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E-raamat: Understanding the Math We Teach and How to Teach It, K-8

  • Formaat: 644 pages
  • Ilmumisaeg: 26-Aug-2025
  • Kirjastus: Stenhouse Publishers
  • Keel: eng
  • ISBN-13: 9781040341490
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  • Formaat: 644 pages
  • Ilmumisaeg: 26-Aug-2025
  • Kirjastus: Stenhouse Publishers
  • Keel: eng
  • ISBN-13: 9781040341490
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"Marian Small has written the kind of book teachers will keep on their closest shelf as they explore and return to the big ideas of mathematics. In her new resource, Understanding the Math We Teach and How to Teach It, Small brings the support and insight teachers need to be able to teach math with clarity and confidence"--

This guide shows K-8 teachers how to teach math, including discussion of the "whys" behind the math that students are expected to learn in most state standards. It discusses how students learn math, what teachers want students to learn, and research on math learning and teaching; focusing instruction on big ideas and mathematical processes; assessment and evaluation; planning instruction, including dealing with math anxiety; and instruction related to specific topics, from numbers to probability. Chapter problems are included throughout, as are activities, teaching tips, student sample responses, tables of common errors and misconceptions, children's literature suggestions to support content topics, and lists of appropriate manipulatives. Annotation ©2019 Ringgold, Inc., Portland, OR (protoview.com)

Marian Small has written the kind of book teachers will keep on their closest shelf as they explore and return to the big ideas of mathematics. In her new resource, Understanding the Math We Teach and How to Teach It, Marian brings the support and insight teachers need to teach math with clarity and confidence.

With this new resource, new and experienced teachers alike will focus on the big ideas and practices in mathematics, deepening your own understanding and content knowledge, learn how to teach those big ideas using a student-centered, problem-solving approach, and anticipate student thinking and explore effective tools, models, and rich mathematical questions that nudge student thinking forward.

This readable and relatable resource will give you a well-founded base of mathematical knowledge, leading to better math instruction that will capture your students&; interest. It is sure to become a trusted treasure you return to again and again.

Preface xx
Acknowledgments xxi
About the Author xxi
Chapter 1 How Students Learn Math and What Math We Want Them to Learn 1(14)
Problem
1(1)
In a Nutshell
1(1)
Research on Mathematical Learning
2(3)
The Importance of Conceptual Understanding
2(1)
A Constructivist Approach
3(1)
Using Manipulatives
4(1)
Mindset
5(1)
What Mathematics We Want Students to Learn
5(5)
Differing Perspectives on What Mathematics Is
5(2)
Differing Perspectives on What Mathematics Is Valued and at What Grade Levels
7(2)
Teaching Developmentally
9(1)
Research on Mathematics Teaching
10(2)
Importance of a Teacher's Pedagogical Content Knowledge
10(1)
Research About Classroom Environment
11(1)
Applying What You've Learned
12(1)
Selected References
13(2)
Chapter 2 Focusing Instruction on Big Ideas and Mathematical Processes 15(20)
Problems
15(1)
In a Nutshell
15(1)
Organizing Content Around Big Ideas
16(5)
Different Approaches to Big Ideas
16(1)
Big Ideas Applying to Many Mathematical Domains
17(3)
Examples of More Domain-Based Big Ideas
20(1)
Focusing Instruction on Proportional Reasoning, Algebraic Reasoning, and Spatial Reasoning
21(4)
Proportional Reasoning
21(2)
Algebraic Reasoning
23(1)
Spatial Reasoning
23(2)
Focusing Instruction on Math Processes and Practices
25(7)
Standards for Mathematical Practice
25(1)
Problem Solving
25(2)
Reasoning
27(2)
Tools and Representations
29(2)
Communication and Discourse
31(1)
Assessing Student Understanding
32(1)
Applying What You've Learned
32(1)
Selected References
32(3)
Chapter 3 Assessment and Evaluation 35(28)
Problem
35(1)
In a Nutshell
36(1)
Introduction
36(2)
Assessment for Learning
