Diversifying Mathematics Teaching: Advanced Educational Content And Methods For Prospective Elementary Teachers

by Sergei Abramovich

Overview

'What one takes away from this book is the notion that there's a lot of potential to do more with these students, and the book stands as a resource for anyone who shares that opinion … Books like Abramovich's are a welcome addition to our options as we try to do our best by these students, and by extension, their future students.'
MAA ReviewsThe book is written to enhance the preparation of elementary teacher candidates by offering teaching ideas conducive to the development of deep understanding of concepts fundamental to the mathematics curriculum they are to teach. It intends to show how the diversity of teaching methods stems from the knowledge of mathematics content and how the appreciation of this diversity opens a window to the teaching of extended content.The book includes material that the author would have shared with teacher candidates should there have been more instructional time than a 3 credit hour master's level course, 'Elementary Mathematics: Content and Methods', provides. Thus the book can supplement a basic textbook for such a course by extending content and diversifying methods.Also, the book can support graduate level mathematics education programs which have problem-solving seminars/courses in their curriculum. The book is well-informed with (available in English) the mathematical standards and recommendations for teachers from Australia, Canada, Chile, England, Japan, Korea, Singapore, and the United States.

Product Details

ISBN-13:	9789813206878
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	04/19/2017
Pages:	284
Product dimensions:	6.20(w) x 9.10(h) x 0.80(d)

Table of Contents

Preface v

Chapter 1 Teaching Elementary Mathematics: Standards, Recommendations and Teacher Candidates' Perspectives 1

