PROBLEM SOLVING

IN THE CLASSROOM

By Carolyn Johnson

University of Georgia

POLYA'S CLASSIC "FOUR PHASES OF THE PROBLEM-SOLVING PROCESS"

PROBLEM SOLVING

"Problem solving has a special importance in the study of mathematics. A primary goal of mathematics teaching and learning is to develop the ability to solve a wide variety of complex mathematics problems...To many mathematically literate prople, mathematics is synonymous with solving problems--doing word problems, creating patterns, interpreting figures, developing geometric constructions, proving theorems, etc. On the other hand, persons not enthralled with mathematics may describe any mathematics activity as problem solving." (Wilson, Fernandez, & Hadaway, 1990)

My self-interest in this topic led me to focus on the different aspects of this subject. I needed this information for my own self use, to strength my skills in mathematics. I felt the need to help other students, who may be lacking in this area, also. I wish to help students in their problem solving experiences, hopefully, making these experiences positive rather than negative ones.

Do we even need problem-solving techniques? George Polya thought so. Since his book, How to Solve It, was published in 1945, Polya has been a leader in the field of study involving methods used in solving mathematical problems. In another of his books, Mathematical Discovery (p.ix), Polya stated that solving a problem means finding a way out of a difficulty, a way around an obstacle, attaining an aim which was not immediately attainable. To him solving problems is an art, like swimming or playing the piano: you learn only by imitation and practice. Therefore, if you want to swim, you have to get into the water: if you want to become a problem solver, you have to solve problems. Polya labeled the study of means and methods of problem solving "heuristics."

Schoenfeld (1989) focused his research on helping the student understand his/her own thought process and guiding him/her to think like a mathematician. He believes that students with mathematical problems tend to look at the problem, then quickly try to solve it. This impulsive approach is in contrast to a mathematician's way of solving a problem. A good math student or mathematician will first try to understand the problem, then carefully plan how to go about solving it. If the plan fails to solve the problem, another plan is devised. In a study of his, over 60% of the students used the impulsive approach. They also lacked the foresight to track their progress along the way, so they failed to see that they were headed down a wrong approach.

Studies by researchers (Eylom & Reif, 1984; Heller & Reif, 1984; Hardiman, Dufresne & Mestrem, 1989) have shown that there are important differences between experts and novices in how they approach and solve problems. One important difference is that experts think about which approach would be best to use, whereas novices think about the superficial aspects of the problem and how to quickly get an answer. As a result, the "surface-feature" group, the students who used superficial aspects of the problem as the basis of their reasoning, had the lowest level of performance. The "principle" group, the students who worked with some principle, had the highest level of performance. These findings were used to suggest that principle-use is related to success in problem solving. Other studies have demonstrated that novices can be taught to think and solve problems like experts. Making students aware of strategies, or principles, that are relevant to problem solving is one of the goals of the Standards of the NCTM. Although the importance of problem solving has been stressed by the NCTM in its Standards, it is often a neglected topic.

Another difference is the understanding, or lack of understanding, of mathematical principles or concepts. In their study, Joan Heller and Harriet Hungate (1985) focused on the crucial aspect of the knowledge of understanding and representing problems to reach correct solutions to problems. Understanding mediates between the problem and its solution, guiding in the selection of method(s) to be used for solving problems. Novices generally do not have the knowledge required to approach problems as the experts. Problem solving requires extensive knowledge of basic concepts and principles. Experts possess an extensive repertoire of familiar patterns and known procedures that are necessary for reliable, consistent performance. Novices have not developed such a repertoire, therefore they lack the "bag of tricks" to pull out the right procedure(s).

