Conceptual Maths E-Book

Peter Mattock

• Herausgeber: Crown House Publishing
• Kategorie: Bildung
• Sprache: Englisch
• Veröffentlichungsjahr: 2023

Seitenzahl: 911

Beschreibung

Written by Peter Mattock,Conceptual Maths: Teaching 'about' (rather than just 'how to do') mathematics in schoolsaims to empower teachers to support students on a comprehensive and coherent journey through school mathematics. Showcasing the best models, metaphors and representations, it provides excellent examples, explanations and exercises that can be used across the curriculum.Concepts are at the heart of the study of mathematics. They are the ideas that remain constant whenever they are encountered, but which combine and build upon each other to create the mathematical universe. It is the structure of each concept that gives rise to the procedures that are used in calculation and problem-solving - and, by learning about these structures, a learner can make sense of how different processes work and use them flexibly as need demands.In his first book,Visible Maths, Peter Mattock focused on the use of representations and manipulatives as images and tools and how this can provide a window into some of these mathematical structures. His aim inConceptual Mathsis to go deeper, beyond the procedures, and to shed greater light on the structures of the subject's different concepts. The book explores how a variety of visual tools and techniques can be used in the classroom to deepen pupils' understanding of mathematical structures, concepts and operations, including: number; addition and subtraction; multiplication and multiples; division and factors; proportionality; functionality; measures; accuracy; probability; shape and transformation; and vectors, among many others. In so doing, Peter equips teachers with the confidence and practical know-how to help learners assimilate knowledge of mathematical concepts into their schema and take their learning to the next level.Containing numerous full-colour diagrams and models to illustrate the conceptual takeaways and teaching techniques discussed,Conceptual Mathsalso includes a glossary covering the key mathematical terms.Suitable for teachers of maths in primary, secondary and post-16 settings

Leseprobe

i

Dedication and thanks

My last book was dedicated to the maths teachers and educational professionals that have been part of my career, to my mum, Lesley, and to my partner (now wife), Rowan. All of you remain an important part of my life and work; please do not think you are forgotten.

This book is dedicated in three parts.

First, to my grandparents, Pat, Ted, Alan and June, whom I know would be proud.

Second, to the boys, Chris, John and Paul. Friends for life.

Finally, to my girls, Erin and Mollie. The lights of my world.

A number of people have allowed me to reproduce their material as part of this book. I would like to thank the following for their contribution:

10TicksAQAbasic-mathematics.comBoss MathsCIMTCorbettmathsDan DraperDesmosDon StewardGo Teach MathsJo MorganJonathan Hall (MathsBot.com)JustMathsKangaroo MathsMath-Aids.ComMaths GenieMath is FunNRICHOnline Math LearningOpen MiddlePixiMathsTransfiniteUKMTVariation Theoryii

Contents

Title PageDedication and thanksIntroduction1.Number2.Addition and subtraction3.Multiplication and multiples4.Division and factors5.Equality/equivalence/congruence6.Proportionality7.Functionality8.Measures9.Accuracy10.Shape11.Transformation and vectors12.Chance13.Charting and graphing14.Data handlingGlossaryBibliographyCopyright

1

Introduction

What is mathematics? Interestingly, although mathematics has been an integral part of the school curriculum for the best part of the last 75 years or so,1 there is little consensus on the answer to this question amongst teachers of mathematics. Some people will say that mathematics is a body of connected knowledge, others that it is a way of behaving and making sense of the world, whilst others will say it is a collection of theorems based on fundamental axioms. Some may see this lack of consensus as problematic – we can hardly ensure that learners of mathematics are getting a consistent experience if their teachers don’t all have the same idea of what they are teaching! However, for me, a consensus amongst maths educators about what mathematics is isn’t as important as what mathematics is not. One thing (in my opinion) that mathematics definitely is not, is a collection of procedures.

