Table Of ContentTeaching to the Math Common Core State Standards
Teaching to the Math Common Core State Standards
Focus on Kindergarten to Grade 5
F. D. Rivera
Department of Mathematics and Statistics
San Jose State University, California, USA
A C.I.P. record for this book is available from the Library of Congress.
ISBN: 978-94-6209-501-4 (paperback)
ISBN: 978-94-6209-502-1 (hardback)
ISBN: 978-94-6209-503-8 (e-book)
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CONTENTS
1. Dear Elementary Majors and Practicing and Beginning
Elementary Teachers: An Introduction 1
1.1 A Blended Multisourced Approach to Learning to Teach Mathematics 4
1.2 Overview of the Remaining Chapters 5
2. Getting to Know the Common Core State Standards
for Mathematical Practice 7
2.1 Content Activity 1: Generating Addition Facts 7
2.2 The Eight Common Core State Standards for
Mathematical Practice 11
2.3 Content Activity 2: Building a Hexagon Flower
Garden Design 11
2.4 Problem Solving Contexts in Elementary School Mathematics 16
2.4.1 Content Activity 3: Different Types of Problems in Elementary
School Mathematics 16
2.5 Representations in Elementary School Mathematics 17
2.5.1 Content Activity 4: A Kindergarten Subtraction Task 17
2.5.2 Content Activity 5: Representing Whole
Numbers in Second Grade 20
2.6 Connections in Elementary School Mathematics 20
2.6.1 Content Activity 6: Repeating Patterns 21
2.7 Reasoning and Proof in Elementary School Mathematics 22
2.7.1 Content Activity 7: Division Problems in Third Grade 22
2.7.2 Activity 8: Even and Odd Numbers in Second Grade 23
2.8 Communication in Elementary School Mathematics 24
2.8.1 Content Activity 9: Addition and Subtraction
Strategies from First to Second Grade 25
2.8.2 Content Activity 10: Generating Addition and
Subtraction Facts in First Grade 27
2.9 Doing Mathematics with an Eye on theContent-Practice
Standards of the CCSSM 28
3. Counting and Cardinality in Kindergarten 31
3.1 Subitizing: Laying the Foundation for Early Multiplicative Thinking 31
3.2 Counting and Writing Numbers and Number Words: Building on
Rhythmic Structures 33
3.3 Comparing Quantities 35
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CONTENTS
3.4 Mapping the Content Standards with the Practice Standards 37
3.5 Developing a Content Standard Progression Table
for the Counting and Cardinality Domain 37
4. Numbers and Operations (Part I) 41
4.1 Place Value in Base Ten: The Basic Mathematical Structure
of Whole Numbers and Decimal Numbers in the CCSSM 42
4.2 Representing Whole Numbers in Base Ten from K to Grade 4 43
4.3 Representing Decimal Numbers in Grade 5 43
4.4 Comparing and Ordering Whole Numbers and
Decimal Numbers from Grades 1 to 5 44
4.5 Rounding Whole Numbers from Grades 3 to 5 45
4.6 Mapping the Content Standards with the Practice Standards 45
4.7 Developing a Content Standard Progression Table for Part I of the
Numbers and Operations in Base Ten Domain 46
5. Operations and Algebraic Thinking 47
5.1 Algebraic Thinking from Kindergarten to Grade 5 47
5.2 The Four Fundamental Operations as General
Methods for Combining and Generating More Numbers 52
5.2.1 Addition 52
5.2.2 Subtraction 54
5.2.3 Multiplication 54
5.2.4 Division 56
5.3 Forming Equivalent Expressions and Understanding
the General Meaning of the Equal Sign 57
5.4 Instantiating Nonsymbolic Algebra Through
Problem Solving in Arithmetical Activity 59
5.5 Pattern Generalization as Instantiations of
Nonsymbolic and Symbolic Algebra 61
5.6 Mapping the Content Standards with the Practice Standards 66
5.7 Developing a Content Standard Progression
Table for the Operations and Algebraic Thinking Domain 66
6. Numbers and Operations in Base Ten (Part II) 67
6.1 Analytical Subitizing in Kindergarten and Sticks
and Circles or Labeled Circles in Grade 1 67
6.2 Adding Whole Numbers in Base Ten from Grades 1 to 4 68
6.3 Subtracting Whole Numbers in Base Ten from Grades 1 to 4 72
6.4 Multiplying Whole Numbers in Base Ten from Grades 3 to 5 76
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CONTENTS
6.5 Dividing Whole Numbers in Base Ten from Grades 3 to 4 79
6.6 Adding, Subtracting, Multiplying, and Dividing Decimal
Numbers in Grade 5 83
6.7 Mapping the Content Standards with the Practice Standards 87
6.8 Developing a Content Standard Progression
Table for the Operations and Number Thinking Part II Domain 87
7. Numbers and Operations – Fractions 89
7.1 Equal Partitioning of Whole Shapes in Halves and Fourths in Grade 1 90
7.2 Equal Partitioning of Whole Shapes in Thirds and