As a teacher trying to guide a student's thinking process, we must always take into consideration the individual differences of students. Students enter your classroom with different learning habits and needs, and different past experiences in math and in the classroom, in general. In their article Turner, Styer, and Daggs (1997) looked at the importance of the learning environment that the teacher creates in his/her classroom. Besides challenging material, the classroom environment that encourages students' interest and thinking greatly enhances positive results from the students. In their research a teacher introduced ideas that were designed to challenge the students. While the students enjoyed the brain teasers, it was difficult for them in that it forced them to confront a misconception (the problems involved fractions). Rather than tell the students that they could think of fractions in more than one way, the teacher encouraged them to think and use different ways of approaching the problem than the way they were familiar with. The teacher supplied questions to help the students redirect their thinking. The teacher frequently asked various students to "help" her with the problem, trying to show the class that they also had the skills that were required to solve the problem.This was done to encourage more creative thinking on the part of the student, and to encourage taking risks. Within the content of taking risks, the student would not be intimidated to try a different strategy if the one they had initially chosen was proving to not be leading to the solution. In another lesson, the teacher specifically designed the problem to be solved by various procedures. It underlying objective was to promote understanding by giving the students the freedom to explore different avenues to explore in solving the problem. The idea also was to force the students to move beyond their conventional and restricted notions of what to do and be willing to risk trying other approaches of problem solving in order to find the solution. The teacher was hoping to convey to the students that she believed that they were capable of thinking mathematically. In this type of atmosphere the students would be more inclined to take risk in using different problem solving strategies, even if it took two or more tries in order to choose the correct one.

Traditionally, classroom lecture has been an ineffective means for developing problem-solving skills. A reason for this is the fact that problem solving is heavily procedural. Therefore learning is probably best learned by actually doing. We know that under certain conditions learning can occur through observation of a model performing the activity. Teachers should be able to communicate problem-solving knowledge, but they must also model the process of working toward a solution. Too often the teacher goes from reading a problem to writing on the board a solution, while skipping the process of how and why each step was done. Also missing are the mistakes and different explorations of different approaches that are all part of the process. Maybe if the teacher modeled these aspects, the students would be able to learn some of the processes by observation. Problem solving is a very individual activity, with each person learning at his own pace.

As a result of his research, Frederiksen (1984) believes that giving students many well-designed homework problems to solve can help improve the students' mathematical performance. This improvement is a result contributed to the opportunity for practice and feedback. Research indicates that many hours of practice are needed for students to be successful. The following benefits of homework were identified in a review of the cognitive aspects of problem solving: (1) recollection of facts and concepts, (2) recognition of patterns, (3) recollection of strategies or procedures that can operate on patterns, (4) use of strategies or procedures automatically, (5) facilitation of work with problems presented in unfamiliar forms, (6) analysis and identification of errors, (7) development of problem-solving skills, (8) development of important cognitive skills that cannot be explicitly taught, and (9) improved speed of problem solving. But the assignments must be carefully designed so that the assigned exercises are not just merely repetitions of each other with different numbers used. Rote methods, without thinking, will be used as a result.

Discussion: To be a good problem solver, the student needs practice, practice, and more practice. Therefore homework is an important aspect in the process of learning to save problem proficiently, though students may not be thrilled about it. Learning-by-doing is an invaluable tool in a student's success in math. Also important in being able to successfully solve problems is the possession of an extensive repertoire of strategies. With a repertoire of strategies, one is able to change to an alternate approach when the approach being used is not leading to the solution.

The following is a listing of Polya's classic "Four phases of the problem-solving process" :

I. Understanding the Problem

(a) Can you state the problem in your own words?

(b) What are you trying to find or do?

(c) What are the unknowns?

(d) What information do you obtain from the problem?

(e) What information, if any, is missing or not needed?

II. Devising a Plan

The following list of strategies, although not exhaustive, is very useful:

(a) Look for a pattern.

(b) Examine related problems and determine if the same technique can be applied.

(c) Examine a simpler or special case of the problem to gain insight into the solution of the original problem.

(d) Make a table.

(e) Make a diagram.

(f) Write an equation.

(g) Use guess and check.

(h) Work backward.

(i) Identify a subgoal.

III. Carrying Out the Plan

(a) Implement the strategy or strategies in step 2 and perform any necessary actions or computations.

(b) Check each step of the plan as you proceed. This may be intuitive checking or a formal proof of each step.

(c) Keep an accurate record of your work.

IV. Looking Back

(a) Check the results in the original problem. (In some cases, this will require a proof)

(b) Interpret the solution in terms of the original problem. Does your answer make sense? Is it reasonable?

(c) Determine whether there is another method of finding the solution.

(d) If possible, determine other related or more general problems for which the techniques will work.