That is not to say that procedures aren’t important in mathematics; simply that if all one learns about mathematics is how to complete procedures, then one hasn’t really learnt a lot about mathematics. Primarily this is because there are (nearly always) many different procedures that will accomplish the same result, with the choice of procedure largely dependent on a mixture of efficiency and what the teacher is familiar with or prefers. Jo Morgan’s excellent A Compendium of Mathematical Methods highlights some of the multitude of procedures that exist for doing things like multiplying large numbers or solving equations.2 But a pupil could learn every method in the book and still not have learnt much about mathematics. To learn about mathematics, one has to go deeper, beyond the procedures, and into the structure of its different concepts. In Mathematics Counts, the Committee of Inquiry into the Teaching of Mathematics in Schools (the Cockcroft Report), states: ‘Conceptual structures are richly interconnected bodies of knowledge, including the routines required for the exercise of skills. It is these which make up the substance of mathematical knowledge stored in the long term memory.’3

Concepts are at the heart of the study of mathematics. They are the ideas that remain constant whenever they are encountered but that combine and build upon each other to create the mathematical universe. The structure of each concept is what gives rise to the procedures and processes that are used in calculation and problem solving. In learning about the structure of each concept, a learner of mathematics can make sense of how different processes are doing what they do, using them flexibly as need demands. A simple image to capture this relationship might look like this:

However, this model ignores two important aspects of mathematics: the interplay between concepts and overarching themes.2

In their .fantastic book Developing Thinking in Algebra, John Mason, Alan Graham and Sue Johnston-Wilder put forward five mathematical themes:

Freedom and constraint.Doing and undoing.Extending and restricting.Invariance and change.Multiple interpretations.4

These themes appear across different mathematical concepts (the last will appear a lot throughout this book) and provide key touchpoints that learners (and teachers) can keep coming back to when they study different concepts. That complicates the model slightly:

But the flowchart above still doesn’t capture the interplay between concepts or how they can come together to build the mathematical universe. One might more accurately adapt the above model to look something like this:

But even this is too simple; some concepts derive from others, some processes bring multiple concepts into play and several themes appear in each concept. The real model is likely to be three-dimensional or higher in order to capture all of the links between all of the themes, concepts and processes. This, of course, explains why it is so difficult to design a curriculum for mathematics even though it is essentially a hierarchical subject – concepts are introduced, then disappear for a while before reappearing in conjunction with other concepts that have been developed in the meantime. It also means that capturing this in text form is incredibly complex, with lots of back and forth between different concepts. In order to support this, each concept will include details such as:3

Concept – what the concept being explored is.Prerequisites for each concept – other concepts or parts of a concept that this concept requires to be secure before the given concept is introduced.Linked concepts – other concepts that will come into play with the given concept when developing certain procedures or other aspects of the concept.Good interpretations – good ways of thinking about/representing the given concept.Good questions to ask – questions that can be asked or tasks that can be given to learners to support understanding of the structure of the concept.Procedures – procedures that mainly arise from, or are associated with, the given concept and how to make their link to the concept explicit.

The first three of the above will typically be listed at the start of each section (as well as highlighted when they appear) and the rest will be explored in the main body of the section. It won’t be necessary to look at all of these details for all concepts, and for some there may be other details that we will look at, but this will be the essence of what will be addressed for each concept.

In order to try and make the exploration of the mathematical concepts more manageable, we will group them into broader topic areas. These are not necessarily the sort of topic areas we might use with children (although some might be), and as explained there will be plenty of crossover from concepts in one area to concepts in another (there would be no matter how you defined the topic area); however, it will allow for a grouping of broadly similar and strongly related concepts.

We will start with an examination of the concept of number, which will include its generalisation into algebra. We will then move on to looking at the standard numerical operations in three parts: addition/subtraction, multiplication and multiples, and then division and factors. We will follow this with an examination of ideas around equivalence and equality before shifting our attention to proportionality and then functionality. From here we will begin to look at concepts in the realm of geometry, including measures, accuracy, shape and transformation. We will finish in the realms of chance, data and graphing/charting.

In my first book, Visible Maths, I concentrated on how the use of representations and manipulatives can provide a window into some of these mathematical structures and can support pupils in creating some of these connections by being able to draw on particular images and tools that could represent mathematical concepts whenever a pupil was working with them. Whilst there will be some crossover in this book, and readers of Visible Maths will find some things familiar, my aim with this book is to go broader, but not necessarily as deep, in all areas. Visible Maths (I hope) really got into the detail of some very specific concepts, at least in terms of the inherent mathematical structures of those ideas and how they can be manipulated. In this book, I will provide more of an overview of more concepts and include teaching ideas that are not necessarily related to the structure (such as good questions or activities that highlight aspects of the concept). My hope is that people will see Visible Maths as a companion to this work, so that in this book they find reference to all the concepts they will be teaching across primary and secondary schools, along with key advice/suggestions for how to ensure they teach this idea in a way that makes its structure explicit so it can be linked with other ideas. In a complementary fashion, in Visible Maths they can delve in depth into certain important concepts and look in detail at how good representations and manipulatives can be used to really get into a concept with pupils.

