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 **

** 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.**

**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/**

**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.**

**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.**

**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.**

**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!**

**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:**

**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.