THE CURRICULUM AND EVALUATION Standards for School Mathematics (NCTM 1989)
states that one of its five general goals is for all students to become
mathematical problem solvers. It recommends that "to develop such abilities,
students need to work on problems that may take hours, days, and even
weeks to solve" (p. 6). Clearly the authors have not taught my students!
When my students first encountered a mathematical problem, they believed
that it could be solved simply because it was given to them in our mathematics
class. They also "knew" that the technique or process for finding the
solution to many problems was to apply a skill or procedure that had been
recently taught in class. The goal for most of my students was simply
to get an answer. If they ended up with the correct answer, great; if
not, they knew that it was "my job" to show them the "proper" way to go
about solving the problem.
I
began to notice an unwritten contract between my students and me when
they experienced difficulties in solving a mathematics problemI would
come to their rescue. At first, I encouraged them to think about things
that we had just learned or advised them to reread the problem. If these
suggestions failed to help an individual or a group of students, I gave
them specific hints that could guide them to find the answer. I might
say, "Use a table to organize your data, and look for patterns." I would
ask them specific questions that I knew would lead directly to the solution
of the problem. When none of these teaching strategies was effective in
helping my students get the answer, I worked the problem on the chalkboard
with the whole class.
I
cannot remember ever taking more than about fifteen minutes on any problem
that I gave my students to solve. Imagine my discomfort and confusion
when I first read the passage from the Curriculum and Evaluation Standards
regarding the need for students to spend hours, days, and weeks on a single
problem. I tried to think of mathematical problems that I could give my
students to keep them thinking and interested for hours, but I was uncomfortably
unsuccessful in this quest.
I began to read articles in mathematics education journals (the Arithmetic
Teacher, Teaching Children Mathematics, Mathematics Teaching in the Middle
School, and the Mathematics Teacher) and attended local professionaldevelopment
meetings in search of a new view of problem solving. At one presentation
on problem solving, the speaker said that she would share a great mathematics
problem that could be used for students of all ages. She put this "problem"
on the overhead projector for us to read:
The Elmwood School caught fire today. After reading this "problem," I
became disappointed and confused. I looked around the room and noticed
that others were chatting about this problem. After discussing my reaction
with those around me, I realized that many other teachers thought the
same way I did. But as we discussed it, I began to get a different view
of problem solving. One person in our group declared, "It was a science
experiment that exploded, causing the fire." Another person stated, "The
sun caused the fire. As it shined through the window of a classroom, it
was caught by a magnifying glass, which was left on a stack of old newspapers."
During this presentation, the speaker helped us see how our own creativity,
reasoning, and prior mathematical knowledge could be used in the problemsolving
process to facilitate our creating meaningful mathematics for ourselves.
I came away with some good ideas on how to proceed with problem solving
in my classes. I also collected good problems from these sources and resolved
to practice patience with my students' problem solving.
As
I planned during the summer, I vowed to integrate problem solving into
all areas of the mathematics curriculum. Armed with the problems I had
gathered and a firm resolve based on my summer reflection, I began the
new year with problem solving as my focus. What occurred during the first
month of school was far from enjoyable.
During the first few weeks of the school year, my students wanted me to
show them how to solve all problems. Furthermore, because I did not give
them hints or strategies on how to solve the problems, they believed that
I was not doing my jobas did some of their parents! I learned that many
of my students had no real problemsolving experiences. As my students
and I have struggled with problem solving over the years, I have developed
some techniques and strategies to help them become more powerful problem
solvers.
GETTING
STARTED MY FIRST SET OF GOALS FOCUSES ON HELPING my students
learn what tools are available for the mathematical problems they encounter.
I introduce a wide variety of problems that facilitate the use of multiple
strategies, manipulatives, and calculators. As my students become familiar
with these tools, I emphasize creativity and novelty in solving mathematics
problems. During the beginning of our problemsolving experiences, I notice
that some students experience anxiety and frustration, which I believe
occurs because I do not give in to their requests to show them "how to
do it." I try to be compassionate but strive to wean them from this behavior.
I find that it is difficult to create the desired atmosphere for problem
solving in a classroom without first changing the students' attitudes
and beliefs about mathematics and problem solving. This attitude shift
begins to occur in my classroom when I demonstrate to my students that
I value problem solving and that patience and persistence are necessary
to be a successful problem solver. Good problems are also essential in
the process of changing my students' beliefs about mathematical problem
solving.