Return

REFERENCES

Eylon, B. & Reif, F. (1984). Effects of knowledge organization on task performance. Cognition and Instruction, 1(1), 5-44.

Frederiksen, N. (1984). Implications of cognitive theory for instruction in problem solving. Review of Educational Research, 54(3), 363-407. In Posamentier, A., Hartman, H., & Kaiser, C. (Eds.), Tips for the mathematics teacher: Research-based strategies to help students learn. Thousand Oaks, California: Corwin Press, 1998.

Hardiman, P., Dugresne, R., & Mestre, J. (1989). The relation between problem categorization and problem solving among experts and novices. Memory and Cognition, 17(5), 627-638.

Heller, J., Hungate, H. Implications for mathematics instruction to research on scientific problem solving. In Edward A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives. Hillsdale, NJ: Lawrence Erlbaum Associates, 1985.

Heller, J. & Reif, F. (1984). Prescribing effective human problem solving processes: Problem description in physics. Cognition and Instruction, 1, 177-216.

Polya, G. Mathematical Discovery. U.S.A.: John Wiley & Sons Inc., 1985.

Schoenfeld, A. H. (1989). Teaching mathematical thinking and problem solving. In L. B. Resnick & L. Klopfer (Eds.), Toward the thinking curriculum: Current cognitive research (pp. 83-103). Alexandria, VA: Yearbook of the Association for Supervision and Curriculum Developments.

Turner, Julianne, Styers, Karen R., & Daggs, Debra (1997). Encouraging mathematical thinking. Mathematics Teaching in the Middle School, 3(1), September, 1997, 66-72.

Wilson, James W., Fernandez, Maria L., & Hadaway, Nelda (1990). Mathematical problem solving, http://jwilson.coe.uga.edu/

PART II

APPLICATIONS, PROBLEMS,

SOLUTIONS, AND MISCELLANEOUS

Usually a problem is stated in words, either orally or written. To solve the problem, one translates the words into an equivalent problem, using mathematical symbols, devise a plan, solve the equivalent problem, then check the answer. Learning to use Polya's 4 phase process and the following diagram are first steps in becoming a good problem solver.

SUGGESTIONS FROM SUCCESSFUL PROBLEM SOLVERS

The following is a list of suggestions that students who have successfully completed a course on problem solving felt were helpful tips:

(1) Accept the challenge of solving a problem.

(2) Rewrite the problem in your own words.

(3) Take time to explore, reflect, and think.

(4) Talk to yourself. Ask yourself lots of questions.

(5) If appropriate, try the problem using simple numbers.

(6) Many problems require an incubation period. If you get frustrated, do not hesitate to take a break---your subconscious may take over. But do return to try again.

(7) Look at the problem in a variety of ways.

(8) Run through your list of strategies to see if one (or more) can help you get a start.

(9) Many problems can be solved in a variety of ways---you only need to find one solution to be successful.

(10) Do not be afraid to change your approach, strategy, and so on.

(11) Organization can be helpful in problem solving. Use the Polya four-step approach with a variety of strategies.

(12) Experience in problem solving is very valuable. Work lots of problems; your confidence will grow.

(13) If you are not making much progress, do not hesitate to go back to make sure that you really understand the problem. This review process may happen two or three times in a problem since understanding usually grows as you work toward a solution.

(14) There is nothing like a breakthrough, a small aha!, as you solve your problems.

(15) Always, always look back. Try to see precisely what was the key step in your solution.

(16) Make up and solve problems of your own.

(17) Write up your solutions neatly and clearly enough so that you will be able to understand your solution if you reread it in 10 years.

(18) Develop good problem-solving helper skills when assisting others in solving problems. Do not give out solutions; instead, provide meaningful hints.

(19) By helping and giving hints to others, you will find that you will develop many new insights.

(20) Enjoy yourself! Solving a problem is a positive experience.

STRATEGIES FOR PROBLEM SOLVING

Looking at Polya's list, you see a list of suggested strategies. A strategy can be thought of as a game plan in solving a problem. One important aspect of being a good problem solver is to have an extensive repertoire of such types of strategies. The following list gives clues as to when you might decide to use a particular strategy rather than another one.

STRATEGY: GUESS AND CHECK

Clues as to when one may select this strategy:

- There is a limited number of possible answers to test.

