Custom Writing Service http://StudyHub.vip/An-Exploration-Of-Strategies-Used-By-St 👈

1. JOURNAL OF MATHEMATICAL BEHAVIOR 15, 263-284 (1996)
An Exploration of Strategies Used by
Students to Solve Problems With Multiple
Ways of Solution
MANUEL SANTOS-TRIGO zyxwvutsrqponmlkjihgfedcbaZYX
Center .for Research and Advanced Studies, Mtkico zyxwvutsrqponmlkjihgfe
INTRODUCTION
What teachers think of mathematics permeates the activities they implement
during instruction. For instance, the type of examples they select to present the
content, the form of evaluation of the students’ learning, and the type of student
interaction allowed during the development of the class are aspects that are
shaped by the way teachers conceptualize mathematics and its learning. Further-
more, what teachers do in the classroom influences directly the students’ learning
of mathematics. Schoenfeld (1989) stated that students often think of mathemat-
ics as a discipline in which one can solve problems quickly or not solve them at
all. They believe that the main activity in learning this subject is to apply the
correct rule or formula to solve problems; that is, students rarely think of mathe-
matics as a subject in which there is room to discuss and defend their ideas, to
formulate conjectures and to test them as a way to improve their ideas, to search
for mathematical connections, or to explore other relationships or even design
their own problems. An important aspect that could help understand how stu-
dents solve problems is to explore what students do when asked to solve prob-
lems that included multiple ways of finding the solution. This type of study will
provide information about to what extent students actually use their mathematical
resources and strategies to solve problems; that is, to what extent they are using
the material studied in their mathematical courses.
Although mathematics teachers may claim that they emphasize problem-
solving activities in their classes, there is indication that students experience
difficulties solving problems that require more than the use of algorithms
(Schoenfeld, 1989). Therefore, there is a need to investigate what types of
Part of this research was completed when the author was at the University of California at
Berkeley as a visiting scholar. The original idea came from a research project directed by Dr. Thomas
Schroeder. The author wants to thank him for his support and advice during the development of this
study.
Correspondence and requests for reprints should be sent to Manuel Santos-Trigo, Dakota 379,
Col. Napoles, 03810, Mexico D.F.
263

2. 264 SANTOS-TRIG0
difficulty students experience during the process of approaching mathematical
problems.
The purpose of this study is to analyze the work of 35 students (Grade 10)
who worked on five problems that could be solved by using different ways. An
important criterion in the selection of the problems was that the content required
to solve the problems had been studied by the students previously. The selection
and the work done by the research team on the five problems became an essential
part of the study. The discussion of the selection process is also an important
objective in this study.
BACKGROUND TO THE STUDY
Although it is well recognized that the main activity involved in learning mathe-
matics is to solve mathematical problems, it is interesting to observe that this
activity could be interpreted and implemented in classrooms differently. For
instance, Schroeder and Lester (1989) identified three different approaches found
in the actual practice of teaching problem solving. One approach may be identi-
fied as teaching ubout problem solving. This approach emphasizes four stages
identified during the process of solving mathematical problems (Polya, 1945).
There is explicit discussion about these stages when solving the problem and
discussion about basic heuristics for solving the problems. Another approach
identified as teaching _/or
problem solving focuses on the use or application of
mathematical content. Therefore, the initial understanding of mathematical con-
tent is a prerequisite to applying it in various contexts. As a consequence,
problem solving emphasizes the applications rather than the understanding of the
mathematical content. The third approach to problem solving is that in which
mathematical content emerges from a problem-solving situation. This situation
requires that the students become actively engaged in the process of making
sense of content. This approach is identified as teaching via problem solving. It
is suggested that this approach has similarities with the process of developing
mathematics.
Although it is common that a problem-solving approach may include a combi-
nation of some of these directions, it is important to note that the way in which
the curriculum is organized often depends on what view is endorsed. For exam-
ple, the British Columbia mathematical curriculum includes a unit called prob-
lem solving, in which the purpose is to discuss the strategies involved in the
process of solving mathematical problems (teaching about problem solving).
Schoenfeld (I 989) suggested that problem solving should be the main activity for
engaging the students in learning and developing mathematics throughout the
entire course. In addition, he pointed out that during the analysis of the students’
problem-solving processes, emphasis should be given to the following four re-
lated components: (a) domain knowledge, which includes definitions, facts, and
procedures used in the mathematical domain: (b) cognitive strategies, which

