Learning and Teaching K-8 Mathematics (with "Understanding Children's Mathematical Thinking" VideoWorkshop CD-ROM), MyLabSchool Edition

An alternative to a comprehensive discussion of theory and research, Teaching K-8 Mathematics focuses on practical applications based on mathematical-based learning theories in K-8 classrooms. An exciting in-classroom video is integrated with the text material to demonstrate examples of actual student work in order to better understand children's mathematical thinking. A totally unique text and video for your math methods course!

Table of Contents:
Most chapters begin with “Thinking About the Mathematics” and conclude with “Let's Review,”  “Homework,”  “Suggested Readings,” and “References.” 1. The basics. Introduction. How We Learn. Theories. Features. Role of the Teacher. Algebra Connections. Middle School Connections. Before we get started. A bit about the NCTM and the NCTM Standards. Technology Connections–The Basics. Classroom Connections. Box 0.1: Classroom Connections (Video enhanced)–Understanding How Children Learn Math. Instruction. Models of Teaching. Problem Solving. Lesson plan components. Box 0.2: Classroom Connections–Providing Interactive Instruction. Questioning. Assessment. Box 0.3: Classroom Connections –Using Assessment of Many Forms. Curriculum. I. GEOMETRY AND SPATIAL THINKING. 2. Thinking about the Mathematics of Geometry and Spatial Thinking. Understanding The Mathematics--Mental Ideas v. Physical Models. Pulling Plane Knowledge from Solid Shapes. Accurate Descriptions. Analyzing a Circle. A Cylinder of a Circle? Concentrating on the Concepts--Relationships among Shapes. Examples and Non-Examples. Rigid Motions. Symmetry. Box 1.1: Classroom Connections–Finding Symmetry in Nature. Practicing Spatial Visualizations. Box 1.2: Middle School Connections–The Net of a Solid Shape. Grasping the Procedures–How to Build Shapes. Children Describe a Square. Drawing a Triangle. 3. How We Learn About Geometry and Spatial Thinking. Using Relevant Learning Theory–The Van Hiele Theory. Level 0 (Visual). Level 1 (Analysis). Level 2 (Informal Deduction). Level 3 (Formal Deduction). Level 4 (Rigor). Characteristics of the Theory. Promoting Development. Issues Specific to Spatial Visualization. Using Relevant Patterns–Extending Pattern Starters. Repeating v. Growing. Recognizing Deep Structure. Using Teaching Aids Appropriately–“Concrete, Pictorial, Symbolic.” Concrete Representations. Pictorial Representations. Symbolic Representations. Concrete Materials v. Level 0 “Thinking.” Box 1.3: Middle School Connections–“Manipulatives Are (Not) for Babies.” Types of Concrete Materials. Cubes. Pattern Blocks. Tangram. Assessment and the Use of Materials. A Chinese Legend. Explore Your Own Culture. Knitting. Box 1.4: Literature Connections–Developing Mathematical Imagery. 4. Role of the Teacher in Geometry and Spatial Thinking Lessons. Analyzing One of The Teaching Models–Discussion. Box 1.5: Technology Connections–Using Digital Technologies as Geometry Discussion Platforms. Cultural Connections–Creating a Mosaic. Highlights of a Lesson Plan Component–Creating Good Questions. Suggestions for Establishing Discourse–Importance of Knowing the Mathematics. Mrs. Hawthorne's Rhombus Lesson. Ms. Steffen's Rectangle Lesson. Comparing Quadrilaterals. The Next Question. Box 1.6: Middle School Connections–Know the Math Well Enough to Ask Questions. Assessment–To Make an Instructional Decision. Process of Assessment. Box 1.7: Classroom Connections–Using the van Hiele Theory to Guide Instruction. Box 1.8: Classroom Connections–Thinking Carefully About Assessment. Ideas for Structuring Class Time–Providing Reflection Time. Worthwile Tasks. Reflection. Advice for Professional Development–Get to Know the Mathematics in Different Ways. Algebra Connections to Geomentry. A Big Idea–Geometric Representations in Algebra. Algebraic Ideas for You to ExploreùSymmetry & Rigid Motions. Variable as a Functional Role. K-8 Algebra Connections--The Concept of Variable. II. WHOLE NUMBERS AND OPERATION. 