probability simulation: what meaning does it have for high school students?

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Probability simulation: What meaning does it have forhigh school students?Gwendolyn M. Zimmermann a & Graham A. Jones ba Glenbrook South High School, West Chicagob Illinois State University,Version of record first published: 26 Jan 2010.

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Probability Simulation: What MeaningDoes It Have for High SchoolStudents?Gwendolyn M. ZimmermannGlenbrook South High School, West ChicagoGraham A. JonesIllinois State University

Abstract: This study investigated high school students' reasoning and beliefs when confronted withcontextual tasks involving the assessment and construction of two-dimensional probability simulations.1

Nine students enrolled in an advanced algebra course, with little formal instruction in probability, engaged inclinical interviews focusing on the simulation tasks. All students showed evidence of being able to recognizeor identify an appropriate probability generator to model a contextual problem. However, their thinking inprobability simulation was constrained by their inability to deal with two-dimensional trials. In assessingthe validity of a given simulation, only one student could identify a flaw that resulted from the use of one-dimensional trials rather than two-dimensional trials. Additionally, when asked to construct a simulation,only two students were able to define an appropriate two-dimensional trial and develop a valid solution.The study also revealed evidence of students' beliefs about probability simulation—some of which could behelpful in informing instruction, others problematic.

Sommaire exécutif: Cette étude analyse les raisonnements et les croyances des étudiants et desétudiantes du secondaire lorsqu'ils sont confrontés à des tâches contextuelles qui consistent à envisager dessimulations de probabilités à deux dimensions. Dans la première tâche, les élèves doivent évaluer la validitéd'une simulation donnée dont le but est de déterminer les probabilités que deux commandes successives depizza soient identiques. Dans la seconde tâche, ils doivent construire une simulation visant à estimer lesprobabilités qu'une même pièce musicale soit passée par une certaine station de radio à deux momentsprécis. L'échantillon se compose de neuf étudiants et étudiantes, expressément choisis parmi les élèves detrois niveaux d'algèbre avancée dans une école secondaire du Midwest américain. Ces étudiants, qui n'avaientguère étudié les probabilités auparavant, ont participé à des entrevues cliniques centrées sur les tâches desimulation.

Six des neuf élèves de cette étude ont été en mesure d'identifier un générateur de probabilités adéquat dansla première tâche, et huit dans la seconde. Cependant, leur raisonnement au cours de cette simulation deprobabilités s'est trouvé limité par leur incapacité d'affronter une épreuve à deux dimensions. Dans lapremière simulation, un seul étudiant a pu identifier une faille résultant de l'utilisation d'un systèmeunidimensionnel. De plus, devant la tâche même de construire une simulation, seuls deux élèves sur neuf ontété en mesure de définir une épreuve à deux dimensions appropriée et de trouver une solution valable. Cinqautres ont manifesté l'intention de définir une épreuve à deux dimensions, mais ont été incapables decombiner entre eux les résultats singuliers et ont fini par retourner aux épreuves unidimensionnelles. Lesdeux autres élèves ont immédiatement transformé la tâche bidimensionnelle en tâche unidimensionnelle.L'incompréhension quant à la nature d'un espace-échantillon à deux dimensions, alliée aux idées erronées surles simulations de probabilités, semblaient constituer des obstacles importants pour ces sept étudiants. Iln'est pas surprenant de constater que seuls les deux élèves capables d'envisager correctement les deuxdimensions étaient en mesure d'expliquer comment calculer les probabilités empiriques de l'événement dansle second problème.