37(1)
Assessment as Learning
37(1)
Assessment of Learning
37(1)
How Assessment Has Changed
38(1)
Characteristics of Good Assessment
38(2)
A Good Assessment Plan
38(2)
Sources of Assessment Data
40(11)
Observations
40(1)
Conversations and Interviews
41(1)
Learning Tasks
42(2)
Assessment Tasks
44(1)
Exit Tickets
44(1)
Performance Tasks
45(1)
Tests and Quizzes
46(3)
Portfolios
49(2)
Keeping Track
51(1)
Assessing Practices and Processes
51(3)
Assessing Problem Solving
51(1)
Assessing Communication
52(2)
Self-Assessment
54(1)
Group Assessment
55(1)
Using Technology in Assessment
55(1)
Interpreting Assessment Results
56(1)
Grading and Reporting
56(2)
Anecdotal Versus Letter Grades or Percents
57(1)
Using Discretion
57(1)
Pitfalls of Percent Grades
57(1)
Including Group Evaluations
58(1)
Evaluations Based on Observation Assessment Data
58(1)
Large-Scale Assessment
58(2)
Preparing Students for Large-Scale Assessment
59(1)
Applying What You've Learned
60(1)
Selected References
61(2)
Chapter 4 Planning Instruction 63(26)
Problem
63(1)
In a Nutshell
63(1)
Math Anxiety: A Special Challenge in Teaching Mathematics
64(1)
Causes of Math Anxiety
64(1)
Diminishing Anxiety in Students
65(1)
Diminishing Anxiety in Teachers
65(1)
Unit Planning and Lesson Planning
65(2)
Year Planning
65(1)
Unit Planning
66(1)
Lesson Planning
67(1)
Lesson Style
67(3)
The Importance of Varied Lesson Styles
67(1)
Different Lesson Styles
68(1)
Specific Lesson Strategies
69(1)
Planning Grouping
70(3)
Heterogeneous and Homogeneous Grouping
70(1)
Group Size
71(1)
Collaboration
71(1)
Suitable Tasks for Group Work
71(1)
Where Groups Work
71(1)
Limitations of Grouping
71(1)
Balancing Whole-Class, Small-Group, and Individual Instruction
72(1)
Planning Support
73(1)
Planning Opportunities for Practice
73(2)
Is Practice Important?
73(1)
What Should Be Practiced?
73(1)
How Much Practice Is Needed?
74(1)
What Should Practice Look Like?
74(1)
Differentiating Instruction: Supporting Individual Differences
75(10)
Moving from the Concrete to the Symbolic
75(2)
Managing Differentiation
77(2)
Guided Groups
79(1)
Students with Learning Disabilities
79(2)
Multilingual Students
81(3)
For Those Seeking Even More
84(1)
Home-School Connections
85(1)
Homework
85(1)
Making Contact with Parents
86(1)
Applying What You've Learned
86(1)
Selected References
87(2)
Chapter 5 Early Number 89(20)
Problem
89(1)
In a Nutshell
89(1)
Numbers in the Child's World
90(1)
Counting Principles
90(1)
Zero and One
91(1)
Counting Strategies
91(3)
Counting On
92(1)
Counting Back
93(1)
Skip Counting
93(1)
A Sense of Number
94(4)
Comparing and Relating Numbers
94(1)
One-to-One Correspondence
94(1)
Position in the Counting Sequence
95(1)
Comparison Language
95(1)
Spatial Comparison
95(1)
Numbers to 10
96(2)
The Teen Numbers
98(1)
Importance of Multiple
Representations
98(1)
Reading and Writing Numerals
99(2)
Common Errors and Misconceptions
101(1)
Appropriate Manipulatives
102(3)
Appropriate Children's Books
105(1)
Assessing Student Understanding
106(1)
Applying What You've Learned
107(1)
Selected References
107(2)
Chapter 6 Early Operations 109(28)
Problem
109(1)
In a Nutshell
109(1)
How the Four Operations Are Related
110(1)
Addition and Subtraction
110(6)
Meanings of Addition
110(1)
Meanings of Subtraction
111(1)
Understanding Addition and Subtraction Situations
112(1)
Solving Addition and Subtraction Number Stories
113(1)
Relating Addition and Subtraction
113(1)
Addition and Subtraction Principles
114(2)
Common Errors and Misconceptions
116(1)
Appropriate Manipulatives
117(1)
Multiplication and Division
118(13)
Meanings of Multiplication
119(2)
Meanings of Division
121(2)
Understanding Multiplication and Division Situations
123(1)
Multiplication and Division Situations
124(1)
Solving Multiplication and Division Problems
124(1)
Relating Multiplication and Division
125(1)
Multiplication and Division Principles
125(6)
Common Errors and Misconceptions
131(1)
Appropriate Manipulatives
132(1)
Appropriate Children's Books
133(1)
Assessing Student Understanding
133(1)
Applying What You've Learned
133(1)
Selected References
134(3)
Chapter 7 Developing Fact Fluency 137(12)
Problem
137(1)
In a Nutshell
137(1)
What Are the Facts?