5. Thinking About the Mathematics of Numbers and Operation. Understanding the Mathematics–Number and Operation. Number is an Amount. Numerals Are Symbols. Numbers Are Ordered. Addition. Subtraction. Multiplication. Division. Concentrating on concepts–Coordinating Counting, Operation, and Number Sense. Counting. Operation. Number Sense. Grasping the Procedures–Building Algorithms from Concepts. Addition. Subtraction. Multiplication. Box 2.1: Middle School Connections–Low-Stress Algorithms. Division. Box 2.2: Middle School Connections–Just Show Me How to Do It. 6. How We Learn About Number & Operation. Understanding Relevant Learning Theory–Number is a Mental Relationship. Number Ideas. Level 1. Level 2. Level 3. Knowing Number Words. Growth Happens in Chunks. Principles of Teaching Number. Principle 1 (Relatioships). Principle 2 (Quantification). Box 2.3: Classroom Connections–(Video Enhanced)–Finding Number Relationships. Principle 3 (Social Interaction). Number Sense. Understanding Relevant Learning Theory–Procedures Build from Solving Problems. Principle 3 (Social Interaction). Number Sense. Understanding Relevant Learning Theory–Procedures Build from Solving Problems Addition. Classifying Addition Problems. Subtraction. Classifying Subtraction Strategies. Learning Addition and Subtraction. Multiplication. Division. Using Relevant Patterns–Skip Counting Lays a Foundation. Calendar as Context. Concrete Representations. Number Line. Connecting Skip-Counting to Operations. Using Relevant Patterns–Figurate Numbers. Cultural Connections–Numeration Systems and Mexican Subtraction. The Dark Ages of Europe. Roman Numerals. Box 2.4: Literature Connections–Comparing Counting Books to Literary Works. Using Teaching Aids Appropriately–“Discrete v. Continuous Materials.” Concrete Representations. Pictorial Representations. Types of Concrete Materials Base-Ten Blocks. Cuisenaire Rods. Chips. 7. Role of the Teacher in Number and Operation Lessons. Analyzing one of The Teaching Models–Concept Attainment. Deductive. Inductive. Highlights of a Lesson Plan Component–Match Evaluation with Objective. Assessment–Evaluating Achievement (Concepts v. Procedures). Box 2.5: Classroom Connections–Dealing with Incorrect Comments. Box 2.6: Classroom Connections (Video Enhanced)–Orchestrating the Sharing of Strategies. Box 2.7: Technology Connections–Communication Tools. Suggestions for Establishing Discourse–Listening to Students to Guide Their Thinking. Ideas for Structuring Class Time–Using Daily Routines for Number Knowledge. Box 2.8: Middle School Connections–Daily Routines of Middle Graders. Advice for Professional Development -Compare Across Grade Levels. Box 2.9: Classroom Connections -Reacting to CGI for the First Time. Algebra Connections to Number and Operation. A big idea–Properties, Formulas, and Equivalence. Properties. Algebraic Ideas for You to Explore–Patterns. Variable as Functional. Variable as a Generalization. Variable as Standing for an Unknown. K-8 Algebra Connections–Arithmetic to Algebra. III. Measurement Ideas. 8. Thinking About the Mathematics of Measurement. Understanding The Mathematics–Geometry and Number. Continuous. Standard Unit. Concentrating on the Concepts–It begins with the Idea of “Size.” Non-standard Units. Size, Comparing and Conservation. Size is Summative. Standard Unit. Iteration of Units. Coverage. Tiling. Part–Whole. Box 3.1: Classroom Connections–Using a Magnified Inch. Coordinating Concepts. Grasping the Procedures–How to Measure Shapes. The Zero (Origin). Counting Spaces. Describing a Part. Formulas. 