Cette recherche a également fait ressortir les idées préconçues des étudiants et des étudiantes sur lessimulations en probabilités. Certaines de ces idées ont été considérées comme utiles à la formation, tandisque d'autres constituaient au contraire un problème. Dans la catégories des idées utiles, mentionnons quequatre élèves considéraient les hypothèses comme nécessaires dans un modèle de simulation, que tous

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croyaient que les probabilités du générateur devaient correspondre aux probabilités contextuelles, et que sixd'entre eux prévoyaient que les probabilités empiriques se rapprocheraient des probabilités théoriques.Quant aux idées sources de problèmes, signalons que pour cinq élèves, les simulations ne pouvaient servirde modèle pour des problèmes réels, que quatre élèves manifestaient certains préjugés liés à la représentativité(Tversky et Kahneman, 1974) et que deux élèves croyaient que le résultat visé devait se manifester dès lapremière tentative, un peu à la manière de la théorie des résultats décrite par Konoid (1991). Ces idéeserronées entravaient souvent les raisonnements des étudiants et des étudiantes, en particulier lorsqu'ilsaffrontaient les complexités d'une épreuve à deux dimensions.

Bien que cette étude soit par nature essentiellement exploratoire, elle a permis d'acquérir des données surles raisonnements et les croyances des étudiants et des étudiantes du secondaire devant les tâches desimulation en probabilités. De telles données s'avéreront certainement fort utiles aux enseignants et auxenseignantes qui voudront créer du matériel didactique centré sur les simulations. En fait, il est souhaitableque d'autres recherches mettent au point des expériences (Cobb, 2000) visant à évaluer la possibilité que cesnouvelles données influencent les pratiques d'enseignement à l'école.

IntroductionAs we are increasingly bombarded with vast numbers of facts and figures, it is imperative that we

be able to engage in statistical reasoning to discriminate among substantive, misleading, and invalidconclusions based on data. There is also a critical need for understanding the basic concepts andprocesses of probability, a requirement for competency in statistical reasoning. In the Principles andStandards for School Mathematics, the National Council of Teachers of Mathematics (NCTM) (2000)recognizes this need when it calls for increased emphasis in statistics and probability across thecurriculum and categorizes statistics and probability among the 'skills necessary to becoming in-formed citizens and intelligent consumers' (p. 48). More specifically, NCTM expects high schoolstudents to 'use simulations to construct empirical probability distribution(s)' (p. 324), and thisexpectation is also echoed in the outline of the College Board's Advanced Placement StatisticsCourse (College Board, 2000, p. 7).2

Although numerous studies have investigated the probabilistic reasoning of students in theelementary and middle grades (Falk, 1983; Fischbein & Gazit, 1984; Jones, Langrall, Thornton, &Mogill, 1997,1999;Piaget&Inhelder, 1951/1975), fewer have examined the probabilistic reasoning ofhigh school students (Fischbein, Nello, & Marino, 1991; Fischbein & Schnarch, 1997; Green, 1983;Shaughnessy, 1977). Moreover, there appears to be no research that has examined high school stu-dents' thinking in relation to probability simulations.

Research questions

The present study, although exploratory in nature, was designed to address the void in themathematics education literature associated with probability simulation. More specifically, the studyattempted to build a picture of the nature and scope of high school students' thinking and beliefswhen they were confronted with problems involving two-dimensional probability simulations. Theresearch questions were as follows: (a) Are high school students able to assess whether a givensimulation process is valid for a two-dimensional contextual probability problem? (b) Are high schoolstudents able to construct a valid simulation process to solve a two-dimensional contextual probabil-ity problem? (c) What is the nature of high school students' probability reasoning and beliefs whenthey deal with contextual problems involving two-dimensional probability?

Theoretical framework

In this section we discuss the theoretical underpinnings that guided our observations and analy-ses of high school students' reasoning and beliefs as they engaged in probability simulation tasks.

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Probability Simulation: What Meaning Does It Have for High School Students?

The key elements of our theoretical framework are represented in Figure 1.

MathematicalConcepts

Students' Cognitionsand Beliefs

Tasks

• Recognition» Solution

Observe fc

Outcomes• Assess validity

of a simulation• Construct a

simulation• Probabilistic

reasoning andbeliefs

Figure 1

In developing the contextual problem tasks, we had to take into account important mathematicalconcepts associated with the simulation of a probability context (e.g., the next two pizza orders are for'no meat'). Yates, Moore, and McCabe (1999) define a simulation as an experimental process thatmodels the probability elements of the context. In elaborating this process, Yates et al. identify anumber of steps in the simulation process. First, the problem should be stated and any assumptionsnoted. Second, random digits (or the outcomes of the probability generator) should be assigned tomodel the outcomes of the problem context. Third, a trial should be defined to meet the conditions ofthe problem. Fourth, this trial should be repeated many times. Finally, the simulation data that havebeen collected should be used to determine the empirical probability. In essence, the process outlinedby Yates et al. provides the mathematical norm, or frame, for assessing students' thinking.

