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Students' Verification Strategies for CombinatorialProblemsMichal Mashiach Eizenberg a; Orit Zaslavsky ba 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', MathematicalThinking and Learning, 6:1, 15 — 36

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Students’ Verification Strategies forCombinatorial Problems

Michal Mashiach EizenbergEmek Yezreel College, Israel

Orit ZaslavskyTechnion—Israel Institute of Technology, Israel

We focus on a major difficulty in solving combinatorial problems, namely, on theverification of a solution. Our study aimed at identifying undergraduate students’ ten-dencies to verify their solutions, and the verification strategies that they employ whensolving these problems. In addition, an attempt was made to evaluate the level of effi-ciency of the students’ various verification strategies in terms of their contribution toreaching a correct solution. 14 undergraduate students, who had taken at least 1course in combinatorics, participated in the study. None of the students had prior di-rect learning experience with combinatorial verification strategies. Data were col-lected through interviews with individual or pairs of participants as they solved, 1 by1, 10 combinatorial problems. 5 types of verification strategies were identified, 2 ofwhich were more frequent and more helpful than others. Students’ verificationsproved most efficient in terms of reaching a correct solution when they were in-formed that their solution was incorrect. Implications for teaching and learningcombinatorics are discussed.

Combinatorics is one of the important areas of discrete mathematics, which is “anactive branch of contemporary mathematics that is widely used in business and in-dustry” (National Council of Teachers of Mathematics [NCTM], 2000, p. 31).Combinatorics provides tools for dealing with people’s everyday experience aswell as their professional practice, and is connected to various strands of mathe-matics and other disciplines, (e.g., computer science, communication, genetics,and statistics). Thus, it is important to include combinatorics as an integral part of

MATHEMATICAL THINKING AND LEARNING, 6(1), 15–36Copyright © 2004, Lawrence Erlbaum Associates, Inc.

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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the mathematics curriculum, from the early elementary grades through to the se-nior high school level (English, 1993; NCTM, 2000).

In spite of the previous, combinartorics is considered one of the more difficultmathematical topics to teach and to learn. Most problems do not have readily avail-able solution methods, and create much uncertainty regarding how to approachthem and what method to employ. There are numerous examples in which two dif-ferent solutions yielding different answers to the same problem may seem equallyconvincing.

Several studies support the assertion that students encounter many difficultiesin solving combinatorial problems and shed light on some factors contributing tothese 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 sim-ple 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 becombined, that is, numbers, letters, people, and objects; Implicit combinatorialmodel, that is, selection, distribution, and partition. The difficulties were examinedmainly through errors associated with three components of the solution process:systematic enumeration, identification of the appropriate type (or operation) ofproblem, and application of the necessary operation. Fischbein and Gazit andBatanero et al. both found that the type (or operation) of the problem, the nature ofthe elements, and instruction have an effect on the problem difficulty and type oferror.

Fischbein and Gazit (1988) also found that before instruction, children in sixthandeighthgradeswereable tosolvevariouscombinatorialproblems intuitively,par-ticularly combination problems (36% of the 6 graders and 50% of the 8 graderssolved correctly the combination problems). More specifically, prior to instruction,permutation problems were the more difficult; whereas combination problems wereeasier. However, after instruction a switch occurred, and permutation problems be-cameeasier.Thiscanbeexplainedby thefact that the formula for thenumberofcom-binations is rather complicated, and once it is introduced, children abandon their in-tuitive empirical strategies. Batanero et al. (1997) described and exemplified 14 typesof common errors, reflecting difficulties in solving combinatorial problems. Theirmain 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 com-plex combinatorial problem. They identified a number of pitfalls in solving theproblem, which are applicable to many other combinatorial problems typical in acombinatorics course. The main pitfalls had to do with the identification of the setof events under question, choice of appropriate notation, systematic method of

16 MASHIACH EIZENBERG AND ZASLAVSKY

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counting, and generalization of a unifying structure and solution for several partic-ular cases.

The previously mentioned studies support the need to address the diffi-culty in solving combinatorial problems. One way to do this is throughmetacognitive skills, specifically, through verification of the solutions. Ver-ifying an answer to a combinatorial problem is a particularly difficult task,because there are no guaranteed ways to ensure the detection of an error. Inaddition, the detection of an error does not necessarily yield a way to reach acorrect solution.

