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Students' Verification Strategies for Combinatorial
Problems
Michal Mashiach Eizenberg a; Orit Zaslavsky b
a
Emek Yezreel College, Israel.
b
Technion-Israel Institute of Technology, Israel.
Online Publication Date: 01 January 2004
To cite this Article: Eizenberg, Michal Mashiach and Zaslavsky, Orit (2004)
'Students' Verification Strategies for Combinatorial Problems', Mathematical
Thinking and Learning, 6:1, 15 — 36
To link to this article: DOI: 10.1207/s15327833mtl0601_2
URL: http://dx.doi.org/10.1207/s15327833mtl0601_2
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MATHEMATICAL THINKING AND LEARNING, 6(1), 15–36
Copyright © 2004, Lawrence Erlbaum Associates, Inc.
Students’ Verification Strategies for
Combinatorial Problems
Michal Mashiach Eizenberg
Emek Yezreel College, Israel
Orit Zaslavsky
Technion—Israel Institute of Technology, Israel
We focus on a major difficulty in solving combinatorial problems, namely, on the
verification of a solution. Our study aimed at identifying undergraduate students’ tendencies to verify their solutions, and the verification strategies that they employ when
solving these problems. In addition, an attempt was made to evaluate the level of efficiency of the students’ various verification strategies in terms of their contribution to
reaching a correct solution. 14 undergraduate students, who had taken at least 1
course in combinatorics, participated in the study. None of the students had prior direct learning experience with combinatorial verification strategies. Data were collected through interviews with individual or pairs of participants as they solved, 1 by
1, 10 combinatorial problems. 5 types of verification strategies were identified, 2 of
which were more frequent and more helpful than others. Students’ verifications
proved most efficient in terms of reaching a correct solution when they were informed that their solution was incorrect. Implications for teaching and learning
combinatorics are discussed.
Combinatorics is one of the important areas of discrete mathematics, which is “an
active branch of contemporary mathematics that is widely used in business and industry” (National Council of Teachers of Mathematics [NCTM], 2000, p. 31).
Combinatorics provides tools for dealing with people’s everyday experience as
well as their professional practice, and is connected to various strands of mathematics and other disciplines, (e.g., computer science, communication, genetics,
and statistics). Thus, it is important to include combinatorics as an integral part of
Requests for reprints should be sent to Orit Zaslavsky, Department of Education in Technology &
Science, Technion–Israel Institute of Technology, Haifa 32000, Israel. E-mail: [email protected]
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16
MASHIACH EIZENBERG AND ZASLAVSKY
the mathematics curriculum, from the early elementary grades through to the senior high school level (English, 1993; NCTM, 2000).
In spite of the previous, combinartorics is considered one of the more difficult
mathematical topics to teach and to learn. Most problems do not have readily available solution methods, and create much uncertainty regarding how to approach
them and what method to employ. There are numerous examples in which two different solutions yielding different answers to the same problem may seem equally
convincing.
Several studies support the assertion that students encounter many difficulties
in solving combinatorial problems and shed light on some factors contributing to
these difficulties (Batanero, Godino, & Navarro-Pelayo, 1997; English, 1991;
Fischbein & Gazit, 1988; Hadar & Hadass, 1981; Kahneman & Tversky, 1973).
Most of the difficulties in solving combinatorial problems were identified in simple one-stage combinatorial problems, which differ along a number of dimensions:
Type (according to Fischbein & Gazit, 1988) or Operation (Batanero et al., 1997),
that is, arrangements, permutations, and combinations; Nature of elements to be
combined, that is, numbers, letters, people, and objects; Implicit combinatorial
model, that is, selection, distribution, and partition. The difficulties were examined
mainly through errors associated with three components of the solution process:
systematic enumeration, identification of the appropriate type (or operation) of
problem, and application of the necessary operation. Fischbein and Gazit and
Batanero et al. both found that the type (or operation) of the problem, the nature of
the elements, and instruction have an effect on the problem difficulty and type of
error.
Fischbein and Gazit (1988) also found that before instruction, children in sixth
and eighth grades were able to solve various combinatorial problems intuitively, particularly combination problems (36% of the 6 graders and 50% of the 8 graders
solved correctly the combination problems). More specifically, prior to instruction,
permutation problems were the more difficult; whereas combination problems were
easier. However, after instruction a switch occurred, and permutation problems became easier. This can be explained by the fact that the formula for the number of combinations is rather complicated, and once it is introduced, children abandon their intuitive empirical strategies. Batanero et al. (1997) described and exemplified 14 types
of common errors, reflecting difficulties in solving combinatorial problems. Their
main finding was that the type of implicit combinatorial model (of the problem) had a
“strong effect on both the problem difficulty and the type of error” (p. 196).
Hadar and Hadass (1981) investigated students’ difficulties in solving one complex combinatorial problem. They identified a number of pitfalls in solving the
problem, which are applicable to many other combinatorial problems typical in a
combinatorics course. The main pitfalls had to do with the identification of the set
of events under question, choice of appropriate notation, systematic method of
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VERIFICATION STRATEGIES FOR COMBINATORIAL PROBLEMS
17
counting, and generalization of a unifying structure and solution for several particular cases.
The previously mentioned studies support the need to address the difficulty in solving combinatorial problems. One way to do this is through
metacognitive skills, specifically, through verification of the solutions. Verifying an answer to a combinatorial problem is a particularly difficult task,
because there are no guaranteed ways to ensure the detection of an error. In
addition, the detection of an error does not necessarily yield a way to reach a
correct solution.
