Mathematics Methods for Elementary and Middle School Teachers

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Chapter 1: Mathematics Education Today (and into the Future).How Do You Think about Mathematics?What's Your Attitude?The Standards Movement in Mathematics Education.The NCTM Principles and Standards (2000.)The Guiding Principles.Equity Principle.Mathematics Curriculum Principle.Teaching Principle.Learning Principle.Assessment Principle.Technology Principle.Recent Changes.The New Curriculum Process Standard Called Representation.Curriculum Standards Condensed in Number.Grade Levels Divided into Smaller Clusters.More on the NCTM Standards.Support for Teaching the NCTM Standards.Curricula Projects.Responses to the Call for Standards.National Council for the Accreditation of T eacher Education.TIMSS.Opposition to the Standards Movement.Where's Mathematics Learning Today?Learning Styles and Children's Preexisting Ma thematical Knowledge.Active Learning.Assessment.The Evaluation Efforts of the NCTM.Assessment of Teachers: Praxisa Exams.Technology.The Internet and the World Wide Web.Calculators.Other Current Trends.What's Next?Summary.Praxis II(t)-Style Questions.Multiple-Choice Format.Constructed-Response Format.Integrating Technology.Video Vignettes.Observations of Good Teaching with the Standards.Internet.WebQuest. Design Your Own Virtual School Website.Chapter 2: Culturally Relevant Mathematics.Multiculturalism: The Giftedness of Many Cultures in Mathematics.Mathematics Simulations Provide Understanding.People in Other Cultures Today.People in Ancient Times.A Numeration System from Ancient Africa.A Numeration System from Ancient Mayan Culture.A New "Look" to Mathematics Provides Understanding.Generalized Procedures Are More Important Than Step-by-Step Rules.The Power of Storytelling.Equity: The Right of All People to Acquire Mathematical Power.Mathematics Seen in Real-World Contexts.Poverty and Its Implication for Mathematics.Poverty and Its Impact on Technology.Substantial Conversation (Oral Discourse). "If-Then" Statements.Variable Reversal Errors.Scaffolding and Modeling: An Answer to Language Problems.Student-Centered Classrooms That Focus on Higher-Order Thinking Skills and Problem Solving.Modeling with the Right Resources.Resources for Contributions of African Americans.Resources for Contributions of Native Americans.Resources for Contributions of Latino Americans.Resources for Contributions of Asian Americans.Resources for Contributions of Women in Mathematics.In Celebration of All Mathematicians.Learning Styles.Summary.Praxis II(t)-Style Questions.Multiple-Choice Format.Constructed-Response Format.Integrating Technology.Internet.WebQuests.Accomplishments of African Americans-Triumph in Many Fields.Animal Tracks (a Native American Approach).Website.Children's Literature.African American.Asian American.Hispanic American.Native American.Chapter 3: The Development of Mathematical Proficiency: Using Learning Research, Assessment, and Effective Instruction.How Students Learn Mathematics-The Brain Research Connection.Learning Is Action (The Brain as Resilient, Growing More Complex): The Cognitivist and Constructivist Theories.Jean Piaget.Piagetian Theory in Practice.Lev Vygotsky.Vygotsky's Theory in Practice.The Current Constructivists.Learning Is Process (The Brain as Highly Diversified, Must Use It or Lose It): The Information Processing Theories.Sensory Pathways.Students with Special Needs.Thought Processing Styles.Perceptual Learning Styles.Gardner's Multiple Intelligences.Many Differing Views.The Use of Assessment in Mathematics Learning.Why Assess?What Is Assessed?What Is Meant by "Performance-Based Assessment"?Who Assesses?International Studies.National Studies: National Assessment of Educational Progress.State Assessment Programs.Students with Special Needs.Local School Districts.Benchmarking to Reach Standards.When and Where to Assess?How to Assess?Questioning.Elaborating Techniques.Student-to-Student Discourse.Observations.Interviews.Diagnostic Interview Technique.Discouraging the "Parroting" Responses of Students.Asking Students to Justify Their