we may well find that serious long division is appropriate for only around half of the cohort, even at Key Stage 3.

1.2.6In exhorting teachers at Key Stage 3 to “build on Key Stage 2” it is only fair to mention that the Key Stage 2 programme of study may prove problematic in some respects. A preliminary indication of the extent of this difficulty may be gleaned from an earlier paper.5 In particular

•a significant amount of material has been included at Key Stage 2 in a way that is likely to prove premature; and

•some of the listed topics which are entirely appropriate in Year 5 and 6 have been specified rather poorly.

Hence one can anticipate that many pupils entering Key Stage 3 will have at best a superficial grasp of some of the listed content from Key Stage 2.

Among the listed topics that are inappropriate and unnecessary in Year 6, many are implicit in the early Key Stage 3 programme of study, so could be safely delayed until Year 7. Some primary schools may recognise this and concentrate on more age-appropriate material—leaving other content to be treated more effectively at Key Stage 3. But many schools will go by the book and will try to cover whatever is listed—with predictable consequences. For both groups, this problematic material will need to be revisited at Key Stage 3 in order to establish a secure platform for progression. Examples of topics which may have been ‘covered’ at Key Stage 2, but which will need serious attention in Years 7 and 8 include:

•the extension of place value to decimals;

•the arithmetic of decimals;

•work with measures—especially compound measures;

•the arithmetic of fractions;

•ratio and proportion;

•the use of negative numbers;

•work with coordinates in all four quadrants;

•simple algebra.

1.3.[Aims p. 2]

Decisions about progression should be based on the security of pupils’ understanding and their readiness to progress to the next stage.

Secondary schools will need to know how this excellent principle of “readiness to progress” has been handled at Key Stage 2. We give just one example of many.

There is a general welcome for the requirement that pupils should learn (i.e. know, and be able to use) their tables. But there is unanimity that this will not be achieved by the end of Year 4 as specified in the official programme of study, and that a more realistic objective may be to expect most pupils to achieve this by the end of Year 5 or Year 6. Hence material listed in Year 5 and Year 6 that depends on ‘prior mastery of tables’ will not be accessible at the expected stage, so will prove unrealistic at that level. (For example, until tables are secure, one is limited in what one can achieve in factorising integers, finding HCFs, working with prime numbers, with short division and long division, with squares and cubes, with equivalent fractions and with cancellation.)

If primary schools feel obliged to try to teach inappropriately ambitious material purely because it is officially listed, this will lead to problems that are entirely avoidable. Thus secondary schools may have to encourage their feeder primary schools to trust their professional judgement in such matters, and to recognise those aspects of the Year 6 programme where work should remain ‘preparatory’, with a serious treatment being delayed until Year 7.

Some of the material that is listed in Key Stage 2 seems inappropriate at that level—partly because we know that it is hard to teach it well even at Key Stage 3. For example, it may make sense at Key Stage 2 to use symbols to summarise familiar formulae: such as re-writing the verbal equation

However, it would be premature to expect most primary pupils to learn more serious elementary algebra (and most primary teachers are in no position to teach it effectively). And while there is every reason to engage pupils at Key Stage 2 in tackling “I’m thinking of a number …” problems, they are best addressed at that age by using ‘inverse mental arithmetic’: that is, where the missing number is discovered by using intelligent, flexible, inverse mental arithmetic, rather than by prematurely trying to formulate such problems algebraically as equations (as suggested by the official Year 6 programme listed under Algebra).

Even where secondary schools liaise effectively with most of their feeder primaries, they should think carefully—as part of ensuring “readiness to progress”—how to consolidate key ideas and techniques from Key Stage 2 in early Key Stage 3, and should be prepared to clear up misunderstandings that may have arisen as a result of material having been introduced prematurely.

A key application of this crucial principle of “readiness to progress” arises because the Key Stage 3 programme of study is now an explicit part of the GCSE specification. Hence decisions about progress through the Key Stage 3 curriculum are bound up with decisions about future GCSE entry. The Key Stage 4 programme of study states explicitly:

Together the mathematical content set out in the Key Stage 3 and Key Stage 4 programmes of study covers the full range of material contained in the GCSE Mathematics qualification. Wherever it is appropriate, given pupils’ security of understanding and readiness to progress, pupils should be taught the full content set out in this programme of study.

In its understated way this both presents a challenge to teach as much of the listed material as possible to as many pupils as possible, and at the same time leaves considerable scope for teachers to use their professional experience to decide where this aspiration may not be “appropriate”.

