1. With George Polya (Hungarian Mathematician 1887 – 1985)
Problem Solving
With George Polya
(Hungarian Mathematician 1887 – 1985)

2. Pólya's four principles
Understand the problem
Devise a plan
Carry out the plan
Review / extend
Problem Solving with Polya

3. First principle: Understand the problem
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. Pólya taught teachers to ask students questions such as:
Do you understand all the words used in stating the problem?
What are you asked to find or show?
Can you restate the problem in your own words?
Can you think of a picture or a diagram that might help you understand the problem?
Is there enough information to enable you to find a solution?
Do you need to ask a question to get the answer?
Problem Solving with Polya

4. Second principle: Devise a plan
Also suggested:
Look for a pattern
Draw a picture
Solve a simpler problem
Use a model
Work backward
Use a formula
Be creative
Use your head / noggin
Pólya mentions (1957) that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:
Guess and check
Make an orderly list
Eliminate possibilities
Use symmetry
Consider special cases
Use direct reasoning
Solve an equation
Problem Solving with Polya

5. Third principle: Carry out the plan
This step is usually easier than devising the plan. In general (1957), 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.
Problem Solving with Polya

6. Fourth principle: Review / extend
Pólya mentions (1957) 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, if these relate to the original problem.
Problem Solving with Polya

7. Problem Solving with Polya
“There is a poetry and beauty in mathematics and every student deserves to be taught by a person that shares that point of view.”
Problem Solving means engaging in a task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understandings. Solving problems is not only a goal of learning mathematics but also a major means of doing so. Students should have frequent opportunities to formulate, grapple with, and solve complex problems that require a significant amount of effort and then be encouraged to reflect on their thinking.
Problem Solving with Polya

8. Problem Solving with Polya
Problem Solving is one of the five Process Standards of NCTM’s Principles and Standards for School Mathematics The following is taken from pages 52 through 55 of that document.
By learning problem solving in mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom. In everyday life and in the workplace, being a good problem solver can lead to great advantages. Problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program. Problem solving in mathematics should involve all five content areas: Number and Operations, Algebra, Geometry, Measurement, and Data Analysis & Probability.
Problem Solving with Polya

9. Problem Solving with Polya
GUESS AND CHECK
Copy the figure at the right and place the digits 1, 2, 3, 4, and 5 in these circles so that the sums across (horizontally) and down (vertically) are the same. Is there more than one solution?
Problem Solving with Polya

10. Problem Solving with Polya
Copy the figure and place the digits 1, 2, 3, 4, and 5 in these circles so that the sums across (horizontally) and down (vertically) are the same. Is there more than one solution?
Emphasize Polya’s four principles, so that that procedure becomes part of what the student knows.
1st. Understand the problem. Have the students discuss it among themselves in their groups of 3, 4 or 5.
2nd. Devise a plan. Since we are emphasiz-ing Guess & Check, that will be our plan.
Problem Solving with Polya

11. Problem Solving with Polya
Copy the figure and place the digits 1, 2, 3, 4, and 5 in these circles so that the sums across (horizontally) and down (vertically) are the same. Is there more than one solution?
3rd. Carry out the plan. It is best if you let the students generate the solutions. The teacher should just walk around the room and be the cheerleader, the encourager, the facilitator. If one solution is found, ask that the students try to find other(s).
Possible solutions:
Things to discuss (it is best if the students tell you these things):
· Actually to check possible solutions, you don’t have to add the number in
the middle – you just need to check the sum of the two “outside” numbers.
· 2 cannot be in the middle, neither can 4. Ask the students to discuss why.
Problem Solving with Polya

12. Problem Solving with Polya
Copy the figure and place the digits 1, 2, 3, 4, and 5 in these circles so that the sums across (horizontally) and down (vertically) are the same. Is there more than one solution?
4th. Look back. Is there a better way? Are there other solutions?
Point out that “Guess and Check” is also referred to as “Trial and Error”. However, I prefer to call this “Trial and Success”, I mean, don’t you want to keep trying until you get it right?
Problem Solving with Polya