3. STUDENT STRATEGIES 265
include heuristic strategies, such as decomposing the problem into simpler prob-
lems, working backwards, establishing subgoals, and drawing diagrams; (c)
metacognitive strategies, which involve monitoring the selection and use of the
strategies while solving the problem; that is, deciding on the types of changes
that need to be made when a particular situation is deemed problematic; and (d)
belief systems, which include the ways that students think of mathematics and
problem solving. The categories of analysis suggested by Schoenfeld are impor-
tant in understanding what aspects of students’ processes used for solving mathe-
matical problems require more attention in mathematical instruction. However,
for the analysis of the data gathered for this study, a model proposed by Perkins
and Simmons (1988) was used. This model includes the cognitive strategies in
each of the categories of frames of knowledge. In addition, it addresses some
pedagogical linkages that could be useful for instructors.
Perkins (1987) pointed out that during the process of solving problems, indi-
viduals use thinking frames to organize and support their thought processes. In
explaining the origin of these frames, Perkins stated that “learners might become
acquainted with a frame through direct instruction, they might invent it for
themselves, or they might ‘soak it up’ from an enriched atmosphere, without the
frame ever taking the form of an explicit representation” (p. 48). He distin-
guished three aspects of learning that influence the thinking process. First is
acquisition, in which the learner initially gets familiar with the thinking frame
and applies it to simple cases. The next aspect is to make this frame automatic;
this stage comes after some practice. Finally, there is the transfer aspect, in
which the learner applies the frame across different contexts that may be different
from the original context of learning.
Perkins and Simmons (1988) discussed an integrative model for explaining
some difficulties that students normally experience in science, mathematics, and
computer science. The model included four frames of knowledge (categories that
distinguish kinds of knowledge): content, problem-solving, epistemic, and in-
quiry frames. This model considers heuristics and metacognitive strategies with-
in any of the four frames and what Schoenfeld identified as beliefs within the
problem-solving or epistemic frame. Perkins and Simmons suggested that heuris-
tics, beliefs, and self-monitoring practices are orthogonal to each of the levels or
frames of knowledge. They stated:
[I]n general, one set of contrasts addresses the form of the knowledge in question-
strategic, background beliefs, autoregulative-whereas the four frames address
what the knowledge in question concerns-matters of content, problem solving,
epistemology, or inquiry. (p. 314)
A brief summary of the frames (content, problem solving, epistemic, and
inquiry) suggested by Perkins and Simmons (1988) could help to differentiate
their main characteristics.

4. 266 SANTOS-TRIG0
The content frame includes the terminology, definitions, and algorithms or
rules related to the content. For example, algebra might include a variable,
expression, equation, solution, and graph as important components of the con-
tent frame. It also includes the corresponding metacognitive strategies associated
with the use of the content. Ways to recall information or to use notation are
associated with this frame.
The problem-solving frame includes specific and general problem strategies,
managerial strategies, and beliefs about problem solving. This involves the solv-
ing of routine (textbook) and nonroutine problems.
The epistemic frame includes a set of criteria used to validate the use or
acceptance of a particular result; that is, the evidence or explanation that clarifies
the use of a particular concept, rule, or procedure.
The inquiry frame includes specific and general beliefs and strategies that are
used to extend or challenge the knowledge of specific content.
IMPORTANCE OF THE STUDY
Problem solving has been identified as an important component for the learning
of mathematics. For example, the National Council of Teachers of Mathematics
(1989) suggested that during the process of learning mathematics, instructors
should include activities that help students to formulate problems, to develop and
apply diverse strategies, and to interpret or make sense of the responses. Lampert
(1990) pointed out that the way in which mathematics is developed as a disci-
pline should permeate the activities to be considered during its learning at school.
Teachers should take into account what students do during the process of solving
problems. She mentioned that “one way to conduct such lessons is to choose and
pose problems that create a context within which students will be inclined to
reveal the assumptions they have about how a piece of mathematics can be
constructed” (p. 125). That is, it is important that students discuss problems in
which the power of several mathematical ideas could be discussed among stu-
dents. She went on to say that “choosing and using ‘good problems’ and institut-
ing appropriate means of classroom communication can be thought of as two of
the tasks that teachers need to do to teach mathematics” (p. 125). The discussion
related to the selection of the problems becomes important because it provides an
example of how several problems could be organized and perhaps discussed
during mathematical instruction. Teachers could also find useful information
about what students may do when solving those problems and be aware of the
need to relate their practice of teaching to more problem-solving activities.
THE INTERVIEWS AND THE PROBLEMS
Thirty-five Grade 10 students from two secondary schools were interviewed. All
the students were volunteers. The interviews lasted from 20 to 45 minutes. The

5. STUDENT STRATEGIES 267
students were asked to think aloud while solving the problems. The interviews
were audiotaped, and the interviewer occasionally asked the students some clari-
fication questions or provided some hints that could be useful in solving the
problems. Using a record sheet, the interviewer also took notes on main steps
that the students followed during the interviews. The problems used for the
interviews were:
1.
2.
3.
4.
5.
A farmer has some pigs and some chickens. He finds that together these
animals have 19 heads and 60 legs. How many pigs and how many chickens
does he have? (farmer problem).
Could you find two whole numbers a and zyxwvutsrqponmlkjihgfedcbaZYXWVU
b whose product is 1 million and
neither a nor zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
b includes a zero on its representation? Are there any other
such pairs? Why or why not? (1 million problem).
A textbook is opened at random. The product of the numbers of the facing
pages is 3,192. To what pages is the book opened? (pages problem).
How many tennis balls do you need to fill your classroom? (tennis balls
problem).
An athletic competition was held at UBC that included 10 events (swim-
ming, cycling, tennis. . . .). A gold medal and three points were given to
each first place winner. A silver medal and two points were given to each
second place finisher, and a bronze medal and one point were given to each
third place finisher. Three teams (A, B, and C) participated in the competi-
tion. Team C won more gold medals than either A or B. The total number of
medals won by Team C is one more than the total number won by B and is
two more than the total won by A. Nevertheless, Team A came in first with
one point more than B and two points more than C. Determine the number of
medals of each type won by each team (competition problem).
It is important to mention that these problems were discussed extensively
among three members of a research group (two graduate students and one pro-
fessor) before they were given to the students. The idea was to identify all the
possible solutions to each problem and to prepare some instruments that could be
useful during the interviews. However, the intention was not to guide the stu-
dents to use a specific way for solving the problems. The students always had
some time to work on the problems alone and the interviewer only intervened
(providing some hints) when no progress had been shown by the students.
WORKING WITH THE PROBLEMS
Although the aim of the study was to investigate how students could solve
specific problems and to relate this information to mathematical instruction, the
problem-formulating process became an important component of the research.
The first stage was the selection of the problems. Here, the intention was to