Verification is considered part of a “looking back” strategy and plays a criticalrole in problem solving (Polya, 1957; Schoenfeld, 1984, 1985; Silver, 1987; Wil-son, Fernandez, & Hadaway, 1993). First and most obvious, by checking solutionsone can catch careless errors, at a local level, and find support for a solution as wellas alternative solutions at a more global level (Schoenfeld, 1985). Polya andSchoenfeld shared the view that by reconsideration and reexamination of the resultand the way in which it was obtained, students not only are likely to become moreconfident and aware of the correctness and of their solution, but also to “consoli-date their knowledge and develop their ability to solve problems” (Polya, 1957, p.15). In addition, looking back at a solution provides an opportunity to investigateconnections within and among problems.

In spite of the importance attributed to the verification process, there is littleevidence regarding the tendency to verify solutions to mathematical problemsand its effect on successful problem solving. In Kantowski’s (1977) study of stu-dents’ tendencies to employ a number of heuristic behaviors in nonroutine ge-ometry problems, she found that students are reluctant to use the “looking back”strategy. A number of studies that looked into the connections between verifica-tion of a solution and success in solving a problem indicate that the use of verifi-cation 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 mathe-matical problem solving with 24 students between 12–14 years old. They lookedinto five elements of the verification process: rereading the problem, checkingcalculations, checking plan, using another method, and redoing the problem. Theauthors found that the use of verification and problem success were moderatelyrelated. These studies support the possible contribution of verification to prob-lem solving.

We are not aware of any studies that have specifically investigated the ways inwhich students and teachers cope with the difficulties associated with the verifica-tion of combinatorial problems. Our study is a step toward understanding students’tendencies and approaches to verifying solutions to combinatorial problems andthe connections to their success in solving these problems.

VERIFICATION STRATEGIES FOR COMBINATORIAL PROBLEMS 17

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THE STUDY

Aims

Similar to the studies investigating students’ solutions to combinatorial problemsprior to formal instruction, we investigated students’ verification strategies thatcome naturally to them, not necessarily based on formal instruction.

In particular, we studied the following research questions about students’verifi-cation of their solutions to combinatorial problems:

1. Towhatextentdostudentsverify their solutions tocombinatorialproblems?2. How successful are they?3. What strategies do they use?

Participants

The participants in this study consisted of 14 undergraduate students all of whomhad completed at least one basic course in combinatorics prior to the study.

The students who participated in the study had responded to an advertisementcalling for candidates to participate in a study dealing with problem-solvingmethods in combinatorics. Twenty-two candidates applied, ages 19 to 24 yearsold. Of those who applied, we selected 16 students who stated that they hadlearned to apply combinatorial principles and operations to problems involvingselection, distribution, and partition models, and felt they knew how to solvesuch problems rather well. These students were interviewed to make sure thatthey had the basic knowledge required for solving the problems included in theresearch instrument. Consequently, we identified two candidates who did notqualify to participate in the study, and did not include them in the study.

Eleven of the 14 participants were undergraduate students in two leading uni-versities in Israel, studying for a bachelor of science degree, with the followingmajors: two in mathematics, three in computer science, four in industrial engi-neering, and two in statistics. All 11 students learned combinatorics toward theirmatriculation exam when they were in their 12th grade of high school, and in ad-dition, took combinatorics as part of a university undergraduate course either indiscrete mathematics or probability. Another three participants were students ina year-long, preuniversity extensive mathematics course, of which a large part in-cluded combinatorics. Of the 14 participants, 3 were women.

Research Instruments

Following Batanero et al.’s (1997) point about the strong connection between theimplicit combinatorial model of a problem and its difficulty, we designed a set of


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10 combinatorial problems varying with respect to their underlying models—a se-lection model, a distribution model, or a partition model (cf. Dubois, 1984). Allproblems required only basic combinatorial tools for solution. However, to fosterthe need to verify the solutions, each problem required the use of a variety of prin-ciples and operations (combinations, arrangements, and permutations) for its solu-tion (see Appendix for the problems and their classification according to the un-derlying models and operations required for their solution). It should be noted thatall the problems were rather common, with no deceptive elements.

Procedure

To be able to interpret, describe, and understand the problem-solving processes ofthe participants, a qualitative method was employed. Data collection was done bythe first author through audiotaped interviews and field-noted observations. Theresearcher met each student 4 times. The first meeting was a short one (about 30min) and the remainder were about 2 hr each, over a period of 2–3 weeks. The firstmeeting was an introductory semistructured interview that aimed at getting ac-quainted with the participants, setting the grounds for the study, gathering some in-formation about their prior experience and attitudes toward combinatorics, andidentifying students who were willing to work in pairs. The participants were en-couraged to work in pairs when engaged in the problem-solving tasks (cf.Schoenfeld, 1985). Thus, in the next three subsequent meetings (2nd–4th meet-ings), 6 students who felt comfortable to work with a peer met the researcher inpairs, and the other 8 continued to meet her individually.