138(1)
Why Is It Important to Learn the Facts?
138(1)
What Does Fact Fluency Mean?
138(1)
Learning Addition and Subtraction Facts
139(3)
Typical Sequence for Acquiring Addition and Subtraction Facts
141(1)
Learning Multiplication and Division Facts
142(3)
Typical Sequence for Acquiring Multiplication and Division Facts
145(1)
What Kind of Practice Is Useful?
145(1)
How Should Facts Be Assessed?
How Should They Not Be Assessed?
145(1)
Common Errors and Misconceptions
146(1)
Appropriate Manipulatives
147(1)
Appropriate Children's Books
147(1)
Assessing Student Understanding
147(1)
Applying What You've Learned
148(1)
Selected References
148(1)
Chapter 8 Representing Larger Whole Numbers 149(24)
Problem
149(1)
In a Nutshell
149(1)
Larger Numbers in the Students' World
150(1)
Numeration Principles
150(6)
Counting Based on Place Value
156(1)
Estimating Numbers
156(2)
Very Big Numbers
157(1)
Rounding
157(1)
Benchmark Numbers
158(1)
Common Errors and Misconceptions
158(1)
Appropriate Manipulatives
159(1)
Number Theory
160(8)
Even and Odd Numbers
160(1)
Multiples
161(1)
Factors
162(4)
Prime and Composite Numbers
166(2)
Common Errors and Misconceptions
168(1)
Appropriate Manipulatives
168(1)
Appropriate Children's Books
169(1)
Assessing Student Understanding
169(1)
Applying What You've Learned
170(1)
Selected References
170(3)
Chapter 9 Estimation and Calculation Strategies with Larger Whole Numbers 173(36)
Problem
173(1)
In a Nutshell
173(1)
Addition and Subtraction
174(12)
Estimating Sums and Differences
174(2)
Varied Approaches to Addition and Subtraction
176(9)
Communicating About Adding and Subtracting
185(1)
Common Errors and Misconceptions
186(1)
Appropriate Manipulatives
187(2)
Multiplication and Division
189(13)
Multiplying and Dividing Using Powers of 10
189(2)
Estimating Products and Quotients
191(2)
Algorithms for Multiplication and Division
193(8)
Communicating About Multiplying and Dividing
201(1)
Common Errors and Misconceptions
202(2)
Appropriate Manipulatives
204(1)
Appropriate Children's Books
205(1)
Assessing Student Understanding
205(2)
Applying What You've Learned
207(1)
Selected References
208(1)
Chapter 10 Fractions 209(38)
Problem
209(1)
In a Nutshell
209(1)
Representing and Comparing Fractions
210(12)
Fractions in the Child's World
210(1)
How the Fraction Meanings Are Equivalent
211(1)
Fraction Definitions and Principles
212(4)
Equivalent Fractions
216(1)
Mixed Numbers and Fractions
217(1)
Visualizing with Fractions
218(1)
What Students Might Learn When Comparing Fractions
218(4)
Relating Fractions to Decimals
222(1)
Common Errors and Misconceptions
222(2)
Appropriate Manipulatives
224(3)
Fraction Operations
227(13)
Adding and Subtracting Fractions
227(6)
Multiplying and Dividing Fractions
233(7)
Common Errors and Misconceptions
240(1)
Appropriate Manipulatives
241(1)
Appropriate Children's Books
242(1)
Assessing Student Understanding
242(1)
Applying What You've Learned
243(1)
Selected References
244(3)
Chapter 11 Decimals 247(22)
Problem
247(1)
In a Nutshell
247(1)
Representing Decimals
248(9)
Decimal Contexts
249(1)
Decimal Principles
249(3)
Equivalent Decimals
252(1)
Equivalent Decimals and Precision
252(1)
Equivalent Fractions and Decimals
252(2)
Rounding and Estimating Decimals
