Elementary Mathematics Is Anything But Elementary: Content and Methods from a Developmental Perspective

539 Pages·2008·21.8 MB·English

by  Kathleen Bahr

LibraryPirate
At the core of Elementary Mathematics Is Anything but Elementary is the
belief that mathematics is an engaging, exhilarating, self-actualizing
endeavor. Everyone is capable of the creative, meaningful work known
as mathematics.
WHAT IS MATH ANXIETY? SOME COMMON MYTHS
ABOUT TEST ANXIETY
■ Math anxiety is usually a result of past
experiences that were negative. As a result ■ Test anxiety cannot be reduced.
of these experiences, any time a person is ■ Any level of test anxiety is bad.
put in a similar situation, he or she gets
■ Students with test anxiety cannot learn
anxious
math.
■ Math anxiety is not related to how smart a
■ Attending class and doing homework
person is.
should reduce all test anxiety.
■ Math anxiety is a learned condition; there-
■ Reducing test anxiety will guarantee better
fore, in most cases, it can be unlearned or
grades.
at least managed.
(From Nolting, Math Study Skills Workbook, 3rd Ed.
■ A person must be willing to change and
Houghton Miffl in, 2008, p. 41.)
fi nd strategies to practice continually that
will help manage math anxiety.
(From Nolting, Math Study Skills Workbook, 3rd Ed.
Houghton Miffl in, 2008, p. 33.)
Use these unique textbook features to combat math anxiety in yourself and
your future students!
■ Conversation in Mathematics. Highlighting a real conversation between a teacher and a
student, this feature shows how students talk about and learn math.
■ Language Tip. The process of learning math requires unique language skills. This feature
facilitates the use of language in the math classroom, including among ELL/multicultural
students.
■ Comprehension Check. Use these checks to ensure your understanding of important
concepts before moving on to the next section.
■ Special Needs Support and Struggling Learner Symptoms. Some students will need
some extra support in the classroom. These features highlight tips and tools for these
students.
■ Caution. Misunderstandings or misconceptions can happen in the math classroom. Use
these tips for working out confusing concepts with your students.
■ Videos. Directly connected to the textbook are videos that show real students, teachers,
and classroom lectures about math concepts.
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Elementary Mathematics
BRIEF
CONTENTS
Is Anything but Elementary
Content and Methods from a Developmental Perspective
Damon L. Bahr
Brigham Young University
Lisa Ann de Garcia
Brigham Young University
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
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Elementary Mathematics Is Anything but © 2010 Wadsworth, Cengage Learning
Elementary: Content and Methods from
ALL RIGHTS RESERVED. No part of this work covered by the copyright herein
a Developmental Perspective
may be reproduced, transmitted, stored, or used in any form or by any means,
Damon L. Bahr, Lisa Ann de Garcia
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Printed in the United States of America
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BBRRIIEEFF
CCOONNTTEENNTTSS
1
Mathematical Literacy: Acquiring the Big Ideas While Doing
Mathematics | 1
2
The Foundation of All Math Learning: Representations of Early
Number Concepts | 21
3
Informing Our Decisions: Assessment and Single-Digit Addition
and Subtraction | 56
4
Deepening Understanding through Communication and
Numeration | 89
5
Learning with Understanding: Concepts and Procedures and
Multidigit Addition and Subtraction | 122
6
Instructional Models: Inquiry-Based Teaching with Single-Digit
Multiplication and Division | 157
7
Lesson Design: Learning How to Create Inquiry Lessons Using
Multidigit Multiplication and Division | 188
8
Fluency through Meaningful Practice: Mathematical Routines
and Algebraic Thinking | 217
9
Instructional Goals, Number Theory, and Integers | 254
10
High- and Low-Level Tasks and Fractional Number Sense | 284
11