In these subsequent meetings, the participants were given 10 combinatorialproblems, one at a time, which they were asked to solve: Those who worked inpairs were asked to do it collaboratively, whereas those who worked individuallywere prompted to think aloud.

To address the main goals of the study, we followed the participants’attempts toverify their solutions either on their own, or in response to the researcher’sprompts. For each problem, there were 2–3 main occasions in the interview wherea participant (or pair) could attempt to verify his or her (or their) solution: (a) Aspart of the initial solution process, by his or her own initiative (we term this asStage V1); (b) After the participant(s) completed the solution, regardless of its cor-rectness or of a prior attempt to verify the solution, the interviewer prompted theparticipant(s) to verify the solution (Stage V2)—note that to avoid possible misin-terpretation of her prompts as hidden messages regarding the correctness of the so-lution, at the very beginning the researcher told the interviewees that the questionsshe would ask during the interview were set ahead of time and would not dependon (or reflect) the correctness of their solutions; (3) when applicable, for solutionsthat remained incorrect after Stage V2, the interviewer disclosed the informationthat the solution was incorrect and prompted the participant(s) to verify their solu-


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tion once again (Stage V3). At the end of the last session, the interviewer asked theparticipant(s) whether they had ever been taught ways in which they could verifytheir solutions to combinatorical problems.

Altogether, 11 series of interviews were conducted with the students (3 withpairs, 8 with individuals), yielding 108 solutions (in only two cases was a studentunable 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 ofverification strategies employed, and (c) the correctness of the solution.

An inductive analysis, with no predetermined classification criteria, of the at-tempts at solution verifications identified 11 types of approach, yielding a five-cat-egory classificatory scheme, which was refined after examination by two experts.

The correctness of a solution was determined first by its final answer: correctand incorrect. To examine more closely the connection between students’attemptsto verify their solutions and the correctness of their solutions, we further coded theincorrect solutions according to the full underlying process, resulting in two levelsof incorrectness: procedurally incorrect and conceptually incorrect. An incorrectsolution was regarded as procedurally incorrect if the method that was used couldlead to a correct solution, yet, in the process of carrying out the plan, either a com-putational error was done or an appropriate formula was wrongly applied.Complimentarily, an incorrect solution was regarded as conceptually incorrect ifthe method employed was inappropriate, in other words, if the student did notidentify the underlying model of the problem or the operations that should be car-ried out or both.

Consequently, we regarded a verification attempt as helpful if it led a student toimprove his or her solution from either a conceptually incorrect one to a procedur-ally incorrect one, or from a conceptually or procedurally incorrect one, to a cor-rect solution. A measure of efficiency of the verification of a solution was ob-tained, for each stage separately, by the percentage of the number of cases in whichthe use of a verification strategy was helpful out of the total number of incorrect so-lutions that were verified at the respective stage.

FINDINGS

The findings pertain to the connections between the tendency to verify a solutionand the correctness of the solution. In addition, for those who verified their solu-


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tions, we characterize the different types of verification strategies employed andanalyze their usefulness in terms of improving the correctness of a solution.

It should be noted that all the participants claimed that they had never beentaught how to verify their solutions to combinatorial problems and that the at-tempts they made to verify their solutions were done intuitively. They felt they hadno 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 leastI never learned how to check what I do. That is, I use formulas that were proven, andwith them I try to solve all kinds of problems. But I don’t have any indication forchecking myself.”

The Tendency to Verify and the Correctnessof the Solutions

At Stage V1 for 66 of the 108 solutions (61%) there were attempts made by stu-dents to verify the solution out of their own initiative. Of these 66 solutions, 34 re-mained incorrect in spite of these attempts. Table 1 presents the distribution of so-lutions that were verified at Stage V1 according to their correctness before andafter they were verified.

Note that only 6 of the 39 incorrect solutions (15%) that were verified at StageV1 were improved as a result of the verification (see Table 1).

As described earlier, at Stage V2 the interviewer asked the students to verifytheir solution regardless of what they did in Stage V1. After this prompt, there werestill 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%) in-correct solutions that were verified at Stage V2 were improved as a result of theverification attempts (see Table 2). In addition, there were 3 unfortunate modifica-tions at Stage V2: Two from a correct to a conceptually incorrect solution and one


TABLE 1The Distribution of Solutions That Were Verified at Stage V1,

by Correctness Before and After the Verification

After Verification

Correctness Correct Procedurally Incorrect Conceptally Incorrect Total

Before VerificationCorrect 27 0 0 27Procedurally incorrect 3 9 0 12Conceptually incorrect 2 1 24 27Total 32 10 24 66

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from a procedurally incorrect to a conceptually incorrect solution (these were fullycorrected at Stage V3).

