Diversifying Mathematics Teaching

The following sections are included:

• Preface
• Contents

The following sections are included:

• Introduction
• Questions as the Major Means of Learning Mathematics
• Answering Questions Both Procedurally and Conceptually
• Connecting Algorithmic Skills and Conceptual Understanding
• Developing Deep Understanding of Mathematics Through Making Conceptual Connections
• Teaching and Learning to Think Mathematically

The following sections are included:

• Introduction
• Rules of Sum and Product
• Tree Diagram and the Rule of Product
• Permutation of Letters in a Word
• Combinations without Repetitions
• Combinations with Repetitions

The following sections are included:

• Introduction
• Constructing a Triangle out of Straws
  • Reflection on the activity with straws
  • A real-life application of the triangle inequality
  • Modifying the S²AC² algorithm to enable linguistic coherency
  • How many triangles can be constructed?
  • Using multicolored straws
    • An equilateral triangle
    • An isosceles triangle with the base being the smaller side
    • An isosceles triangle with the base being the larger side
    • A scalene triangle
• Two Types of Representation as Means of Transition from Visual to Symbolic
• Signature Pedagogy of Elementary Mathematics Teacher Education
• Towards Rich Interpretations of Manipulative Representations
  • Manipulative representation as text
  • From “brothers” to Pascal’s triangle
• Learning to Move from One Type of Symbolism to Another and Back
• Perimeter and Area Using Square Tiles
• The Importance of Teacher Guidance in Using Manipulative Materials by Students
• Conceptualizing Base-Ten System Using Manipulative Materials
• Modeling as a Way of Creating Isomorphic Relationships

The following sections are included:

• Introduction
• W⁴S Principle and the Duality of Its Affordances
• W⁴S Principle in Teaching Primary School Mathematics
• Comparing Non-Unit Fractions Using Area Model
  • Comparing fractions being close to each other
  • Comparing fractions that are a unit fraction short of the whole
• From Comparison of Fractions to Arithmetical Operations Using Area Model
  • Fractions as part-whole and divisor-dividend models
  • The concept of common denominator
  • Reducing a fraction to the simplest form
  • Using unit fractions as benchmark fractions
  • Multiplying fractions using area model
• Dividing Fractions Using Area Model
  • Partition model for division supports contextualization
  • The importance of unit in solving word problems with fractions
  • The meaning of “invert and multiply” rule
• Ratio and Proportion
• Percent and Decimal as Alternative Representations of a Fraction
• Multiplying and Dividing Decimal Fractions

The following sections are included:

• Introduction
• Partition of Integers into Summands
• Activities with Towers Motivate Introduction of Algebraic Notation
• Ferrers-Young Diagrams
• Recursive Definition of P(n, m) Informed by Ferrers-Young Diagrams
• Making Mathematical Connections
• Recursive Definition of Q(n, m) Informed by Ferrers-Young Diagrams
• Connection to Triangular Numbers Opens a Window to a New Concept

The following sections are included:

• Introduction
• The Basic Task
• Background Information: Triangular and Trapezoidal Numbers
  • Activities
  • Solutions to the tasks
  • Trapezoidal numbers
• Conceptually Oriented Discussion of the Basic Problem
  • Grouping the sums by the number of addends
  • Grouping the sums by the first addend
  • Partitioning integers into the sums
• Discourse Motivated by Multiple Ways of Creating Sums of Consecutive Natural Numbers
  • Clarifying the meaning of the word special in the context of arithmetic
  • The first encounter with a special property of the sums of two addends
  • Moving from novice to expert practice in revealing special properties
  • Exploring the sums of four consecutive integers
• Proof of the Conjecture about Trapezoidal Numbers
• Sums in Pairs of Odds and Evens
  • Learning to generalize from special cases
  • Comparing triangles to trapezoids with the top row greater than two
• Mathematical Knowledge Used for Teaching Young Children
• How Many Trapezoidal Representations Does a Number Have and How Can One Find Them?

The following sections are included:

• Introduction
• Geoboard Explorations
• Towards a Double-Application Environment
• Guided Exploration on a Computational Geoboard
• Transition to a Spreadsheet
• Preparing Data for Empirical Induction
• Abstracting from Numbers to Equations Using First-Order Symbols
• Visual and Symbolic Deduction of Pick’s Formula
• Moving to a New Learning Site
• Measuring vs. Counting
• Encountering Limitation of the Environment
• From Particular to General through Visualization
• Communicating about Mathematics

The following sections are included:

• Introduction
• Basic Concepts and Tools of the Probability Strand
  • Randomness and sample space
  • A more complicated example of constructing a sample space
  • Different representations of a sample space
• Fractions as Tools in Measuring Chances
• Bernoulli Trials and the Law of Large Numbers
• A Problem of Chevalier De Méré
• A Modification of the Problem of De Méré
• Wagering for the Odds/Evens in a Game of Chance
• Paradoxes in the Theory of Probability
  • Bertrand’s Paradox Box problem
  • Monty Hall Dilemma
• Probabilistic Perspective on Partitioning Problems
  • A problem of tossing three dice
  • Unordered partitions of integers into unequal summands
• Experimental Probability
  • Experimental probability calls for a long series of observations
  • Comparing experimental and theoretical probabilities when tossing a fair coin
  • Calculating relative frequencies for the problems of De Méré
• Exploring Irreducibility of Fractions through the Lenses of Probability

The following sections are included:

• Introduction
• The Pedagogy of Using Counter-Examples
  • The role of linguistic constraints
  • A counter-example as a motivation for further learning
• Providing Explanation through Counter-Examples
• Constructing a Counter-Example: an Illustration
  • From modeling with fractions to algebraic generalization
  • From a counter-example to its conceptualization
  • A family of jumping fractions found by a teacher candidate
  • Conceptualizing the teacher candidate’s choice of seven
• A Counter-Example and Empirical Induction
• Counter-Example as a Tool for Conceptual Development
• Counter-Example in Explaining the Meaning of Negative Transfer
  • Inequalities
  • Counting matchsticks
• Transition from Combinations without Repetitions to Combinations with Repetitions
• Missing Fibonacci Numbers

The following sections are included:

• Bibliography
• Index