254(1)
Reading and Writing Decimals
255(1)
Comparing Decimals
255(2)
Decimal Operations
257(5)
Adding and Subtracting Decimals
257(2)
Multiplying and Dividing Decimals
259(3)
Common Errors and Misconceptions
262(1)
Appropriate Manipulatives
263(3)
Appropriate Children's Books
266(1)
Assessing Student Understanding
266(1)
Applying What You've Learned
267(1)
Selected References
268(1)
Chapter 12 Ratio and Proportion 269(24)
Problem
269(1)
In a Nutshell
269(1)
Proportional Reasoning
270(1)
Ratio
270(7)
Equivalent Ratios
271(3)
Types of Ratio Problems
274(3)
Rates
277(2)
Using Unit Rates
278(1)
Fermi Problems
278(1)
Role of Proportional Reasoning in Other Strands
279(2)
Measurement
279(1)
Geometry
280(1)
Data
280(1)
Probability
280(1)
Algebra
281(1)
Financial Literacy
281(1)
Percent
281(6)
Principles for Percent
281(6)
Common Errors and Misconceptions
287(1)
Appropriate Manipulatives
288(1)
Appropriate Children's Books
289(1)
Assessing Student Understanding
289(1)
Applying What You've Learned
290(1)
Selected References
291(2)
Chapter 13 Extending the Number System to Negative and Irrational Numbers 293(22)
Problem
293(1)
In a Nutshell
293(1)
Introducing Integers
294(1)
Integer Contexts
294(1)
Reading and Writing Integers
294(1)
Comparing Integers
295(1)
The Zero Property
295(1)
Adding Integers
296(2)
Subtracting Integers
298(2)
Multiplying Integers
300(3)
Dividing Integers
303(2)
Rational Numbers
305(1)
Exponents
305(3)
Using Powers of 10 to Give a Sense of Number Size
306(1)
Order of Operations Including Exponents
307(1)
Rational Versus Irrational Numbers
308(1)
Common Errors and Misconceptions
309(1)
Appropriate Manipulatives
310(2)
Appropriate Children's Books
312(1)
Assessing Student Understanding
312(1)
Applying What You've Learned
313(1)
Selected References
314(1)
Chapter 14 Patterns and Algebra 315(38)
Problem
315(1)
In a Nutshell
315(1)
Patterns
316(8)
Types of Patterns
316(3)
Describing and Extending Patterns
319(3)
Creating Patterns
322(1)
Mathematical Situations Rich in Patterns
322(1)
Using Patterns to Develop Mathematical Concepts
323(1)
Common Errors and Misconceptions
324(1)
Appropriate Manipulatives
325(2)
Algebra
327(12)
Moving from Pattern to Algebra
327(2)
Algebra as Generalizing Number Relationships
329(1)
Algebraic Notation
330(3)
Describing Functions
333(1)
Using Graphs to Describe Relationships
334(5)
Algebraic Manipulation
339(6)
Common Errors and Misconceptions
345(2)
Appropriate Manipulatives
347(1)
Appropriate Children's Books
348(1)
Assessing Student Understanding
349(1)
Applying What You've Learned
349(1)
Selected References
350(3)
Chapter 15 3-D and 2-D Shapes 353(48)
Problem
353(1)
In a Nutshell
354(1)
Fundamental Aspects of Geometry
354(1)
Development of Geometric Thinking
355(2)
Identifying and Classifying Shapes
357(1)
Identification of 2-D and 3-D Shapes
357(1)
Geometric Attributes and Properties
358(3)
Comparing Shapes
358(1)
Sorting and Patterning with Shapes
359(2)
Types of 3-D Shapes
361(2)
Polyhedrons, Spheres, Cones, and Cylinders
361(1)
Components of 3-D Shapes
362(1)
Using Properties to Classify 2-D Shapes
363(4)
Polygons and Circles
363(1)
Components of 2-D Shapes
363(4)
Exploring Properties
367(1)
Planes
367(1)
Lines, Segments, and Rays
368(1)
Symmetry
369(4)