At the end of Stage V2, 63 solutions remained incorrect. As described earlier, atthe following stage (Stage V3) the interviewer informed the participants that theirsolutions were still incorrect and repeated her request to verify their solutions. Thistime the number of solutions that were verified increased: 54 of the 63 incorrect so-lutions (86%) were verified.

As indicated in Table 3, the verification attempts at Stage V3 were more ef-fective than in the preceding stages: 18 solutions (33%) were improved (see Ta-ble 3). Note that these include three solutions that had been wrongly modified atStage V2. Yet, although the participants knew at this stage that their solution wasincorrect, for 41 solutions they were still not able to find where they wentwrong.


TABLE 2The Distribution of Solutions That Were Verified at Stage V2,

by Correctness Before and After Stage V2

After Stage V2


Before Stage V2

Correct 30 0 2 32Procedurally incorrect 3 11 1 15Conceptually incorrect 1 3 32 36Total 34 14 35 83

TABLE 3The Distribution of Solutions That Were Verified at Stage V3, by

Correctness Before and After Stage V3

After Stage V3


Before Stage V3

Correct — — — —Procedurally incorrect 6 7 3 16Conceptually incorrect 7 5 26 38Total 13 12 29 54

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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 so-lution was verified several times within and across stages. There were 76 solutionsthat were verified at more than one stage, and 19 solutions that were verified inmore than one way within a stage. As described earlier, our analysis yielded afive-category scheme, as follows:

Verification by:

1. Reworking the solution;2. Adding justifications 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 answers.

We turn to a detailed description of these strategies, including some interviewexcerpts illustrating their use.

Strategy 1: Verification by Reworking the Solution

The participants who used this strategy reworked their solution by going overand checking all or parts of it a second time, without adding any substantial justifi-cations to their solution. This kind of checking focused on various aspects of thesolution, such as checking their calculations or the extent to which their originalplan for solution was carried out. In several cases, this strategy served as a spring-board for a more profound strategy of verification.

Strategy 2: Verification by AddingJustifications to the Solution

The participants who used this strategy added justifications to their solution tosupport it. The justifications referred to either a particular step in the solution or toa more global aspect of the solution. Generally, the justifications were of three mu-tually related types: One type was directed to clarify and support some (or all) spe-cific parts of the solution. A second type aimed at justifying in a more global waythe model (or formula) used to solve the problem, by showing how the conditionsand nature of the problem matched the model. A third type of justification wasbased on an analogy to another, more familiar or previously solved problem, thesolution of which was known to the participant. This strategy was particularlyhelpful for improving procedurally incorrect solutions, in which the general solu-


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tion method was appropriate; however, there were some steps in which an error oc-curred in applying it.

Example 2.1. This example illustrates a justification of the model that wasused. It is taken from a solution of a pair of students, Gal and Yuval, who wereworking on Problem 8 at Stage V1, and had solved it correctly. After reaching theanswer: 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 overall the constraints of the problem and made sure that the model they used satisfiedthese conditions.

Problem 8: A teacher has 8 pupils and 4 different pieces of candy. In howmany ways can the teacher distribute the candies to the pupils, if each pupilmay 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 1gets them, 2 gets them, 3 gets them, and 4 gets them. You willget confused with all the numbers. We leave it. This is right. It isclear that it is right. You satisfied all the conditions. This is thechecking: Each pupil can get more than 1 piece of candy. Youcovered this limitation. [According to our solution] you choosefor a piece of candy to go to a pupil, and then it could be that 2pieces go to the same pupil. So you covered the possibility thateach pupil can get more than 1 piece of candy. Also, not allpieces have to be distributed. You took this as a possibility. Thisis definitely right.

Example 2.2. This example illustrates a comparison to a similar problem. It istaken from a solution of a pair of students, David and Omer, who were working onProblem 6 at Stage V2, and had solved it correctly. They solved problem 6 using thesame operation and principle as they previously had done successfully for a similarproblem—problem 5, as follows:( ) [ ]3

5 4 43 3 2 3 360⋅ − ⋅ + = . Although, initially

they treated problem 6 as a selection problem, at the verification stage they looked atit asadistributionproblem,andwere thusable tocompareacross the twoproblems.

Problem 6: Four executives were offered 5 different types of insurance poli-cies. In how many ways can the executives choose each an insurance policy,so that altogether exactly 3 different types of insurance policies are chosen?


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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 inclu-sion-exclusion worked there [in Problem 5] it should also workhere. 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. Thisis w(0). What is w(1)?

