1. CHAPTER 1 School Mathematics in a Changing World
Tina Rye Sloan To accompany Helping Children Learn Math10e, Reys et al.
©2012 John Wiley & Sons

2. Focus Questions
1. What is mathematics? 2. What determines the mathematics currently taught? 3. What is the role of the NCTM’s six principles in school mathematics? 4. What resources are available to help you continue developing your knowledge of mathematics and the learning and teaching of mathematics?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10thth Edition, © 2012

3. What is Mathematics? a study of patterns and relationships
a way of thinking
an art
a language
a tool
Master 1-1: What is Mathematics?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10thth Edition, © 2012

4. NCTM (2000) Content Standards
Figure 1-1 shows how the National Council of Teachers of Mathematics (NCTM) envisions the distribution of mathematics content from pre-kindergarten through Grade 12.
Source: Principles and Standards for School Mathematics, p. 30, copyright 2000 by the National Council of Teachers of Mathematics. All Rights Reserved.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10thth Edition, © 2012

5. What Determines the Mathematics Being Taught?
Needs of the subject
Needs of the child
Needs of society
Master 1-2: Factors That Influence the Mathematics Curriculum
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10thth Edition, © 2012

6. NCTM Principles for School Mathematics
Equity Principle
Curriculum Principle
Teaching Principle
Learning Principle
Assessment Principle
Technology Principle
Master 1-3: Principles for School Mathematics
Principles and Standards for School Mathematics
National Council of Teachers of Mathematics, 2000
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10thth Edition, © 2012

7. CHAPTER 2 Helping Children Learn Mathematics with Understanding
To accompany Helping Children Learn Math10e, Reys et al.
©2012 John Wiley & Sons

8. Focus Questions
1. How can we create a supportive classroom climate for the diverse learners in our classroom? 2. What is procedural knowledge and how is it different from conceptual knowledge? 3. How do behaviorist approaches to learning differ from constructivist approaches to learning? 4. What are four recommendations for helping children make sense of mathematics based on what is known about how children learn mathematics?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012

9. HOW CAN WE SUPPORT THE DIVERSE LEARNERS IN OUR CLASSROOMS?
Create a Positive Learning Environment
Avoid Negative Experiences That Increase Anxiety
Establish Clear Expectations
Treat All Students as Equally Likely to Have
Aptitude for Mathematics
Help Students Improve Their Ability to Retain
Mathematical Knowledge and Skills
Master 2-1: Strategies that Support All Learners
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10th Edition, © 2012 9

10. Procedural Knowledge-skillful use of mathematical rules or algorithms
HOW CAN WE HELP CHILDREN ACQUIRE BOTH PROCEDURAL KNOWLEDGE AND CONCEPTUAL KNOWLEDGE?
Procedural Knowledge-skillful use of mathematical rules or algorithms
Conceptual Knowledge-understanding meaning of mathematical concepts
Adding is
putting
together
Master 2-2: Procedural and Conceptual Knowledge
Add then
subtract…
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11. Two Theories of Learning: Behaviorism and Constructivism
HOW DO CHILDREN LEARN
MATHEMATICS?
Two Theories of Learning:
Behaviorism and Constructivism
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10th Edition, © 2012

12. Behaviorism
Behavior can be shaped by reinforcement of drill and practice.
Specific skills need to be learned in a
fixed order.
Master 2-3: Behaviorism and Constructivism
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10th Edition, © 2012 12

13. Behaviorism (cont.) Clear objectives help students and teachers.
Edward L. Thorndike
B.F. Skinner
Robert Gagne
Master 2-3: Behaviorism and Constructivism
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14. Constructivism
Learners actively create or invent (construct) their own knowledge.
 Students create (construct) new mathematical knowledge by reflecting on their physical and mental actions.
Master 2-3: Behaviorism and Constructivism
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10th Edition, © 2012 14

15. Constructivism (cont.)
Learning reflects a social process in which children engage in dialogue and discussion with themselves as well as others as they develop intellectually.
William Brownell,
Jean Piaget,
Jerome Bruner,
Zoltan Dienes
Master 2-3: Behaviorism and Constructivism
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10th Edition, © 2012 15

