Problem-Solving for Educators

Welcome to our Problem Solving Course!  

“Mathematics is a living subject which seeks to understand patterns that permeate both the world around us and the mind within us. Although the language of mathematics is based on rules that must be learned, it is important for motivation that students must move beyond rules to be able to express things in the language of mathematics.” (Schoenfeld, 1992). We are training our students to be compliant and not innovative. With all our prowess and access to technology, our students still lack the fundamental skills of thinking for themselves  This program is designed to train educators to teach problem-solving skills to our students because our current educational system does not currently do that.

There are three primary learning objectives for this course: 

This course will focus on a research-based, problem-solving process developed by George Polya in 1945. He identifies four-basic strategies we all use when problem solving. 

  1. Understand the Problem
  2. Devise a Plan
  3. Carry Out the Plan
  4. Look Back

Once the content has been reviewed, students will take an assessment and must achieve an 80% or above to earn a certificate for the course. Once students have satisfactorily met the learning objectives and have completed the course, they will have an opportunity to provide feedback to the designers of this program so we can continue to make improvements and make this course better. 

 

Thank you for choosing us for your professional development and for supporting the growth and achievement of your students. 

Understand the Problem

Understanding the Problem: Overview

This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don’t understand it fully, or even in part. Polya taught teachers to ask students questions such as:

  1. Do you understand all the words used in stating the problem?
  2. What are you asked to find or show?
  3. Can you restate the problem in your own words? 
  4. Can you think of a picture or diagram that might help you understand the problem?
  5. Is there enough information to enable you to find a solution?
  6. What is known or unknown?

Be sure to consider the terminology and notation used in the problem. If time allows it, write down a problem similar to the one you are working. Sometimes it's good to interpret things in a different context. 

Retrieved from learnlogic.net 

Devise a Plan

Devise a Plan: Overview

Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You must start somewhere so try something. How are you going to attack the problem? You will find choosing a strategy increasingly easy. A partial list of strategies is included:

  • Guess & Check
  • Make an orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation
  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backwards
  • Use a formula
  • Be ingenious

Once you understand what the problem is, if you are stumped or stuck, set the problem aside for a while. Your subconscious mind may keep working on it. Moving on to think about other things may help you stay relaxed, flexible, and creative rather than becoming tense,frustrated, and forced in your efforts to solve the problem.

Carry Out the Plan

Carry Out the Plan: Overview

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don’t be misled, this is how mathematics is done, even by professionals.

Once you have an idea for a new approach, jot it down immediately. When you have time, try it out and see if it leads to a solution. If the plan does not seem to be working, then start over and try another approach. Often the first approach does notwork. Do not worry, just because an approach does not work, it does not mean you did it wrong. You actually accomplished something, knowing a way does not work is part of the process of elimination!

Once you have thought about a problem or returned to it enough times, you will often have a flash of insight: a new idea to try or a new perspective on how to approach solving the problem. The key is to keep trying until something works!

Look Back

Look Back/Verification: Overview

Polya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked, and what didn’t. Doing this will enable you to predict what strategy to use to solve future problems. Once you have a potential solution, check to see if it works.

  1. Did you answer the question?
  2. Is your result reasonable? 
  3. Double check to make sure that all of the conditions related to the problem are satisfied.
  4. Double check any computations involved in finding your solution.


If you find that your solution does not work, there may only be a simple mistake. Try to fix or modify your current attempt before scrapping it. Remember what you tried—it is likely that at least part of it will end up being useful. Is there another way of doing the problem which may be simpler? (You need to become flexible in your thinking. There usually is not one right way.) Can the problem or method be generalized so as to be useful for future problems?

Remember, problem solving is as much an art as it is a science‼

Review & Assessment

Reminders!

Review of Possible Strategies

  1. Draw pictures
  2. Use a variable and choose helpful names for variables or unknowns 
  3. Be systematic
  4. Solve a simpler version of the problem 
  5. Guess and check. Trial and error. Guess and test. (Guessing is OK.)
  6. Look for a pattern or patterns 
  7. Make a list.

Which method is NOT one of the methods prescribed by George Polya?

  • Devise a Plan
  • Look Back
  • Edit and Revision
  • Carry Out the Plan

According to Polya, what is a strategy we can use to devise a problem-solving plan?

  • Use a model
  • Use my notes
  • Use my neighbor
  • Use the internet

In what year did Polya invent this method?

  • 2017
  • 1914
  • 1975
  • 1945

What should you do after carrying out your plan?

  • Devise a Plan
  • Understand the Problem
  • Look Back/Review Your Answers
  • Move On to the Next Problem.

What are the four steps of Polya's problem-solving process?

  • Understand the Problem
  • Move on to the Next Problem
  • Devise a Plan
  • Carry Out the Plan
  • Ignore the Plan
  • Look Back/Verify Your Answers

Feedback

Congratulations you finished the course!

Congratulations!

Thank you for finishing this course. 

The course had appropriate communication, demonstration, and computation capabilities.

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The course objectives were clearly stated and used understandable terms.

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The course materials were both useful and easy to follow.

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The course relates directly to my current job responsibilities.

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I would recommend this course to other teammates.

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