David: Out of the 3 types of insurance policies that we chose, 1 typedidn’t get anyone. 4 balls to 3 cells, when 1 cell is empty, anycell can be empty. W(2) is when 2 cells are empty. Now youwant to divide 4 executives to 3 types [of policies], so that foreach type there is at least 1 executive.

Omer: Right, because in my world 34 it could be that all 4 fall into onetype. We must subtract all the cases in which 1 type [of the 3] isnot 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 Evaluatingthe Reasonableness of the Answer

The participants who used this strategy looked at the final result that they hadobtained and tried to examine its reasonableness either by an intuitive estimate, ormore commonly, by calculating the size of the outcome-space. In a number ofcases, the participant noticed that the result she or he had reached was larger thanthe outcome space, which did not make sense. In some cases this strategy led to theidentification of wrong answers, however, it was not helpful in locating the specificerroneous considerations and steps in an incorrect solution.

Example 3.1. This example illustrates an evaluation of reasonableness ofthe answer. It is taken from a solution of one student, Harry, who was working onProblem 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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sample size, thus it must be wrong, and consequently, turned to another solutionmethod.

Problem 7: In how many ways can we choose 4 people out of 6 married cou-ples, so that at least one married couple is chosen?

Harry: The size of my sample is choosing 4 people out of 12 people, [cal-culates ( )4

12 495= ] that is 495. This cannot be. This means that

100% I will choose a married couple. This cannot be. Why? I needto choose 4 people out of 6 couples. I can choose 4 couples, and outof each couple I can choose 1 person. Then, I can be sure that therewill not be a married couple. So there must be a mistake up there.

Strategy 4: Verification by ModifyingSome Components of the Solution

The participants who used this strategy made one of the following modificationsto their original solution: They either altered the representation they had used in theirsolution or tried to apply the same solution method by using smaller numbers. Thosewho used the former approach tried to represent the situation of the problem in a dif-ferent 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 at-tempt did not prove helpful for them, because they repeated the same considerationsand arguments as in their original solution, failing to identify any faulty step. Thosewho used the latter approach, that is, used smaller numbers, implicitly assumed thatusing smaller numbers does not change the given problem in any significant way, al-though this implicit assumption is not necessarily true. On the contrary, usingsmaller numbers for a combinatorial problem in an unskillful way may lead to anonisomorphic problem. However, when the use of smaller numbers is done withoutchanging the nature of the problem, it may prove helpful both for identifying errorsand for correcting the solution, as suggested in Example 4.2.

Example 4.1. This example illustrates a use of another representation. It istaken from a solution of a pair of students, David and Omer, who were working onproblem 9 at Stage V1, and had solved it correctly, obtaining ( )3

7 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 sothat each kid can get no more than one gift?


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Omer: Here, look at it this way: In how many ways can I arrange the 3discs 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: ( )3

7 .

At Stage V2, after the interviewer prompted them to verify their solution, Omerreferred 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 partitionsthe length of which is 7 and have to choose 3 cells. I have ( )3

7

ways to do it, because after all there is no meaning to the order ofthe selection.

Example 4.2. This example illustrates using smaller numbers. As previouslymentioned, some students tried to apply this approach but were unsuccessful in us-ing it. To illustrate its potential, we bring an example taken from a solution of an ex-pert professor of mathematics, Allan, who was working on Problem 7 (see Appen-dix)aspartofa largerstudyaddressingsimilar issues (MashiachEizenberg,2001).

Allan had initially solved this incorrectly, obtaining 285 (instead of 255) as ananswer. His reasoning was that we can choose either 2 couples or 1 couple and 2‘singles.’ Thus, according to Allan, there were( )2

6 ways to choose exactly 2 cou-

ples out of 6 couples, and there were ( )6 210⋅ ways to choose one couple and 2 ‘sin-

gles.’ Allan’s mistake was in the counting of the 2 ‘singles.’ His initial reasoningsuggested that there were 6 ways to choose a couple, and then( )2

10 ways to choose

2 ‘singles’ of the remaining 10. In his reasoning he neglected to take into accountthat the 2 people that were selected (at random) of the 10 may be a married coupleand not 2 ‘singles.’Allan used the following verification strategy in response out ofhis own initiative.

Allan: I think there is no big difference if instead of 6 couples in theproblem there would be 3 couples. To keep the same principle Ican’t decrease the number of people to choose, that is, I still


FIGURE 1 Omer’s sketch.

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need to choose 4 people, because I need to maintain at least 2couples. Otherwise it changes the problem.Let’s call the couples (A,1), (B,2) and (C,3). Then the possiblecases 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 canhave (A,1) and 2 singles. Oh, now I see my mistake.I chose [in the original problem] 2 out of the remaining 10, butthey must not be married. So I need to fix this.[returning to the smaller number problem] The possibilities areeither (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[original] problem it’s not but ( )2

10 less 5 couples, that is there

are ( )[ ]6 5210⋅ − ways to choose exactly one couple.