In preparing ourselves to construct simulation tasks and to observe students' mental actions asthey encountered these tasks, we reviewed the probability literature pertaining to students ' cognitionand beliefs. Although we found no research that addresses high school students' thinking in relationto probability problems involving simulation, we did find substantial literature on students' reasoningand beliefs about theoretical probability (Batanero & Serrano, 1999; Fischbein & Schnarch, 1997;Jones et al., 1997; Shaughnessy, 1992; Watson, Collis, & Moritz, 1997) as well as recent studies onstudents' reasoning with respect to probability modelling (Benson, 2000; Benson & Jones, 1999).Even though both of these studies involved small samples, they have particular relevance to thisresearch in that Benson and Jones found that only three of their seven students (ranging from Grade2 through post-secondary levels) were able to construct a valid model for a contextual probleminvolving two-dimensional probability. Two of the students who could construct a valid model wereof university age, and the third was in Grade 12. Benson and Jones concluded that students' ability toconstruct two-dimensional models seemed to be closely related to their knowledge of two-dimen-sional theoretical probability. (By 'theoretical probability' they meant that the probability of an eventis determined by analysing the sample space using symmetry, number, or simple geometric measures).While the Benson and Jones's study provides a glimpse of students' thinking in relation to probabil-ity simulation, it should be noted that it did not look at the complete simulation process, only the firststep—that is, modelling the contextual situation.

A review of the probability literature on students' beliefs revealed that researchers used varyingterms in referring to the dispositions subjects brought to studies on probability. The most commonterms used were 'intuition' and 'beliefs.' In order to cast as wide a net as possible in relation tostudents' beliefs, we used Schoenfeld's (1985) characterization of beliefs as a lens through which anindividual sees and approaches mathematics. When viewed according to Schoenfeld's characteriza-tion, the literature reveals that beliefs play a major role (Fischbein & Gazit, 1984; Fischbein et al., 1991 ;

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Fischbein & Schnarch, 1987; Garfïeld & Ahlgren, 1988; Piaget & Inhelder, 1975; Shaughnessy, 1992)in students' reasoning about probability. That is, students' conceptions of probability often involvethe use of judgement heuristics, such as representativeness, availability, and outcome approach, thatare grounded in beliefs (Konoid, 1991; Konoid, Pollatsek, Well, Lohmeier, & Lipson, 1993; Tversky &Kahneman, 1974). According to Shaughnessy (1992), students' belief in judgement heuristics such asrepresentativeness, availability, and outcome approach indicates that they may not have developeda cognitive framework in which to understand the mathematical relevance of probability experiments.If this were the case, it would certainly have implications for students' understanding of probabilitysimulations.

In investigating how students dealt with probability simulations, we designed our research tosee if they could handle two complementary tasks (Figure 1). On the one hand, we wanted to seewhether they could recognize whether a probability simulation was valid or not On the other hand, wewanted to determine whether the students could construct a valid simulation solution. Both recogni-tion and construction tasks were two-dimensional experiments. That is, they involved the repetitionof a one-dimensional experiment with replacement (e.g., two pizza phone orders are taken, each orderbeing either 'meat' or 'not meat' according to given probabilities). In order to recognize or constructa valid simulation of a two-dimensional problem, students need to realize that each outcome is anordered pair and, ipso facto, that each trial in their simulation requires the generation of two outcomeson the probability generator. Recall that Benson and Jones (1999) found that only three of the sevenstudents in their study were capable of modelling a two-dimensional task, let alone dealing with theentire simulation process.