Verification is considered part of a “looking back” strategy and plays a critical
role in problem solving (Polya, 1957; Schoenfeld, 1984, 1985; Silver, 1987; Wilson, Fernandez, & Hadaway, 1993). First and most obvious, by checking solutions
one can catch careless errors, at a local level, and find support for a solution as well
as alternative solutions at a more global level (Schoenfeld, 1985). Polya and
Schoenfeld shared the view that by reconsideration and reexamination of the result
and the way in which it was obtained, students not only are likely to become more
confident and aware of the correctness and of their solution, but also to “consolidate their knowledge and develop their ability to solve problems” (Polya, 1957, p.
15). In addition, looking back at a solution provides an opportunity to investigate
connections within and among problems.
In spite of the importance attributed to the verification process, there is little
evidence regarding the tendency to verify solutions to mathematical problems
and its effect on successful problem solving. In Kantowski’s (1977) study of students’ tendencies to employ a number of heuristic behaviors in nonroutine geometry problems, she found that students are reluctant to use the “looking back”
strategy. A number of studies that looked into the connections between verification of a solution and success in solving a problem indicate that the use of verification is related to problem success (e.g., Cai, 1994; Hembree, 1992; Lucangeli,
& Cornoldi, 1997; Malloy & Jones, 1998). For example, Malloy and Jones
(1998) investigated the connection between verification and success in mathematical problem solving with 24 students between 12–14 years old. They looked
into five elements of the verification process: rereading the problem, checking
calculations, checking plan, using another method, and redoing the problem. The
authors found that the use of verification and problem success were moderately
related. These studies support the possible contribution of verification to problem solving.
We are not aware of any studies that have specifically investigated the ways in
which students and teachers cope with the difficulties associated with the verification of combinatorial problems. Our study is a step toward understanding students’
tendencies and approaches to verifying solutions to combinatorial problems and
the connections to their success in solving these problems.
18
MASHIACH EIZENBERG AND ZASLAVSKY
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THE STUDY
Aims
Similar to the studies investigating students’ solutions to combinatorial problems
prior to formal instruction, we investigated students’ verification strategies that
come naturally to them, not necessarily based on formal instruction.
In particular, we studied the following research questions about students’ verification of their solutions to combinatorial problems:
1. To what extent do students verify their solutions to combinatorial problems?
2. How successful are they?
3. What strategies do they use?
Participants
The participants in this study consisted of 14 undergraduate students all of whom
had completed at least one basic course in combinatorics prior to the study.
The students who participated in the study had responded to an advertisement
calling for candidates to participate in a study dealing with problem-solving
methods in combinatorics. Twenty-two candidates applied, ages 19 to 24 years
old. Of those who applied, we selected 16 students who stated that they had
learned to apply combinatorial principles and operations to problems involving
selection, distribution, and partition models, and felt they knew how to solve
such problems rather well. These students were interviewed to make sure that
they had the basic knowledge required for solving the problems included in the
research instrument. Consequently, we identified two candidates who did not
qualify to participate in the study, and did not include them in the study.
Eleven of the 14 participants were undergraduate students in two leading universities in Israel, studying for a bachelor of science degree, with the following
majors: two in mathematics, three in computer science, four in industrial engineering, and two in statistics. All 11 students learned combinatorics toward their
matriculation exam when they were in their 12th grade of high school, and in addition, took combinatorics as part of a university undergraduate course either in
discrete mathematics or probability. Another three participants were students in
a year-long, preuniversity extensive mathematics course, of which a large part included combinatorics. Of the 14 participants, 3 were women.
Research Instruments
Following Batanero et al.’s (1997) point about the strong connection between the
implicit combinatorial model of a problem and its difficulty, we designed a set of
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VERIFICATION STRATEGIES FOR COMBINATORIAL PROBLEMS
19
10 combinatorial problems varying with respect to their underlying models—a selection model, a distribution model, or a partition model (cf. Dubois, 1984). All
problems required only basic combinatorial tools for solution. However, to foster
the need to verify the solutions, each problem required the use of a variety of principles and operations (combinations, arrangements, and permutations) for its solution (see Appendix for the problems and their classification according to the underlying models and operations required for their solution). It should be noted that
all the problems were rather common, with no deceptive elements.
Procedure
To be able to interpret, describe, and understand the problem-solving processes of
the participants, a qualitative method was employed. Data collection was done by
the first author through audiotaped interviews and field-noted observations. The
researcher met each student 4 times. The first meeting was a short one (about 30
min) and the remainder were about 2 hr each, over a period of 2–3 weeks. The first
meeting was an introductory semistructured interview that aimed at getting acquainted with the participants, setting the grounds for the study, gathering some information about their prior experience and attitudes toward combinatorics, and
identifying students who were willing to work in pairs. The participants were encouraged to work in pairs when engaged in the problem-solving tasks (cf.
Schoenfeld, 1985). Thus, in the next three subsequent meetings (2nd–4th meetings), 6 students who felt comfortable to work with a peer met the researcher in
pairs, and the other 8 continued to meet her individually.
In these subsequent meetings, the participants were given 10 combinatorial
problems, one at a time, which they were asked to solve: Those who worked in
pairs were asked to do it collaboratively, whereas those who worked individually
were prompted to think aloud.