It should be noted that selecting numbers for the smaller-number problem is notat all straightforward and entails deep considerations, as Allan articulated, to keepthe general structure of the problem in tact.

Strategy 5: Verification by Using a DifferentSolution Method and Comparing Answers

The participants who used this strategy employed a completely different solu-tion method for the problem. In these cases, the new solution method led to eitherthe same result that they had reached or to a different one. Table 4 presents the dif-ferent cases according to the correctness of the first and second solution methods,the difference between the results obtained in each way, and consequently, the de-cisions the participants made based on this strategy of verification.


TABLE 4Decisions Made by Participants Employing Verification Strategy 5

First Method Second Method Comparison of Results Choice of Solution Total

Incorrect (N=43) Correct (N=13) Different Second 12First 1

Incorrect (N=30) Same Same 3Different Second 17

First 9None 1

Correct (N=14) Correct (N=12) Same Same 12Incorrect (N=2) Different Second 2

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Example 5.1. This example illustrates using a different solution method. Itis taken from a solution of one student, Yariv, who was working individually onProblem 6 (see Appendix) at Stage V2, and had solved it incorrectly. His solutionmethod looked at the problem as one in which the executives choose the insurancepolicies, and went as follows:( )2

4 5 4 3⋅ ⋅ ⋅ , which is basically correct. However, he

made a wrong calculation, obtaining 320 (instead of 360) as an answer. Using thefollowing verification strategy, namely, a different solution method, he got a differ-ent answer—the correct one (360). Consequently, to be completely convincedwhich 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 insurancepolicies. Let’s say I choose 3 types of policies out of the 5, that is

( )35 . Now we need to distribute them to 4 executives.

I choose( )24 [2 of the executives that will have the same type of

policy], and multiply this by 3 [there are now 3 types of policies]multiply by 2 [the other 2 executives can switch policies]. Thus:

( ) ( )35

24 3 2 360⋅ ⋅ ⋅ = .

Before I got 320, but this [solution] is what counts. [goes back tohis 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. Itis taken from a solution of one student, Harry, who was working on Problem 8 (seeAppendix) at Stage V3, and had solved it incorrectly, in the following way:

8 4 6810

4i

i=∑ = , . The explanation he gave was that we distribute either 0, 1, 2, 3, or 4

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 andthe inner arrangement of the candies increases the number of ways to distribute thecandy. Using a different solution method, he obtained the correct solution, wentback 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 anykid or not given at all. That’s why it [refers to his original solu-tion] is not right.So [the answer is] 94 = 6,561. [turns back to his first solution andwrites:]


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Now Harry is puzzled, because he got 6,564 instead of 6,561. He tries to findwhere this difference came from:

Ok. I got 2 different results. First, what possibilities are re-peated? I have 3 extra ones. What does this 4 represent? [Refersto the 4 in the 1 4⋅ at the beginning of his long computation]. Itrepresents the possibility that i = 0, that the teacher distributes 0candies. In how many ways can he do it? Only 1 way, not 4. Thisis 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 in-correct solution. There were other cases, in which this verification strategy wasemployed. As seen in Table 4, Strategy 5 was used altogether in 57 cases. Therewere 43 cases with an incorrect solution, and 14 cases with a correct solution, justbefore this verification mode was employed. Of the 43 cases with an incorrect so-lution, in 12 cases this strategy proved helpful in correcting their solutions.

Interestingly, there were two pairs of students and one individual student whoreached the same incorrect result with two different solution methods, thus theiranswers 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 con-ducted, Mashiach Eizenberg, 2001).

Of the 14 cases with a correct solution, 12 cases remained correct in both solu-tion methods. However, Strategy 5—verification by using a different solutionmethod 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 cor-rect to an incorrect solution. Fortunately, in the following stage (V3) another switchwas 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 toboth correct and incorrect solutions. Table 5 presents the distribution of the verifi-cation attempts by the different verification strategies. It should be noted that thedistribution of the verification attempts of only the incorrect solutions was similarto 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 onthe extent to which each strategy was helpful in leading the participants to improv-ing their solutions. Altogether, there were 32 improved solutions as a result of em-


� � � � � � � �4 4 4 41 2 3 41 4 8 64 512 4,096 6,564� � � � � � � � � �

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ploying a verification strategy (note that some improvements are attributed to morethan one verification strategy). Table 6 presents the distribution of the improvedcases by the different verification strategies. As shown in Table 6, the most effi-cient verification strategies were Strategy 2 (in 12 cases) and Strategy 5 (in 13cases). Note that there were 5 cases in which a combination of verification strate-gies led to an improved solution, in each of which Strategy 5 was the primary strat-egy leading to the correct solution, whereas the others served only to detect an er-ror and motivate the search for an alternative solution. For example, one studentdetected an error through Strategy 3, identified the kind of error through Strategy4, and finally corrected the solution using Strategy 5.

