# Learntofish's Blog

## Problem solving skills

Posted by Ed on November 30, 2009

In the following I will describe what problem solving skills are and list some books.

Many students have difficulties understanding a problem but they don’t recognize why exactly they can’t solve it. They sit in front of it and have absolutely no clue on how to proceed. The famous mathematician George Polya asked himself if it is possible to teach something like “problem solving skills”. And the result is his little book entitled “How to Solve It”.

He describes four steps on how to approach a problem:
1. Understanding the problem:
First. You have to understand the problem.
What is the unknown? What are the data? What is the condition?
2. Devising a plan:
Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
3. Carrying out the plan:
Carrying out your plan of the solution, check each step
4. Looking Back
Fourth. Examine the solution obtained.
Can you check the result? Can you check the argument?
Can you derive the solution differently? Can you see it at a glance?

The steps don’t sound impressive but I can assure you they are a mighty tool. Example:
a) An object with mass 0.2kg is thrown out of the window from the third floor where each floor has a height of 4 meters. Calculate the velocity when the object reaches the ground!
b) Calculate the velocity for a mass of 0.3kg, 0.4kg, 0.5kg, 0.6kg!
Step 1:
What data is given? => mass = 0.2kg. Height = 3*4 meters
What are the unknowns? => Velocity
Step 2:
What is the connection between the data and the unknown?
=> Velocity, height and mass. Hmm…looks like it has something to do with potential and kinetic energy.
Step 3:
Potential energy is calculated by m*g*h and kintetic energy by 1/2*m*v^2.
Many students plug in the numerical values into the equation mgh=1/2mv^2 and solve for v.
Step 4:
Since in step 3 many students plug in the numerical values before solving for v they might not recognize immediately
that the mass doesn’t play any role. But of course, they will notice the same v for m= 0.3kg, 0.4kg, 0.5kg, 0.6kg. And if they look back in step 4 they might discover that solving for v first (i.e. before plugging in the numerical values) the mass cancels out..

You have probably already noticed: Problem solving is about asking the right questions, like “What is the data, what are they asking for…”. This is what makes good problem solvers successful.

Another book I’d like to recommend is “Thinking Mathematically” by Mason, Burton and Stacey.
In the same line as Polya’s book it teaches you to use a systematic approach (by asking the right questions). Last time I wrote about “How to Write Math Proofs”. And of course, if you want to be a mathematician you have to understand the language of mathematics, namely proofs. But before moving to the level of proofs you have to consider the more basic level of problem solving.
Personally, I like this book more than that of Polya because it contains more problems between the chapters. Polya’s book has problems only in the end and is more of a reference.

Recreational maths is a mean to train your problem solving skills. For this, I recommend “The Moscow Puzzles: 359 Mathematical Recreations (Math & Logic Puzzles)” by Boris A. Kordemsky. I am currently reading this book and enjoy it very much.

Each of the books above can be purchased for less than 13 Dollars or 14 Euro.