16. HOW CAN WE HELP CHILDREN MAKE SENSE OF MATHEMATICS?
Several characteristics and stages of thinking exist; children progress through stages as they mature.
Recommendation #1: Teachers should teach to the developmental characteristics of students.
Master 2-4: How Children Learn
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17. HOW CAN WE HELP CHILDREN MAKE SENSE OF MATHEMATICS?
Learners are actively involved in the learning process.
Recommendation #2: Teachers should actively involve students.
Master 2-4: How Children Learn
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10th Edition, © 20012 17

18. HOW CAN WE HELP CHILDREN MAKE SENSE OF MATHEMATICS?
Learning proceeds from the concrete to abstract.
Recommendation #3: Teachers
should move learning from
concrete to abstract.
Master 2-4: How Children Learn
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10th Edition, © 2012 18

19. HOW CAN WE HELP CHILDREN MAKE SENSE OF MATHEMATICS?
Learners need opportunities for talking and communicating their ideas with others.
Recommendation #4: Teachers should use communication to encourage understanding.
Master 2-4: How Children Learn
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10th Edition, © 2012 19

20. CHAPTER 3 Planning For and Teaching Diverse Learners
To accompany Helping Children Learn Math10e, Reys et al.
©2009 John Wiley & Sons

21. Focus Questions
1. What questions must an elementary mathematics teacher answer when planning for teaching? 2. Why does the teacher plan mathematics lessons so carefully? 3. What levels of plans does the teacher create? 4. What are three types of lessons used to teach mathematics, and what is the purpose of each? 5. How can the teacher meet the needs of all students? 6. How does the teacher integrate planning with assessing and analysis?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012

22. PREPARING TO TEACH: QUESTIONS TO ASK BEFORE PLANNING BEGINS
Do I understand the mathematics I am teaching?
Where are my students developmentally?
What do my students know?
Master 3-1: Preparing to Teach
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10th Edition, © 2012 22

23. Preparing to Teach (cont.)
What kinds of tasks will I give my students?
How will I encourage my students to communicate?
Master 3-1: Preparing to Teach
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10th Edition, © 2012 23

24. Preparing to Teach (cont.)
What materials will we use?
Master 3-1: Preparing to Teach
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10th Edition, © 2012 24

25. Effective Mathematical Tasks
Are often authentic in that they come from the students’ environment
Are challenging yet within students’ reach
Pique the students’ curiosity
Encourage students to make sense of mathematical ideas
Encourage multiple perspectives and interrelated mathematical ideas
Nest skill development in the context of problem solving
(Reys and Long, 1995)
Master 3-2: Good Mathematical Tasks
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10th Edition, © 2012 25

26. Include questions that help students:
work together to make sense of mathematics.
rely more on themselves to determine whether something is mathematically correct.
learn to reason mathematically.
learn to conjecture, invent, and solve problems.
connect mathematics, its ideas, and its applications.
Master 3-3: Questions
Professional Standards for Teaching Mathematics
National Council of Teachers of Mathematics, 1991, pp. 3,4
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10th Edition, © 2012 26

27. Manipulatives The teacher should be certain that:
manipulatives have been chosen to support the lesson's objectives.
students have received orientation concerning the manipulatives and classroom procedures.
the lesson involves active participation of each student.
the lesson plan includes procedures for evaluation that reflect an emphasis on the development of reasoning skills.
Master 3-4: Manipulatives
Ross and Kurtz, 1993, p. 256
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10th Edition, © 2012 27

28. Teaching English Language Learners (cont.)
Use teaching strategies and groupings that reduce the anxiety of students.
Assign activities in the classroom that offer students opportunities for active involvement.
(Herrell, 2000, p. xiv)
Master 3-11: Teaching English Language Learners
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10th Edition, © 2012 28

29. Meeting the Needs of All Students: Potential Barriers for Students with Special Needs
Memory: visual memory, verbal/auditory memory, working memory
Self-regulation: excitement/relaxation, attention, inhibition of impulses
Visual Processing: visual memory, visual discrimination, visual/spatial organization, visual-motor coordination
Master 3-12: Potential Barriers for Students with Special Needs
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10th Edition, © 2012 29