The research questions for the study were predicated on the expectation that we would be ableto document theoretical knowledge on the following: (a) students' reasoning as they assessed thevalidity of a two-dimensional, contextual probability problem; (b) students' reasoning as they con-structed a probability simulation; and (c) students' beliefs as they worked on probability simulationtasks. We expected that students would demonstrate not only conceptions and beliefs that werealready identified in the literature but also conceptions and beliefs that were unique to the simulationtasks.

Method

The high school students in this study were enrolled in an advanced algebra course3 and hadreceived little formal probability instruction. Using a researcher-designed assessment protocol, stu-dents were interviewed one-on-one by the first author. The clinical interview was designed in such away (Goldin, 1997) as to provide the flexibility necessary to explore students' thinking as they at-tempted each of the tasks in the protocol.

Participants

The participants of this study were nine students from a high school located in the US Midwest.Students were purposefully sampled (Miles & Huberman, 1994); three students were drawn from eachof the three tracks of'Advanced Algebra' at the school. These tracks were hierarchically ordered onthe basis of prior mathematical achievement; hence the nine students were representative of the rangeof mathematical achievement in this course. While the selection of the three students from each trackwas essentially random, we did check with the teacher to ensure that each of the chosen students wasable to communicate their mathematical thinking orally in an effective manner. The students, repre-sented by pseudonyms, were Alicia and Hanna, in Grade 11, and Beth, in Grade 12, from a lower-trackAdvanced Algebra class; Christy, Doug, and Gino, in Grade 11, from a middle-track Advanced Alge-bra class; and Ellen and Ivan, in Grade 10, and Frank, in Grade 11, from a higher-track Advanced

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Algebra class. The mean age of the nine students was 16.7 years, and their ages ranged from 15.4 to18.1 years.

Procedure

Each student responded to the assessment protocol in an interview setting. The interviews wereaudio- and videotaped and were conducted in a quiet area by the first author. Students were seated ata table across from the interviewer and were provided with appropriate probability generators: col-oured chips, dice, spinners, a random number table, and a graphing calculator. They were also givenpaper and pencils. In each interview the researcher handed a copy of the problem to the student, readit aloud to the student, and then gave him or her time to think about the problem and do any writing heor she wished. When the student was ready, he or she responded to the researcher's questions. Thesame procedure was followed for the second problem. Students were not under time pressure, and allstudents completed the two tasks in 30 minutes, while some completed them in less than 20 minutes.

Instrumentation

The research questions and theoretical framework guided the design of the assessment protocol,which comprised two two-dimensional contextual probability problems involving simulation (seeTable 1). The table illustrates the tasks given to the students and the follow-up questions they wereasked. The contexts of the problems were chosen to be relevant to the life of a teenager.

Table 1 : Summary of simulation tasks

Situation Task

1a. Stan was given the following problem.The Pizza Wagon has determined that 60 percent of theirphone orders for pizza contain meat (sausage, pepperoni, etc.)and the remaining 40 percent of their phone orders are for pizzaswith no meat (cheese, veggie, etc.). What is the probability thatthe next two phone orders for pizza are each with meat?

To simulate the Pizza Wagon's situation, Stan used colored chips.Stan chose 6 red chips to each represent an order for pizza withmeat, and he chose 4 green chips to each represent an order forpizza without meat. To simulate the actual order, Stan put all 10chips into a bag, shook the bag, and drew out one chip. Herecorded the color, put the chip back, and then repeatedthe simulation.

1b. Stan conducted his experiment 50 times and his resultswere as follows:

Red chip 22 timesGreen chip 28 times

2. The school radio station plays three types of music: hip-hop,alternative, and country. The DJ uses a format such that theprobability he plays hip-hop is 0.4, that he plays alternative is0.4, and that he plays country is 0.2. If you turn your radio on at10:00 am and then again at 2:30 p.m., what is the probabilitythat both times you hear a hip-hop song?

Remember that the Pizzawagon is trying to determine theprobability that the next 2 pizzashave meat. Do you think that Stan'ssimulation would enable him todetermine the probability that thenext two pizzas have meat?• Why do (don't) you think Stan

would be able to determinethe probability that the nexttwo pizzas have meat?