In recent years there has been, in some circles of mathematics education, a strong move back to trying to secure ‘procedural fluency’ prior to developing ‘conceptual understanding’. 4Many influential maths teachers are suggesting that learners can gain greater insight into the structure of a concept if they have first reached the point where they are very comfortable with the procedures associated with the concept. However, I am sceptical of this for two reasons:

Much of my experience of maths education to date has been of a very instrumental approach in which pupils are often practising procedures for much of the time in the classroom that they don’t spend listening to their teacher.5 Whilst I can see ways to improve this practice so that learners take more from the experience, I don’t see it providing the gains in pupil understanding or in their motivation to continue studying mathematics education.As I have intimated, the procedures attached to different concepts actually arise from the structure of the concept itself. To rely on knowledge of the procedure to provide understanding of the concepts seems to be backwards in approach. In addition, because there are many different procedures associated with each concept, it would seem to be time-intensive to have to study many of them to the point of automaticity before being able to use this experience to gain a window into the underpinning structure. Instead, first securing the structure and then exploiting the structure to gain insight into the associated procedures would seem much more logical.

This move back towards procedural fluency is generally attributed to a reaction against the perceived dominance of constructivist approaches to education6 (which are thought to be linked with a more discovery-based approach to learning) in the late 1990s and first decade of the 21st century. Some feel that this has led to learners being held back as they haven’t had their learning directed adequately by a teacher. For me, though, if this is the case, the cure is not to move to teaching procedures. Teaching structure is compatible with both constructivist and didactic approaches to education because it concerns itself with the content to be taught rather than how to go about teaching it. That is what I aim to show in this book: how exposing learners to mathematical structure can ensure they achieve both procedural fluency and conceptual understanding, whether your preferred pedagogy is to teach it explicitly or to offer learners activities to discover this structure through inquiry. Hopefully, in reading this book, teachers will become familiar with the underlying structure for the key concepts in school-level mathematics and will then be able to use this knowledge to support learners in making sense of the content they study. Whether you support learners in constructing that sense for themselves or explicitly teach good ways of making sense of concepts, ensuring learners can make sense of mathematics concepts puts them in a much better place to see the connections between the things they study.

The point about making connections is important. In the last few years, cognitive science has had increasing exposure to teachers and is influencing practice on a larger scale. One of the key ideas in cognitive science is that of a schema. A schema represents our knowledge in a particular area, and how it is connected. If we wish to learn something new in that area, we have to be able to connect it to our existing schema. By teaching about the structure of concepts, these connections become much easier to highlight because the concept is recognisable every time it reappears.7 What I would hope is that, having read this book, teachers feel able to support learners in recognising the structure behind different mathematical concepts and help them assimilate or accommodate new knowledge of the concept into their schema.

1 M. McCourt, A Brief History of Mathematics Education in England, Emaths [blog] (29 December 2017). Available at: https://www.emaths.co.uk/index.php/blog/item/a-brief-history-of-mathematics-education-in-england.

2 J. Morgan, A Compendium of Mathematical Methods (Woodbridge: John Catt Educational Ltd, 2019).

3 W. Cockcroft, Mathematics Counts: Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr W. H. Cockcroft [Cockcroft Report] (HMSO, 1982), p. 71. Available at: http://www.educationengland.org.uk/documents/cockcroft/cockcroft1982.html.

4 J. Mason, A. Graham and S. Johnston-Wilder, Developing Thinking in Algebra (London: SAGE Publications, 2005).

5 R. R. Skemp, Relational Understanding and Instrumental Understanding, Mathematics Teaching, 77 (1976), 20–26. Available at: https://www.lancsngfl.ac.uk/secondary/math/download/file/PDF/Skemp%20Full%20Article.pdf.

6 Learning Theories, Constructivism (n.d.). Available at: https://www.learning-theories.com/constructivism.html.

7 Learning Theories, Schema Theory (n.d.). Available at: https://www.learning-theories.org/doku.php?id=learning_theories:schema_theory.