Teaching Mathematics at Secondary Level TONY GARDINER To access digital resources including: blog posts videos online appendices and to purchase copies of this book in: hardback paperback ebook editions Go to: https://www.openbookpublishers.com/product/340 Open Book Publishers is a non-profit independent initiative. We rely on sales and donations to continue publishing high-quality academic works. Teaching Mathematics at Secondary Level Tony Gardiner http://www.openbookpublishers.com ⃝c 2016 Tony Gardiner This work is licensed under a Creative Commons Attribution 4.0 International license (CC BY 4.0). This license allows you to share, copy, distribute and transmit the work; to adapt the work and to make commercial use of the work providing attribution is made to the author (but not in any way that suggests that they endorse you or your use of the work). Attribution should include the following information: Tony Gardiner, Teaching Mathematics at Secondary Level. Cambridge, UK: Open Book Publishers, 2016. http://dx.doi.org/10.11647/OBP.0071 Further details about CC BY licenses are available at http://creativecommons.org/licenses/by/4.0/ All external links were active on 19/01/2016 and archived via the Internet Archive Wayback Machine: https://archive.org/web/ Every effort has been made to identify and contact copyright holders and any omission or error will be corrected if notification is made to the publisher. Digital material and resources associated with this volume are available at http://www.openbookpublishers.com/isbn/9781783741373 This is the second volume of the OBP Series in Mathematics: ISSN 2397-1126 (Print) ISSN 2397-1134 (Online) ISBN Paperback 9781783741373 ISBN Hardback: 9781783741380 ISBN Digital (PDF): 9781783741397 ISBN Digital ebook (epub): 9781783741403 ISBN Digital ebook (mobi): 9781783741410 DOI: 10.11647/OBP.0071 Cover photo: Un phare en coquille (2007) by TisseurDeToile, https://www.flickr.com/photos/tisseurdetoile/2141698014/in/photolist- CC BY-NC-SA All paper used by Open Book Publishers is SFI (Sustainable Forestry Initiative) and PEFC (Programme for the Endorsement of Forest Certification Schemes) Certified. Printed in the United Kingdom and United States by Lightning Source for Open Book Publishers Contents About the author i Introduction and summary iii I. Background: Why focus on Key Stage 3? 1 II. The general advice in the Key Stage 3 programme of study 7 1. Aims 8 2. Working mathematically 25 III. The listed subject content for Key Stage 3 77 1. Number (and ratio and proportion) 77 2. Algebra 146 3. Geometry and measures 195 4. Probability and Statistics 254 IV. A sample curriculum for all—written from a humane mathematical viewpoint 271 1. Very Brief version 272 2. Brief version 279 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 Erdos¨ 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. iv Tony Gardiner 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 Teaching Mathematics at Secondary Level v 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. vi Tony Gardiner • 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.