At the beginning of the year, I start with a problem that is easy to understand
and that can be approached in various ways. We call one such problem that
has generated productive discussions in my classes the "shoe store problem":
A
man goes into a shoe store and buys a pair of shoes for $5 and pays with
a counterfeit $20 bill. The shoe store owner does not realize it. Not
having change for the $20 bill, he runs to the grocer next door. The grocer
gives the shoe store owner four $5 bills for the bogus $20. The shoe store
owner returns to his store and gives the man the shoes and $15 change.
Later the grocer comes back to the shoe store owner with the FBI and informs
him that the $20 bill was bogus. So the shoe store owner gives the grocer
$20, and the FBI keeps the bogus bill. How much did the shoe store owner
lose? (adapted from Sobel and Matelesky 1988).
I recommend that my students read the problem in a number of different
ways. First we read the problem aloud as a whole group. Then we read it
again and have certain groups read alternate sentences to help students
interpret the problem in a variety of ways. This step leads into a brainstorming
activity in which students offer what they believe is important information
in the problem. We do not discredit any ideas at this point; we only list
possible relevant and irrelevant information. For example, my students
usually list all participants involved, the order of the events, the amounts
of money, and any other information that they believe is important in
the shoe store problem. The purpose of this part of the activity is to
generate direction for the smallgroup work that follows.
After
a few minutes of brainstorming, students work in small groups of three
or four to generate strategies that can be used to solve the problem.
They carry out their methods until the group members are satisfied with
their solution or all agree that no solution is possible. As they are
working, I rotate among the groups and ask questions to stimulate their
thinking and communication. As they attempt to solve this problem in their
small groups, it is more important to me that students exchange points
of view about their own thinking than that they conform to someone else's
thinking. However, each group must at some point agree on the most convincing
approach and be prepared to defend its conclusion to the whole class.
After
students have had sufficient time to agree on an interpretation, a method,
and a solution for the problem, the groups reconvene into one large group
to share their results. Here the insightful discussions usually take place.
As they are sharing, I keep track of the types of strategies used by groups
and their solutions and ask questions that require students to reconsider
their solutions or their methods for solving the problem. For the shoe
store problem, students typically try to present a "logical sequential"
argument or one that contains visual informationacting it out, charts,
diagrams, and so onin an attempt to convince others in the class. A
typical student argument follows: "The shoe store owner gives the man
$15 change and the shoes, then must give the $20 back to the grocer and
the bogus bill to the FBI. He is therefore out $15 + $20, or $35, plus
the shoes." As these arguments are shared, students frequently use pictures
or actions to illustrate the "flow" of money in their solutions.
One
reason that I like the problem is the significant disagreement that it
creates among my students. As students convince their peers of the logic
of their solutions, I restate these arguments as closely as possible,
often with some confusion in my voice. When I use this approach, I notice
that many students in the class nod in agreement with my expressed confusion.
The most difficult teacher "action" for me during this period of our development
together is not answering students' questions. My students frequently
look to me to verify their answers or settle their disagreements. Answering
their questions was how I used to think that I was helping my students.
I have found that this help is only superficial and temporary.
Rather,
by asking questions that keep my students focused and thinking about central
mathematical ideas, I give them the boost they need to develop confidence
in their own abilities to think mathematically. I ask questions like these:
"Which solution makes the most sense to you after considering these arguments?
Are these arguments convincing to you? How would you defend your thinking
to those who may not agree with you?" As my students become more independent
and creative, I ask questions that encourage them to investigate their
own problems and ideas in mathematics, for example, "How could you use
what you have found in this problem to solve a related problem, if, for
example, a part of the original problem was changed in some way? How does
changing parts of this problem affect your problemsolving process and
solution?" I can tell when my students are progressing because they begin
asking me these types of questions before I ask them.
Waiting
for the students to take charge of verifying a solution is still a bit
uncomfortable for me, but contributions from student observers reflecting
on the solution presented occur spontaneously. When I gave them time to
respond in one class, a student suggested from the corner of the room
that "the $20 given to the grocer by the shoe store owner was not lost."
The room then erupted with talk from all groups until a different student
blurted out, "Let's act it out." After such reenactments and further discussion,
I review all previous arguments by asking new questions for all my students
to consider, such as, "How much was the shoe store owner out after he
got back from the grocery store and gave the man his change and shoes?"
By reviewing students' arguments and asking them questions, I affirm the
viability of their techniques and encourage them to think about different
ways of viewing the problem.
This
method helps not only those who are confused by a problem or someone else's
line of reasoning but the presenters as they reconsider their own thinking.