Reflection Symmetry of 2-D and 3-D Shapes
369(2)
Rotational Symmetry of 2-D and 3-D Shapes
371(2)
Representing Shapes
373(7)
Modeling Shapes
373(4)
Drawing 3-D Shapes
377(1)
Drawing 2-D Shapes
378(2)
Decomposing and Composing Shapes
380(5)
Composing Shapes
381(2)
Decomposing Shapes
383(2)
Congruence and Similarity
385(3)
Congruence of 2-D Shapes
386(2)
Congruence of 3-D Shapes
388(1)
Similarity of 2-D Shapes
388(3)
Determining Similarity of 2-D Shapes
389(1)
Constructing Similar 2-D Shapes
389(2)
Common Errors and Misconceptions
391(3)
Appropriate Manipulatives
394(3)
Appropriate Children's Books
397(1)
Assessing Student Understanding
397(1)
Applying What You've Learned
398(1)
Selected References
399(2)
Chapter 16 Location and Movement 401(26)
Problem
401
In a Nutshell
40(362)
Location and Movement
402(1)
Developing Positional Vocabulary
402(2)
Using Dance, Song, and Play
402(1)
Word Walls
402(2)
Maps and Coordinate Grids
404(2)
Drawing and Interpreting Maps
404(1)
Working with Grids
404(2)
Transformations
406(12)
Euclidean Transformations-Slides, Flips, and Turns
406(1)
Transformations on Simple Grids
407(1)
Transformations on Coordinate Grids
407(1)
Slides
407(2)
Flips
409(3)
Turns
412(2)
Coding
414(1)
Non-Euclidean Transformations-Dilations
414(2)
Tessellations
416(1)
Constructions
417(1)
Common Errors and Misconceptions
418(3)
Appropriate Manipulatives
421(1)
Appropriate Children's Books
422(1)
Assessing Student Understanding
422(1)
Applying What You've Learned
423(1)
Selected References
424(3)
Chapter 17 The Nature of Measurement, with a Focus on Length and Area 427(50)
Problem
427(1)
In a Nutshell
427(1)
The Nature of Measurement
428(2)
Addressing Measurement Principles in the Classroom
429(1)
Stages of Measurement Instruction
430(2)
Systems of Measurement
432(1)
The Imperial System of Measurement
432(1)
The Metric System of Measurement
433(1)
SI-Derived and Other Units
434(1)
Metric Prefixes
434(1)
Measuring Length
434(12)
Introducing Length Concepts
435(1)
Length: The Definition/Comparison Stage
435(1)
Length: The Nonstandard Unit Stage
436(3)
Length: The Standard Unit Stage
439(5)
Measuring Perimeter
444(2)
Common Errors and Misconceptions
446(3)
Appropriate Manipulatives
449(2)
Measuring Area
451(17)
Introducing Area Concepts
451(1)
Area: The Definition/Comparison Stage
451(1)
Area: The Nonstandard Unit Stage
452(5)
Area: The Standard Unit Stage
457(3)
Formulas for Calculating Areas of Shapes
460(7)
Pythagorean Theorem
467(1)
Common Errors and Misconceptions
468(3)
Appropriate Manipulatives
471(2)
Appropriate Children's Books
473(1)
Assessing Student Understanding
474(1)
Applying What You've Learned
474(1)
Selected References
475(2)
Chapter 18 Volume, Mass, Time, and Angles 477(52)
Problem
477(1)
In a Nutshell
477(1)
Measuring Volume and Mass
478(1)
Introducing Volume
478(13)
Volume: The Definition/Comparison Stage
479(1)
Volume: The Nonstandard Unit Stage
480(4)
Volume: The Standard Unit Stage
484(2)
Relating Units of Volume
486(1)
Solving Problems Involving Volume
487(4)
Introducing Mass Concepts
491(6)
Mass: The Definition/Comparison Stage
492(1)
Mass: The Nonstandard Unit Stage
493(2)
Mass: The Standard Unit Stage
495(2)
Common Errors and Misconceptions
497(3)
Appropriate Manipulatives