IMPLICATIONS FOR TEACHINGAND LEARNING COMBINATORICS

Our findings support the assertion that combinatorics is a complex topic—only 43of the 108 initial solutions were correct. The findings also strengthen the merit ofencouraging students to verify their solutions to combinatorial problems. It seems


TABLE 5The Distribution of the Number of Attempts to Verify a Solution

by the Type of Verification Strategy Employed

Types of Verification Strategies Number of Attempts

1. Reworking the solution 832. Adding justification to the solution 573. Evaluating the reasonability of the answer 114. Modifying some components of the solution 115. Using a different solution method and comparing 57Total 219

TABLE 6Distribution 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 Number of Cases

1. Reworking the solution 22. Adding justification to the solution 123. Evaluating the reasonability of the answer 34. Modifying some components of the solution 25. Using a different solution method and comparing 13Total 32

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that, when encouraged to do so, there are students who are capable, to a certain ex-tent, of finding efficient ways to verify their solutions without prior direct instruc-tion. This is true particularly when students know that their solution is incorrect.However, the findings indicate that many of the students who made attempts toverify their incorrect solutions, whether out of their own initiative or in response tothe interviewer’s prompts, were not able to come up with efficient verificationstrategies 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 aspart 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 in-crease students’ awareness of the need to verify and provide them with ways to doso. We propose that the verification strategies that were identified within our studyshould be explored by students, addressing the potential and the limitation of eachone.

As shown earlier, Strategy 1 (i.e., verification by reworking the solution) wasthe most frequently used, however, it turned out to be one of the least efficient strat-egies in terms of helping students shift toward an improved solution. The low effi-ciency level of Strategy 1 is in accordance with the view that merely going over asolution of a mathematical problem is insufficient for verifying it (Polya, 1957;Schoenfeld, 1985). Strategy 2 (i.e., verification by adding justifications to the solu-tion) turned out to be considerably helpful, especially in clarifying and supportingthe various steps in a solution. This strategy was particularly helpful in detectingminor errors.

Unlike Strategy 2, Strategy 3 (i.e., verification by evaluating the reasonabilityof the answer) was not frequently used, probably because estimating an expectedoutcome in a combinatorial problem is extremely hard to do. Fischbein andGrossman (1997) found that when asked to estimate such results, students usuallygave lower estimates than the actual number. Thus, Strategy 3 was helpful in de-tecting that an error had occurred only in cases when the answer that was obtainedwas larger than the size of the outcome-space and when the student(s) applyingthis 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 so-lution method by using smaller numbers. However, this requires deep structuralconsiderations that need to be dealt with. We speculate that although it may seemnatural to students to employ this strategy (as indeed some tried to), applying itcorrectly needs direct and systematic learning. Support for our claim was found inthe recommendations of three professors of mathematics who teach both under-graduate and graduate combinatorics courses. In advocating the potential of Strat-egy 4, one professor stated, “I think that maybe the simplest way to verify it [thesolution] is to write the smallest numerical example that conserves the content ofthe problem” (Mashiach Eizenberg, 2001, p. 154). Clearly, maintaining the general


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structure of the problem is not a trivial task. It is one of the main difficulties thatHadar and Hadass (1981) identified in solving a combinatorial problem.

Finally, we turn to Strategy 5 (i.e., verification by using a different solutionmethod), which was both frequent and rather helpful. In most cases, using a gen-uinely different solution method is likely to be helpful in detecting an error.However, the limitation of this strategy is manifested in (not so frequent) casesin which the same incorrect answer can be reached by two different solutionmethods. This occurs when what appear to be two different autonomous solu-tion methods are in fact interconnected in a way that the incorrectness of oneimplies the incorrectness of the other. For example, if, for a given solutionmethod, 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 directsolution will be repeated for the complementary one. In our study, there werethree such cases that raised the participants’ degree of confidence in their in-correct solution.

To illustrate the limitation of Strategy 5, we bring a quote from a student, whosuggested to his partner at Stage V3 (after learning from the interviewer that theirsolution was incorrect) to verify their solution by solving the problem in a differentway. The student argued, “If we reach the same result in a different way then theanswer is right and the question is wrong.” To apply Strategy 5, students need togain experience in essentially different multiple approaches to solving combina-torial problems. Thus, teaching this type of verification strategy is tightly con-nected to teaching problem solving methods in combinatorics.

