Techniques of Problem Solving

Agenda

• Problem Solving and IB
• Polya and Problem Solving
• Practice

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Problem Solving and IB

Problem Solving and IB

• Part of the “IB Mathematical Toolkit”
• Investigative
• Improving problem solving
• Improving mathematical modelling
• Improving mathematical communication

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Polya and Problem Solving

Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematics problems.

George Polya, How to Solve It, 1945.

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Polya and Problem Solving

• Types of problems to solve?
• In this course (and most of your schooling)
• Not wicked problems
• The problems will be:
• Well-defined
• Self-contained
• (Mostly) solvable
• (Usually) related to your course

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Polya and Problem Solving

• How to solve problems?
• Recognize that you can solve it
• Mindset - you may not know how to solve it - yet
• There may require one or more insights before solving is possible
• There may require specific application of techniques

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The Learning

Pit

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Polya Problem Solving Process

• Understand the problem
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• Devise a plan
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• Carry out the plan
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• Revise if not done
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• Reflect
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• Many of these steps are unconscious! Be conscious of the process!

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Polya Problem Solving Process

• Understand the problem
• What is (are) the goal(s)?
• What are we trying to achieve?
• What are the conditions? Data? Unknowns?
• Draw a picture (or many)
• Devise a plan
• How do we get there? One step? Multiple steps?
• Carry out the plan
• Is each step correct? Is this provable?
• Revise if not done
• Don’t give up after multiple failures
• Reflect
• Are there other approaches?
• Check for correctness and generalize
• Many of these steps are unconscious! Be conscious of the process!

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Understanding the Problem

• Establish goals
• Do we need to add anything to the problem to help understand it?
• Variables, diagrams, extensions
• Can you explain the problem to someone else?
• Use “rubber duck debugging” if no one is available to listen
• Can you solve a simpler, related problem?
• This might give clues to the original

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Devise a Plan

• Where are the signposts of your journey?
• What are the subgoals?
• You might not know everything yet!
• That’s ok!
• Requires flexibility of thinking
• Need ideas of what might work
• Might need to learn new things first

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Carry Out The Plan

• Requires techniques and fluency
• Correct implementation
• Self-correction
• The final result is wrong if the steps have mistakes

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Revise

• What if we’re stuck?
• You might need several attempts
• Problem solving is rarely linear
• Try different approaches!

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Reflect

• Yay, we’re done!
• Not quite 🙃
• For complicated problems, there are many different approaches
• Some methods work better than others
• Some methods improve our understanding in different ways
• Now that you’ve finished one approach, try a different one!
• How do they connect to other branches of mathematics?
• Don’t just finish the problem to get it done
• The problem is there to improve your learning

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Practice

Practice

• Go to the Jamboard Unit 2 Day 2 Techniques of Problem Solving

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CREDITS

Special thanks to all the people who made and released these awesome resources for free:

• Presentation template by SlidesCarnival
• Photographs by Unsplash
• "Rubber Duck" by Paul Gravestock is licensed under CC BY-NC-ND 2.0
• "Can get there from here" by Audra B is licensed under CC BY-NC-ND 2.0
• "Second Floor Plans" by Smalloy is licensed under CC BY-NC-SA 2.0
• "The Thinker" by ajk408 is licensed under CC BY-NC-SA 2.0

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