Answers.Keeping a Student Elaborating on a Procedure or Reason.Student Self-Assessment.Math Journals and Diaries.Students with Special Needs.Student Portfolios.Creating and Scoring Performance-Based Assessments.Creating One's Own Assessment Tasks.Using Scoring Guides (Rubrics).Instruction to Develop Mathematical Proficiency. Lesson Plans to Develop Mathematical Proficiency.A Research Base for Quality Lesson Planning.Lesson Study Using the Video Vignettes: Actual Teachers in Classrooms.Homework in Light of the New Approaches to Instruction.Appropriate Use of Technology.Evaluation of Student Work to Develop Mathematical Proficiency.Summary.Praxis II(t)-Style Questions.Multiple-Choice Format.Constructed-Response Format.Integrating Technology.Internet Websites.Chapter 4: Middle School Mathematics.Middle School Students.Motivation.Attribution Theory.Goal Theory.Self-Determination Theory.Intrinsic and Extrinsic Motivation. The Teacher's Role in Motivating Middle School Students.Other Considerations.Children's Literature as a Motivational Tool.Special Students.Students At-Risk.Students with Special Needs.Students with Learning Disabilities.Gifted and Talented Students.Students Who Are Underserved.Middle School Teachers.Middle School Mathematics Curricula. Three Views.NSF-Funded Curricula Development.Connected Mathematics Project (CMP).Mathematics in Context (MiC).MathScape.Math Thematics.Pathways to Algebra and Geometry.Integrated Mathematics, Science, and Technology (IMaST).Criteria for Standards-Based Mathematics Curricula.The Role of Technology.Graphing Calculators.Computer Software.Summary.Praxis II(t)-Style Questions.Multiple-Choice Format.Constructed-Response Format.Integrating Technology.Internet.WebQuest.Prove the Pythagorean Theorem!Websites.Geometer's Sketchpad.Graphing Calculator Activities.Spreadsheet.Planetary Adjustments.Children's Literature.Chapter 5: Problem Solving.The Standards in Problem Solving.The Complexity of Problem Solving.Problem Solving as an Educational Method.Problem Solving as a Broad Definition.Problem Solving as an Environment in and of Itself.The Methods, Kinds, and Processes of Problem Solving.Methods of Interpreting Problem Solving.Problem Solving as a Goal.Problem Solving as a Process.Problem Solving as a Skill.Kinds of Problem Solving.Problem Solving for Open-Ended Questions.Problem Solving for Discovery Questions.Problem Solving for Guided Discovery Questions.Kinds of Problems.Processes for Problem Solving.Polya's Process.Cognitively Guided Instruction.Share and Compare.Problem-Solving Strategies.Acting Out the Problem.Estimation and Check.Looking for Patterns.Drawing Pictures, Graphs, and Tables.Modeling.Developing Formulas and Writing Equations.Working Backward.Flowcharting.Simplifying the Problem.Problem-Solving Strategies for Teachers: Effective Teaching of Problem Solving.Kinds of Cooperative Grouping Activities.Think-Pair-Share.Grid Approach.Weekly Problem Approach.Problem Solving and Students with Special Needs.Assessment.Rating Group Problem Solving.Rubrics, Checklists, and Self-reflection.Putting it All Together: A Student's Work with Problem Solving.Summary.Praxis II(t)-Style Questions.Multiple-Choice Format.Constructed-Response Format.Integrating Technology.Video Vignette.Doubling. Internet.WebQuest.Extrapolate and Interpolate: A Math WebQuest.Websites.Children's Literature.Chapter 6: Early Childhood Mathematics-Number Readiness.Early Pre-Number Understandings.Emergent Mathematics.Patterns and Combinations-Precursors to Number Operations and Algebra Skills.Developing Patterning Skills.Constructions-Precursors to Geometry and Measurement.Assessment of Pre-Number Understandings.Classification.Assessment Task. Teaching Ideas.Class Inclusion.Assessment Task. Teaching Ideas.Assessment Task.Teaching Ideas.Seriation.Assessment Task.Teaching Ideas.Number Conservation.Assessment Task.Teaching Ideas.Equivalence of Sets.Assessment Task.Teaching Ideas.Implications for the Curriculum.Building the Concept of Number.Number Relationships.One-to-One Correspondence.More, Less, and Same.Rote Counting.Rational Counting.Counting Sets.Numeral-Set Association.Developing Meaning for Numbers.Writing Numerals.Teaching Ideas.Readiness for Operations.Developing Meaning for Operations.Building Part-Whole Understanding.New Curriculum for Ages 3 to 6.Good Beginnings-More than Number.Technology and Young Children.Calculators.Computers.Putting It All Together-Assessing Real Student Work.Summary.Praxis II(t)-Style Questions.Multiple-Choice Format.Constructed-Response Format.Integrating Technology.Internet.WebQuest.The Animal Habitat-A First Grade Mathematics WebQuest.Websites.Children's Literature.Chapter 7: Operations and Number Sense.Where's the Mathematics?