• If you don't think it does, howwould you change it?

• Using the outcomes from Stan'sexperiment, would you be ableto determine the probability thatthe next two phone orders forpizza have meat?

• If yes, how? If no, why not?

• How would you simulate thissituation to determine theprobability that both times youhear hip-hop?

• Why don't you do a few cases toshow me how you would do it?

• How many times would you do it?• Would the solution change if

you did the experiment 50 times,1000 times, or 100,000 times?

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Beginning with Situation la, each student was given a copy of the Pizza Wagon probabilityproblem, in which a fictional character, Stan, designed a simulation to determine the probability thatthe next two phone orders were for pizza with meat. Each student was asked to determine the validityof 'Stan's' simulation process. In Situation lb, each student was shown the results Stan obtainedfrom his experiment and asked if the relevant probability could be determined based on Stan's out-comes. The intent of Situation la was to determine the extent to which students were able to assessthe validity of a simulation design. The purpose of Situation lb was twofold: (a) to determine whetherstudents were able to recognize the inappropriateness of the outcomes as they were recorded, and (b)to ascertain whether they could determine whether or not the recorded results allowed them tocalculate the required empirical probability. It should be noted that the outcomes were recorded assingle outcomes and not as pairs, as was required to solve the two-dimensional pizza problem.

In Situation 2, students were asked to consider a probability problem involving a school radiostation with a specified airtime format for three different types of music: hip-hop, alternative, andcountry. After reading the problem, each student was asked to design a simulation to determine theprobability that hip-hop would be playing at two specified times when the radio was turned on. Ifstudents had difficulty constructing a simulation, a prompt was given to remind them of the previousproblem, which had involved using chips to model the simulation. After the student had constructeda simulation for the problem, they were asked to demonstrate both the simulation and the collection ofdata. A major objective of this part of the protocol was to determine whether students could constructa simulation that incorporated a valid probability generator, an appropriately defined trial, and aproperly determined empirical probability. The students were also asked how increasing the numberof trials would affect the empirical probability.

During assessment interviews, the protocol was followed consistently from student to student.Wherever appropriate, general probes such as 'why or why not?' 'tell me more,' or 'could you saythat in another way?' were used to encourage students to clarify and elaborate their responses.

Data sources and analysis

Data on students' thinking in relation to the simulation problems were collected from threesources: (a) students' audiotaped responses to the two tasks, (b) students' videotaped responses tothe tasks, and (c) researcher field notes.

Following the interviews, the first researcher generated transcripts for each of the interviews andattached to the transcripts summaries of the field notes she had taken during each interview. Thedouble-coding procedure described by Miles and Huberman (1994) was used to generate codes fromthe transcripts of the nine students. Using this procedure, each researcher independently producedtwo sets of codes, one set for probability reasoning and the other for probability beliefs. With respectto probability reasoning codes such as 'correspondence,' 'one-dimensional model,' and 'with re-placement' were generated. With respect to probability beliefs, codes generated by the researchersincluded 'assumptions,' 'representativeness,' and 'success on the first trial.' When the two research-ers had completed their independent coding of the transcripts, they reached consensus on a set ofcommon codes and then used these common codes to analyse the complete corpus of data.

Miles and Huberman's 'three part analysis' (1994, pp. 10-11) was used to analyse the data. In thefirst part of this process, data reduction, we used the common codes to generate images and impres-sions of each student's reasoning and beliefs. In the second part, we constructed, for each task, datadisplays (matrices) that compared students' thinking (columns) on each of the simulation steps(rows). The final part of the process involved conclusion drawing and verification, that is, generat-ing emergent trends and patterns for probability reasoning and beliefs for the two tasks and the ninestudents. Each researcher verified these patterns by rechecking the data corpus and the data displaysto see whether there was sufficient evidence to warrant the patterns.

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Results

Pizza problem

The primary goal of the first problem was to determine if students could recognize whether agiven simulation was valid or not. Table 2 categorizes students' reasoning according to each of thesimulation steps for the pizza task.

