This textbook is for prospective teachers of middle school mathematics. It reflects on the authors’ experience in offering various mathematics education courses to prospective teachers in the US and Canada. In particular, the content can support one or more of 24-semester-hour courses recommended by the Conference Board of the Mathematical Sciences (2012) for the mathematical preparation of middle school teachers. The textbook integrates grade-appropriate content on all major topics in the middle school mathematics curriculum with international recommendations for teaching the content, making it relevant for a global readership.
The textbook emphasizes the inherent connections between mathematics and real life, since many mathematical concepts and procedures stem from common sense, something that schoolchildren intuitively possess. This focus on teaching formal mathematics with reference to real life and common sense is essential to its pedagogical approach.
In addition, the textbook stresses the importance of being able to use technology as an exploratory tool, and being familiar with its strengths and weaknesses. In keeping with this emphasis on the use of technology, both physical (manipulatives) and digital (commonly available educational software), it also explores e.g. the use of computer graphing software for digital fabrication. In closing, the textbook addresses the issue of creativity as a crucial aspect of education in the digital age in general, and in mathematics education in particular.
Author(s): Sergei Abramovich, Michael L. Connell
Series: Springer Texts in Education
Publisher: Springer
Year: 2021
Language: English
Pages: 454
City: Cham
Preface
Contents
1 Teaching Middle School Mathematics: Standards, Recommendations and Teacher Candidates’ Perspectives
1.1 Introduction
1.2 The Importance of Content Knowledge
1.3 The Importance of Pedagogical Content Knowledge
1.4 The Need for Teachers with ‘Deep Understanding’ of Mathematics
1.5 Learning to Address Misconceptions Through Mathematical Connections
1.6 Making Conceptual Connections
1.7 Using Technology in the Classroom
1.8 Conclusion
2 Modeling Mathematics in the Digital Era
2.1 Introduction
2.2 Modeling Mathematics and Mathematical Modeling
2.2.1 Modeling Mathematics: Selecting a Referent or Object to Act upon
2.2.1.1 Initial Modeling Considerations
2.2.1.2 Cakes and Pizzas
2.3 Modeling Mathematics
2.3.1 Modeling Equivalent Surface Area Functions—Stamping Functions
2.3.2 Modeling Correlations—Burning the Candle
2.3.3 Modeling Systems of Equations—The Flagpole Factory
2.3.4 Data-Tables Revisited
2.4 The Problem Solving Board
2.5 Conclusion
2.6 Activity Set
3 Reasoning and Proof
3.1 Introduction
3.2 Elementary Reasoning at the Concrete Operational Stage
3.3 Supporting Plausible Mathematical Reasoning and Formal Proofs with CARE
3.3.1 Commonplaces
3.3.2 Authority
3.3.3 Reason
3.3.4 Experience
3.4 CARE and Rotating Squares
3.5 Basic Elements of Plausible Reasoning
3.6 Basic Elements of Propositional Logic
3.6.1 Conjunction, Disjunction and Implication
3.6.2 Application of the Logical Conjunction Truth Values in Spreadsheet Modeling
3.6.3 Application of the Logical Disjunction Truth Values in Spreadsheet Programming
3.6.4 De Morgan Rules
3.6.5 Converse, Inverse and Contrapositive Statements
3.6.6 Modus Ponens: [(p ⇒ q) ∧ p] ⇒ q
3.6.7 Modus Tollens: [(p ⇒ q) ∧ \negq] ⇒ \negp
3.7 Conclusion
3.8 Activity Set
4 Modeling Mathematics with Fractions
4.1 Introduction
4.2 Transition from Arithmetic of Integers to Arithmetic of Fractions
4.3 The Meaning of Multiplying and Dividing Fractions
4.3.1 Multiplying Two Fractions
4.3.1.1 Multiplying Two Proper Fractions
4.3.1.2 Multiplying Two Improper Fractions
4.3.1.3 Multiplying Proper and Improper Fractions
4.3.1.4 Reflection on Three Cases Discussed in Sects. 4.3.1.1–4.3.1.3
4.3.2 Dividing Fractions
4.3.2.1 Dividing Whole Numbers as a Window into Fractions
4.3.2.2 Dividing Two Fractions as Finding a Missing Factor
4.3.2.3 Solving Word Problems Using Fractions
4.4 Conceptual Meaning of the Invert and Multiply Rule: From Integers to Fractions
4.4.1 The Case of Dividing Integers
4.4.2 The Case of Dividing Fractions