-Conceptual Understanding of Operations and Properties.Prerequisites for Students: Readiness to Learn Basic Facts.Meaning of Operations.Number Conservation.Class Inclusion and Reversibility.Number Meaning.Concrete to Symbolic.Prerequisites for Teachers: Overview of Operations and Properties. The Operation of Addition.Combining.Static.The Operation of Subtraction.Take-Away.Comparison. Missing Addend. Properties of Addition and Subtraction.The Operation of Multiplication.Repeated Addition.Combinations.Arrays. The Operation of Division.Measurement Division (Repeated Subtraction).Partitive Division .Properties of Multiplication and Division.Thinking Strategies for Building Procedural Fluency and Flexibility with Basic Facts.Commutativity.Counting On by One, Two, and Three. Near Doubles.Bridging to 10. Thinking Strategies for the Basic Facts of Subtraction.Fact Families.Using 10.Part-Whole.Counting Back and Counting On.Thinking Strategies for the Basic Facts of Multiplication.Skip Counting.Finger Multiplication.Doubles.Friendly Facts.Split a Factor.Thinking Strategies for the Basic Facts of Division.Missing Factor.Mastering the Facts to Build Strategic Competence.Basic Facts--Pros and Cons.Practice Devices.Build Confidence.Assess Mastery.Assess Strategy Preferences.Assessment to Help Students with Special Needs.Field-Dependent and Field-Independent Learners.Both Learners Working Together with Basic Fact Tables.Correcting Common Misconceptions.Failure to Perform the Operation.Confusion with Multidigit Algorithms.Anxiety Over Speed Testing.Failure to Handle Symbolic Notation.Failure to Understand Commutativity.Faulty Reasoning 1: Position Counting.Faulty Reasoning 2: Subtrahend/Divisor Substitution.Faulty Reasoning 3: Missing the Answer by One.Faulty Reasoning 4: Trying to Do Work Without Manipulatives.Teacher Assessment of Student Work: Steps in Analytic Thinking.Steps in Analytic Thinking.Summary.Praxis II(t)-Style Questions.Multiple-Choice Format.Constructed-Response Format.Integrating Technology.Internet.Websites.Children's Literature.Chapter 8. Numeration and Number Sense.Early Understandings of Numeration.The Three Components of Number.The Importance of Place Value. Where's the Mathematics?-The Structure of Numeration Systems.The Base 10 Number System.Decimals.The Relative Magnitude of Numbers.Any Base Number System.Strategies for Building Conceptual Understanding of Numeration and Number Sense.The Nature of Place Value.Counting Activities.Numeration Models.Total Value and Expanded Notation.Understanding and Interpreting Large Numbers.The Relative Magnitude of Numbers.Exponential, Scientific, and Calculator Notation.Reading Numbers.Rounding.Understanding and Interpreting Decimals.Representing Decimals.Comparing Decimals.Estimation with Decimals.Assessment to Help Students with Special Needs.Correcting Common Misconceptions.Whole Number Numeration.Decimals.Nonalignment of the Decimal Points.Attention to Decimals Point as Place Holder.Annexing Zeroes.Name Value Confused with Place Value.Field-Dependent Learners.Field-Independent Learners.Putting It All Together with Real Student Work.Summary.Praxis II(t)-Style Questions.Multiple-Choice Format.Constructed-Response Format.Integrating Technology.Internet.Websites.Children's Literature.Chapter 9: AlgorithmsChapter 10: Fractions.Chapter 11: Percent, Ratio, and Rate.Chapter 12: Geometry.Chapter 13: Measurement.Chapter 14: Algebra.Chapter 15: Probability and Statistics.

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