9. How We Learn About Measurement. Understanding Relevant Learning Theory. Compare and Order. Transitivity. Mixing Units. Visualizing. Box 3.2. Middle School Connections–Measurements from the Real World. Using van Hiele. More About the Geoboard. Phases of Learning. Phase 1–Inquiry/Information. Phase 2–Directed Orientation. Phase 3–Explication. Phase 4–Free Orientation. Phase 5–Integration. Ms. Hawthorne Measures a Bus. Finding Angle Measure. Using Measuring Devices. Using Relevant Patterns–Formulas, Skip-Counting, and Similarity. Developing General Formulas. Using Skip-Counting. Generalizing Ratios of Similar Shapes. Cultural Connections–Measuring Time and Visualizing Congruence. The Hopi Calendar. The Scrap Bag. Measuring Bushels. Box 3.3: Literature Connections–Illustrating Cultures and Solving Problems. Box 3.4: Classroom Connections–Using Literature to Pose Different Problems. Box 3.5: Classroom Connections–Using Literature that Already Describes Problems. Using Teaching Aids Appropriately–Measuring Devices and Discrete Manipulatives. Measuring Devices. Types of Concrete Materials. Square Tiles and Cubes. Geoboards. 10. Role of the Teacher in Measurement Lessons. Analyzing one of The Teaching Models–Presentation and Partnered Work. Box 3.6: Middle School Connections–Writing Their Own Problems. Highlights of a Lesson Plan Component–Content Differentiation: Acceleration v. Enrichment. Tiered Assignments. Compacted Curriculum. Modifying Roles. Guided Exploration. Box 3.7: Classroom Connections–Dealing with High-Ability Students. Box 3.8: Classroom Connections–Luke's Enrichment. Suggestions for Establishing Discourse–Importance of Sharing Thinking. Measuring a Puddle. Ideas for Structuring Class Time–Student Presentations and Assessment of the Presentation. Box 3.9: Technology Connections– Presenting Measurement Projects. Box 3.10: Classroom Connections (Video Enhanced Video clip 3.1)–Sharing a Good Plan. Box 3.11: Middle School Connections–Have Fun Along with Them and Create Wonder. Advice for Professional Development–Any Activity You Choose Should Match Your District Goals. Algebra Connections to Measurement. A big idea–Making Generalizations About Measurements. Algebraic Ideas For You to Explore–The Pythagorean Theorem. Variable as a Generalization and an Unknown. K-8 Algebra Connections–Relating the Area of a Triangle to Other Shapes. IV. Rational Numbers and Proportions. 11. Thinking About the Mathematics of Rational Numbers and Proportions. Understanding The Mathematics–What Exactly Are Rational Numbers? Concentrating on the Concepts–Connections Among Rational Number Genres. Sharing. Congruent Parts. Equivalent Fractions. Contextualized Situations. The whole Unit. Box 4.1: Classroom Connections–Illuminating Fraction Concepts. Grasping the Procedures–How to Operate and Move Between Genres. Whole Number Knowledge. Addition and Subtraction. Selecting Contexts. Multiplication. Division. Finding Equivalent Fractions. Revisiting Procedures. 12. How We Learn About Rational Numbers and Proportions. Understanding Relevant Learning Theory. Personal Knowledge. Representations and Intuition. Technical Symbols. Building Knowledge. Designing Activities. Begin with Contexts. Share Strategies. Using Tools. Building Knowledge. Partitioning Whole Units. Ms. Hawthorne Shares Cookies. Revisit Ms. Hawthorne. Oddness. Using Relevant Patterns–Skip-Counting in Ratio Tables. Ratio Tables. Cultural Connections– Using and Making Unexpected Ratios. Counting Innings. Drumming Ratios. Using Teaching Aids Appropriately–Discrete v. Continuous. Continuous. Discrete. Role of Context. Types of Concrete Materials Fraction Circles. Paper Folding. Chips. Base-Ten Blocks. Using Metrics. Box 4.2–Literature Connections–Letting Characters Physically Illustrate Proportions. 