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Learning and Teaching Mathematics Without a Textbook
Investigative approaches all of which have been used in many different classrooms
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 Learning and Teaching Mathematics Without a Textbook
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Learning and Teaching Mathematics Without a Textbook
Reviews
This book offers alternative ways for students to work on mathematics without needing to do exercises from a textbook.
This book is about learning mathematics using investigative approaches without needing to use a textbook as the main resource. More specifically it offers alternative ways, for students to work on mathematics, to the commonly used method of carrying out exercises from a textbook. It is a collection of starting points and extension ideas, all of which have been used in many different classrooms.
“I have written up this collection of ideas in order to support teachers to help their students engage with the intrigue, beauty, surprise and fascination of mathematics. I seek to demonstrate that it is not only feasible to engage with mathematics in such ways, it is wholly desirable. The ideas offered are not intended to form a ‘scheme’ or to try to prescribe how they might be used. I merely present them with an intention that other teachers will have an opportunity to think about how they might adapt them to their own practice.”
ISBN 1 898611 15 7
Introduction
Rationale and Strategies
Activities x+y=7
Diagonal through a rectangle
The 4-Square Perimeter Problem
Volume
Stars and Statistics Do we meet?
Borders
Cuboids and surface area
Dual Tessellations
Fibonacci
Arithmogons and Pyramids
The 1,2,3,7,12 problem
A number of Number Ideas
I choose to use investigative approaches to learning because I believe this is one important way we all learn. For students to learn mathematics by exploring ideas, independent of graded exercises in textbooks, means tapping into a basic human condition of enquiry. This natural response, to be inquisitive, to have a desire to find out about something is something we spend a lot of our time doing. We have a natural fascination for wanting to know how and why something works, to see the view beyond the immediate horizon. We pose problems, tell jokes (knock, knock...), set conundrums, amaze friends with our latest `conjuring' trick, learn how to juggle and solve crossword puzzles; some people go orienteering, some do jig-saw puzzles; some see rock-climbing as a 3-dimensional jig-saw puzzle. Such activities contain some element of problem-solving and to varying degrees we are all problem-solvers. Through solving problems a `need' arises to work mathematically, to figure something out, to experience an `aha' moment, to practise and sharpen skills and to consolidate an understanding of conceptual strictures.
To teach mathematics without a textbook also means wresting control away from authors of schemes and texts. They cannot know about individual teachers' classrooms and the particular ways in which they function. To teach mathematics without a textbook opens up opportunities for teachers to construct and use a wide range of strategies. In turn this places ingenuity and professional integrity at the centre of one of the most creative and potentially exciting and exacting occupations - teaching.
I am not, therefore, writing a textbook and have no intention to try to guide the reader through specific, narrow, step-by-step, chapter-by-chapter approaches to learning mathematics. I offer no `exercises' although 1 suggest ways that students might be able to set up and solve each other’s problems; these can arise naturally from the ideas offered.
I am writing an eclectic mix of ideas, of starting points and extension tasks that I have found useful in my teaching. There is little that is `new' within the pages of this book and I make no claim to originality; that the ideas suggested are of my own invention.
Well, perhaps in the following pages there may be just one or two problems I have `invented' or which are pieces of `original' thought!
So why should any teacher wish to read what I have to offer? Firstly, on very many occasions, when working with teachers at conferences and through in-service work in schools, I have been asked if `my' ideas are written down anywhere. Of course they are and in various publications. Many of the ideas I use stem from marvellous resource books such as:
The latter is a seminal text formerly published by Tarquin that, sadly, is out of print.
My first intention, therefore, is to attempt to collect together many of the ideas I have used into a single publication. Of course the moment it goes to print another idea will have sprung to mind, making this book automatically in need of up-dating... or perhaps it will be followed by a second collection of ideas.
Secondly, my thirty years of teaching experience, mainly in secondary mathematics departments but also in primary schools and latterly with undergraduates and postgraduate students, has enabled me to use the ideas in classrooms from a wide range of contexts. From 1986 to 1995, prior to moving into teacher education, I led a mathematics department in an 1 1-16 comprehensive school and taught mathematics without using a textbook with mixed-ability groups across the age range. During this time, and because I did not buy any textbooks, I was able to exercise a choice to spend capitation on other and, in my opinion, more valuable resources such as:
I did purchase resource books such as the Brian Bolt collection, Lorraine Mottershead, and Martin Gardner, to name but a few.
Thirdly, the ideas 1 suggest are ones I have used during INSET in schools and at conferences throughout England and as far afield as Gran Canaria, Poland and Eritrea.
I mention this not to offer a miniature curriculum vitae, but because these experiences of working in other countries lead me to generalise that `good' or `interesting' ideas can be adapted for use in all classrooms in different contexts and different cultures.
I have chosen not to reference the ideas in this book to any national scheme or curriculum initiative. This is because I believe that effective mathematics teaching is a craft, an art and a science learnt over time and developed through the ideas we accumulate, happen upon, try out, plan into lessons, adopt and adapt. We learn to teach in the light of experience and according to a wide variety of circumstances and energy levels. We gain ownership of how we teach by taking responsibility for deciding what works well and having choices to change and develop ways we work. Through ownership we achieve greater autonomy and this is fundamental to our professional integrity. I believe we are driven, in part, by a desire to get better at how we teach and how to foster effective (and affective) learning. As such this book would resonate with many of the guiding principles in publications such as Mathematics Counts: Cockroft Report (1982), Mathematics from 11 to 16: HMI Curriculum Matters 3 (1985) and Non-Statutory Guidance: NCC (1989)
Mathematics is a beautiful, exciting and mind-bending discipline and I have purposefully stayed away from using, so-called, real-life contexts within which to frame mathematics.
This is because I do not think that students believe or take seriously the contexts that arc frequently offered in textbooks in order to provide a `reason' for why students should learn mathematics. I offer no contexts such as builders and numbers of paving slabs, farmers and fences, ladders and walls, tax and VAT or where to stand when the tree falls down! This does not mean that real-life information sources have no place in the mathematics classroom; just the contrary. My argument is that when a ‘real-life’ context appears in a textbook it automatically becomes out of date, unreal and a pseudo con-text, an excuse almost for doing mathematics. Using real sources of information such as newspapers, timetables, calenders is a very different prospect and can be valuable source material, particularly when working on mathematics in cross-curriucular ways. However, the context I offer is problem solving; through engagement with such problems I suggest the beauty and the intrigue of mathematics arises.
