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Mathematical Problem Solving

The goal of any mathematics course is to introduce you to a variety of methods used to solve real-world problems. In this way, you are able to practice solving problems and, along the way, improve your critical thinking skills.

It may seem that solving a mathematical problem involves nothing more than knowing the right formula and following the correct steps. However, that is only part of the process. Solving problems is more of an art than a science which means that human intuition and imagination play an extremely important role in the process. Through these human abilities, we link experience to knowledge and skills resulting in a synergy that leads to a solution.

In 1945, mathematician George Polya (1887-1985) published a book titled How To Solve It in which he demonstrated his approach to solving problems. Here are his principles of problem solving:

Polya’s First Principle: Understand the Problem

To solve a problem, you have to understand the problem.

  • Do you understand all the words used in stating the problem?
  • What is the problem looking for?
  • What data or information has been provided in the problem?
  • Are there any special conditions mentioned in the problem that we need to consider in the solution?
  • Can you restate the problem in your own words?
  • Can you draw a picture or make a diagram that could help you solve the problem?
  • Is there enough information in the problem to help you find a solution?

Polya’s Second Principle: Devise A Plan

There are many different ways to solve a problem.  The way you choose is based upon your own creativity, level of knowledge, and skills. Basically, solving the problem involves finding the connections that exist between the data you’ve been provided and the unknown you need to find.

Here is the assignment:

Think of a problem that involved several steps that you had to follow to reach a solution.  If you can’t think of a problem, look at any of the problems in Chapter 1 in the book and pick one that you can explore here.

Prepare a 200-300 word multiple paragraph response defining the problem and the steps needed to reach a solution.

Then in your responses to at least two classmates, address the following items:

  1. Comment on the problems they demonstrated and the steps they employed to reach solutions. 
  2. What would you have done the same or different? 
  3. Do you agree with the solution? 
  4. Can you suggest a different approach to solving the same problem?

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