- You want to gain a better understanding of the problem.

- You have a good idea of what the answer is.

- You can systematically try possible answers.

- Your choices have been narrowed down by the use of other strategies.

- There is no other obvious strategy to try.

STRATEGY: USE A VARIABLE (write an equation)

Clues as to when one may select this strategy:

- A phrase similar to "for any number" is present or implied.

- A problem suggests an equation.

- A proof or a general solution is required.

- A problem contains phrases such as "consecutive," "even," or "odd" whole numbers.

- There is a large number of cases.

- A proof is asked for in a problem involving numbers.

- There is an unknown quantity related to known quantities.

- There is an infinite number of numbers involved.

- You are trying to develop a general formula.

STRATEGY: LOOK FOR A PATTERN

The "Look for a pattern" strategy may be appropriate when:

- A list of data is given.

- A sequence of numbers is involved.

- Listing special cases helps you deal with complex problems.

- You are asked to make a prediction or generalization.

- Information can be expressed and viewed in an organized manner, such as in a table.

STRATEGY: MAKE A LIST (make a table)

The strategy may be appropriate when:

- Information can easily be organized and presented.

- Data can easily be generated.

- Listing the results obtained by using "Guess and Test."

- Asked "in how many ways" something can be done.

- Trying to learn about a collection of numbers generated by a rule or formula.

STRATEGY: DRAW A PICTURE (make a diagram)

This strategy may be appropriate when:

- A physical situation is involved.

- Geometric figures or measurements are involved.

- You want to gain a better understanding of the problem.

- A visual representation of the problem is possible.

STRATEGY: SOLVE A SIMPLER PROBLEM

This strategy may be appropriate when:

- The problem involves complicated computations.

- The problem involves very large or very small numbers.

- You are asked to find the sum of a series of numbers.

- A direct solution is too complex.

- You want to gain a better understanding of the problem.

- The problem involves a large array or diagram.

STRATEGY: SOLVE A SIMPLER OR SPECIAL CASE OF THE PROBLEM

Clues as to when this strategy may be appropriate:

- You can find an equivalent problem that is easier to solve.

- A problem is related to another problem you have solved previously.

- A problem can be represented in a more familiar setting.

- A geometric problem can be represented algebraically or vice versa.

- Physical problems can easily be represented with numbers or symbols.

STRATEGY: WORK BACKWARD

Clues when this strategy may be appropriate:

- The final result is clear and the initial portion of a problem is obscure.

- A problem proceeds from being complex initially to being simple at the end.

- A direct approach involves a complicated equation.

- A problem involves a sequence of reversible actions.

STRATEGY: IDENTIFY SUBGOALS

Clues as to when this strategy may be appropriate:

- A problem can be broken down into a series of simpler problems.

- The statement of the problem is very long and complex.

- You can say, "If I only knew..., then I could solve the problem."

- There is a simple, intermediate step that would be useful.

- There is other information that you wished the problem contained.

PROBLEMS

March 24, 2000, in my Math 3200, Proofs class, we, just for fun, looked at the following problem:

In spite of appearances the following is not a spelling lesson but a multiplication problem in which each digit has been replaced by a different letter.

FUR x DOG = AGER + DRIP_ + FUR__ = RIPER. (To see this better, rewrite problem in a column; the blanks after DRIP and FUR represent place-holders). What two numbers were multiplied?

The class consisted of upper-class sophomores through college seniors. Rather than being concern with getting the correct answer, I sat and observed the way the class approached the problem. I was interested in which "strategy" the majority would use (this exercise was done orally). The strategy used by all who participated was the Guess and Check. A number was suggested by a student, then the class checked it to see if it would help in the solution. This activity lasted the whole class time. The majority of the students did not actively participate, but the ones who did showed some rather impressive intuitive guessing. D was the first letter to be solved, D= 1. The next letter solved was P, P= 0. Then the professor began doing "case studies", which is a glorified version of "guess and check." For the first case study, we said, "If R= 2, then G + I = 8. Then G= 6, therefore, I= 2. But we had already said that R= 2, so this case study did not work. We continued on in like fashion until we arrived at case study 4. For this one, we said, "If R= 5, then G+ I= 5 or 15, and that G had to be odd. With this we had G= 3 and I= 2. This worked! The answer was: A= 2; D= 1; E= 6; F= 3; G= 7; I= 8; O= 4; P= 0; R= 5; and U= 9, giving the result of 58,065. So at this point we had gotten an answer, but the question then was, "Are there any more solutions?" So to be totally through with the problem, we had to continue on with case studies to check to see if there existed any more solutions. We did not have enough time in class to complete the work. Can you come up with another solution? Give it a try!