To address the main goals of the study, we followed the participants’ attempts to
verify their solutions either on their own, or in response to the researcher’s
prompts. For each problem, there were 2–3 main occasions in the interview where
a participant (or pair) could attempt to verify his or her (or their) solution: (a) As
part of the initial solution process, by his or her own initiative (we term this as
Stage V1); (b) After the participant(s) completed the solution, regardless of its correctness or of a prior attempt to verify the solution, the interviewer prompted the
participant(s) to verify the solution (Stage V2)—note that to avoid possible misinterpretation of her prompts as hidden messages regarding the correctness of the solution, at the very beginning the researcher told the interviewees that the questions
she would ask during the interview were set ahead of time and would not depend
on (or reflect) the correctness of their solutions; (3) when applicable, for solutions
that remained incorrect after Stage V2, the interviewer disclosed the information
that the solution was incorrect and prompted the participant(s) to verify their solu-
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MASHIACH EIZENBERG AND ZASLAVSKY
tion once again (Stage V3). At the end of the last session, the interviewer asked the
participant(s) whether they had ever been taught ways in which they could verify
their solutions to combinatorical problems.
Altogether, 11 series of interviews were conducted with the students (3 with
pairs, 8 with individuals), yielding 108 solutions (in only two cases was a student
unable to generate any solution).
Data Analysis
All interviews were transcribed. Each problem solution, for each individual or pair,
at each stage of the interview, was coded according to the following constructs: (a)
whether there was an attempt to verify the solution, (b) when applicable, the types of
verification strategies employed, and (c) the correctness of the solution.
An inductive analysis, with no predetermined classification criteria, of the attempts at solution verifications identified 11 types of approach, yielding a five-category classificatory scheme, which was refined after examination by two experts.
The correctness of a solution was determined first by its final answer: correct
and incorrect. To examine more closely the connection between students’ attempts
to verify their solutions and the correctness of their solutions, we further coded the
incorrect solutions according to the full underlying process, resulting in two levels
of incorrectness: procedurally incorrect and conceptually incorrect. An incorrect
solution was regarded as procedurally incorrect if the method that was used could
lead to a correct solution, yet, in the process of carrying out the plan, either a computational error was done or an appropriate formula was wrongly applied.
Complimentarily, an incorrect solution was regarded as conceptually incorrect if
the method employed was inappropriate, in other words, if the student did not
identify the underlying model of the problem or the operations that should be carried out or both.
Consequently, we regarded a verification attempt as helpful if it led a student to
improve his or her solution from either a conceptually incorrect one to a procedurally incorrect one, or from a conceptually or procedurally incorrect one, to a correct solution. A measure of efficiency of the verification of a solution was obtained, for each stage separately, by the percentage of the number of cases in which
the use of a verification strategy was helpful out of the total number of incorrect solutions that were verified at the respective stage.
FINDINGS
The findings pertain to the connections between the tendency to verify a solution
and the correctness of the solution. In addition, for those who verified their solu-
21
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VERIFICATION STRATEGIES FOR COMBINATORIAL PROBLEMS
tions, we characterize the different types of verification strategies employed and
analyze their usefulness in terms of improving the correctness of a solution.
It should be noted that all the participants claimed that they had never been
taught how to verify their solutions to combinatorial problems and that the attempts they made to verify their solutions were done intuitively. They felt they had
no formal knowledge of how to do so. For example, one participant claimed,
“This is the problem in combinatorics that you can’t check. I have no tools, or at least
I never learned how to check what I do. That is, I use formulas that were proven, and
with them I try to solve all kinds of problems. But I don’t have any indication for
checking myself.”
The Tendency to Verify and the Correctness
of the Solutions
At Stage V1 for 66 of the 108 solutions (61%) there were attempts made by students to verify the solution out of their own initiative. Of these 66 solutions, 34 remained incorrect in spite of these attempts. Table 1 presents the distribution of solutions that were verified at Stage V1 according to their correctness before and
after they were verified.
Note that only 6 of the 39 incorrect solutions (15%) that were verified at Stage
V1 were improved as a result of the verification (see Table 1).
As described earlier, at Stage V2 the interviewer asked the students to verify
their solution regardless of what they did in Stage V1. After this prompt, there were
still 25 of the 108 solutions (23%) for which no attempt was made to verify them,
whereas 83 solutions were verified. As shown in Table 2, only 7 of the 51 (14%) incorrect solutions that were verified at Stage V2 were improved as a result of the
verification attempts (see Table 2). In addition, there were 3 unfortunate modifications at Stage V2: Two from a correct to a conceptually incorrect solution and one
TABLE 1
The Distribution of Solutions That Were Verified at Stage V1,
by Correctness Before and After the Verification
After Verification
Correctness
Before Verification
Correct
Procedurally incorrect
Conceptually incorrect
Total
Correct
27
3
2
32
Procedurally Incorrect
0
9
1
10
Conceptally Incorrect
0
0
24
24
Total
27
12
27
66
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MASHIACH EIZENBERG AND ZASLAVSKY
TABLE 2
The Distribution of Solutions That Were Verified at Stage V2,
by Correctness Before and After Stage V2
After Stage V2
Correctness
Before Stage V2
Correct
Procedurally incorrect
Conceptually incorrect
Total
Correct
Procedurally Incorrect
30
3
1
34
0
11
3
14
Conceptally Incorrect
2
1
32
35
Total
32
15
36
83
from a procedurally incorrect to a conceptually incorrect solution (these were fully
corrected at Stage V3).