6. 268 SANTOS-TRIG0
select problems for the students that were difficult but solvable. The content of
the problems also had to be related to the content the students had studied in their
mathematics classes. It is important to mention that problem selection was not an
easy task. Several sources of problems were examined and some of the problems
selected initially were discarded after being discussed. This was because they did
not include several strategies in their solution or because the mathematical con-
tent required to solve them was not part of the curriculum. Other problems were
reworded as a result of a pilot phase; that is, when the students showed serious
difficulties in understanding the statement of the problem, the problem was
reworded or changed.
The discussion of problem-solving know-how was another important part of
the study. Finding several solutions to the problems helped us identify several
strategies that we did not see at the beginning. It also helped design the guide that
was used by the interviewer when the students had major difficulties solving the
problems. It was also useful in designing the record sheet used to capture the
information from the interviews.
The analysis of particular solutions was very important when categorizing the
work done by the students. Ideas associated with the different qualities of the
solutions were identified in discussing the possible solutions. These ideas were
contrasted with the work shown by the students. That is, the work done to the
problems before the interviews helped clarify the potential of the problems and
organize the process used by the students.
An example of the type of methods that were discussed during the problem-
formulating process is illustrated with the farmer problem. It is important to
mention that for each problem, several ways of finding the solution were identi-
fied. The idea was to explore the potential of the problem and design some
research instruments, but it was not expected that the students necessarily had to
use one of the anticipated solutions. The work done with the problems was
important in order to understand and categorize the ideas shown by the students.
For example, working on the anticipated solution provided us information related
to potential approaches that students could pursue and also was useful to think of
possible questions that could help students overcome some difficulties. In fact,
the instruments given in Appendix A and Appendix B were designed taking into
account the work done on the problems previous to the interviews.
Problem Statement
A farmer has some pigs and some chickens. He finds that together these animals
have 19 heads and 60 legs. How many pigs and how many chickens does he
have‘?
Anticipated Solutions
1. A pictorial approach involves the use of figure drawings or diagrams to
represent the problem. The student may actually draw some diagrams to repre-

7. STUDENT STRATEGIES zyxwvutsrqponmlkjihgfedcbaZYXW
269
sent chickens and pigs and use then as a reference for adding or eliminating more
of them according to the number of legs, (see Figure 1).
2. A guess and test approach could be selected for solving the problem. This
approach could take several directions in accordance with the way the guesses
are selected. What may determine the direction of the type of guesses is the
rationale used in selecting the first guesses. For example, the student could use:
a. A trading approach focuses on examining a number of either chickens or
pigs and then trading them in accordance with the number of legs. For
example, the student may start by considering 19 pigs and calculating
their number of legs and then decreasing the number of pigs by one and
adding one chicken to compensate for the number of heads. Repeating
this procedure eventually leads to the solution.
b. A table construction approach could help the problem solver select num-
bers systematically. For example, by starting with the extreme cases (only
chickens or only pigs) and making the next choice according to that
information (see Figure 2).
c. A counting approach could involve starting with any number of chickens
and pigs, for example, 10 chickens and 9 pigs. Counting the total number
of legs, one finds 20 + 36 = 56; now, four legs are missing. Then the next
guess could be 9 chickens and 10 pigs; this leads to 18 + 40 = 58 legs.
For this case, 2 legs are missing. Then the next guess will be considering
8 chickens and 11 pigs, which produces the desired solution.
3. The correspondence approach could also be selected to solve this problem.
The idea is to make a correspondence between the number of legs and heads.
Two similar ways are discussed next.
a. Suppose all the chickens stand on one leg and all the pigs on their hind
legs. There are half the legs now; that is, 30 legs. In this number, the head zyxwvutsrq
Figure zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR
1.

