# Problem solving

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## INTRODUCTION

Looking closer at the math of the Indian mathematician SRINIVASA RAMANUJAN I came to see a name I recognized. I remembered Gorge Polya but could not remember why he was so important for me. I looked through my old lessons in google drive and found lesson 87 held April 14 2015, with this doc https://docs.google.com/presentation/d/1Aj2WHzaxP-vmt0jvWWnFfUzzzguIJyj0Rx6DmQLcRoE/edit?usp=sharing

I understood that it was Polya’s problem-solving strategy, that I tried to use with my math students. I discovered that when my pupils had the objects in front of them, they could easily solve problems. They needed to visualize the object talked about in the books. So I  had my pupils from grade 4 to visualize the problems with simple sketches (like a piece of chocolates that had a cost or another value, e.g. 1/8.

I decided to make similar share like math.berkeley.edu but I let Polya’s word stay as they are in the original book in pdf format.

Melvin writes:

“In 1945 George Polya published the book How To Solve It which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. In this book, he identifies four basic principles of problem-solving. “

# INDEX

### 1. Polya’s First Principle: Understand the problem

• First. You have to understand the problem.
• What is the unknown? What are the data? What is the condition?
• Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
• Draw a figure. Introduce suitable notation.
• Separate the various parts of the condition. Can you write them down?

### 2. Polya’s Second Principle: Devise a plan

• Have you seen it before? Or have you seen the same problem in a slightly different form?
• Do you know a related problem? Do you know a theorem that could be useful?
• Look at the unknown! Try to think of a familiar problem having the same or a similar unknown.
• Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary elements in order to make its use possible?
• Could you restate the problem? Could you restate it still differently? Go back to definitions.

• If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?
• Did you use all the data? Did you use the whole condition? Have you taken into account all the essential notions involved in the problem?

### 3. Polya’s Third Principle: Carry out the plan

• Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

### 4. Polya’s Fourth Principle: Look back

• Can you check the result? Can you check the argument?
• Can you derive the solution differently? Can you see it at a glance?
• Can you use the result, or the method, for some other problem?

## conclusion

I followed mainly these steps:

### 1. Polya’s First Principle: Understand the problem

Draw a figure. Introduce suitable notation.

Draw images of the tools in question

I listened to  Ann-Louise Ljungdahl who emphasized helping students with their inner vision. If you can visualize a problem then it is easier to solve it.

So I asked my students to write down a fact sheet, containing:

• the data and units you got
• the formulas that may be applied to this kind of problem.
• Draw geometrical figures if these are part of the problems.
• for smaller kids with candy and price problems, I asked them to draw the candies.

One way to understand the problem is to be able to write an answer before the start calculating e.g “The banana will cost  ____ Euro that is less than 3 Euro.

I asked them also to evaluate what a reasonable answer could with an interval     <10 Euro  < X  > 100 Euro and what unit it should be.

### 3. Polya’s Third Principle: Carry out the plan

• Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

The factsheet I mentioned above became a important strategy part of my standard problem-solving plan.

### 4. Polya’s Fourth Principle: Look back

• Can you check the result? Can you check the argument?
• Can you derive the solution differently?

I always asked my students to check the answer. Putting the answer into the original problem, e.g. the result in the equation.

I asked to check if the answer is reasonable. That a kilo of banana costs 1000 Euro, is that reasonable?

There is no one telling you what is right. You have to derive the solution differently to see that you get the same answer.

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## Notebook of a pluralist

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