At the end of Stage V2, 63 solutions remained incorrect. As described earlier, at
the following stage (Stage V3) the interviewer informed the participants that their
solutions were still incorrect and repeated her request to verify their solutions. This
time the number of solutions that were verified increased: 54 of the 63 incorrect solutions (86%) were verified.
As indicated in Table 3, the verification attempts at Stage V3 were more effective than in the preceding stages: 18 solutions (33%) were improved (see Table 3). Note that these include three solutions that had been wrongly modified at
Stage V2. Yet, although the participants knew at this stage that their solution was
incorrect, for 41 solutions they were still not able to find where they went
wrong.
TABLE 3
The Distribution of Solutions That Were Verified at Stage V3, by
Correctness Before and After Stage V3
After Stage V3
Correctness
Before Stage V3
Correct
Procedurally incorrect
Conceptually incorrect
Total
Correct
Procedurally Incorrect
Conceptally Incorrect
Total
—
6
7
13
—
7
5
12
—
3
26
29
—
16
38
54
VERIFICATION STRATEGIES FOR COMBINATORIAL PROBLEMS
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Students’ Verification Strategies
Altogether, 219 attempts at solution verification were made at stages V1, V2, V3.
Except for 3 solutions, each solution was verified at least once. In many cases, a solution was verified several times within and across stages. There were 76 solutions
that were verified at more than one stage, and 19 solutions that were verified in
more than one way within a stage. As described earlier, our analysis yielded a
five-category scheme, as follows:
Verification by:
1.
2.
3.
4.
5.
Reworking the solution;
Adding justifications to the solution;
Evaluating the reasonability of the answer;
Modifying some components of the solution;
Using a different solution method and comparing answers.
We turn to a detailed description of these strategies, including some interview
excerpts illustrating their use.
Strategy 1: Verification by Reworking the Solution
The participants who used this strategy reworked their solution by going over
and checking all or parts of it a second time, without adding any substantial justifications to their solution. This kind of checking focused on various aspects of the
solution, such as checking their calculations or the extent to which their original
plan for solution was carried out. In several cases, this strategy served as a springboard for a more profound strategy of verification.
Strategy 2: Verification by Adding
Justifications to the Solution
The participants who used this strategy added justifications to their solution to
support it. The justifications referred to either a particular step in the solution or to
a more global aspect of the solution. Generally, the justifications were of three mutually related types: One type was directed to clarify and support some (or all) specific parts of the solution. A second type aimed at justifying in a more global way
the model (or formula) used to solve the problem, by showing how the conditions
and nature of the problem matched the model. A third type of justification was
based on an analogy to another, more familiar or previously solved problem, the
solution of which was known to the participant. This strategy was particularly
helpful for improving procedurally incorrect solutions, in which the general solu-
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MASHIACH EIZENBERG AND ZASLAVSKY
tion method was appropriate; however, there were some steps in which an error occurred in applying it.
Example 2.1. This example illustrates a justification of the model that was
used. It is taken from a solution of a pair of students, Gal and Yuval, who were
working on Problem 8 at Stage V1, and had solved it correctly. After reaching the
answer: 94 = 6,561, they decided to verify their answer out of their own initiative.
Basically, Gal was convinced that they solved the problem correctly. He went over
all the constraints of the problem and made sure that the model they used satisfied
these conditions.
Problem 8: A teacher has 8 pupils and 4 different pieces of candy. In how
many ways can the teacher distribute the candies to the pupils, if each pupil
may get more than one piece, and not all pieces need to be distributed?
Yuval: If you want, we can check this [the answer].
Gal: In my way? It’s a very long way. You have the possibility that 1
gets them, 2 gets them, 3 gets them, and 4 gets them. You will
get confused with all the numbers. We leave it. This is right. It is
clear that it is right. You satisfied all the conditions. This is the
checking: Each pupil can get more than 1 piece of candy. You
covered this limitation. [According to our solution] you choose
for a piece of candy to go to a pupil, and then it could be that 2
pieces go to the same pupil. So you covered the possibility that
each pupil can get more than 1 piece of candy. Also, not all
pieces have to be distributed. You took this as a possibility. This
is definitely right.
Example 2.2. This example illustrates a comparison to a similar problem. It is
taken from a solution of a pair of students, David and Omer, who were working on
Problem 6 at Stage V2, and had solved it correctly. They solved problem 6 using the
same operation and principle as they previously had done successfully for a similar
problem—problem 5, as follows: 53 ⋅ 3 4 − 3 ⋅ 2 4 + 3 = 360. Although, initially
( )[
]
they treated problem 6 as a selection problem, at the verification stage they looked at
it as a distribution problem, and were thus able to compare across the two problems.
Problem 6: Four executives were offered 5 different types of insurance policies. In how many ways can the executives choose each an insurance policy,
so that altogether exactly 3 different types of insurance policies are chosen?
VERIFICATION STRATEGIES FOR COMBINATORIAL PROBLEMS
25
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Omer: We have to arrange 4 people in 3 companies.
David: 4 balls in 3 cells.
Omer: It’s like dropping off, for example, 4 people in 3 stops. If inclusion-exclusion worked there [in Problem 5] it should also work
here. Different people … different stops.
David: And this you have 5 over 3 possibilities to do it.
Omer: We have 4 executives, and each 1 has 3 choices. So it is 34. This
is w(0). What is w(1)?