8. 270 SANTOS-TRIG0 zyxwvutsrqponmlkjihgfedcbaZYXWVUT
chickens Pi@ legs
19 0 36
0 19 76
10 9 56
8 11 60
Figure 2.
of a chicken is counted only once, whereas the head of a pig is counted
twice. Subtracting all the heads (19) from 30 gives the remaining heads,
which will be pig heads. That is 11 pigs. (30 - 19 = 11) and 8 chickens.
b. Another variant of the correspondence approach is to suppose that all the
animals stand on two legs; then there will be 38 legs touching the ground
and (60 - 38) = 22 legs in the air. These 22 legs must be pigs’ legs.
Because only two of each pig’s legs are in the air, then there are 11 pigs.
4. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
A semialgebraic approach might be selected to solving this problem. For
example, the student could represent c = # of chickens and p = # of pigs; then
write a formula like c + p = 19 orp = 19 - c. Then the students may use this
representation as a base to explore the likely combinations that might satisfy the
expression when using the number of legs (see Figure 3).
5. Algebraic approaches could also lead to the solution. One way is to
represent the given information in a system of equations. This system of equa-
tions involves two equations with two unknowns that can be solved by routine
procedures.
number of chickens = x
number of pigs = y
number of heads x + y = 19--------------------( 1)
number of legs 2x + 4y = 60-------------------(2)
Multiplying (1) by 2 and subtracting (2) gives:
2y = 22; then y = 11 and x = 8. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO
C 19 - c
4 19 -4
6 19-6
8 19 - 8
2 (c) + 4 (19 - p) legs
2 (4) + 4 (15) = 68
2 (6) + 4 (13) = 62
2 (8) + 4 (11) = 60
Figure 3.

9. STUDENT STRATEGIES zyxwvutsrqponmlkjihgfedcbaZYXWVU
271
+oblems
rhe farmer problem
The one million
oroblem
The page problem
The tennis ball
problem
The competition
problem
Methods of possible Strategies Content
Solution
Use of diagrams or Basic operation with
*Pictorial actual pictures, trial integers. Equation of
*Guess and Test and error, organized first degree or system
*Correspondence list or table, of two equations.
*Algebraic comparing, and
equations
Use of a list or table, Multiplication and
Trial divisors thinking of a simpler division of integers,
*Prime factor problem (10, 100, or factoring, exponents,
*Simpler problem 1000). and prime numbers
Consecutive
numbers,
‘Guess and test Estimation, trial and multiplication of
*Factorization error, symbolic integers, factorization,
*Square root representation, and meaning of square
*Algebraic equation. root, quadratic
equation
*Guessing and test Estimation, trial and Estimation, volume,
*Dividing error, thinking of a area, length, units,
‘Layer simpler problem, and and basic operations.
*Formula unit iteration.
Use of subgoals, Basic operation with
*Sequential organized list or table, integers, equations,
arrangement trial and error, and relation between
‘Interacting symbolic the whole and its
information representation, and parts.
(Subgoals) coordinating
conditions. zyxwvutsrqponmlkjihgfedcbaZYXWVUT
Figure 4.
The students may also decide to use an algebraic representation that involves
only one variable. For example, x could represent the number of chickens and
(19 - x) the number of pigs; this leads to 2x + 4( 19 - x) = 60, which represents

10. 272 SANTOS-TRIG0
a linear equation. This equation is the same as 2~ + 76 - 4x = 60, which gives
x = 8.
The main features of the problems are summarized in Figure 4.
RESULTS FROM THE INTERVIEWS
For the farmer problem, the most common approach used by the students was to
represent the problem algebraically. The students spent almost no time attempt-
ing to make sense of the information given in the problem and immediately
began to write down the data. Some students introduced different variables for
representing the chickens, pigs, heads, and legs. This type of representation was
confusing for the students, but after they reexamined the problem, in general,
they were able to solve it. The interviewer then asked the students whether or not
there was another way to solve this problem. They recognized that there was
another way, “the long way,” in which they could use guessing and testing. When
the students were asked to try this method, it was observed that they were not
systematic, and often they even tried extreme cases, such as 19 chickens or 10
pigs. Not one of the students used a table or a diagram for solving this problem
and they relied on the use of a calculator to check even small numbers. One
student added 19 and 60 and divided the result by 2 and 4, obtaining 34.5 and
17.2, respectively; then the student claimed that there were 34.5 chickens and
17 pigs.
It was observed that the students did not spend time trying to understand the
problem. They read the problem and immediately started to work on some
calculations without identifying the main data of the problem and their relation-
ships. The difficulties that they experienced while trying to use the information
often pushed them to reread the problem and to think of a more organized
strategy. In accordance with Perkins and Simmons (1988), it was clear that the
students associated the information of the problems with some operations but
they lacked the basis to select a plan. The metacognitive aspect related to the use
of the operations was not present in the students’ work; that is, the students were
fluent while operating with the numbers, but they did not relate the meaning of
the operation to the information naturally. It was also observed that it was
difficult for the students to realize that some of their work did not match the
information or conditions of the problem; that is, they had not developed some
sort of strategies that could help them to challenge their own work (epistemic
frame).
Some of examples of the students’ work are presented next.
A farmer has some pigs and some The interviewer gave the written prob-
chickens. He finds that together these lem to the student and asked her to work
animals have 19 heads and 60 legs. loudly
How many pigs and how many chick-
ens does he have?