1.1 Introduction 1

1.2 Questions as the Major Means of Learning Mathematics 3

1.3 Answering Questions Both Procedurally and Conceptually 6

1.4 Connecting Algorithmic Skills and Conceptual Understanding 8

1.5 Developing Deep Understanding of Mathematics Through Making Conceptual Connections 11

1.6 Teaching and Learning to Think Mathematically 13

Chapter 2 Counting Techniques 17

2.1 Introduction 17

2.2 Rules of Sum and Product 19

2.3 Tree Diagram and the Rule of Product 20

2.4 Permutation of Letters in a Word 21

2.5 Combinations without Repetitions 24

2.6 Combinations with Repetitions 27

Chapter 3 Counting and Reasoning with Manipulative Materials 33

3.1 Introduction 33

3.2 Constructing a Triangle out of Straws 35

3.2.1 Reflection on the activity with straws 38

3.2.2 A real-life application of the triangle inequality 40

3.2.3 Modifying the S²AC² algorithm to enable linguistic coherency 41

3.2.4 How many triangles can be constructed? 43

3.2.5 Using multicolored straws 45

3.2.5.1 An equilateral triangle 46

3.2.5.2 An isosceles triangle with the base being the smaller side 47

3.2.5.3 An isosceles triangle with the base being the larger side 48

3.2.5.4 A scalene triangle 49

3.3 Two Types of Representation as Means of Transition from Visual to Symbolic 49

3.4 Signature Pedagogy of Elementary Mathematics Teacher Education 51

3.5 Towards Rich Interpretations of Manipulative Representations 51

3.5.1 Manipulative representation as text 51

3.5.2 From "brothers" to Pascal's triangle 54

3.6 Learning to Move from One Type of Symbolism to Another and Back 56

3.7 Perimeter and Area Using Square Tiles 58

3.8 The Importance of Teacher Guidance in Using Manipulative Materials by Students 63

3.9 Conceptualizing Base-Ten System Using Manipulative Materials 65

3.10 Modeling as a Way of Creating Isomorphic Relationships 68

Chapter 4 We Write What We See (W⁴S) Principle 71

4.1 Introduction 71

4.2 W⁴S Principle and the Duality of Its Affordances 73

4.3 W⁴S Principle in Teaching Primary School Mathematics 74

4.4 Comparing Non-Unit Fractions Using Area Model 78

4.4.1 Comparing fractions being close to each other 78

4.4.2 Comparing fractions that are a unit fraction short of the whole 80

4.5 From Comparison of Fractions to Arithmetical Operations Using Area Model 81

4.5.1 Fractions as part-whole and divisor-dividend models 81

4.5.2 The concept of common denominator 83

4.5.3 Reducing a fraction to the simplest form 84

4.5.4 Using unit fractions as benchmark fractions 84

4.5.5 Multiplying fractions using area model 87

4.6 Dividing Fractions Using Area Model 89

4.6.1 Partition model for division supports contextualization 89

4.6.2 The importance of unit in solving word problems with fractions 91

4.6.3 The meaning of "invert and multiply" rule 94

4.7 Ratio and Proportion 96

4.8 Percent and Decimal as Alternative Representations of a Fraction 98

4.9 Multiplying and Dividing Decimal Fractions 100

Chapter 5 Partitioning Integers into Like Summands 105

5.1 Introduction 105

5.2 Partition of Integers into Summands 106

5.3 Activities with Towers Motivate Introduction of Algebraic Notation 116

5.4 Ferrers-Young Diagrams 117

5.5 Recursive Definition of P(n, m) Informed by Ferrers-Young Diagrams 117

5.6 Making Mathematical Connections 120

5.7 Recursive Definition of Q(n, m) Informed by Ferrers-Young Diagrams 122

5.8 Connection to Triangular Numbers Opens a Window to a New Concept 125

Chapter 6 Hidden Curriculum of Mathematics Teacher Education 129

6.1 Introduction 129

6.2 The Basic Task 130

6.3 Background Information: Triangular and Trapezoidal Numbers 132

6.3.1 Activities 132

6.3.2 Solutions to the tasks 133

6.3.3 Trapezoidal numbers 137

6.4 Conceptually Oriented Discussion of the Basic Problem 139

6.4.1 Grouping the sums by the number of addends 139

6.4.2 Grouping the sums by the first addend 142

6.4.3 Partitioning integers into the sums 142

6.5 Discourse Motivated by Multiple Ways of Creating Sums of Consecutive Natural Numbers 143

6.5.1 Clarifying the meaning of the word special in the context of arithmetic 143

6.5.2 The first encounter with a special property of the sums of two addends 144

6.5.3 Moving from novice to expert practice in revealing special properties 146

6.5.4 Exploring the sums of four consecutive integers 148

6.6 Proof of the Conjecture about Trapezoidal Numbers 151

6.7 Sums in Pairs of Odds and Evens 152

6.7.1 Learning to generalize from special cases 153

6.7.2 Comparing triangles to trapezoids with the top row greater than two 156

6.8 Mathematical Knowledge Used for Teaching Young Children 158

6.9 How Many Trapezoidal Representations Does a Number Have and How Can One Find Them? 159

Chapter 7 Informal Geometry 163

7.1 Introduction 163

7.2 Geoboard Explorations 165

7.3 Towards a Double-Application Environment 172

7.4 Guided Exploration on a Computational Geoboard 174

7.5 Transition to a Spreadsheet 178

7.6 Preparing Data for Empirical Induction 180

7.7 Abstracting from Numbers to Equations Using First-Order Symbols 182

7.8 Visual and Symbolic Deduction of Pick's Formula 183

7.9 Moving to a New Learning Site 187

7.10 Measuring vs. Counting 188

7.11 Encountering Limitation of the Environment 190

7.12 From Particular to General through Visualization 191

7.13 Communicating about Mathematics 194

Chapter 8 Probability as a Blend of Theory and Experiment 197

8.1 Introduction 197

8.2 Basic Concepts and Tools of the Probability Strand 200

8.2.1 Randomness and sample space 200

8.2.2 A more complicated example of constructing a sample space 202

8.2.3 Different representations of a sample space 202

8.3 Fractions as Tools in Measuring Chances 205

8.4 Bernoulli Trials and the Law of Large Numbers 207

8.5 A Problem of Chevalier De Méré 209

8.6 A Modification of the Problem of De Méré 210

8.7 Wagering for the Odds/Evens in a Game of Chance 212

8.8 Paradoxes in the Theory of Probability 214

8.8.1 Bertrand's Paradox Box problem 214

8.8.2 Monty Hall Dilemma 215

8.9 Probabilistic Perspective on Partitioning Problems 217

8.9.1 A problem of tossing three dice 217

8.9.2 Unordered partitions of integers into unequal summands 218

8.10 Experimental Probability 219

8.10.1 Experimental probability calls for a long series of observations 219

8.10.2 Comparing experimental and theoretical probabilities when tossing a fair coin 220

8.10.3 Calculating relative frequencies for the problems of De Méré 222

8.11 Exploring Irreducibility of Fractions through the Lenses of Probability 224

Chapter 9 Using Counter-Examples in the Teaching of Elementary Mathematics 227

9.1 Introduction 227

9.2 The Pedagogy of Using Counter-Examples 228

9.2.1 The role of linguistic constraints 228

9.2.2 A counter-example as a motivation for further learning 231

9.3 Providing Explanation through Counter-Examples 231

9.4 Constructing a Counter-Example: an Illustration 233

9.4.1 From modeling with fractions to algebraic generalization 233

9.4.2 From a counter-example to its conceptualization 234

9.4.3 A family of jumping fractions found by a teacher candidate 235

9.4.4 Conceptualizing the teacher candidate's choice of seven 237

9.5 A Counter-Example and Empirical Induction 238

9.6 Counter-Example as a Tool for Conceptual Development 240

9.7 Counter-Example in Explaining the Meaning of Negative Transfer 241

9.7.1 Inequalities 242

9.7.2 Counting matchsticks 244

9.8 Transition from Combinations without Repetitions to Combinations with Repetitions 246

9.9 Missing Fibonacci Numbers 248

Bibliography 251

Index 265