David: Out of the 3 types of insurance policies that we chose, 1 type
didn’t get anyone. 4 balls to 3 cells, when 1 cell is empty, any
cell can be empty. W(2) is when 2 cells are empty. Now you
want to divide 4 executives to 3 types [of policies], so that for
each type there is at least 1 executive.
Omer: Right, because in my world 34 it could be that all 4 fall into one
type. We must subtract all the cases in which 1 type [of the 3] is
not chosen, and all the cases in which 2 types are not chosen.
David: To add all the cases in which 2 types are not chosen [w(2)].
Omer: That’s it. It’s ok.
Strategy 3: Verification by Evaluating
the Reasonableness of the Answer
The participants who used this strategy looked at the final result that they had
obtained and tried to examine its reasonableness either by an intuitive estimate, or
more commonly, by calculating the size of the outcome-space. In a number of
cases, the participant noticed that the result she or he had reached was larger than
the outcome space, which did not make sense. In some cases this strategy led to the
identification of wrong answers, however, it was not helpful in locating the specific
erroneous considerations and steps in an incorrect solution.
Example 3.1. This example illustrates an evaluation of reasonableness of
the answer. It is taken from a solution of one student, Harry, who was working on
Problem 7 at Stage V1, and had solved it incorrectly, obtaining 495 as an answer.
Using the following verification strategy he realized that his answer is equal to the
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MASHIACH EIZENBERG AND ZASLAVSKY
sample size, thus it must be wrong, and consequently, turned to another solution
method.
Problem 7: In how many ways can we choose 4 people out of 6 married couples, so that at least one married couple is chosen?
Harry: The size of my sample is choosing 4 people out of 12 people, [calculates 12
= 495] that is 495. This cannot be. This means that
4
( )
100% I will choose a married couple. This cannot be. Why? I need
to choose 4 people out of 6 couples. I can choose 4 couples, and out
of each couple I can choose 1 person. Then, I can be sure that there
will not be a married couple. So there must be a mistake up there.
Strategy 4: Verification by Modifying
Some Components of the Solution
The participants who used this strategy made one of the following modifications
to their original solution: They either altered the representation they had used in their
solution or tried to apply the same solution method by using smaller numbers. Those
who used the former approach tried to represent the situation of the problem in a different way, mostly by using some visual symbols (e.g., circles, squares, blocks, etc.)
to represent the different components of the problem. Unfortunately, this kind of attempt did not prove helpful for them, because they repeated the same considerations
and arguments as in their original solution, failing to identify any faulty step. Those
who used the latter approach, that is, used smaller numbers, implicitly assumed that
using smaller numbers does not change the given problem in any significant way, although this implicit assumption is not necessarily true. On the contrary, using
smaller numbers for a combinatorial problem in an unskillful way may lead to a
nonisomorphic problem. However, when the use of smaller numbers is done without
changing the nature of the problem, it may prove helpful both for identifying errors
and for correcting the solution, as suggested in Example 4.2.
Example 4.1. This example illustrates a use of another representation. It is
taken from a solution of a pair of students, David and Omer, who were working on
problem 9 at Stage V1, and had solved it correctly, obtaining 73 = 35 as an an-
( )
swer. They used the following verification strategy out of their own initiative.
Problem 9: In how many ways can we distribute 3 identical gifts to 7 kids so
that each kid can get no more than one gift?
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VERIFICATION STRATEGIES FOR COMBINATORIAL PROBLEMS
27
Omer: Here, look at it this way: In how many ways can I arrange the 3
discs in this [sketches a row with 7 cells, see Figure 1]. In fact,
we need to choose 3 places out of 7. That’s it: 73 .
( )
At Stage V2, after the interviewer prompted them to verify their solution, Omer
referred to what they did at Stage V1 and supported it, as follows:
Omer: So that’s it, this is actually the proof. We take a row of partitions
the length of which is 7 and have to choose 3 cells. I have 73
( )
ways to do it, because after all there is no meaning to the order of
the selection.
Example 4.2. This example illustrates using smaller numbers. As previously
mentioned, some students tried to apply this approach but were unsuccessful in using it. To illustrate its potential, we bring an example taken from a solution of an expert professor of mathematics, Allan, who was working on Problem 7 (see Appendix) as part of a larger study addressing similar issues (Mashiach Eizenberg, 2001).
Allan had initially solved this incorrectly, obtaining 285 (instead of 255) as an
answer. His reasoning was that we can choose either 2 couples or 1 couple and 2
‘singles.’ Thus, according to Allan, there were 62 ways to choose exactly 2 couples out of 6 couples, and there were 6 ⋅
( )
( ) ways to choose one couple and 2 ‘sin10
2
gles.’ Allan’s mistake was in the counting of the 2 ‘singles.’ His initial reasoning
ways to choose
suggested that there were 6 ways to choose a couple, and then 10
2
( )
2 ‘singles’ of the remaining 10. In his reasoning he neglected to take into account
that the 2 people that were selected (at random) of the 10 may be a married couple
and not 2 ‘singles.’ Allan used the following verification strategy in response out of
his own initiative.
Allan: I think there is no big difference if instead of 6 couples in the
problem there would be 3 couples. To keep the same principle I
can’t decrease the number of people to choose, that is, I still
FIGURE 1
Omer’s sketch.