11. STUDENT STRATEGIES 273
She started to read it. She immediately
divided 60 by 4 and 19 by 2 and wrote
I5 and 9
She mentioned that there were 15 pigs
and 9 chickens
The student then realized that she was
not taking into account the information.
She then said that there were only 19
animals and perhaps she could have 10
chickens and 9 pigs. However, when
she counted the number of legs, she
found that this answer did not fit the
conditions. Here she then worked using
trial and error by subtracting and adding
a unit, but she took 19 as fixed number.
The interviewer asked her to explain the
answer.
Here the interviewer suggested that she
could check the information of the
problem and the results.
At this time she felt confident and
started to work very quickly.
For the 1 million problem, the students experienced difficulty accepting the
existence of such factors. They represented the problem algebraically as a X b =
1,OOO,OOO
and then they isolated a = 1,000,000/b. They assigned several values
to b and with the use of a calculator they found values for a. After several trials,
some mentioned that such factors did not exist. None of the students was able to
make progress in this problem alone. When the interviewer suggested trying
smaller numbers than l,OOO,OOO, some of the students were able to find the
pattern by trying 10 = 2 X 5, 100 = 4 X 25, and so on. Only one student
decided to factor l,OOO,OOOand found that there were sequences of twos and
fives; however, he was not able to see on his own that he had found the desired
factors. For the second part of the problem, no one (on his or her own) was able
to explain why there were no other pairs. Two students mentioned that because
for 10 and 100 it was not possible to find other such representations, then the
same could be applied to 1,OOO,OOO.
When the interviewer asked them how they
were sure that for 100 there were no other pairs, they listed all the pairs whose
products give 100 instead of checking the prime factors. It was observed that the
students tried to solve this problem by selecting some concrete numbers. These
numbers did not work and they thought that the problem did not have a solution.
This showed that the students experienced serious difficulty in recognizing the
consistency of plausibility of the information of the problem; that is, it was not
clear for the students that checking for three or four examples does not mean that
the numbers could not exist. Again, strategies related to the epistemic frame
were not present in the students’ work.
Some samples of the students’ work are presented next.
Could you find two whole numbers a The interviewer explained to him that
and b whose product is I million and the idea was to explore the types of
neither a nor b includes a zero in its strategies that he would use in order to

12. 214 SANTOS-TRIG0
representation? Are there any other
such pairs? Why or why not? (1 million
problem).
Rob read the problem and started to try
some numbers with the use of the calcu-
lator. He tried 963 x 56 = 53,928;
99,987 x 89 = 9,799&M; 9,999 x 89
= 88,911; and 99,999 X 89 =
899,911.
Rob then suggested working with IO
and he wrote: 10 = 5 X 2; Although he
hesitated to continue, he then wrote 100
= 25 x 4; 1,000 = 125 x 8; 10,000 =
625 x 16. Rob at this point mentioned
that the number of zeros was the expo-
nent of 5 and 2. Hence, he wrote
1,000,000 = 56 x 26.
Rob, using a hand calculator, found the
two numbers. He mentioned that the
problem was not easy until he found
the pattern.
Rob mentioned that he was sure that
there was not another pair because the
other representation involves zeros.
solve the problem. He was told that his
work was not going to be evaluated but
analyzed afterward in order to identify
his strategies. Rob mentioned that it
was going to be difftcult to get the num-
ber by this method but that at that mo-
ment he could not think of something
else.
The interviewer then suggested to con-
sider a smaller number than 1 million.
The interviewer then asked to explain
why those numbers did not involve
zeros in their representations.
Rob was asked about the next part of
the problem.
His response to this part was based on
observing that the numbers 10, 100,
( had only 2 and 5 as factors.
For the pages problem, the students represented the problem algebraically;
however, they experienced difficulty solving the quadratic equation’ involved in
the representation. It was interesting to observe that some students wanted to
isolate the variable by dividing the two sides of the equation by the variable; that
is, from x2 + x = 3,192 to x + 1 = 3,192/x without realizing that they were
dealing with a quadratic equation. When the interviewer asked the students if
they could use other methods, they responded “probably by using the long way”
(guess and test). When they were trying this guess-and-test approach, some of
the students were using guesses in which the numbers were not consecutive even
when originally they had represented the problem correctly. Not one of the
students used the square root of 3,192 as the approximation of the page numbers.
In general, they found products of consecutive numbers larger and smaller than
3,192 and explored all the possibilities between these numbers. It seemed that it
was difficult for the students to think of another method of solution even when
they had not shown much progress. They stuck to one approach for some time
‘Although the students were familiar with the quadratic equation, the Grade 10 curriculum only
includes particular ways of finding the roots such as factoring.