500(2)
Measuring Time
502(1)
Introducing Time Concepts
503(8)
Time: The Definition/Comparison Stage
505(1)
Time: The Nonstandard Unit Stage
505(1)
Time: The Standard Unit Stage
506(4)
Measuring and Calculating Elapsed Time
510(1)
Common Errors and Misconceptions
511(2)
Appropriate Manipulatives
513(1)
Measuring Angles
513(2)
Introducing Angle Concepts
515(6)
Angles: The Definition/Comparison Stage
516(1)
Angles: The Nonstandard Unit Stage
516(2)
Angles: The Standard Unit Stage
518(3)
Common Errors and Misconceptions
521(2)
Appropriate Manipulatives
523(2)
Appropriate Children's Books
525(1)
Assessing Student Understanding
525(1)
Applying What You've Learned
526(1)
Selected References
527(2)
Chapter 19 Data 529(58)
Problem
529(1)
In a Nutshell
529(1)
Data Organization
530(4)
Sorting and Classifying
530(1)
Recognizing Attributes
530(1)
Describing a Sorting Rule
531(2)
Venn Diagrams
533(1)
Common Errors and Misconceptions
534(1)
Appropriate Manipulatives
535(1)
Data Collection
536(3)
Why People Collect Data
536(1)
Asking Good Survey Questions
536(1)
Choosing Data Collection Topics
537(1)
Data Sources
538(1)
Factors Influencing Data Collection
538(1)
Common Errors and Misconceptions
539(1)
Appropriate Manipulatives
540(1)
Data Display
540(20)
Data Display Formats
540(17)
Appropriate Use of Data Display Formats
557(3)
Common Errors and Misconceptions
560(5)
Appropriate Manipulatives
565(2)
Data Analysis
567(8)
Reading Graphs
567(3)
Drawing Inferences from Graphs
570(2)
Misleading Graphs
572(3)
Common Errors and Misconceptions
575(1)
Statistics
576(5)
Measures of Central Tendency
576(3)
Measures of Data Spread
579(1)
Data Shape
580(1)
Common Errors and Misconceptions
581(1)
Appropriate Manipulatives
582(1)
Appropriate Children's Books
582(1)
Assessing Student Understanding
582(1)
Applying What You've Learned
583(1)
Selected References
584(3)
Chapter 20 Probability 587(20)
Problem
587(1)
In a Nutshell
587(1)
Fundamental Notions About Probability
588(1)
Probability Misconceptions
588(1)
Early Work with Probability
589(1)
Using a Probability Line
589(2)
Outcomes, Events, and Sample Space
591(1)
Simple and Compound Events
591(1)
Independent and Dependent Compound Events
591(1)
Probabilities as Ratios and Fractions
592(1)
Experimental Probability
592(3)
Sample Size
593(1)
Making Predictions Based on Experimental Results
594(1)
Theoretical Probability
595(4)
The Transition from Experimental to Theoretical for Compound Events
595(2)
Reconciling Theoretical and Experimental Probability
597(1)
Determining Theoretical Probability
597(2)
Common Errors and Misconceptions
599(2)
Appropriate Manipulatives
601(2)
Appropriate Children's Books
603(1)
Assessing Student Understanding
603(1)
Applying What You've Learned
604(1)
Selected References
605(2)
Index 607
Marian Small is the former Dean of Education at the University of New Brunswick in Canada. She has been a professor of mathematics education and worked in the field for close to 40 years. Dr. Small is a regular speaker on K12 mathematics throughout Canada and around the world. Her focus is on teacher questioning to get at the important math, to include and extend all students by appropriately differentiating instruction, and to focus on critical thinking and creativity.