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28
MASHIACH EIZENBERG AND ZASLAVSKY
need to choose 4 people, because I need to maintain at least 2
couples. Otherwise it changes the problem.
Let’s call the couples (A,1), (B,2) and (C,3). Then the possible
cases of exactly 2 couples are: (A,1) & (B,2), or (A,1) & (C,3),
or (B,2) & (C,3). Now, if I want to have exactly 1 couple, I can
have (A,1) and 2 singles. Oh, now I see my mistake.
I chose [in the original problem] 2 out of the remaining 10, but
they must not be married. So I need to fix this.
[returning to the smaller number problem] The possibilities are
either (A,1) and B & C, or (A,1) and 2 & 3, or (A,1) and B & 3,
or (A,1) and C & 2. That is less 2 couples. So going back to the
less 5 couples, that is there
[original] problem it’s not but 10
2
( )
[( ) − 5] ways to choose exactly one couple.
are 6 ⋅
10
2
It should be noted that selecting numbers for the smaller-number problem is not
at all straightforward and entails deep considerations, as Allan articulated, to keep
the general structure of the problem in tact.
Strategy 5: Verification by Using a Different
Solution Method and Comparing Answers
The participants who used this strategy employed a completely different solution method for the problem. In these cases, the new solution method led to either
the same result that they had reached or to a different one. Table 4 presents the different cases according to the correctness of the first and second solution methods,
the difference between the results obtained in each way, and consequently, the decisions the participants made based on this strategy of verification.
TABLE 4
Decisions Made by Participants Employing Verification Strategy 5
First Method
Incorrect (N=43)
Correct (N=14)
Second Method
Comparison of Results
Choice of Solution
Total
Correct (N=13)
Different
Incorrect (N=30)
Same
Different
Correct (N=12)
Incorrect (N=2)
Same
Different
Second
First
Same
Second
First
None
Same
Second
12
1
3
17
9
1
12
2
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VERIFICATION STRATEGIES FOR COMBINATORIAL PROBLEMS
29
Example 5.1. This example illustrates using a different solution method. It
is taken from a solution of one student, Yariv, who was working individually on
Problem 6 (see Appendix) at Stage V2, and had solved it incorrectly. His solution
method looked at the problem as one in which the executives choose the insurance
policies, and went as follows: 42 ⋅ 5 ⋅ 4 ⋅ 3, which is basically correct. However, he
( )
made a wrong calculation, obtaining 320 (instead of 360) as an answer. Using the
following verification strategy, namely, a different solution method, he got a different answer—the correct one (360). Consequently, to be completely convinced
which of the two solutions was the correct one, he went back to his first solution,
went over it carefully and detected his error.
Yariv: I’ll try to think about it maybe from the angle of the insurance
policies. Let’s say I choose 3 types of policies out of the 5, that is
5
3 . Now we need to distribute them to 4 executives.
( )
I choose
( ) [2 of the executives that will have the same type of
4
2
policy], and multiply this by 3 [there are now 3 types of policies]
multiply by 2 [the other 2 executives can switch policies]. Thus:
5
4
⋅ 3 ⋅ 2 = 360.
3 ⋅ 2
( )( )
Before I got 320, but this [solution] is what counts. [goes back to
his first solution and checks the calculation].
Great! I got here also 360, so I’m sure.
Example 5.2. This example illustrates using a different solution method. It
is taken from a solution of one student, Harry, who was working on Problem 8 (see
Appendix) at Stage V3, and had solved it incorrectly, in the following way:
4
∑8
i
= 4,681. The explanation he gave was that we distribute either 0, 1, 2, 3, or 4
i=0
pieces of candies, and each piece of candy can be given to one of the 8 kids. In this
(erroneous) method, he did not take into account that the pieces are different and
the inner arrangement of the candies increases the number of ways to distribute the
candy. Using a different solution method, he obtained the correct solution, went
back to his previous solution, and corrected it, as follows:
Harry: So I can look at it as if each piece of candy can be given to any
kid or not given at all. That’s why it [refers to his original solution] is not right.
So [the answer is] 94 = 6,561. [turns back to his first solution and
writes:]
30
MASHIACH EIZENBERG AND ZASLAVSKY
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1 × 4 + 8 × (14 )+ 64 × (24 )+ 512 × (34 )+ 4,096 × (44 ) = 6,564
Now Harry is puzzled, because he got 6,564 instead of 6,561. He tries to find
where this difference came from:
Ok. I got 2 different results. First, what possibilities are repeated? I have 3 extra ones. What does this 4 represent? [Refers
to the 4 in the 1⋅ 4 at the beginning of his long computation]. It
represents the possibility that i = 0, that the teacher distributes 0
candies. In how many ways can he do it? Only 1 way, not 4. This
is good enough for me.
[He is now satisfied that he detected the error in his long sum]
Examples 5.1 and 5.2 illustrate how Strategy 5 was helpful in correcting an incorrect solution. There were other cases, in which this verification strategy was
employed. As seen in Table 4, Strategy 5 was used altogether in 57 cases. There
were 43 cases with an incorrect solution, and 14 cases with a correct solution, just
before this verification mode was employed. Of the 43 cases with an incorrect solution, in 12 cases this strategy proved helpful in correcting their solutions.
Interestingly, there were two pairs of students and one individual student who
reached the same incorrect result with two different solution methods, thus their
answers remained incorrect leaving them more confident of their wrong solutions
(as part of a larger study, a systematic trace of their degree of confidence was conducted, Mashiach Eizenberg, 2001).
