Teaching Through Problem Solving

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Teaching Through Problem Solving

• NOTES
• __________________________________________________
  ___________________
• Adapted from
• Van de Walle (2007). Elementary and Middle
  School Mathematics Teaching Developmentally,
  (6th Ed.). Pearson Allyn Bacon, pp. 37-70.

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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
• Allows entry point for a wide range of students

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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

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Hierarchy of Thinking

• Upper two levels
• Higher-Order

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

• Any task or activity for which a student has
  no prescribed or memorized rules or methods, nor
  is there a perception by the student that there
  is a specific correct solution method.
• (Hiebert et al., 1997)

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Features of a Good Math Problem

• Must begin where the students are
• Task must take into consideration the current
  understanding of the students
• Problematic or engaging aspect of the problem
  must be due to the mathematics the students are
  to learn
• Mathematics of the problem should be the focus
  not the context or creation of a product
• Must require justifications and explanations of
  answersshould be an important part to problem
  solutions

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Example of Problem-Based Task

• The following task might be used in grades 5-6
  as part of the development of fraction concepts
• __________________________________________________
  ______
• Place an X on the number line about where
  11/8 would be. Explain why you put the X where
  you did. Perhaps you will want to draw and label
  other points on the line to help explain your
  answer.

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Types of Problems

• Process Problems
• Requires solution processes other than
  computational procedures
• Translation Problems
• Include one- and two-step story problems
  typically found in textbooks
• Application Problems
• Generally uses computation once data is gathered
  and a decision about a solution strategy has been
  made
• Puzzles

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• U.S. teachers usually spend a small portion
  of a lesson explaining or reviewing an idea and
  then have students go through a set of exercises.
  Lessons in this format follow the
  explain-then-practice pattern. Teachers find
  themselves going from desk to desk re-teaching
  and explaining to individuals. This is in
  contrast to a lesson built around a single
  problemwhich is the typical approach for
  student-centered problem-based lessons.
• __________________________________________________
  ____
• Teaching through problem solving suggests a
  simple three part structure for lessons.
• Before
• During
• After

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Problem-Solving Lesson
Three-Part Format

• Before--Introducing
• During--Exploring
• After--Summarizing

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Before Phase

• Be sure the task is understood by students
• Establish expectations for how students are to
  work and what products are to be prepared
• Prepare students mentally for the task by
  activating student prior knowledge related to the
  problem

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During Phase

• Let gogive students a chance to work without
  guidance.
• Observe interactions and listen for group ideas,
  strategies, and work procedures to use in class
  discussions
• 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

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After Phase

• Engage class in productive discussionencourage
  student-student dialogue and not just
  student-teacher dialogue that may exclude the
  class by having students ask questions also
• Listen actively without evaluationhave students
  explain methods, solutions, and justifications
  for answers
• Summarize main ideas and identify problems that
  could be used for future explorations
• Some Questions to Consider
• How did you organize your work?
• 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?

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Planning for Instructionwhat to consider

• Selecting appropriate tasks and materials
• Identifying sources of problems
• Clarifying the teachers role
• Organizing and implementing instruction
• Changing the difficulty of problems by altering
  the --Problem Context --Problem Structure

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Selecting Effective Tasks

• Sources of Effective Tasks
• Textbooks
• Standards-Based Curricula
• National Council of Teachers of Mathematics
  JournalMathematics Teaching in the Middle School
• National Council of Teachers of Mathematics
  Navigation Series books
• Internet Lesson Plans

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Steps for Task Selection

• 1. How is the activity done?
• Actually do the activity to determine how
  students might do the activity, what materials
  are needed, what is recorded or written down, and
  what misconceptions may emerge.
• 2. What is the purpose of the activity?
• Determine what mathematical ideas are developed
  by the activity. Are these ideas concepts or
  procedural skills? What connections can be made
  to other topics?
• 3. Will the activity accomplish its purpose?
• 4. What must the teacher do?
• --In the before part of the lesson?
• --In the during part of the lesson?
• --In the after part of the lesson?

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Planning a Problem-Based Lesson

• Content and Task Decisions
• Begin with the Math!
• Articulate clearly the ideas you want students to
  learn as a result of the lesson
• Consider your students
• What do the students already know and understand
  about the topic
• Decide on task
• Keep it simplegood tasks need not be elaborate
• Predict what will happen
• Think about what your students will likely do
  with the taskmodify task as needed based on your
  predictions

• Teaching Actions
• Articulate student responsibilities
• Think about how students will supply information
  about what they learned
• Plan the Before portion of lesson
• Think about the During portion of the lesson
• Think about the After portion of the lesson

Last, Write the Lesson Plan
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Variations of the Three-Part Lesson

• MinilessonsTasks that do not require the entire
  class period--think-pair-share strategy is useful
• Workstations and GamesCan be set-up around the
  room without the need to distribute and collect
  materials to allow students to work on different
  tasks and concepts
• Problem-Solving MenuA 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.

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Four Step Guide for Solving a Problem

• Understand the Problem
• Read and explore
• Plan
• Devise a strategy to use to solve the problem
• Carry out the Plan
• Solve the problem
• Look Back
• Reflect on answer and justify solution

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Some Problem-Solving Strategies

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

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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

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Diversity in the Classroom

• In addition to using a problem-based approach,
  consider the following to attend to the diversity
  in the classroom.
• Be sure that problems have multiple entry points
• Plan differentiated tasks
• Use heterogeneous groupings
• Make accommodations and modifications for
  English-language learners
• Listen carefully to students

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Drill or Practice?

• Practice
• Different problem-based tasks or experiences,
  spread over numerous class periods, each
  addressing the same basic ideas
• Drill
• Repetitive, non-problem-based exercises designed
  to improve skills or procedures already acquired

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What does practice provide?

• Increased opportunity to develop conceptual ideas
  and more elaborate and useful connections
• Opportunity to develop alternative and flexible
  strategies
• Greater chance for all students to understand,
  not just a few
• Clear message that math is about figuring things
  out and making sense

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What does drill provide?

• Increased facility with a strategy but only with
  a strategy already learned
• Focus on a singular method and an exclusion of
  flexible alternatives
• False appearance of understanding
• Rule-oriented view of what mathematics is about

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When is drill appropriate?

• An efficient strategy for the skill to be drilled
  is already in place
• Automaticity with the skill or strategy is a
  desired outcome

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Homework

• Practice as Homework
• Allows students to practice at home
• Communicates to parents that math is sense-making
  and has value
• Drill as Homework
• Keep it short
• Provide a key
• Do not grade based on correctness
• Do not use class time going over drill homework