13. STUDENT STRATEGIES 275
and gave up easily. Santos-Trigo (1990) showed that students often expect to be
told what method to use to solve problems. An example of a student working on
this problem is shown next.
A textbook is opened at random. The
product of the numbers of the facing
pages is 3,192. To what pages is the
book opened? (pages problem).
Martha started immediately using trial
and error with nonconsecutive num-
bers. Some of the numbers were 32 X
95, 32 x 99, 33 x 95.
She reread the problem and realized that
the numbers were consecutive, then
started with 32 X 33, 60 X 61, 55 X
54, 57 x 58, 55 x 56.
She mentioned “breaking up” 3,192 but
without going further.
Then she tried 56 X 53 and continued
guessing until she got 56 X 57.
The interviewer asked her about the
meaning of those numbers.
At this point, she noticed that 57 X 58
was too high and 55 X 56 was too low.
However, she could not see other possi-
bilities on her own.
The interviewer asked her to think of
other methods to solve this problem.
When she was asked to use the square
root of 3,192 she got 56.49.
She did not see that the square root
could suggest 56 X 57.
For the tennis balls problem, the students experienced difficulty in estimating
the dimensions of the classroom and the tennis balls. All the students asked the
interviewer to provide the dimensions and when they were asked to estimate,
some of the students asked for a meter stick. The most common approach was to
divide the volume of the classroom by the volume of a tennis ball. However, the
students experienced difficulty in equating the units, that is, from cm3 to m3 and
vice versa. When the students were asked to use other methods to estimate the
number, they used the layer approach, in which they approximated the number of
balls needed to cover the floor and then multiplied that number by the number of
layers that might fit in the classroom. However, this number was normally
different and they did not feel confident in this approximation. Two of the
students thought of using different units instead of using the actual tennis ball,
such as small cube or box, but no one divided the classroom in smaller parts for
estimating the number. The estimation provided by the students (for the same
room) included 768,000; 60,000; 73,728; 24,840; 162,000,OOO; 636,050; and
126,562 tennis balls. The range of these numbers showed that the students
experienced difficulties in accepting the sense of their responses. In addition, it
was observed that the students relate a problem to a set of numerical data, and it
was difficult for them to get the data from their environment. It seems that when
the problem fails the model that they expect, it becomes more difficult to think of

14. 276 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
SANTOS-TRIG0
the solutions. Besides, it was difficult for them to see different ways to get the
estimation and that these different ways could be used to check their solutions.
For the medals problem, some students experienced difficulty recognizing that
the total number of medals and points given in the competition was necessary to
determine the total number of medals and points won by each team. Although
some students got confused when representing the information that related the
number of medals and points, in general, they were able to represent it alge-
braically. When the students determined the number of points and medals won by
each team, they experienced difficulty in using the information that was neces-
sary to reduce the number of combinations in order to find the type of medals
won by each team; for example, only two students used the information “Team C
won more gold medals than either Team A or B” without any hint from the
interviewer. In addition, the students did not display the information that they
had in a form in which they could check the number of points and medals easily.
As a result, the students often considered some guesses more that once. For
example, some students recognized that Team C could not have five or six gold
medals initially, but after some time they again checked these cases.
One student determined the total number of medals and points won by each
team. Then, he started to explore the possible combinations for finding the type
of medals won by each team. At this stage, he stopped and mentioned that the
problem did not make any sense. The interviewer asked him to explain why and
he responded that if there were three teams, then for each event each team had to
win a medal, and, therefore, it was not possible that a team won more than 10
medals. It was interesting to observe that this student expressed this concern after
he had worked for some time on the problem.
For this problem, it was important to write down (table form or list) the given
information and take into account this information to make some progress. The
students worked on specific parts of the problem and failed to recognize their
relationships. Reading the problem several times gave the students some indica-
tions of how to find the connections. However, the intervention of the inter-
viewer who asked the students to put the information together was necessary.
This helped them to advance and see the relationships.
DISCUSSION OF THE RESULTS AND RECOMMENDATIONS
There was an indication that some students experienced difficulties in identifying
the key information of the problems and they were not organized in presenting
the information that could be used to solve the problem. They believed that the
most efficient way for solving the problems was the algebraic approach, even
when they often struggled in representing the information. They used guess and
test as the last resort and failed to use simpler problems as a means to solve the
problems. It is suggested that the students need to discuss this type of problems

15. STUDENT STRATEGIES 277
during class with other classmates and the teacher. The discussion should involve
problems in which the students defend their ideas about how to approach them
and the sense of their solutions. It is necessary to engage the students in solving
problems in which they have to estimate the data and the sense of the solutions.
For example, for the tennis balls problems, the students commented that they
were not confident in their responses.
The students failed to monitor their processes while working on the problems;
they often used information that was not consistent with something that they had
used previously. For example, for the pages problem they recognized that the
numbers had to be consecutive, but when they were searching for the numbers,
they often tried any number. Similar situations occurred in the medals problem;
the students checked some of the combinations more than once because they
failed to keep records of their work or because they did not display the informa-
tion in an organized way (list or table).
Teachers should pay more attention to the ways that students make sense of
problems. For example, some students may repeat or reword what is required to
find in a specific problem, but they may fail to use or make sense of the
information in order to solve the problem. For example, some students recog-
nized that for the pages problem the numbers have to be consecutive; however, in
the process of searching for the numbers, they often tried any pair of numbers. A
similar phenomenon occurred in the medal problem; the students often tried
guesses that they had previously checked and discarded as possible solutions. It
seems that students need to be aware of the use of metacognitive strategies while
solving mathematical problems.
In the analysis in which the frames suggested by Perkins and Simmons (1988)
were considered, it was found that much work needs to be done in activities
related to mathematical instruction. For example, in the content frame, some
students got confused using basic algebraic operations and representing the vari-
ables and using metacognitive strategies to monitor their work. They also lacked
the use of cognitive strategies, such as looking for patterns, using simpler prob-
lems, or drawing diagrams (problem-solving frame). They failed to recognize
when a series of algebraic transformations did not lead to any useful representa-
tion or when their representations were inconsistent with the information given in
the problem. They often failed to use mathematical arguments to explain whether
or not a conjecture was valid (epistemic frame). As a consequence, they never
intended to explore other extensions of the problem on their own (inquiry frame).
The results showed that the use of these type of problems could provide
information about the difficulties that students experience while trying to solve
the problems. It may be suggested that asking these problems to the students
could be an important activity to implement regularly in the classroom. Another
main recommendation that emerges from this study is that mathematical instruc-
tion should include activities in which the four frames are discussed and used