Of the 14 cases with a correct solution, 12 cases remained correct in both solution methods. However, Strategy 5—verification by using a different solution
method and comparing answers—led to two unfortunate decisions (in Stage V2),
where a wrong result was obtained the second time, causing a switch from a correct to an incorrect solution. Fortunately, in the following stage (V3) another switch
was done, in both cases returning to the initial correct solution.
Some Comparisons between the Verification Strategies
As mentioned earlier, there were altogether 219 verification attempts, applied to
both correct and incorrect solutions. Table 5 presents the distribution of the verification attempts by the different verification strategies. It should be noted that the
distribution of the verification attempts of only the incorrect solutions was similar
to the one in Table 5. The most frequently used strategies were Strategy 1 (38%),
Strategy 2 (26%), and Strategy 5 (26%).
A further analysis of the use of the different verification strategies focused on
the extent to which each strategy was helpful in leading the participants to improving their solutions. Altogether, there were 32 improved solutions as a result of em-
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VERIFICATION STRATEGIES FOR COMBINATORIAL PROBLEMS
31
TABLE 5
The Distribution of the Number of Attempts to Verify a Solution
by the Type of Verification Strategy Employed
Types of Verification Strategies
1. Reworking the solution
2. Adding justification to the solution
3. Evaluating the reasonability of the answer
4. Modifying some components of the solution
5. Using a different solution method and comparing
Total
Number of Attempts
83
57
11
11
57
219
ploying a verification strategy (note that some improvements are attributed to more
than one verification strategy). Table 6 presents the distribution of the improved
cases by the different verification strategies. As shown in Table 6, the most efficient verification strategies were Strategy 2 (in 12 cases) and Strategy 5 (in 13
cases). Note that there were 5 cases in which a combination of verification strategies led to an improved solution, in each of which Strategy 5 was the primary strategy leading to the correct solution, whereas the others served only to detect an error and motivate the search for an alternative solution. For example, one student
detected an error through Strategy 3, identified the kind of error through Strategy
4, and finally corrected the solution using Strategy 5.
IMPLICATIONS FOR TEACHING
AND LEARNING COMBINATORICS
Our findings support the assertion that combinatorics is a complex topic—only 43
of the 108 initial solutions were correct. The findings also strengthen the merit of
encouraging students to verify their solutions to combinatorial problems. It seems
TABLE 6
Distribution of the Number of Cases in Which the Use of a Verification
Strategy was Helpful in the Process of Improving a Solution,
by Type of Verification Strategy
Types of Verification Strategies
1. Reworking the solution
2. Adding justification to the solution
3. Evaluating the reasonability of the answer
4. Modifying some components of the solution
5. Using a different solution method and comparing
Total
Number of Cases
2
12
3
2
13
32
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MASHIACH EIZENBERG AND ZASLAVSKY
that, when encouraged to do so, there are students who are capable, to a certain extent, of finding efficient ways to verify their solutions without prior direct instruction. This is true particularly when students know that their solution is incorrect.
However, the findings indicate that many of the students who made attempts to
verify their incorrect solutions, whether out of their own initiative or in response to
the interviewer’s prompts, were not able to come up with efficient verification
strategies and were thus neither able to detect an error nor to correct their solution.
This state of affairs calls for the need to explicitly teach verification strategies as
part of the teaching of combinatorial problem solving. Our study offers a variety of
(more and less useful) verification strategies, which could serve as a basis to increase students’ awareness of the need to verify and provide them with ways to do
so. We propose that the verification strategies that were identified within our study
should be explored by students, addressing the potential and the limitation of each
one.
As shown earlier, Strategy 1 (i.e., verification by reworking the solution) was
the most frequently used, however, it turned out to be one of the least efficient strategies in terms of helping students shift toward an improved solution. The low efficiency level of Strategy 1 is in accordance with the view that merely going over a
solution of a mathematical problem is insufficient for verifying it (Polya, 1957;
Schoenfeld, 1985). Strategy 2 (i.e., verification by adding justifications to the solution) turned out to be considerably helpful, especially in clarifying and supporting
the various steps in a solution. This strategy was particularly helpful in detecting
minor errors.
Unlike Strategy 2, Strategy 3 (i.e., verification by evaluating the reasonability
of the answer) was not frequently used, probably because estimating an expected
outcome in a combinatorial problem is extremely hard to do. Fischbein and
Grossman (1997) found that when asked to estimate such results, students usually
gave lower estimates than the actual number. Thus, Strategy 3 was helpful in detecting that an error had occurred only in cases when the answer that was obtained
was larger than the size of the outcome-space and when the student(s) applying
this strategy compared these two numbers.
Strategy 4 (i.e., verification by modifying some components of the solution)
could be very powerful (see Example 4.2), particularly when applying the same solution method by using smaller numbers. However, this requires deep structural
considerations that need to be dealt with. We speculate that although it may seem
natural to students to employ this strategy (as indeed some tried to), applying it
correctly needs direct and systematic learning. Support for our claim was found in
the recommendations of three professors of mathematics who teach both undergraduate and graduate combinatorics courses. In advocating the potential of Strategy 4, one professor stated, “I think that maybe the simplest way to verify it [the
solution] is to write the smallest numerical example that conserves the content of
the problem” (Mashiach Eizenberg, 2001, p. 154). Clearly, maintaining the general
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VERIFICATION STRATEGIES FOR COMBINATORIAL PROBLEMS
33
structure of the problem is not a trivial task. It is one of the main difficulties that
Hadar and Hadass (1981) identified in solving a combinatorial problem.