In the process of looking for an alternative technique or perspective
for the problem, my students either come away with confidence in their
original thinking or shift toward a more convincing perspective.
It
is important during these initial experiences with problem solving that
my students view problems from many different perspectives and present
and listen to convincing arguments. After initial arguments to the shoe
store problem are presented and reviewed, I call for any new solutions
or different arguments not yet presented. By asking for further discussion,
I want my students to understand that the real, important issue in problem
solving is considering a variety of possible strategies in solving problems.
Furthermore, students must learn that becoming a good problem solver requires
that each person alone must decide whether his or her thinking about a
given problem is correct. This requirement makes me reluctant to contribute
my personal views on individual problems until I believe that they will
not overly influence my students' thinking or confidence to do mathematics.
The rule that I use for knowing when to respond with my viewpoint during
our problemsolving activities involves listening to my students' comments.
If they regularly listen and openly discuss one another's ideas in class,
and offer arguments to support or refute these ideas, I will begin to
offer my thinking about a particular problem. If after I offer my thinking
on a problem, students respond to me in the same way that they respond
to their classmates, I will continue to contribute my ideas. This type
of student response indicates that my students' confidence in doing their
own mathematics is strong. I have found that the length of time I spend
doing these initial activities varies with my students' prior experiences
and age and with my ability to communicate these new problemsolving goals
consistently.
Eventually
my students figure out that I am really interested in the process of problem
solving. Unfortunately, they also begin to believe that anything they
say or do is correct and that answers to mathematical problems are not
important or necessary. It is common during this time for my students
to say, "He isn't saying I'm wrong, I must be right! As long as I write
down anything, he'll give me full credit." Piaget believed that all students
will eventually arrive at the truth of mathematical ideas if they argue
long enough, but therein lies the problem (Inhelder and Piaget 1958).
If I tell my students that their answers are wrong, I become their mathematical
power. If I do not tell them that some of their answers are wrong, they
seem to be satisfied with their first attempts.
I
find that my students pursue problems more rigorously when I require them
to create arguments to communicate and to convince others that their thinking
and solutions are correct. I have each group present its version of a
problem to the class; afterward I ask such questions as"Does anyone have
another way to think about this problem?" When my students ask me to tell
them which answer or strategy is correct, I ask, "Which answer do you
think it is? Why do you believe that that solution or strategy is correct?"
This response does not satisfy some students in my classes at first, and
they insist that I verify the correct answer. I tell them that if they
have any doubt about the solution and method they used, their problemsolving
process is not complete. Anyone who has doubt remaining should find yet
another view of the problem and solution.
Doubt usually lingers with the shoe store problem. After much discussion
and some encouragement to seek another way to view this problem, one group
presented a simpler problem and a workingbackward approach. They stated,
"Instead of thinking about how much money the shoe store owner lost, consider
who in the problem ended up with money and how much they have. From this
we can say how much the shoe store owner lost. The grocer gave $20 and
got that back, so he is not ahead.
The
FBI only has the counterfeit bill. The only other person in the story
(besides the owner) is the person who wanted to 'buy' the shoes. He left
the store with the shoes and $15." These varied presentations, discussions,
and reflections help us understand what problem solving is all about.
My
goal is accomplished when my students and I interact as equals in the
problemsolving process. When this situation occurs, I am viewed as one
among many members of our community of problem solvers. My suggestions
and ideas are valuable, but no more so than those of any other member
of our community. My role is to introduce problems and questions that
stimulate mathematical thinking. I am successful in stimulating thinking
if my students ask questions that I have not previously considered. For
example, I gave my students a problem about finding the area of a set
of isosceles triangles. During the process of solving this problem, one
group conjectured that the product of a leg with onehalf the base was
always greater than the area of these triangles. I had not previously
encountered, or could not remember, this conjecture. At first I felt the
need to be able to verify this conjecture immediately, but instead I devised
questions that helped the students and me carefully consider the impact
that this conjecture might have on other things that we know about area
and isosceles triangles: "How are isosceles triangle special? How does
the height of an isosceles triangle compare with either of its legs? How
is finding the area of an isosceles triangle related to the length of
its sides?"
When
my students are confident in their problemsolving abilities, they automatically
create mathematical arguments to support or refute hypotheses and understand
the need to share these arguments with the whole class. Needless to say,
this phase does not occur overnight. It takes first, a teacher willing
to participate in the problemsolving process with students, and then,
time, persistence, and good problems.
Finding
a few good problems to use in my classes helped me begin changing the
way in which I taught and interacted with my students. I include a few
of my favorite problems in the hope that they may be useful.