16. 278 SANTOS-TRIG0
throughout the entire course. The students must engage in open discussion with
their classmates (small groups, pairs) and have the opportunity to explain what
they do while solving the problems. Schoenfeld (1989) suggested that a micro-
cosm of mathematical practice in which the students have the opportunity to
express and defend their ideas, to speculate, to guess, and to reflect on possible
approaches for understanding mathematical ideas or solving mathematical prob-
lems could be the initial point for attaining this goal. The students engaged in
such practice will be exposed to an intellectually demanding environment that
should eventually improve their ways of understanding mathematics.
An important recommendation that emerged from this study is that it is
necessary to discuss the problems in detail before presenting them to the stu-
dents. It is also important to give problems that involve different contexts and
types of data. For example, the students should be aware of the existence of
problems in which they have to provide the data or even design or formulate their
own problems.
INSTRUCTIONAL IMPLICATIONS
The results of this study may have direct implications for mathematical instruc-
tion. For example, the difficulties shown by the students while trying to under-
stand the statement of the problems suggest that teachers should encourage their
students to discuss the main ideas of the problem. That is, class activities should
include examples in which students realize the importance of understanding the
sense of the problem before going further. Another aspect that teachers should
pay attention to during instruction is emphasis on the need to explore various
ways to solve problems, as well as the need to characterize the qualities of
different methods of solution. For example, it is important that students recog-
nize that a problem often could be solved using trial and error, for example, in a
better way than using algebra. In addition, teachers should discuss during their
instruction problems that can be solved by different methods (geometrical, alge-
braic, or numerical). The idea here is that teachers use these problems to explore
the processes used by the students while trying to solve them. As Easly (1977)
stated, “teachers would have to understand rather well the process of cognitive
development and listen to and observe children carefully so as to grasp with
reasonable accuracy what kind of mental operations they are bringing to bear on
a given task” (p. 21).
Students were consistent in trying to find a rule or formula that they could use
to solve the problem. This suggests that teachers should discuss regularly exam-
ples in which these means could not be useful for getting to the solution of the
problem; that is, to show the students that applying a formula or algorithm is not
the only way to find the solution of a problem. It seems that in order to develop

17. STUDENT STRATEGIES 279
the students’ mathematical disposition to learn mathematics is important to pro-
vide a class environment in which students consistently are asked to (a) work on
tasks that offer diverse challenges; (b) discuss the importance of using diverse
types of strategies including the metacognitive strategies; (c) participate in small-
and whole-group discussions; (d) reflect on feedback and challenges that emerge
from interactions with the instructor and other students; (e) communicate their
ideas in written and oral forms; and (f) search for connections and extensions of
the problems. These learning activities play a crucial role in helping students see
mathematics as a dynamic discipline in which they have the opportunity to
engage in mathematical discussions and thus value the practice of doing mathe-
matics (Santos-Trigo, 1995).
Finally, this study illustrates the importance of discussing the problems among
colleges. As a result, diverse methods or strategies could be identified. This
discussion should also be promoted among students as a part of instruction.
Students should select or design their own problems to be discussed during the
development of the class. They may contribute to developing a view of mathe-
matics that reflects the proper activities of the discipline.
REFERENCES
Easly, Jack (1977). On clinical studies in mathematics education. Columbus: Ohio State University,
Information Reference Center for Science, Mathematics, and Environmental Education.
Lampert, Magdalene (1990). Connecting mathematical teaching and learning. In Elizabeth Fennema,
Thomas P. Carpenter, & Susan J. Lamon (Eds.), Integrating research on teaching and
learning mathematics (pp. 121-152). Albany: State University of New York Press.
National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for
school mathematics. Reston, VA: Author.
Perkins, David N. (1987). Thinking frames: An integrative perspective on teaching cognitive skills.
In Joan B. Baron & Robert J. Stemberg (Eds.), Teaching thinking skills: Theory and practice
(pp. 41-61). New York: W.H. Freeman.
Perkins, David N., & Simmons, Rebecca (1988). Patterns of misunderstanding: An integrative model
for science, math, and programming. Review zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ
of Educational Research, 58, 303-326.
Polya, George (1945). How to solve it. Princeton: Princeton University Press.
Santos-Trigo, Manuel (1990). College students methods for solving mathematics problems as result
of instruction based on mathematics problem solving. Unpublished doctoral dissertation,
University of British Columbia, Vancouver, B.C. Canada.
Santos-Trigo, Manuel (1995, June). On mathematical problem solving instruction: Qualities of some
learning activities. Paper presented at the Function Research Group at the University of
California, Berkeley, CA.
Schoenfeld, Alan H. (1989). Ideas in the air: Speculations on small group learning, environmental
and cultural influences on cognition, and epistemology. International Journal of Educational
Research, 13(l), 71-88.
Schroeder, Thomas L., & Lester, Frank K. (1989). Developing understanding in mathematics via
problem solving. In Paul R. Trafton & Albert P. Schulte (Eds.), New directions for elemen-
tary school mathematics Yearbook (pp. 3 I-42). Reston, VA: National Council of Teachers of
Mathematics.