Table 2: Student responses to the 'pizza problem'

Student Doug Beth Alicia Frank Ellen Christy Gino Hanna Ivan

Model• Recognizes

probabilitygenerator

Yes

TrialRecognizes that Noit is not a 2-D trial

Acceptsdistribution ofoutcomes

Yes No No Yes* Yes

No No No No Weak

No No No No

Repetition of trial

Recognizes thatyou can'tcalculate a validexperimentalprobability

No

No

Yes# Yes# Yes# Yes# Yes#

No Yes

No Weak

No

No

Yes# Yes#

Yes*

Yes

Yes

* Indicates that the student also discussed a second probability generator that was equivalent to thegiven one

- Indicates that the student did not address this in the course of the interview# Indicates that the justification is invalid, based on problematic beliefs

Six of the nine students recognized that the probability generator modelled the situation. Beth'sjustification of the appropriateness of the probability generator was reflective of this type of reason-ing: 'I think that the six chips for order with meat would represent like the 60% of the phone orders forpizza that contained meat, and then the four green chips for the 40% and then 10 representing 100% ofthe pizzas.' Like most other students, she established a correspondence between generator andproblem outcomes and between generator and problem probabilities.

Only one of the nine students was able to recognize that the results did not reflect a two-dimensional trial. When shown the results of the pizza problem, Ivan responded,

No, that's not going to get what he got the next two phone orders 'cause that's just working one at atime ... He'd have to get it to record like, put it in, pull out a chip. Then put it back, shake it up and thenpull out another one. And then it's whether they're both red or whether they're not both red that is theprobability for whether you both get meat.

Ivan recognized that the trial should produce an ordered pair, the first element of the pair beingthe colour of the first chip drawn and the second element being the colour of the second chip drawn,after the first chip had been replaced. With the possible exception of Christy and Hanna, who are

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discussed later in this section, the remaining students did not see the need to consider two consecu-tive outcomes in order to generate the target event—'two meat.' Instead, what seemed to bother fiveof the students was the nature of the outcomes. That is, the distribution of green and red chips did notmatch the probabilities of 40% and 60% stated in the problem. Because the distribution did not matchthe students' expected distribution, they considered the simulation to be invalid.4

Student expectations that the simulation outcomes should match the parent probabilities alsoinfluenced their ability to calculate the empirical probability for the pizza problem. Of the eight stu-dents who said it was not possible to calculate the probability using the data presented, Ivan was theonly one who recognized that the data was not recorded as two-dimensional trials. The remainingseven students held various problematic beliefs, most salient of which was that the outcomes did notmatch up. Given their problematic beliefs, it did not make sense for them to calculate a probability.Ellen sums up what these students thought: 'I don't think so because they don't seem to match up ...you'd think that it'd be more likely for someone to call and ask for meat with pizza where these answersdon't seem to reflect that.' Doug was the only student who thought you could calculate a probabilitybased on the stated results. However, in stating that there was 'a 44% chance of getting meat,' heseemed to be unaware that he was supposed to find the probability that the next two orders were formeat

Christy and Hanna were the only students to consider the role of repetition as it related to thepizza problem. Christy seemed to believe that repetition was unnecessary: 'It doesn't make sense touse 50 times worth of pulling out chips in proportion to one phone call.' She may have been sayingthat 50 times is too many trials and that a smaller number of trials would have sufficed; or she mayhave been suggesting the need to look at a second phone call. In other words, Christy may haveinterpreted the outcomes as trials for the first phone call and did not understand why someone woulddraw a chip 50 times for the first call when the problem required an outcome of two consecutive phonecalls. Hanna stated her belief more clearly: 'he did it 50 times and we are only looking for two times, thetwo phone orders.' Both of these students were at least questioning the validity of the data and werethe only ones apart from Ivan who appeared to have some notion of the need for a two-dimensionaltrial.