13. Problem Solving with Polya
Below is an exercise to assign to consolidate students’ understanding of the previous example:
Put the numbers 2, 3, 4, 5, and 6 in the circles to make the sum across and the sum down equal to 12. Are other solutions possible? List at least two, if possible.
Problem Solving with Polya

14. Problem Solving with Polya
Put the numbers 2, 3, 4, 5, and 6 in the circles to make the sum across and the sum down equal to 12. Are other solutions possible? List at least two, if possible.
Problem Solving with Polya

15. Problem Solving with Polya
Put the numbers 2, 3, 4, 5, and 6 in the circles to make the sum across and the sum down equal to 12. Are other solutions possible? List at least two, if possible.
SOLUTION: One possibility: 2
Other solutions possible
Have students suggest those. 6
Problem Solving with Polya

16. Problem Solving with Polya
MAKE AN ORGANIZED LIST
Three darts hit this dart board and each scores a 1, 5, or The total score is the sum of the scores for the three darts. There could be three 1’s, two 1’s and 5, one 5 and two 10’s, and so on. How many different possible total scores could a person get with three darts?
Problem Solving with Polya

17. Problem Solving with Polya
Three darts hit this dart board and each scores a 1, 5, or The total score is the sum of the scores for the three darts. There could be three 1’s, two 1’s and 5, one 5 and two 10’s, and so on. How many different possible total scores could a person get with three darts?
1st. Understand the problem. Gee, I hope so.  But let students talk about it just to make sure.
2nd. Devise a plan. Again, it would be what we are studying: Make an organised or orderly list. Emphasize that it should be organised. If students just start throwing out any combinations, they are either going to list the same one twice or miss some possibilities altogether.
Problem Solving with Polya

18. Problem Solving with Polya
Three darts hit this dart board and each scores a 1, 5, or The total score is the sum of the scores for the three darts. There could be three 1’s, two 1’s and 5, one 5 and two 10’s, and so on. How many different possible total scores could a person get with three darts?
# of 1’s
# of 5’s
# of 10’s
Score 3 2 1 7 12 11 16 21 15 20 25 30
3rd. Carry out the plan.
There are 10 different possible scores.
4th. Look back.
Point out the there are other ways to “order” the possibilities.
Problem Solving with Polya

19. Student Consolidation Exercise
List the 4-digit numbers that can be written using each of 1, 3, 5, and 7 once and only once. Which strategy did you use?
Problem Solving with Polya

20. Problem Solving with Polya
List the 4-digit numbers that can be written using each of 1, 3, 5, and 7 once and only once. Which strategy did you use?
SOLUTION:
24 possible 4-digit numbers.
Problem Solving with Polya

21. Problem Solving with Polya
DRAW A DIAGRAM
In a stock car race, the first five finishers in some order were a Ford, a Corolla, a Peugeot, a Barina, and a Daihatsu.
The Ford finished seven seconds before the Corolla.
The Peugeot finished six seconds after the Barina.
The Daihatsu finished eight seconds after the Barina.
The Corolla finished two seconds before the Peugeot.
In what order did the cars finish the race? What strategy did you use?
Problem Solving with Polya

22. Problem Solving with Polya
In a stock car race, the first five finishers in some order were a Ford, a Corolla, a Peugeot, a Barina, and a Daihatsu. The Ford finished seven seconds before the Corolla. The Peugeot finished six seconds after the Barina. The Daihatsu finished eight seconds after the Barina. The Corolla finished two seconds before the Peugeot. In what order did the cars finish the race? What strategy did you use?
1st. Understand the problem.
Let students discuss this.
2nd. Devise a plan.
We will choose to draw a diagram to be able to “see” how the cars finished.
3rd. Carry out the plan.
Make a line and start to place the cars relative to one another so that the clues given are satisfied. We are also using guess and check here.
Problem Solving with Polya