Curriculum Integration and Fraction Computation | 316
12
Primary- and Second-Language Issues in Mathematics and
Decimals, Percentages, and Ratios | 340
13
Exceptional Learners and Geometric and Spatial Reasoning | 371
14
Teaching Measurement in a Meaning-Centered Classroom | 408
15
Technology Integrations and Data Analysis and Probability | 450
Appendix A | 483
Appendix B | 485
Appendix C | 488
Glossary | 495
References | 498
Index | 505
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CONTENTS
1
CHAPTER
Mathematical Literacy: Acquiring the Big Ideas
While Doing Mathematics | 1
PEDAGOGY Processes That Characterize Doing Mathematics | 3
Learning Theory | 3
Application to the Learning and Teaching of Mathematics | 4
Problem Solving | 4
Communication | 6
Reasoning and Proof | 7
Representation | 9
Connections | 12
CONTENT Curriculum Focal Points and Big Ideas in Mathematics Grade
by Grade | 14
2
CHAPTER
The Foundation of All Math Learning: Representations
of Early Number Concepts | 21
PEDAGOGY Mathematical Tools: Concrete and Abstract Representations | 23
Learning Theory | 23
Application to the Learning and Teaching of Mathematics | 24
What Are Tools? | 24
Constructing Meaning for Tools | 27
Making Connections between the Concrete and the
Abstract | 30
Misconceptions, Pitfalls, and Diffi culties Associated
with Tool Use | 33
CONTENT Early Number Concepts | 34
Big Ideas and Focal Points | 34
Number Sense | 36
Developmental Phases for Numbers | 36
Emergent Phase | 37
Matching Phase | 38
Quantifying Phase | 44
Assessments | 50
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vi Contents
3
CHAPTER
Informing Our Decisions: Assessment and Single-Digit
Addition and Subtraction | 56
PEDAGOGY Assessment for Instruction | 58
Learning Theory | 58
Application to the Learning and Teaching of Mathematics | 58
Backward Design | 58
Four Important Assessment Terms | 59
Traditional Assessments | 61
Alternative Assessments | 63
CONTENT Single-Digit Addition and Subtraction | 70
Big Ideas and Focal Points | 70
Operation Sense | 70
Problem Types | 72
Problem-Solving Strategies | 74
Learning the Basics of Addition and Subtraction | 77
Practicing for Quick Recall | 81
4
CHAPTER
Deepening Understanding through Communication
and Numeration | 89
PEDAGOGY Collaborating and Refl ection through Communication
and Discourse | 92
Learning Theory | 92
Application to the Learning and Teaching of Mathematics | 92
Collaboration through Accountable Talk | 92
Teacher and Student Roles | 97
Refl ection through Writing | 100
Assessing Teacher and Student Communication | 101
CONTENT Numeration: Understanding Place Value | 102
Big Ideas and Focal Points | 103
Number Systems | 103
Hindu-Arabic Number System | 104
Understanding Place Value | 106
Working with Larger Numbers | 116
Assessments | 118
5
CHAPTER
Learning with Understanding: Concepts and Procedures
and Multidigit Addition and Subtraction | 122
PEDAGOGY Concepts and Procedures | 125
Learning Theory | 125
Application to the Learning and Teaching of Mathematics | 126
Conceptual versus Procedural Knowledge | 126
A Balanced Set of Beliefs about Mathematics | 127
Types of Concepts and Related Procedures | 132
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Contents vii
Constructing Mental Concepts | 133
Developing Procedural Fluency | 137
Standard versus Alternative Algorithms | 138
CONTENT Multidigit Addition and Subtraction | 141
Big Ideas and Focal Points | 141
Invented Algorithms | 143
Standard Algorithms | 147
Assessments | 148
Using Inquiry to Teach Multidigit Addition
and Subtraction | 150
6
CHAPTER
Instructional Models: Inquiry-Based Teaching
with Single-Digit Multiplication and Division | 157
PEDAGOGY Instructional Models | 159
Learning Theory | 159
Behaviorism | 159
Constructivism | 160
Application to the Learning and Teaching of Mathematics | 160
Components of a Direct Instruction Lesson | 160
Components of an Inquiry Lesson | 162
CONTENT Multiplication and Division: Groups of and Shared By | 168
Big Ideas and Focal Points | 168
Understanding Multiplication and Division | 169