30. Potential Barriers for Students with Special Needs (cont.)
Language Processing: expressive language, vocabulary development, receptive language, auditory processing
Related academic skills: reading, writing, study skills
Motor Skills: writing legibly, aligning columns, working with small manipulatives, using one-to- one correspondence, writing numerals
(Karp and Howell, 2004, p. 120)
Master 3-12: Potential Barriers for Students with Special Needs
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10th Edition, © 2012 30

31. Meeting the Needs of All Students: Nine Types of Adaptations
Size
Time
Level of Support
Input
Difficulty
Output
Participation
Alternate Goals
Substitute Curriculum
Master 3-13: Adaptations
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10th Edition, © 2012 31

32. CHAPTER 4 Assessment: Enhanced Teaching and Learning
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10th Edition, © 2012

33. Assessment
Assessment should support the learning of important mathematics and provide useful information to teachers and students.
(NCTM, 2000, p.22)
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012

34. Focus Questions
1. How are assessments of learning (summative assessment) and assessments for learning (formative assessment) alike and different—in characteristics and in when or how they are used? 2. How do the four phases of classroom assessment help teachers inform their instruction? 3. What different methods can teachers use to gather information about their students’ abilities, dispositions, and interests, and what do each of these methods communicate to students about what is valued in teaching and learning mathematics?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012

35. Two Types of Assessment
1. Summative
2. Formative
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012

36. Summative Assessment
Assessment of Learning - Summative assessment provides evidence of student achievement for purposes of public reporting and accountability.
Ex. tests, end-of year exams, standardized tests.
Master 4-1: Two Types of Assessment
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10th Edition, © 2012 36

37. Formative Assessment
Assessment for Learning – Formative Assessment documents students’ achievement as well as guides instructional decisions and helps students learn.
Ex. homework, in-class assignments, performance assessments, teacher observations, classroom tests.
Master 4-1: Two Types of Assessment
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10th Edition, © 2012 37

38. Four Phases of Assessment
Plan
Assessment
Gather
Evidence
Interpret
Use
Results
Master 4-3: Purposes and Phases of Assessment
Assessment Standards for School Mathematics
National Council of Teachers of Mathematics, 1995
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10th Edition, © 2012 38

39. Four Purposes of Assessment
Making Instructional Decisions
Monitoring Students' Progress
Evaluating Students' Achievement
Evaluating Programs
Assessment Standards for School Mathematics
National Council of Teachers of Mathematics, 1995
Master 4-3: Purposes and Phases of Assessment
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9th Edition, © 2009 39

40. Assessment Standards for School Mathematics
Shifts in Making Instructional Decisions toward:
integrating assessment with instruction
using evidence from a variety of assessment formats and contexts
using evidence of every student's progress toward long- range goals in instructional planning
Master 4-2: Assessment Shifts
Assessment Standards for School Mathematics
National Council of Teachers of Mathematics, 1995
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10th Edition, © 2012 40

41. Assessment Standards for School Mathematics
Shifts in Monitoring Students’ Progress toward:
assessing progress toward mathematical power
communicating with students about performance in a continuous, comprehensive manner
using multiple and complex assessment tools
students learning to assess their own progress
Master 4-2: Assessment Shifts
Assessment Standards for School Mathematics
National Council of Teachers of Mathematics, 1995
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10th Edition, © 2012 41

42. Assessment Standards for School Mathematics
Shifts in Assessing to Evaluate Students’ Achievement toward:
comparing students’ performance with performance criteria
assessing progress toward mathematical power
certification based on balanced, multiple sources of information
profiles of achievement based on public criteria
Master 4-2: Assessment Shifts
Assessment Standards for School Mathematics
National Council of Teachers of Mathematics, 1995
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10th Edition, © 2012 42

43. Ways to Assess Observation Questioning Interviewing Performance Tasks
Self-Assessment and Peer Assessment
Master 4-4: Ways to Assess
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10th Edition, © 2012 43