18. 280 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
SANTOS-TRIG0
APPENDIX A: HINTS AND FOLLOW-UP QUESTIONS
Understanding the Problem
General understanding of Can you explain in your own words what the prob-
the problem situation: lem is telling you?
What do you know.? What do you have to find?
Understanding the relation- What number of legs could correspond to each
ship between the number head?
of heads and legs
Planning and Carrying Out the Plan
Getting started: Do you have any idea what strategy you could use
to solve this problem.?
Choosing and using “esti-
mation”:
Choosing and using “guess-
and-test,“:
Choosing and using “elim-
inating possibilities”:
Choosing and using an al-
gebraic approach;
Looking Back
Seeing connections or oth-
er ways to solve this prob-
lem
Could you think of a number of chicken and pigs.7
What numbers should you try?
What numbers should you try next.7
Would 19 chickens (pigs) be a good guess? Why or
why not.7 Is a table helpful to organize your
choices?
Could you use algebra to solve this problem?
What variable would you use? What would it rep-
resent.?
What equation can you write to represent the prob-
lem :’
How can this equation be solved for x?
How would you check that your solution is cor-
rect.?
Could you think of another approach to solve this
problem?
What about if the animals stand on only two legs?
How many legs would be in the air and whose legs
would these be.7 zyxwvutsrqponmlkjihgfedcbaZYXWVUT
Why P
APPENDIX B: INTERVIEW RECORD SHEET
Evaluation of Students’ Problem Solving Processes Problem 10-l
Name(s) SChOOl

19. STUDENT STRATEGIES 281
Birthdate Grade Cot==- Teacher
Interviewer Date Tit0
Problem: A farmer has some pigs and some chickens. He finds that together these animals
have 19 heads and 60 legs. How many pigs and how many chickens does he have?
Tools to be available:
Student Page
Calculator -- 5-function
Plain paper for calculating, diagramming, etc.
Pencils or pens (different colors for student and interviewer
UNDERSTANDING
[] Problem readily understood Evidence:
Difficulty with [] “differentiating the number of legs”
[] Other:
[] “relation between head and legs’
Questions and hints, comments:
STRATEGY SELECTION
A Pictorial Approach used [] Initially [] Exclusively [] Sometimes [] Led to solution
[] With hint [] Without hint
Drawing the legs systematically [] With hint [] Without hint

20. 282 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
SANTOS-TRIG0
Guess and Test Approach zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED
used [] Initially [] Exclusively [] Sometimes [] Led to solution []
Without hint [] With hint:
Guesses are reasonable: [] Always [] Sometimes [] Seldom
Trading Approach [] Always [] Sometimes [] Seldom [] With hint [] Without hint
Table construction [] Always [] Sometimes [] Seldom [] With hint [] Without hint
Counting Approach [] Always [] Sometimes [] Seldom [] With hint [] Without hint
Information from last guess used in next guess: [] Always [] Sometimes [] Seldom
[] Possibilities ruled out (e.g., 19 chickens or pigs ...) [] Without hint [] With hint:
Comments, Evidence:
Sequence of guesses, Comments:
Correspondence Approach used [] Initially [] Exclusively [] Sometimes [] Led to solution []
With hint [] Without hint
comments
Semi-Algebraic Approach used [] Initially [] Exclusively [] Sometimes [] Led to solution []
Without hint [] With hint:
Representation of variables [] c [] 19-c [] Other []With Hint [] Without hint
Comments

21. STUDENT STRATEGIES 283
Algebraic Approach used [] Initially [] Exclusively [] Sometimes [] Led to solution
[] With hint [] Without hint
Variables utilized [] x [] y representation of [] one equation [] a system of equations
[] method used to solve the system of equations
MONITORING
[JAlternative plans named and considered Comments:
[] Progress with chosen strategy(ies) monitored Evidence, Comments:
[] Mathematical connections considered and/ or discussed Evidence:
[] Reasonableness of results checked; solution verified Comments:

22. 284 SANTOS-TRIG0
OVERVIEW [] Student(s) solved the problem essentially on their own
[] Student(s) solved the problem, but needed help from interviewer
[] Student(s) did not solve the problem, even with help from interviewer
Time to Time spent extending the problem
Solution: min or looking back: mill
Describe order in which students used different approaches. Estimate time(s)