Constructing Equivalent Shares in Grade 2 92
7.3 Unit Fractions and Equivalent Fractions with Denominators
2, 3, 4, 6, and 8 as Equal Shares and Numbers on a Number
Line and Whole Numbers as Improper Fractions in Grade 3 95
7.4 Numerical Equivalence of Fractions Involving Denominators
2, 3, 4, 5, 6, 8, 10, 12, and 100, All Improper Fractions,
Fraction Addition and Subtraction with Like Denominators,
Simple Fraction Multiplication, and Decimal Fraction
Addition and Comparison in Grade 4 99
7.5 Fraction Addition, Subtraction, Multiplication,
and Simple Division in Grade 5 104
7.6 Mapping the Content Standards with the Practice Standards 109
7.7 Developing a Content Standard Progression
Table for the Number and Operations – Fractions Domain 109
8. Geometry 111
8.1 Recognizing and Composing Shapes in Kindergarten and Grade 1 111
8.2 Defi ning and Reasoning about Shapes in Grades 2, 3, and 5 114
8.3 Lines and Angles in 2D Shapes in Grade 4 114
8.4 The Coordinate Plane in Grade 5 116
8.5 Mapping the Content Standards with the Practice Standards 117
8.6 Developing a Content Standard Progression
Table for Geometry Domain 117
9. Measurement and Data 119
9.1 Measuring from Kindergarten to Grade 5 119
9.1.1 Qualitative Measurements in Kindergarten 119
9.1.2 Measuring Lengths in Grades 1 and 2 120
9.1.3 Finding Perimeters and Measuring Lengths with
Halves and Fourths of an Inch, Areas, and Estimated Liquid
Volumes and Masses of Objects in Standard Units in Grade 3 121
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CONTENTS
9.1.4 Measuring Angles in Grade 4 123
9.1.5 Measuring Volumes of Cubes and Other
Right Rectangular Prisms in Grade 5 124
9.1.6 Conversion Problems in Grades 4 and 5 125
9.2 Telling and Writing Time in Grades 2 and 3 125
9.3 Money in Grades 2 and 4 126
9.4 Classifying, Representing, and Interpreting Data
from Kindergarten to Grade 5 127
9.4.1 Forming Categories Involving Discrete
Objects in Kindergarten 127
9.4.2 Structuring Data from Kindergarten to Grade 5 127
9.5 Mapping the Content Standards with the Practice Standards 127
9.6 Developing a Content Standard Progression Table
for the Measurement and Data Domain 128
10. Content-Practice Assessment 129
10.1 General Conceptions Surrounding Assessment 131
10.2 Norm- and Criterion-Referenced Tests 135
10.3 Principles of Effective Classroom Assessments 135
10.4 Formative Assessments 136
10.5 Summative Assessments 136
10.5.1 General Conceptions Surrounding Summative Assessments 136
10.5.2 The Smarter Balanced Assessment (SBA) 137
10.6 Projects and Portfolios as Alternative Summative Assessments 140
10.7 Math Journals and Lesson Investigations as Alternative Formative
Assessments 141
10.8 An Assessment Project 142
11. Content-Practice Learning 145
11.1 Defi ning Learning 145
11.2 Changing Views of Learning and Their Effects in the Elementary
Mathematics Curriculum 146
11.3 Math Wars: Debating About What and How Students Should Learn
Mathematics 147
11.4 Understanding Piagetian and Vygotskian Views of Learning in
Mathematics and Finding a Way Out of Extreme
Views of Learning 147
11.5 Learning Progressions in School Mathematics 148
11.6 Learning from Neuroscience 149
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CONTENTS
12. Content-Practice Teaching 151
12.1 Describing (Good) Teaching 151
12.2 Teaching Models in Elementary School Mathematics Settings 154
12.2.1 E-I-S-Driven Teaching 154
12.2.2 C-R-A Sequenced Teaching 154
12.2.3 Van Hiele Sequenced Teaching 155
12.2.3 Culturally-Relevant Teaching 155
12.2.4 SDAIE-Driven Teaching 156
12.2.5 Differentiated Instruction 156
12.2.6 Flip Teaching 157
12.3 Teaching with Concrete and Virtual Manipulatives,
Computer-Based Learning Tools, and Video Games and Apps 157
12.4 Teaching Mathematics with Guide Questions 158
12.5 Content-Practice Unit Planning 159
12.6 Content-Practice Lesson Planning 165
12.7 A Planning Project 167
13. Orchestrating a Content-Practice Driven Math Classroom 175
13.1 Persistence and Struggles in Math Classrooms 175
13.2 Fostering Persistent Content-Practice Learners 176
13.3 Developing Effective Collaborative Content-Practice
Learning Through Complex Instruction 177
13.4 Other Collaborative Content-Practice Learning Techniques 180
13.5 Developing an Optimal Content-Practice Learning
Environment for All Elementary Students 180
13.6 Dealing with Potential Behavior Problems 181
13.7 Assigning Homework, Grading and Testing, and Seating 183
13.8 A Classroom Management Plan Project 185
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