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Mathematical Principles for Problem Solving

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  • 0:54 Always Principle
  • 1:59 Counterexample Principle
  • 3:17 Order Principle
  • 5:00 Splitting Hairs Principle
  • 6:10 Analogies Principle
  • 6:58 Lesson Summary
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Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

Solving problems is not just a simple, straightforward process. There are a few principles that can help you as you approach any problem solving scenarios. This lesson covers those principles with examples.

Problem Solving Principles

Problem solving in math - and in all other areas of your life, really - is one of the trickiest things to do. Many problems are similar, but must be solved in different ways. Some look very different from each other, but should be solved in similar ways. How can you know how to solve a problem, what to do to get an answer, where to even start?

Well, there are five principles that you can keep in mind to help you attack any problem solving situation you may find:

  • The Always Principle
  • The Counterexample Principle
  • The Order Principle
  • The Splitting Hairs Principle
  • The Analogies Principle

In this lesson, I will discuss these principles as well as give examples for each.

Always Principle

The always principle is fairly simple. Basically it states that something is true, 100% of the time, with no exceptions.

The sun always rises in the East; always. No exceptions. If it is morning and you are looking at the sun, you are looking to the East. Likewise, if it is late afternoon and you are facing the sun, you are facing west. No exceptions.

In math, there are a few examples of the always principle:

  • The product of two negative numbers is always a positive: -2 * -3 = +6
  • Negative numbers are always less than positive numbers: -3 < 2
  • Dividing by 0 always ends in an 'undefined' result

So, when you approach a problem, look for any 'always scenarios' that might be present.

Counterexample Principle

Do you have a friend or relationship who is always late? Have you given up on them ever being on time? If that person ever showed up somewhere on time, then it would be a counterexample to your concept that they are always late. A counterexample is an example that disproves a statement or theory. So, one act of showing up on time would forever disprove that your friend was 'always' late.

Again, in math we have some statements that we think are 'always' statements, but if even a single counterexample can be found, then we have disproved the 'always.' Take prime numbers for example. We know that prime numbers are numbers which can only be divided by themselves and 1. Some examples are 3, 5, 11, 13 and 17. What do you notice about each of these? I notice that they are all odd numbers. It would be easy to state that 'all prime numbers are odd.'

However, there is a counterexample: 2 is a prime number but is not odd. When solving problems, watch out for counterexamples to commonly held ideas.

Order Principle

A great example of how order matters is the old riddle about having to get a chicken, a fox and a bag of grain across a river one at a time. You cannot leave the fox with the chicken (the fox would eat it). You cannot leave the chicken with the grain (the chicken would eat the grain). So, how do you do it? First you take the chicken across. Then go back for the grain. When you get to the other side, leave the grain and pick up the chicken again for the return. Next, swap the chicken and the fox, leaving the chicken on the original side. Drop off the fox with the grain and return for the chicken. There, you've done it! But, it is clear that the order really did matter a great deal here.

The order principle states that order usually does matter. The most well-known order principle in math is the order of operations, which gives the order in which to conduct mathematical operations: PIMDAS, parenthesis, indices, multiplication, division, addition, subtraction, which is the order in which mathematical problems should be solved.

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