Those pupils who should progress comfortably to GCSE Higher tier may be able to swallow the complete Key Stage 3 programme by the end of Year 9. But those who may land up taking Foundation tier GCSE will often benefit from proceeding more slowly through Key Stage 3 in order to establish a solid foundation for those parts of the Key Stage 4 programme which they might subsequently manage to cover, and perhaps master. In other words, schools would seem to be free to interpret the Key Stage 3 programme as part of GCSE, and to allow some material to spill over into Year 10 where this seems appropriate. Those pupils heading for Foundation tier are far more likely to achieve mastery of some of this material if they are allowed to proceed more steadily (e.g. taking four years rather than three), than if they are forced to cover the material prematurely, and then have to repeat it.

1.4.[Aims p. 2]

Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content in preparation for Key Stage 4. Those who are not sufficiently fluent should consolidate their understanding, including through additional practice, before moving on.

The second sentence reinforces the comments made at the end of 1.3 above.

The first sentence advises against acceleration. It also highlights the fact that each listed topic can be treated on many levels, and states the important general principle that those who grasp a basic concept should be faced with more challenging variations on the same material before they move ahead. This is an extension of the idea of “readiness to progress”: namely that

before allowing pupils to progress to more advanced topics, we should routinely expect a much deeper understanding on the part of those who might one day proceed further.

At present we routinely let down large numbers of pupils by failing to establish a sufficiently robust mastery of important basic ideas. For example, the very first item under Number (Subject content p. 5: see Part III, section 1) states that pupils should

understand and use place value for decimals, measures and integers of any size.

Other requirements under the sub-heading Number relate to calculating with fractions, working with percentages, and simple algebra. But the evidence is that, even when teaching such basic material we in England have expected far too little—including from our more able pupils. Consider the following items, given to Year 9 pupils in around 50 different countries as part of the major international comparison TIMSS 2011:6

1.4AWhich fraction is equivalent to 0.125?

1.4BWhich number is equal to ?

A: 0.8B: 0.6C: 0.53D: 0.35

1.4C

A: 0.043B: 0.1043C: 0.403D: 0.43

1.4DThe fractions and are equivalent. What is the value of …?

A: 6B: 7C: 11D: 14

1.4EWhich of these number sentences is true?

1.4FWhich shows a correct method for finding ?

1.4GWrite in decimal form rounded to 2 decimal places.

1.4HSimplify the expression

Show your work.

Success rates are never easy to interpret. But it seems sensible to compare the success rates for Year 9 pupils in England with those in Russia, in Hungary, in the USA, and in Australia rather than with countries from the Far East (for the released items and the corresponding results, see http://timss.bc.edu/timss2011/international-released-items.html). We note that:

•in Russia, children start school only at age 7, and in Hungary at age 6;

•the primary curriculum in Russia may include the idea of fractional parts, and the link with decimals, but calculation with fractions would seem to begin only in secondary school;

•tasks 1.4A–1.4F are multiple-choice questions with just four options, and some of the options could never be obtained as a result of making a mistake (which suggests that the English success rates for 1.4A–1.4C are already embarrassing).

1.4ARussia 86%,USA 76%,Hungary 74%,Australia 67%,England 62%;

1.4BRussia 84%,USA 83%,Australia 70%,Hungary 67%,England 59%;

1.4CRussia 83%,Australia 68%,Hungary 63%,USA 63%,England 57%;

1.4DRussia 62%,USA 55%,Hungary 49%,Australia 45%,England 43%;

1.4ERussia 58%,Hungary 53%,Australia 36%,USA 36%,England 33%;

1.4FRussia 63%,Australia 34%,Hungary 33%,USA 29%,England 28%;

1.4GRussia 39%,Australia 31%,Hungary 29%,USA 29%,England 24%;

1.4HRussia 35%,Hungary 34%,USA 19%,Australia 14%,England 9%.

The implication of these comparisons would seem to be that we in England

•are failing to achieve basic competence even for our more able pupils,

•that we routinely allow (or even encourage) pupils to move on to some “higher level” before basic material has been properly understood, and

•that we need to slow down and routinely use slightly harder and more varied problems to probe and strengthen pupils’ understanding before they move on in this way.

This inference was supported by the recent ICCAMS study which set a sample of 15 year olds in English schools problems that had been used in a similar study in the late 1970s. We give just two examples:

1.4JOn the motorway my car can go 41.8 miles on each gallon of petrol. How many miles can I expect to travel on 8.37 gallons? [Six calculations involving 41.8 and 8.37 were given, and the relevant calculation was to be ‘circled’, not implemented.]

30 years ago 54% of 14 year olds managed to circle 8.37 × 41.8; now only 33% manage this.

1.4KSix tenths written as a decimal is 0.6. How would you write eleven tenths as a decimal?

30 years ago 36% managed to write 1.1; now just 16% of 14 year olds respond correctly.