44. Ways to Assess Work Samples Portfolios Checklists Writings
Teacher-Designed Written Tests
Standardized Achievement Tests
Student Self Assessment
Master 4-4: Ways to Assess
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9th Edition, © 2009 44

45. Communicating Assessment Information
Teachers have three main audiences to whom assessment will be communicated:
To Students
To Parents or Guardians
To School Administration
Master 4-5: Keeping Records and Communicating about Assessment
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10th Edition, © 2012 45

46. CHAPTER 5 Mathematical Processes & Practices
To accompany Helping Children Learn Math10e, Reys et al.
©2012 John Wiley & Sons

47. Focus Questions
What five processes are identified in Principles and Standards for School Mathematics as key to an active vision of learning and doing mathematics?
What eight mathematical practices are highlighted in the Common Core State Standards for Mathematics (CCSSI, 2010)?
How is teaching mathematics through problem solving different from simply teaching students to solve problems?
For young children, what does mathematical reasoning involve and how does it help them make sense of mathematical knowledge and relationships? How can elementary children be encouraged to communicate their mathematical thinking?
What connections are important to aid elementary children in learning mathematics? What are three major goals for representation as a process in elementary school mathematics?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012

48. NCTM Process Standards
Problem Solving
Reasoning and Proof
Communication
Connections
Representations
Principles and Standards for School Mathematics
(NCTM, 2000)
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10th Edition, © 2012

49. NCTM Process Standards
Instructional programs from pre-kindergarten through grade 12 should enable students to:
Problem Solving
build new mathematical knowledge through problem solving
solve problems that arise in mathematics and in other contexts
apply and adapt a variety of appropriate strategies to solve problems
monitor and reflect on the process of mathematical problem solving
Master 5-1: Process Standards
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10th Edition, © 2012 49

50. NCTM Process Standards
Instructional programs from pre-kindergarten through grade 12 should enable students to:
Reasoning and Proof
Recognize reasoning and proofs as fundamental aspects of mathematics
Make and investigate mathematical conjectures
Develop and evaluate mathematical arguments and proofs
Select and use various types of reasoning and methods of proof
Master 5-1: Process Standards
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51. NCTM Process Standards
Instructional programs from pre-kindergarten through grade 12 should enable students to:
Communication
organize and consolidate their mathematical thinking through communication
communicate their mathematical thinking coherently and clearly to peers, teachers, and others
analyze and evaluate the mathematical thinking and strategies of others
use the language of mathematics to express mathematical ideas precisely
Master 5-1: Process Standards
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52. NCTM Process Standards
Instructional programs from pre-kindergarten through grade 12 should enable students to:
Connections
recognize and use connections among mathematical ideas
understand how mathematical ideas interconnect and build on one another to produce a coherent whole
recognize and apply mathematics in contexts outside of mathematics
Master 5-1: Process Standards
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53. NCTM Process Standards
Instructional programs from pre-kindergarten through grade 12 should enable students to:
Representations
create and use representations to organize, record, and communicate mathematical ideas
select, apply, and translate among mathematical representations to solve problems
use representations to model and interpret physical, social, and mathematical phenomena
Master 5-1: Process Standards
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10th Edition, © 2012 53

54. How Can Teachers Support Mathematics Learning with the Process Standards?
For each standard, list specific instructional practices you plan to include in your classroom.
Problem Solving -encourage sense making, nonroutine problems
Reasoning and Proof -encourage conjectures and explanation of ideas
Master 5-2: Supporting Mathematics Learning with the Process Standards
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10th Edition, © 2012 54

55. How Can Teachers Support Mathematics Learning with the Process Standards? (cont’d)
Communication-work individually and in small groups, use whole class discussion, and writing
Connections-connect to real life and other subjects
Representations-provide a variety of materials, have students use objects, symbols, pictures and look for various representations/solutions
Master 5-2: Supporting Mathematics Learning with the Process Standards
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10th Edition, © 2012 55