13. Role of the Teacher in Rational Number and Proportions Lessons. Analyzing One of The Teaching Models–Problem Solving (Discovery). Ms. Steffen's Experiment. Asking Good Questions. Solving Division Problems. Box 4.3: Technology Connections–Using Digital Images to Make Mathematics Come Alive. Box 4.4: Middle School Connections–Problem Solving Can Ground Algorithms. Assessment–To Monitor Progress Toward a Goal. Box 4.5: Middle School Connections–Select Homework Thoughtfully. Box 4.6: Classroom Connections (Video Enhanced Video clip 4.1) –Assessing Problem Solutions Using a Rubric. Highlights of a Lesson Plan Component–Importance of Context (Lesson Set-up). Evaluating Contexts. Suggestions for Discussion–Listening for Benchmark Numbers (0, 1/2, 1) Ideas for Structuring Class Time–To Create Shared, Rich Experiences Using Proportions. Box 4.7–Middle School Connections–Appropriate Tasks Build Proportional Thinking. Advice for Professional Development–Fraction Scavenger Hunt. Algebra Connections to Rational Numbers and Proportional Thinking. A Big Idea–Proportional Thinking. Algebraic Ideas For You to Explore–Proportional Thinking. K-8 Algebra Connections–Recognizing General Strategies: The Case of Proportions. V. Data Analysis. 14. Thinking About the Mathematics of Data Analysis. Understanding The Mathematics–Statistics and Probability. Statistics. Sample v. Population. General Statements. Types of Data. Averages. Basic variance. Box 5.1: (Video Enhanced Video clip 5.1)–Doing Data Analysis With Physical Materials. Prediction v. Description. Prediction v. Inference. Concentrating on the Concepts–What Generalizations Really Mean. Statistics. Data. Sample. Generalization. The Need for Probability. Probability. Experimental Probability. Theoretical Probability. Likelihood. Grasping the Procedures–Organizing Data and Finding Probabilities. Data Organization. Box 5.2: Classroom Connections–Teaching Students to Move From Data to Representations of Data. Making Picture Representations. Ms. Hawthorne's Weather Data. Box 5.3: Classroom Connections (Video Enhanced Video clip 5.2)–Creating Picture Graphs With Physical Objects. Probability. Fairness. Expected Value. Odds. 15. How We Learn About Data Analysis. Understanding Relevant Learning Frameworks–Friel, Curcio and Bright. (Statistics), and Jones Langrall, Thornton and Mogill (Probability). Underlying Facts. The Importance of Student-Generated Data. Statistics Framework. Organizing Concepts. Making Interpretations Based on a Sample. Probability Framework. Monitoring Development Through Likelihood. Monitoring Development Through Randomness. Comparing Probabilities. Monitoring Development Through Sampling. Confronting Enigmas. Box 5.4 Classroom Connections–Guiding Students' Uses of Vocabulary. Using Relevant Patterns–Organized Counting (Pascal's Triangle). Cultural Connections–Organizing Data and Simulating Game Situations. Organizing Astronomical Information. Analyzing Dice Games. Great Grandmother's Long Walk. Box 5.5: Literature Connections–Using Stories About Chance and Organized Counting. Using Teaching Aids Appropriately–“Using Student Generated Data.” Types of Concrete Materials. Two-Color Counters. Dice. Spinners. 16. Role of the Teacher in Data Analysis Lessons. Analyzing The Teaching Models–Presentation, Problem Solving, Partnered Learning, Discussion and Concept attainment. Box 5.6: Technology Connections–The Power of the Spreadsheet. Highlights of A Lesson Plan Component–Matching Materials to Objective. Creating the Situation. Selecting Supportive Materials. Guiding the Data Analysis. Assessment–To Guide Instructional Decisions. Suggestions for Establishing Discourse–Using Student Comments to Analyze Teaching. Ideas for Structuring Class Time–To Defend View Points. Box 5.7: Classroom Connections–Using Sports to Create Interest. Box 5.8: Middle School Connections–Wise Consumers of Statistical Knowledge. Advice for Professional Development–(Contact Your Local Education-Support Agency to Find the Technology Available in Your District.) Algebra Connections to Data Analysis. A Big Idea–Using Models to Represent Quantitative Relationships. Box 5.9: Middle School Connections–Algebra as Social Justice. Algebraic Ideas for You to Explore–Linear Regression. Algebra in Unusual Places. K-8 Algebra Connections–A Grain of Rice.