4.4.3 The Invert and Multiply Rule as a Change of Unit
4.5 Unit Fractions
4.5.1 Unit Fractions as Benchmark Fractions
4.5.2 Representation of a Unit Fraction as a Sum of Two Like Fractions
4.5.3 Representation of 1/2 as a Sum of Three Different Unit Fractions
4.5.4 Egyptian Fractions and the Greedy Algorithm
4.6 From Long Division to Decimal Representation of Fractions
4.7 Rational Numbers in Non-Decimal Bases
4.7.1 Conversion of Integers into Non-Decimal Bases
4.7.2 Conversion of Common Fractions into Non-Decimal Bases
4.7.3 Visual Representation of Conversion of Common Fractions in Different Bases
4.8 Integer Sequences as Sources of Fractions
4.9 Continued Fractions
4.9.1 Euclidean Algorithm
4.9.2 Continued Fraction Representation of Common Fractions
4.10 Connecting Fractions to Quadratic Equations
4.11 Conclusion
4.12 Activity Set
5 Decimal and Percent Representation of Rational Numbers
5.1 Introduction
5.2 Developing Numerical Fluency
5.3 Transitioning from Numerical Fluency to the Base-Ten Notational System
5.4 Decimals and Decimal Fractions
5.5 Decimals and Common Fractions
5.6 Decimal and Percent
5.7 Repeating Decimals
5.8 From Representation to Operations
5.8.1 Comparison
5.8.2 Addition and Subtraction
5.8.3 Multiplication and Division
5.9 Conclusion
5.10 Activity Set
6 Ratio and Proportion
6.1 Introduction
6.2 Ratios
6.2.1 Part-Part Ratios
6.2.2 Part-Whole Ratios
6.2.3 Reflecting Quotients as Ratios
6.2.4 Reflecting Rates as Ratios
6.3 Proportions
6.3.1 Scaling
6.3.2 Proportions and Conversions
6.3.3 The Golden Ratio
6.3.4 The Silver Ratio
6.3.5 Metallic Means
6.4 Conclusion
6.5 Activity Set
7 Geometry and Measurement
7.1 Introduction
7.2 Geometry
7.2.1 Geometric Thinking and Van Hiele Levels
7.2.1.1 Level 0—Visualization
7.2.1.2 Level 1—Analysis
7.2.1.3 Level 2—Informal Deduction (Relationships)
7.2.1.4 Level 3—Formal Deduction
7.2.1.5 Level 4—Rigor
7.2.1.6 Summary
7.2.2 Developing Geometric Vocabulary
7.2.3 Deductive Proofs
7.2.4 Transformations
7.2.5 Tessellations
7.2.6 Parallel Lines
7.2.7 Pythagorean Theorem, Pythagorean Triples, and the Distance Formula
7.2.8 Triangles and Introductory Trigonometry
7.2.9 Quadrilaterals
7.3 Measurement
7.3.1 Application of Scaling Factors in Measurement
7.3.2 Circumference, Area of a Circle, and Π
7.4 Conclusion
7.5 Activity Set
8 Combinatorics
8.1 Introduction
ch8Chap8
8.3 Tree Diagrams
8.4 Permutation of Letters in a Word
8.5 Combinations Without Repetition
8.5.1 Developing Counting Techniques
8.5.2 Counting Rectangles on a Checkerboard
8.5.3 Why Are There 256 Representations of 9 Through a Sum of Ordered Integers?
8.6 Combinations with Repetition
8.7 The Sum of Powers of Integers as a Combinatorial Tool
8.8 Conclusion
8.9 Activity Set
9 Conceptual Approach to the Ideas of Middle School Algebra
9.1 Introduction
9.2 From Early Algebra to Graphing and Problem Posing with Technology
9.2.1 Three Problems from Different Grade Levels
9.2.2 Problem Reformulation
9.2.3 Solution of Problem 9.5 Using Kid Pix
9.2.4 Opening a Window to Traditional School Algebra
9.3 Different Types of Generalization in School Algebra
9.3.1 Algebraic Generalization of the First Kind
9.3.2 Algebraic Generalization of the Second Kind
9.3.3 From Summation of Odd Numbers to Summation of Squares
9.3.4 The Sum of Cubes in the Multiplication Table
9.4 Conceptual Generalizations of the First Kind
9.4.1 Patterns of Elimination Leading to Polygonal Numbers
9.4.2 Developing Subsequences of Fibonacci Numbers
9.5 Parameterization as Conceptual Generalization of the Second Kind
9.6 Insufficiency of Generalization by Empirical Induction
9.7 Numeric Tables as a Context for Developing Algebraic Skills
9.7.1 Finding the Sum of All Numbers in the n \times n Addition Table
9.7.2 Finding the Sum of All Numbers in the n \times n Multiplication Table
9.7.3 Exploring Divisibility of Numbers in Addition and Multiplication Tables
9.7.3.1 Even Numbers in Addition Tables
9.7.3.2 Even Numbers in Multiplication Tables
9.7.3.3 Multiples of Three in Addition Tables
9.7.3.4 Multiples of Three in Multiplication Tables
9.8 Solving Algebraic Word Problems Through Conceptual Strategies/Shortcuts
9.9 Inequalities