56. Eight Common Core State Standards for Mathematical Process

57. CHAPTER 6 Helping Children with Problem Solving
Tina Rye Sloan To accompany Helping Children Learn Math10e, Reys et al.
©2012John Wiley & Sons

58. Focus Questions
What is the difference between solving problems and practicing exercises?
What does it mean to teach math through problem solving? What “signposts” for teaching guide this approach?
What types of problems can be used in teaching through problem solving?
What strategies for problem solving are helpful for elementary students?
Why is looking back such an important phase in problem solving? What questions should students learn to ask themselves when they are solving problems and reflecting on their solutions?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012

59. Which terms are synonyms?
Problem Types
Problem-involves a situation in which the solution route is not immediately obvious
Exercise-a situation in which the solution route is obvious
Routine problem-the application of a mathematical procedure in the same way it was learned
Non-routine problem-the choice of mathematical procedures is not obvious
Which terms are synonyms?
Master 6-1: Problem Types
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60. Are these problems or exercises?
Problem Types (cont’d)
Are these problems or exercises?
15 rows of stamps. 8 stamps in each row. How many stamps?
24 packs of baseball cards.
8 cards in a package.
How many baseball cards?
Master 6-2: Is this a problem?
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10th Edition, © 2012 60

61. Try this. Is this a problem for you?
Problem Types (cont’d)
Try this. Is this a problem for you?
Use the numerals 0,1,2,3,4,5,6,7,8 to form a 3 by 3 square. The sum of the numbers in every row is 12. The sum of the numbers in every column is 12.
Master 6-2: Is this a problem?
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62. Try this. Is this a problem for you?
Problem Types (cont’d)
Try this. Is this a problem for you?
Begin with the digits 1, 2 ,3, 4 ,5, 6, 7, 8, and 9. Use each digit at least once and form three four-digit numbers with the sum of 9636.
  ___ ___ ___ ___ +___ ___ ___ ___ + ___ ___ ___ ___ = 9636
Master 6-2: Is this a problem?
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10th Edition, © 2012 62

63. Signposts for Teaching Mathematics Through Problem Solving
Signpost 1: Allow Mathematics to Be Problematic for Students Signpost 2: Focus on the Methods Used to Solve Problems Signpost 3: Tell the Right Things at the Right Time
Master 6-3: Signposts for Teaching Mathematics Through Problem Solving
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64. Factors for Success in Problem Solving
Instruction should build on what children already know.
Engaging children in problem solving should not be postponed until after they have “mastered” computational skills.
Children should be taught a variety of problem-solving strategies to draw from.
Master 6-4: Factors for Success in Problem Solving
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10th Edition, © 2012 64

65. Factors for Success in Problem Solving
Children’s problem-solving achievements are related to their developmental level. Thus, they need problems at appropriate levels of difficulty.
Factors which contribute to children’s difficulties with problem solving include knowledge, beliefs and affects, control, and sociocultural factors.
Master 6-4: Factors for Success in Problem Solving
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10th Edition, © 2012 65

66. Choosing Appropriate Problems
Consider including problems that:
Ask students to represent a mathematical idea in various ways.
Ask students to investigate a numeric or geometric concept.
Require students to estimate, or to decide on the degree of accuracy required, or to apply mathematics to practical situations.
Master 6-5: Choosing Appropriate Problems
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67. Polya’s Model of Problem Solving
Understand the problem.
Devise a plan for solving it.
Carry out your plan.
Look back to examine your solution.
Master 6-6: Problem Solving Strategies
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10th Edition, © 2012 67

68. Problem-Solving Strategies
Act It Out Make a Drawing or Diagram Look for a Pattern Construct a Table Guess and Check Work Backward Solve a Simpler but Similar Problem
Master 6-6: Problem Solving Strategies
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10th Edition, © 2012 68

69. The Importance of Looking Back
Some of the most important learning that results from problem solving occurs after the problem has been solved, when students look back at the problem, at the solution, and at how they found the solution.
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70. The Importance of Looking Back
Look back at the problem.
Look back at the answer.
Look back at the solution process.
Look back at one’s own thinking.
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10th Edition, © 2012