WHAT STRATEGY WOULD YOU USE?

PROBLEM: The largest angle of a triangle is 9 times the smallest. The third angle is equal to the difference of the largest and the smallest. What are the measures of the angles?

What strategy would you first choose? Look back at the list of clues if that might help you decide.

How about using the "guess and check" strategy? If you are a really good guesser and you have a whole lot of time, this method might eventually work. I don't believe that this would be a very efficient method to use. Let's look another strategy. What about "solve a simpler problem". As one of the clues, it states that the problem involves complicated computations. I don't think that this would fall into that category. Let's move on to another choice. This trial and error method is an important part of finding the correct path to use. Sometimes you may be able to correctly choose the correct one at the beginning. But please do not be afraid to move on to another strategy if the one you thought would work doesn't. So, let's move on and examine the "use a variable (write an equation)" strategy. I would say that this one would be correct (but remember that while one strategy may be the most efficient and the best choice, other strategies may solve the problem, just not as easily or efficiently as the "best" one). Why did I decide on the "use a variable" strategy as the one to be used? Looking at the clues under this strategy, the second (a problem suggests an equation) and the seventh (there is an unknown quantity related to known quantities) clues seemed to be applicable. Let's try and see if it will work, or if we need to search for another strategy.

x= the smallest angle

9x= the largest angle

9x- x = the third angle

Information that must be remembered is the fact that the total angular measurement of a triangle is equal to 180 degrees. With this information we can set up an equation:

x + 9x + (9x - x) = 180

18x = 180

x= 10 degrees

9x = 90 degrees

9x - x = 80 degrees

Is it feasible that this is the correct answer? Yes. Why? Because the total sum of the angles equals to 180 degrees. So far, so good. Now to see if it truly is correct, substitute 10 for x in the equation. After doing the arithmetic, we see that 10, 80, and 90 are correct.

PROBLEM: Can you cut a pizza into 11 pieces with four straight cuts?

Let's follow Polya's four-step problem-solving process as we go through the solution for this problem.

I. Understanding the Problem-

(a) Can you state the problem in your own words? Yes.

(b) What are you trying to find or do?- using only four straight cuts, we are to slice a pizza into 11 pieces.

(c) What are the unknowns?- do all pieces have to be the exact same size?

(d) What information do you obtain from the problem?- just that there is one pizza, four straight cuts, and 11 pieces.

(e) What information, if any, is missing or not needed?- I think that the missing information is the fact of whether or not all 11 pieces have to be equal or if they can be different sizes. And also, do all the cuts have to go through the center of the pizza?

II. Devising a Plan-

After looking over the list of different plans, I chose the "make a diagram" (draw a picture). I need to actually draw a representation of the problem to visualize what is being asked.

III. Carrying out the Plan-

(a) Implement the strategy in step 2 and perform any necessary actions or computations-

After drawing the picture, we would probably need to incorporate a second plan, also, that is, the "guess and check" plan. The above is one "guess" which did not give us the results that we wanted. Therefore, we will try another guess. This is definitely a tricky problem. After many guess- and-checks, I found a solution. This doesn't mean that it is the only solution, maybe it is, maybe not. The following is a solution.

See, I told you that it was tricky, at least it was to me. I would not want to be the person to receive pieces 9, 10, or 11 if I were really hungry! Would you?

PROBLEM: In a dart game, three darts are thrown. All hit the target. What scores are possible?