9.9.1 Algebraic Inequalities as Tools of Digital Fabrication
9.9.2 Digitally Fabricating Points, Segments and Arcs
9.9.2.1 Fabricating a Point
9.9.2.2 Fabricating a Segment
9.9.2.3 Fabricating an Arc
9.9.2.4 Fabricating Triangles
9.9.3 Arithmetic Mean-Geometric Mean Inequality
9.10 Algebraic Recreations
9.11 Conclusion
9.12 Activity Set
10 Patterns and Functions
10.1 Introduction
10.2 From Patterns to Functions
10.3 From Visual to Symbolic
10.4 Patterns Structured by Colors
10.4.1 From Colors to Functions
10.4.2 From Empirical Conjectures to Mathematical Induction Proof
10.4.3 Two-Color Pattern Guided by Consecutive Odd Numbers
10.4.3.1 The Case of the Last R
10.4.3.2 The Case of the First R
10.4.3.3 The Case of the Last B
10.4.3.4 The Case of the First B
10.4.3.5 Adressing the Query: Which Color Does the 1225th Position Have?
10.4.4 More Patterns Defined by an Arithmetic Sequence
10.5 Finding Patterns Formed by Functions
10.5.1 Developing Formula for {\usertwo f}_{{\bf {1}}}{{\bf (}}{\usertwo n}{{\bf ,}}\,{\usertwo m}{{\bf )}}
10.5.2 The Cases of the Position of the First R and the First/Last B
10.5.3 Alternative Verification of the Obtained Results
10.5.3.1 The Case of the Position of the First R
10.5.3.2 The Case of the Position of the Last B
10.5.3.3 The Case of the Position of the First B
10.6 The Case of Three Colors
10.7 The Case of p Colors
10.7.1 Pattern Guided by Arithmetic Sequence with Difference One
10.7.2 Pattern Guided by Arithmetic Sequence with Difference Two
10.7.3 Pattern Guided by Arithmetic Sequence with Difference Three
10.7.4 Pattern Guided by Arithmetic Sequence with Difference Four
10.7.5 Pattern Guided by Arithmetic Sequence with Difference m
10.8 From Generaization to Computerization
10.9 Conclusion
10.10 Activity Set
11 Financial Literacy and Blockchain
11.1 Introduction
11.2 Financial Literacy
11.2.1 Earning
11.2.2 Budgeting
11.2.2.1 First Steps Toward a Budget
11.2.2.2 A 50-30-20 Budgeting Strategy
11.2.2.3 Budgeting to Buy a Car
11.2.3 Checking
11.2.4 Interest
11.2.4.1 Simple Interest
11.2.4.2 Compound Interest
11.2.4.3 Additional Applications of the Compound Interest Formula
11.2.5 Credit and Borrowing
11.2.6 Mortgages
11.2.7 Savings
11.2.8 Credit Reports
11.2.9 Planning and Paying for College
11.2.10 Insurance
11.2.10.1 Life Insurance
11.2.10.2 Health Insurance
11.2.10.3 Long-Term Disability
11.2.11 Investments
11.2.12 Common Financial Pitfalls
11.3 Blockchain
11.3.1 Blockchain Basics
11.3.2 Blockchain Security Concerns
11.3.3 Consensus Mechanisms and Incentivation
11.3.4 Cryptocurrency and Other Applications
11.3.5 Advantages of Blockchain
11.3.6 Disadvantages of Blockchain
11.4 Conclusion
11.5 Activity Set
12 Probability and Statistical Data Analysis
12.1 Introduction
12.2 Experiments with Equally Likely Outcomes
12.3 Randomness and Sample Space
12.4 Different Representations of a Sample Space
12.5 Fractions as Tools in Measuring Chances
12.6 Explorations with Addition and Multiplication Tables
12.6.1 Computational Experiments with Pairs of Random Integers
12.6.2 Selecting Even Numbers in Addition Tables
12.6.3 Selecting Even Numbers in Multiplication Tables
12.6.4 Selecting Multiples of Three in Addition Tables
12.6.5 Selecting Multiples of Three in Multiplication Tables
12.6.6 Theoretical Probabilities and Their Monotone Convergence
12.7 Bernoulli Trials and the Law of Large Numbers
12.8 Classic Problems that Motivated Theoretical Development
12.9 A Modification of the Problem of De Méré
12.10 Experimental Probability Requires a Long Series of Observations
12.11 Probability Experiments with Spinners and Palindromes
12.12 Monty Hall Dilemma as a Paradox in the Theory of Probability
12.13 Transition from Probability Theory to Statistical Data Analysis
12.14 Graphic Representations of Numeric Data
12.15 Measures of Central Tendency
12.16 Measures of Dispersion
12.17 The Binomial Distribution and the Normal Curve
12.18 The z-Score and the Standard Deviation
12.19 Bivariate Analysis
12.20 Conclusion
12.21 Activity Set
Appendix
Chapter 9: Conceptual Approach to the Ideas of Middle School Algebra
Chapter 10: Patterns and Functions
Chapter 12: Probability and Statistical Data Analysis
Bibliography