Finally, we turn to Strategy 5 (i.e., verification by using a different solution
method), which was both frequent and rather helpful. In most cases, using a genuinely different solution method is likely to be helpful in detecting an error.
However, the limitation of this strategy is manifested in (not so frequent) cases
in which the same incorrect answer can be reached by two different solution
methods. This occurs when what appear to be two different autonomous solution methods are in fact interconnected in a way that the incorrectness of one
implies the incorrectness of the other. For example, if, for a given solution
method, the alternative one relies on the principle of complement of an event,
it is likely that the same faulty considerations that were employed for the direct
solution will be repeated for the complementary one. In our study, there were
three such cases that raised the participants’ degree of confidence in their incorrect solution.
To illustrate the limitation of Strategy 5, we bring a quote from a student, who
suggested to his partner at Stage V3 (after learning from the interviewer that their
solution was incorrect) to verify their solution by solving the problem in a different
way. The student argued, “If we reach the same result in a different way then the
answer is right and the question is wrong.” To apply Strategy 5, students need to
gain experience in essentially different multiple approaches to solving combinatorial problems. Thus, teaching this type of verification strategy is tightly connected to teaching problem solving methods in combinatorics.
Of interest, the verification strategies that were found helpful did not rely on recalling formulas, but mostly on an intuitive empirical approach. Thus, it seems important that teachers build on students’ informal approaches and be aware of the
frequent danger of formal instruction on success in solving combinatorial problems, as indicated in Fischbein and Gazit’s study (1988).
The differences that were found with respect to the efficiency of the various
strategies may serve as a springboard for teaching combinatorial verification strategies with a focus on metacognitive processes, in general, and on the use of more
efficient verification strategies, in particular. We suggest that through more successful verification experiences students are likely to become aware of the potential of verifying their solutions, and hopefully, will be motivated to verify their solutions on their own initiative.
REFERENCES
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Cai, J. (1994). A protocol-analytic study of metacognition in mathematical problem solving. Mathematics Education Research Journal, 6, 166–183.
Dubois, J. G. (1984). Une systematique des configurations combinatoires simples. [A systematic for
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English, L. D. (1993). Children’s strategies for solving two-and three-dimensional combinatorial problems. Journal for Research in Mathematics Education, 24, 255–273.
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Fischbein, E., & Grossman, A. (1997). Schemata and intuitions in combinatorial reasoning. Educational Studies in Mathematics, 34, 27–47.
Hadar, N., & Hadass, R. (1981). The road to solving a combinatotial problem is strewn with pitfalls. Educational Studies in Mathematics, 12, 435–443.
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Macmillan.
Selection Model
With replacement
and with order
Without
replacement and
with order
Without
replacement and
without order
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The Operation by
Model
APPENDIX
The Research Instrument
Classification of the Ten Problems Used in the Study by Combinatorial Models and Operations
Permutations
Arrangements With Replacement
Arrangements Without
Replacement
Problem 6: Four executives were
offered 5 different types of
insurance policies. In how many
ways can the executives choose
each an insurance policy, so that
altogether exactly 3 different types
of insurance policies are chosen?
Combinations Without
Replacement
Problem 6: Four executives were
offered 5 different types of
insurance policies. In how many
ways can the executives choose
each an insurance policy, so that
altogether exactly 3 different types
of insurance policies are chosen?
Problem 4: How many 5 digit
numbers with 5 different digits can
we form without 0, so that if the
digits 5 and 6 appear in the
number, they are not adjacent to
each other?
Problem 2: There are 3 red balls, 3
white balls and 2 black balls in a
bowl. In how many ways can we
choose (without replacement) 4
balls, so that there is at least one
ball of each color?
Problem 7: In how many ways can
we choose 4 people out of 6
married couples, so that at least
one married couple is chosen?
35
(continued)
Distribution Model
Different objects
into different
cells with order
Different objects
into different
cells without
order
Identical objects
into different
cells
Partition Model
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36
The Operation by
Model
Appendix (Continued)
Permutations
Problem 1: In how many ways can we
seat in one row, 2 men, 2 women,
and a dog, one next to the other, so
that the 2 men do not sit next to
each other, and the 2 women do not
sit next to each other?
Problem 3: In how many ways can
we seat in a circle, 3 men and 3
women, one next to the other, so
that none of the men sit next to
each other and none of the
women sit next to each other?
Arrangements With
Replacement
Arrangements Without
Replacement
Combinations Without
Replacement
Problem 8: A teacher has 8 pupils
and 4 different pieces of candy.
In how many ways can the
teacher distribute the candies to
the pupils, if each pupil may get
more than one piece, and not all
pieces need to be distributed?
Problem 5: In how many ways can
we drop off 5 bus passengers in 3
stops, so that in each stop at least
one passenger gets off the bus?
Problem 5: In how many ways can
we drop off 5 bus passengers in 3
stops so that in each stop at least
one passenger gets off the bus?
Problem 9: In how many ways can
we distribute 3 identical gifts to
7 kids so that each kid can get no
more than one gift?
Problem 10: Six boys and 6 girls
are divided into 2 groups (not
necessarily of the same size). In
how many ways can they be
grouped so that in each group,
there is the same number of girls
as boys?