Before trying to devise a plan, let's review Polya's list again. Looking at "I. Understanding the Problem," ask yourself, "Do I really understand what is being asked to solve?" What are you trying to find? It seems that the problem is asking for all the possible combinations of scores that can be obtained. Question to think about: Do you see any restrictions given as to where the darts may land? That is, can more than one dart land in the same circle or does each have to land in a separate circle? So, what do you think would be a good first approach? Remember that I said "first approach" because we may have to come up with a second or even third approach before we actually achieve the final answer. Well, one strategy we do not have to worry about is the "draw a picture," since the problem did it for us. I think that using the "guess and check" approach would not be a good way; it doesn't seem to need an equation to be solved; there's nothing to work backwards from; and there doesn't seem to be any simpler cases to look at. I think that we should first try the "make a list" (make a table). With this we can add all the different combinations that can be gotten with the three darts. Let's see:

0 + 1 + 4 = 5

1 + 4 + 16 = 21

0 + 1 + 16 = 17

16 +1 + 1 = 18(assuming that the darts can land in the same circle), and so on. The whole list of possible combinations seem to be: 48, 36, 33, 32, 24, 21, 20, 18, 17, 16, 12, 9, 8, 6, 5, 4, 3, 2, 1, and 0. The problem did not restrict us from having more than one dart landing in the same circle. Some of these combinations would not be valid if the problem had given this as a restriction.

PROBLEM: In three years, Chad will be three times my present age. I will then be half as old as he. How old am I now?

At first glance this problem seems to be a bit tricky. But, before we panic, we will follow Polya's list and confidently attack the problem head-on, not giving in to any fear of failure. Right? Sure!

Seeing "three times" and "half as old" indicates the multiplication and division operations being used. The person's age is being varied from the present age to what it will be in the future. I think the word "varied" gives us a very good hint as to what approach should be used. I would guess that "write an equation" (use a variable) would be the one to first try.

x = present age of the person

3(x) = Chad's age in 3 years

x + 3 = 1/2[3x]

x + 3 = 3/2(x)

2x + 6 = 3x

x = 6. Therefore, solving the equation gave us the answer that the person is 6 years old at this time.

PROBLEM: An eastbound bike enters a tunnel at the same time that a westbound bike enters the other end of the tunnel. The eastbound bike travels at 10 kilometers per hour, the westbound bike at 8 kilometers per hour.A fly is flying back and forth between the 2 bikes at 15 kilometers per hour, leaving the eastbound bike as it enters the tunnel. The tunnel is 9 kilometers long. How far has the fly traveled in the tunnel when the bikes meet? (now, how many problems have you ever had to solve involving a fly flying? none, I hope!)

There are certainly a lot of numbers in this problem. I believe that it would be a safe bet that the "guess and check" strategy would not work very well. For myself, I would need to first "draw a picture" just to get a better grasp about what the problem is asking. After drawing the picture, study it to see if you can come up with the second strategy needed in order to solve the problem. "Working backwards" would be a futile attempt. I don't know of another "simpler case" that we could use. What about "write an equation?" This may be what we need later on, but for now I think that "identifying a subgoal" would be the better approach. If we can determine the time that the fly is traveling, we should be able to determine the distance the fly travels. The "subgoal" is to find the time that the fly travels. Another "subgoal" would be to determine the length of time that it takes the 2 bikes to meet. Now we would use the strategy of "writing an equation."

Let t be the time in hours it takes for the bikes to meet. The eastbound bike travels 10(t) kilometers, and the westbound bike travels 8(t) kilometers.

10t + 8t = 9

18t = 9

t = 1/2, so the time that it takes for the bikes to meet is 1/2 hour-this reaches a subgoal. Since the fly travels for 1/2 hour, the fly travels 15x 1/2 = 7 1/2 kilometers.

PROBLEM: A child has a set of 10 cubical blocks. The lengths of the edges are 1 cm, 2 cm, 3 cm, ...,10 cm. Using all the cubes, can the child build 2 towers of the same height by stacking one cube upon another? Why or why not?

Hum, maybe we need to add to Polya's list another strategy called " move on to the next problem." (just kidding) So what strategy do you think will work? I wish that I could hear your input. Initially I think that "draw a picture", "guess and check" and/or "make a table" might work.But remember that I said that there may be more than one way to solve the problem, but generally there will be a way (or combination) that would be more efficient in the use of our time and brain power. The strategy of choice is "restate into an equivalent problem." This equivalent problem could be stated as, "Can 1- 10 be put into 2 sets (2 towers) whose sums are equal?" Does this help you have a better grasp of the problem? Hopefully, it does. The answer is "No." Why? because if the sums are equal in each set and if these two sums are added together, the sum would be even. But the sum of 1-10 is 55, an an odd number.