Problem Types and Solution Strategies | 171
Exploring Multiplication and Division | 173
Basic Facts | 177
Using Inquiry to Teach Multiplication and Division | 181
7
CHAPTER
Lesson Design: Learning How to Create Inquiry Lessons
Using Multidigit Multiplication and Division | 188
PEDAGOGY Creating Inquiry Lessons | 190
Learning Theory | 190
Application to the Learning and Teaching of Mathematics | 192
Steps for Planning an Inquiry Lesson | 192
Lesson Examples | 195
Planning a Unit | 197
CONTENT Multidigit Multiplication and Division | 199
Big Ideas and Focal Points | 199
Multidigit Problems | 199
Assessments | 208
Using Inquiry to Teach Multidigit Multiplication
and Division | 209
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viii Contents
8
CHAPTER
Fluency through Meaningful Practice: Mathematical
Routines and Algebraic Thinking | 217
PEDAGOGY Developing Fluency with Mathematical Routines | 219
Learning Theory | 219
Procedural Fluency | 220
Algorithms | 220
Mental Math Activities | 221
Application to the Learning and Teaching of Mathematics | 221
What Are Routines? | 221
Types of Routines | 222
Designing a Routine Series | 230
Planning Routines for the Year | 230
Assessing Routines | 232
CONTENT Algebraic Thinking and Reasoning | 232
Big Ideas and Focal Points | 232
What Is Algebra? | 234
Algebraic Reasoning | 235
Making Conjectures and Generalizations | 245
Justifi cation and Proof | 248
Integrating Arithmetic and Algebra | 249
Using Inquiry to Teach Algebraic Concepts | 249
9
CHAPTER
Instructional Goals, Number Theory, and Integers | 254
PEDAGOGY Instructional Goals | 256
Learning Theory | 256
Application to the Learning and Teaching of Mathematics | 257
Content Goals | 258
Process Goals | 259
Disposition Goals | 260
Social Goals | 261
Choosing Goals and Objectives | 261
CONTENT Number Theory and Integers | 262
Big Ideas and Focal Points | 263
Number Theory | 263
Integers | 271
Using Inquiry with Number Theory and Integers | 279
10
CHAPTER
High- and Low-Level Tasks and Fractional Number Sense | 284
PEDAGOGY High- and Low-Level Tasks | 285
Learning Theory | 285
Application to the Learning and Teaching of Mathematics | 287
What Is a Task? | 287
Outcomes of a Problematic Task | 289
Designing and Selecting a Problematic Task | 291
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Contents ix
Analyzing the Level of a Task | 294
Maintaining the Intended Level of a Task | 295
Teacher and Student Roles during a Task | 296
CONTENT Fractional Number Sense | 298
Big Ideas and Focal Points | 298
Part–Whole Fractions | 300
Part–Set Fractions | 302
Fractions as Division | 302
Other Fraction Notions | 303
Implications in the Classroom | 305
Using Inquiry to Teach Fractional Number Sense | 310
11
CHAPTER
Curriculum Integration and Fraction Computation | 316
PEDAGOGY Curriculum Integration | 318
Learning Theory | 318
Application to the Learning and Teaching of Mathematics | 319
Reading and Literature Integrations | 319
Fine Arts Integrations | 320
Science and Social Studies Integrations | 321
CONTENT Fraction Computation | 322
Big Ideas and Focal Points | 322
Addition and Subtraction of Fractions | 323
Multiplication of Fractions | 325
Division of Fractions | 330
Implications in the Classroom | 332
Using Inquiry to Teach the Computation of Fractions | 335
12
CHAPTER
Primary- and Second-Language Issues in Mathematics
and Decimals, Percentages, and Ratios | 340
PEDAGOGY Primary- and Second-Language Issues in Mathematics | 342
Learning Theory | 342
Application to the Learning and Teaching of Mathematics | 343
Language Acquisition | 343
Learning the Language of Math | 344
Irregularities with Math Vocabulary | 346
Explicit Teaching of Vocabulary | 347
Environments That Support Both Language and Concept
Development | 351
Discourse Structures | 351
CONTENT Decimals, Percentages, and Ratios | 353
Big Ideas and Focal Points | 353
Decimals | 355
Percentages | 359
Ratios and Proportions | 363
Using Inquiry to Teach Decimals, Percentages, and Ratios | 366
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