23. Problem Solving with Polya
In a stock car race, the first five finishers in some order were a Ford, a Corolla, a Peugeot, a Barina, and a Daihatsu. The Ford finished seven seconds before the Corolla. The Peugeot finished six seconds after the Barina. The Daihatsu finished eight seconds after the Barina. The Corolla finished two seconds before the Peugeot. In what order did the cars finish the race? What strategy did you use?
The order is: Ford, Barina, Corolla, Peugeot, Daihatsu.
4th. Look back.
Not only do we have the order of the cars, but also how many seconds separated them.
Problem Solving with Polya

24. Student Consolidation Exercise:
Four friends ran a race:
· Matt finished seven seconds ahead of Ziggy.
· Bailey finished three seconds behind Sam.
· Ziggy finished five seconds behind Bailey.
In what order did the friends finish the race?
Problem Solving with Polya

25. Problem Solving with Polya
Four friends ran a race: Matt finished seven seconds ahead of Ziggy. Bailey finished three seconds behind Sam. Ziggy finished five seconds behind Bailey. In what order did the friends finish the race?
The order was: Sam, Matt, Bailey, and Ziggy.
Problem Solving with Polya

26. Problem Solving with Polya
MAKE A TABLE
Dreamworld has a special package for large groups to attend their amusement park: a flat fee of $20 and $6 per person. If a club has $100 to spend on admission, what is the most number of people who can attend?
Problem Solving with Polya

27. Problem Solving with Polya
Dreamworld has a special package for large groups to attend their amusement park: a flat fee of $20 and $6 per person. If a club has $100 to spend on admission, what is the most number of people who can attend? 12 13 14 15 16
Problem Solving with Polya

28. Problem Solving with Polya
Dreamworld has a special package for large groups to attend their amusement park: a flat fee of $20 and $6 per person. If a club has $100 to spend on admission, what is the most number of people who can attend?
$20 + $6 x 12 = $92
13 $20 + $6 x 13 = $98
14 $20 + $6 x 14 = $104
15 $20 + $6 x 15 = $110
16 $20 + $6 x 16 = $116
Problem Solving with Polya

29. Problem Solving with Polya
Dreamworld has a special package for large groups to attend their amusement park: a flat fee of $20 and $6 per person. If a club has $100 to spend on admission, what is the most number of people who can attend?
$20 + $6 x 12 = $92
13 $20 + $6 x 13 = $98
14 $20 + $6 x 14 = $104
15 $20 + $6 x 15 = $110
16 $20 + $6 x 16 = $116
Problem Solving with Polya

30. Problem Solving with Polya
Dreamworld has a special package for large groups to attend their amusement park: a flat fee of $20 and $6 per person. If a club has $100 to spend on admission, what is the most number of people who can attend?
$20 + $6 x 12 = $92
13 $20 + $6 x 13 = $98 
Problem Solving with Polya

31. Problem Solving with Polya
Quotes by Polya:
What is good education? Systematically giving opportunity to the student to discover things by himself.
He also gave the wise advice:-
If you can't solve a problem, then there is an easier problem you can solve: find it.
Problem Solving with Polya

32. Problem Solving with Polya
Quotes from Polya:
Mathematics in the primary schools has a good and narrow aim and that is pretty clear in the primary schools. ... However, we have a higher aim. We wish to develop all the resources of the growing child. And the part that mathematics plays is mostly about thinking. Mathematics is a good school of thinking. But what is thinking? The thinking that you can learn in mathematics is, for instance, to handle abstractions. Mathematics is about numbers. Numbers are an abstraction. When we solve a practical problem, then from this practical problem we must first make an abstract problem. ... But I think there is one point which is even more important. Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems.
Problem Solving with Polya

33. Problem Solving with Polya
Quotes from Polya:
Teaching is not a science; it is an art. If teaching were a science there would be a best way of teaching and everyone would have to teach like that. Since teaching is not a science, there is great latitude and much possibility for personal differences. ... let me tell you what my idea of teaching is. Perhaps the first point, which is widely accepted, is that teaching must be active, or rather active learning. ... the main point in mathematics teaching is to develop the tactics of problem solving.
Problem Solving with Polya

34. Problem Solving with Polya
References / Sources:
Problem Solving with Polya