PROBLEM: A street vendor had a basket of apples. Feeling generous one day, he gave away 1/2 of his apples plus one to the first stranger he met, 1/2 of the remaining apples plus one to the next stranger he met, and 1/2 of his remaining apples plus one to the third stranger he met. If the vendor had one left for himself, how many apples did he start with?

Once again, let's take a look at Polya's list. How about "look for a pattern?" At first that appears that might be the one. There seems to be a pattern to his giving, but we have nothing to begin with in order to apply the pattern. So the "look for a pattern" would not be a good choice. "Examine a simpler case" would not apply. I don't think there is a simpler case. "Guess and check" would not be very efficient, if at all feasible. Let's move on. What about "work backwards?" A clue for this is that the solution is simpler than the first of the solution. The solution is "one", since the vendor is suppose to have only one apple left for himself. So we already know the finish. Now we need to get to the beginning. I think that this strategy would be the one to choose.

End result: one apple for the vendor

Next to last step: 4 (1 + 1= 2 and 2x 2=4)

Previous step: 10 (4 + 1=5 and 2x5=10)

First step: the vendor started out with 22 apples (10 + 1=11 and 2x11= 22)

So we see that the vendor began with 22 apples before he decided to give all but one away.

PROBLEM: A student had to do an experiment for his science class. The project was to find the results of a container of water cooling after being heated to the temperature of 190 degrees. That is, how quickly did the water's temperature drop within a period of 30 minutes, recording the decrease in temperature in one-minute intervals?

Your first reaction may be that this is for the department of science, not math. This shows how math is important in other facets of our education, not just self-contained within the math classroom. Math has to be used to solve this science project. Now that we have gotten over that hurdle, let's proceed on to our problem-solving task. (1) Do you understand what is actually being asked for; (2) Do you have all the information needed to solve this problem? What else do you need to know in order to complete this project?

With regards to understanding, it seems that we are being asked to find the decreasing temperatures of a container of water after it has been heated to 190 degrees and is allowed to stand in room temperature while cooling down. We are to record the decrease in temperature in one-minute intervals. What else do we need to know? What about the container that holds the water. Don't you think that would have an effect upon the cooling process? I would think so. I guess that can be part of the project also, to see if the type of container has any effect in the cooling. We will try to keep constant the quantity of water, the beginning temperature, and the thermometer, the measuring instrument, being used. We will use different types of containers: a glass container, a stainless steel, and a heavy plastic container. Therefore we will have three sets of data to collect and record. The question is, "What do we do with the data?" Remember that we will have 30 recordings for each of the three containers, giving us 90 pieces of data to handle. That is alot. So what should our first strategy be (I would think that more than one strategy will be used throughout the project). First of all, what do we do with all this information once it is obtained? We definitely can't work backwards. I wouldn't think that an equation would do us any good at this point, maybe later on. Drawing a picture wouldn't help our situation at this point. How about listing our data? We definitely will have to do that in some manner. So let's try the strategy of "Make a table." Sounds reasonable. After obtaining our data, let us use the EXCEL program for our listing and calculating the results.

 minutes temperature 0 190 1 188.109468412342 2 186.237747928284 3 184.384651374217 4 182.549993438941 5 180.733590655136 6 178.935261381007 7 177.15482578213 8 175.392105813461 9 173.646925201533 10 171.919109426832 11 170.20848570634 12 168.51488297626 13 166.838131874907 14 165.178064725773 15 163.534515520761 16 161.90731990358 17 160.296315153313 18 158.701340168142 19 157.122235449239 20 155.558843084817 21 154.011006734336 22 152.478571612871 23 150.961384475633 24 149.459293602645 25 147.972148783567 26 146.499801302678 27 145.042103924002 28 143.598910876588 29 142.170077839927 30 140.755461929526