Of interest, the verification strategies that were found helpful did not rely on re-calling formulas, but mostly on an intuitive empirical approach. Thus, it seems im-portant that teachers build on students’ informal approaches and be aware of thefrequent danger of formal instruction on success in solving combinatorial prob-lems, as indicated in Fischbein and Gazit’s study (1988).

The differences that were found with respect to the efficiency of the variousstrategies may serve as a springboard for teaching combinatorial verification strat-egies with a focus on metacognitive processes, in general, and on the use of moreefficient verification strategies, in particular. We suggest that through more suc-cessful verification experiences students are likely to become aware of the poten-tial of verifying their solutions, and hopefully, will be motivated to verify their so-lutions on their own initiative.

REFERENCES

Batanero, C., Godino, J. D., & Navarro-Pelayo, V. (1997). Effect of the implicit combinatorial model oncombinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32,181–199.


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Cai, J. (1994). A protocol-analytic study of metacognition in mathematical problem solving. Mathe-matics Education Research Journal, 6, 166–183.

Dubois, J. G. (1984). Une systematique des configurations combinatoires simples. [A systematic forsimple combinatorial configurations]. Educational Studies in Mathematics, 15, 37–57.

English, L. D. (1991). Young children’s combinatoric strategies. Educational Studies in Mathematics,22, 451–474.

English, L. D. (1993). Children’s strategies for solving two-and three-dimensional combinatorial prob-lems. Journal for Research in Mathematics Education, 24, 255–273.

Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents.Zentralblatt fuer Didaktitk der Mathematik, 5, 193–198.

Fischbein, E., & Grossman, A. (1997). Schemata and intuitions in combinatorial reasoning. Educa-tional Studies in Mathematics, 34, 27–47.

Hadar, N., & Hadass, R. (1981). The road to solving a combinatotial problem is strewn with pitfalls. Ed-ucational Studies in Mathematics, 12, 435–443.

Hembree, R. (1992). Experiments and relational studies in problem solving: A meta-analysis. Journalfor Research in Mathematics Education, 23, 242–273.

Kahneman, D., & Tversky, A. (1973). Availability: A heuristic for judging frequency and probability.Cognitive Psychology, 5, 207–232.

Kantowski, M. G. (1977). Processes involved in mathematical problem solving. Journal for Researchin Mathematics Education, 8, 163–180.

Lucangeli, D., & Cornoldi, C. (1997). Mathematics and metacognition: What is the nature of the rela-tionship? Mathematical Cognition, 3(2), 121–139.

Malloy, C. E., & Jones, M. G. (1998). An investigation of African American students’ mathematicalproblem solving. Journal for Research in Mathematics Education, 29, 143–163.

Mashiach Eizenberg, M. (2001). Novices and experts coping with control on the solution of combina-torial problems. Unpublished doctoral dissertation, Technion, Israel.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathemat-ics. Reston, VA: Author.

Polya, G. (1957). How to solve it (2nd ed.). New York: Doubleday.Schoenfeld, A. H. (1984). Heuristic behavior variables in instruction. In G. A. Goldin & C. E.

McClintock (Eds.), Task variables in mathematical problem solving (pp. 431–454). Philadelphia:Franklin Institute Press.

Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic Press, Inc.Silver, E. A. (1987). Foundations of cognitive theory and research for mathematics problem-solving in-

struction. In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 33–60).Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

Wilson, J. W., Fernandez, M. L., & Hadaway, N. (1993). Mathematical problem solving. In P. S. Wilson(Ed.), Research ideas for the classroom, high school mathematics (pp. 57–78). New York:Macmillan.


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35

APPENDIX

The Research Instrument

Classification of the Ten Problems Used in the Study by Combinatorial Models and Operations

The Operation by

Model Permutations Arrangements With Replacement

Arrangements Without

Replacement

Combinations Without

Replacement

Selection Model

With replacement

and with order

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 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?

Without

replacement and

with order

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?

Without

replacement and

without order

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?

(continued)

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36 Appendix (Continued)

The Operation byModel Permutations

Arrangements With

Replacement

Arrangements Without

Replacement

Combinations Without

Replacement

Distribution Model

Different objects

into different

cells with order

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?


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?

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?

Different objects

into different

cells without

order


we drop off 5 bus passengers in 3

stops, so that in each stop at least

one passenger gets off the bus?


we drop off 5 bus passengers in 3

stops so that in each stop at least

one passenger gets off the bus?

Identical objects

into different

cells


we distribute 3 identical gifts to

7 kids so that each kid can get no

more than one gift?

Partition Model 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?

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