Problem Solving and Reasoning Problems, Strategies, and Lesson Design

1
Problem Solving and Reasoning Problems,
Strategies, and Lesson Design

• TAMU-QUMethods for Teaching Mathematics

2
Value of Teaching With Problems

• Places students attention on ideas and sense
  making
• Develops mathematical power
• Develops the belief in students that they are
  capable of doing mathematics and that is makes
  sense
• Provides ongoing assessment data that can be used
  in instructional decisions

3
Hierarchy of Thinking

• Upper two levels
• Higher-Order

CREATIVE
Upper Three Levels --Reasoning
CRITICAL
BASIC
RECALL
4
A problem is defined by

• Any task or activity for which a student is
  seeking some goal for which a suitable course of
  action is not immediately apparent. A
  mathematical problem requires the use of math
  skills, concepts, or processes to arrive at the
  goal.

5
Criteria for Mathematical Problems

• Perplexing situation that the student understands
• Student is interested in finding a solution
• Student is unable to proceed directly toward a
  solution
• Solution requires use of
• mathematical ideas

6
Problem-Solving Strategies

• Look for a Pattern
• Make an Organized List
• Draw a Picture
• Guess and Check
• Write an Equation
• Construct a Table
• Act it Out
• Use Objects
• Work Backward
• Solve a Simpler (or similar) Problem

7
Strategy for solving a problem

• Understand the Problem
• Read and Explore
• Plan
• Devise a Plan
• Carry out the Plan
• Solve the Problem
• Look Back
• Reflect on Answer

8
Cooperative Problem-Solving Lesson
Three-Part Format

• Introducing
• Exploring
• Summarizing

9
Introducing

• Present or review concepts
• Pose a part of the problem or a similar but
  smaller one
• Present the problem to be solve
• Discuss to make sure students understand what
  they are to do

10
Exploring

• Observe the interaction, listening for use in
  later discussion to groups ideas, strategies,
  and work procedures
• Offer assistance when needed, either when all
  members of a group raise their hands or if a
  group is not working
• Provide an extension for groups that finish more
  quickly than others

11
Summarizing

• Have groups report their processes, both group
  procedures and strategies used
• Have groups present solutions
• Generalize from the solutions
• Questions to Consider
• How did you organize the work in your group?
• What problems did you encounter?
• What strategies did you use? Which were most
  helpful?
• How did you arrive at your solution?
• What other strategy might have been used instead?

12
Independent Problem Solving
The Menu

• A Menu is a collection of activities for a
  student to do. A menu can provide class work
  activities for several days, a week, or for a
  longer period of time. The tasks on the menu or
  not hierarchical and do not conceptually build
  upon each other.

13
How much to tell and not to tell

• Three types of information teachers should
  provide to their students
• Math conventions
• Alternative methods
• Clarification of students methods

14
GoalsTeaching about Problem Solving

• Strategies and Process
• Develop problem analysis skills
• Develop and select strategies
• Justify solutions
• Extend of generalize problems
• Metacognition
• Monitor and regulate actions
• Disposition
• Gain confidence and belief in abilities
• Be willing to try and persevere
• Enjoy doing mathematics

15
Common Arguments of Problem Solving

• There is not enough time.
• The tests we have to give mainly test arithmetic.
• Parents expect the school to teach arithmetic.
• Problem solving is too hard for students who have
  difficulty with the basic skills.
• Some teachers are not good in math and do not
  feel comfortable teaching what they do not
  understand.
• What do you think?