The above illustrates our "table", the first strategy that we are using to work our problem. Using the EXCEL program allows us to enter our data, enter a formula, and the program does our calculations. The formula that we are using is the exponential formula for growth or decay. We will use the "decay" aspect of the formula, because the temperature is decreasing (decaying). The chart illustrates the "ideal" decline in temperature. The following chart will show the actual results obtained from our gathered data using the glass container to hold the water.

 minutes temperature 0 190 1 185 2 180 3 177 4 168 5 162 6 160 7 155 8 150 9 148.5 10 145 11 144.5 12 140 13 140 14 136 15 135 16 132.5 17 130 18 127.5 19 125.5 20 125 21 125 22 123 23 120 24 120 25 118 26 117 27 116 28 115 29 112.5 30 112

Our next set of data was gathered while using a stainless-steel container to hold the water.

 minutes temperature 0 190 1 182.5 2 178.5 3 175 4 170 5 167.5 6 162.5 7 160 8 157.5 9 155 10 152 11 150 12 147.5 13 145 14 145 15 142.5 16 140 17 138 18 136 19 135 20 134.5 21 132.5 22 131 23 130 24 129.5 25 127.5 26 127 27 125 28 125 29 124 30 122.5

For our final set of data, the heavy plastic container was used to hold the heated water. The following illustrates the results that were obtained.

 minutes temperature 0 190 1 182.5 2 178.5 3 175 4 170 5 167.5 6 162.5 7 160 8 157.5 9 155 10 152 11 150 12 147.5 13 145 14 145 15 142.5 16 140 17 138 18 136 19 135 20 134.5 21 132.5 22 131 23 130 24 129.5 25 127.5 26 127 27 125 28 125 29 124 30 122.5

Now let's group our line charts together so that it will be easier to compare the results.

The vertical axis represents "temperature" and the horizontal axis represents the "minutes." This chart shows all of the results together, making it easier to compare the outcomes. The purple cubes represents the "theoretical" results. This means that a formula was used in deriving the results. This would represent the "ideal" situation in performing the experiment. Since we did not work in an ideal set-up, with ideal and accurate measuring devises, the "obtained" data show what we actually got. The dark blue string represents the cooling with the water in the heavy plastic container. The dark brown string of balls represents the results from the stainless-steel container. And the last one, the light blue string of stars, represents the data while using the glass container. So now we can see that theoretically the water should not have cooled off quite as much as it did in each case. The heavy plastic container held the water's heat better that the other two. The stainless-steel container did not hold the heat as well as the plastic, while the glass container lost the most heat. So we can make the conjecture that the container holding the water does affect the heat loss of the water. Can you make another conjecture after seeing the results of the project?

PROBLEM: The following problem was presented to a class of tenth graders. The teacher had a dog who stayed in his backyard which was shaped like a right triangle. The teacher's dilemma was that when he was away for a short time he wanted Fido to guard the yard. But because he did not want to risk Fido getting out of the yard, the teacher wanted to put a leash on the dog. The problem was where to put the stake that held the leash so that Fido could still reach each corner of the yard. Where should the stake be placed?

After thought-provoking questions such as: "Do you really own a dog?; "Only a math teacher would have a triangle-shaped yard, or at least noticed that it was triangular; and " What kind of dog is it?, the students were told to come up with a plan to solve the problem. The students had available to them compasses, rulers, calculators, and the GSP software on the computers.

What do you think would be a good approach? First, do we understand what is being asked? It seems that we are to find a point in the triangular-shaped backyard where the teacher could secure the dog leash, allowing the dog to reach each corner of the yard. Ok, we understand what is being asked. Now what? Seems that we are to devise a plan. What should that be? I would need to draw a picture to see what I was working with. Since the GSP program is available, that would be great to use. Using GSP would allow us to see what we are attempting to solve. One strategy that we can use with GSP is the "guess and check" plan. The idea is to find the shortest total length of the three line segments (you will see when you click over to the GSP). Working with the GSP, we can move the point representing the "dog" around until we obtain the smallest total length. Let's go ahead and click over to GSP to see what I'm talking about. Click here . Were you successful in manipulating the "dog" so that the shortest lenght was obtained? I hope that you were successful. That was fun.

I hope that you enjoyed exploring the world of problem solving in mathematics. This work is just a minute look into the extensive research that has been and continues to be done on the subject.