Thinking Through Mathematics

Thinking Through Mathematics for secondary schools provides teachers of mathematics with a number of thinking skills activities which develop both generic thinking strategies and more specialised mathematical thinking

Teaching mathematics to children can be difficult, especially with the current climate of failing mathematics grades, due to candidates being "required to now exhibit a degree of familiarity with a much wider but shallower curriculum" (Reform 2006).

The activities included in Thinking Through Mathematics will help you to:

•  encourage pupils to discuss and share their mathematical understanding with each other, and with you
• vary the mathematical diet of teaching and learning for you and your pupils
• develop pupils' reasoning and thinking skills so that they are able to take a more active role in mathematics lessons
• enable pupils to realise that mathematics is not always about getting one right answer
• make mathematics a more enjoyable experience for all.

The structure of the book

Part One
Aims to provide you with a general introduction as to why and how the book was written. It consists of the following chapters:

Introduction outlines the purpose of the book and underpinning principles.

• Chapter 1 considers the benefits of teaching thinking in the mathematics classroom.

• It provides a big picture to which you can relate the exemplars that follow.

• Chapter 2 introduces the overall structure of the book and provides detailed informationabout the exemplars and strategies included in this book and the layout of the exemplars and guidance on how to use the book.

• Chapter 3 presents an overview table of the activities and their links to mathematical topic areas.

Part Two
Contains 21 exemplars, grouped under eight thinking skills strategies. Each of the strategies is demonstrated with at least one activity which was developed for use in the mathematics classroom and trialled with different age and attainment groups.

Below is a summary of the strategies and activities:

Strategy 1 - Odd one out
Odd one out is a powerful strategy that supports classification and understanding of the properties and defining attributes of things.

Exemplars in strategy 1 include activities linking to:

• algebra

• fractions, percentages and decimals

• shapes and angles

Strategy 2 - Collective memory
Collective memory begins to make pupils think harder about how they see and make sense of visual information - maps, sketches, photographs and diagrams.

Exemplars in strategy 2 include activities linking to:

• memory techniques
• revision

 Strategy 3 - Classfication
Classifying data or objects is a fundamental cognitive skill. It encourages pupils to identify key characteristics and justify their choice.

Exemplars in strategy 3 include activities linking to:

• angle properties of triangles

• identifying compound shapes

• three-dimensional shapes

• graphs

• number sequences

Strategy 4 - Living graphs
Living graphs give the figures and the lines of a graph real context and allow students to make connections between the abstraction of the graph on the page and the people and events that lie behind it.

Exemplars in strategy 4 include activities linking to:

• interpreting distance-time graphs

• identifying changes on a gradient

Strategy 5 - Mysteries
The strategy is designed to encourage pupils to deal with ambiguity through addressing a question which has no single correct answer and where they are not sure what information is relevant - like in real life.

Exemplars in strategy 5 include activities linking to:

• area

• surveys

Strategy 6 - Relational diagrams
The use of relational diagrams in this strategy gives a visual representation of the relationships between the data being discussed in the diagrams. This is a good tool for clarifying understanding or clearing up misunderstandings.

Exemplars in strategy 6 include activities linking to:

•  geometrical reasoning

• number properties

Strategy 7 - Taboo theory
This strategy is based on the board game whereby the explainer in each team has got to get across a keyword to the rest of their team without using any of the taboo words.

Using cards produced by The Association of Teachers of Mathematics (ATM) entitled 'Fourbidden', you and your students play a version of this game.

Exemplars in strategy 7 include activities linking to:

• various words and topics

Strategy 8 - Sequencing
Sequencing is a strategy that promotes reasoning.

In this context reasoning includes:

• explaining why an answer must be correct

• constructing chains of deductions

• understanding the difference between

• a practical demonstration and a proof

Exemplars in strategy 8 include activities linking to:

• calculating and reasoning

• triangle theorems

• creating and resolving equations

• angles in a triangle.

Part Three
This section of the book is intended to elaborate and expand on points made in the exemplars. These points draw your attention to why the teachers did things is a particular way, and to how you might adapt that to your own teaching and learning contexts.

Chapters included in Part Three are:

Chapter 4 describes principles which underpin teaching thinking and the generic features of thinking skills lessons. This includes consideration of the close correlation between principles of teaching thinking and those promoted in Teaching and Learning in the Foundation Subjects (DfES, 2003).

Chapter 5 looks at implications of teaching thinking for the roles of teacher and learner.

Chapter 6 provides a few rules of thumb and some practical suggestions for you to get started.

Chapter 7 is to a refl ect upon what we think now about thinking skills and mathematics. It summarises strengths of infusing thinking skills into mathematics lessons and points out challenges facing mathematics educators.

Chapter 8 concentrates on the development that is made for us as teachers using these strategies in our classrooms.

ISBN: 1 899587 49 4

Editor: David Leat

Authors: Sally Taverner & David Wright