Radio problem

The reason for including the radio problem was to determine whether or not students were ableto construct a valid simulation of the problem. As Table 3 indicates, the first step in the simulationprocess is to state assumptions. While five of the nine students made explicit statements aboutreplacing or not replacing chips that had been drawn, two other students made this assumptiontacitly, as demonstrated by their use of a with-replacement simulation. Four students made otherassumptions. Gino, Hanna, and Ivan made the initial assumption that a 24-hour period would beconsidered. Frank went even further. He commented,

Chances are most people are just waking up, all that. We'll pretend this is the weekend. Chances arethat the DJ is going to be playing, you know it could be any song. So you have no clue what's going tobe played. So I think that just drawing once for 10 chips is going to work for 10:00. Two-thirty getskind of questionable because if the same DJ has been on for four hours, he may get a little bored and trysomething else. So maybe the probability lessens.

The second step in a simulation is the construction of a probability generator. Eight of the ninestudents were able to correctly assign a probability generator, and, in fact, two of them identified asecond generator that also modelled the situation. Frank typified their reasoning when he said,

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Point 4 is just four out of 10 so we'll just take four blue chips and our probability for alternative(music) is the same thing. So we'll just use a different colour. We'll use red for another four chips (hip-hop music), and then we use two for country. I know that I can lower this and take two off here, twooff here, and one off here.

That is, he suggested moving from a generator ratio of 4:4:2 to 2:2:1.

Table 3: Student responses to the 'radio problem'Student

Assumptions•Replacement•Others

Model•Constructprobabilitygenerator

Trial• Construct2-D trial

Doug

! With

Yes

Weak

Repetition Approachesof trial theoretical

Explains howto calculateexperimentalprobability

Based on1-D trial

Beth

;

No

Weak

No

No

Alicia

Without

Yes

1-D

No

No—'there

Isn't one"

Frank

Yes

Yes*

Some

Approachestheoretical

No

Ellen Christy

With

Yes Yes-

Yes Weak

Approaches Weaktheoretical

Yes Based on1-D trial

Gino

WithYes

Yes

1-D

Weak

No

Hanna

Yes

Yes

Weak

Weak

Based ontwo

1-D trials

Ivan

WithYes

Yes

Yes

Approachestheoretical

Yes

* Indicates that the student discussed a second probability generator equivalent to their first- Indicates that the student did not address this in the course of the interview

When asked to design a valid trial for the simulation, Ellen and Ivan were the only students ableto clearly produce a valid two-dimensional trial. Ellen's simulation, pulling chips out of an envelope,typifies the thinking of these two students. When asked to explain what she was doing, she saidpulling out the first chip was 'like the first time I turned on the radio at 10:00 am. The second time wouldbe like the second time I turn on the radio. I'd do this about 100 times.' Five of the remaining sevenstudents (Doug, Beth, Frank, Christy, and Hanna) were unable to generate a valid two-dimensionaltrial, even though they seemed to have some understanding that a two-dimensional trial was needed.They realized that they had to draw one chip to represent 10:00 a.m. and one to represent 2:30 p.m., butthey were never able to form in their minds the pairs of possible outcomes that could arise from suchdrawings. Doug was an extreme example of this. He began to model a two-dimensional trial, but wasunhappy with his results: '[draws a chip] blue is alternative [music] and then we put it back ... andthen [drawing another chip] red is hip-hop. So, that didn't work. I'm completely confused on this one.'On probing, he seemed to be confused because he did not get two hip-hop songs; yet he seemedoblivious to the fact that he had obtained a valid two-dimensional outcome, that is, an outcome(ordered pair) that belonged to the sample space. The other four students who demonstrated someintent to form a two-dimensional trial were also unable to generate pairs of outcomes and, as aconsequence, came up with two one-dimensional trials rather than one two-dimensional trial. Theremaining two students, Alicia and Gino, reasoned only from a one-dimensional perspective. In thecase of Alicia, her preoccupation with one-dimensional trials may have been a manifestation of thefact that she held a belief similar to Doug's: 'there is a problem if the target element does not appear inthe first trial.'

When probed about the effect that increasing the number of trials had on the empirical probabil-ity, four of the students indicated that increasing the number of trials would result in an empirical

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probability that would approach the real (theoretical) probability. Frank expressed this more explicitlythan the others when he said, 'I think that the more times you do something in terms of probability likethis, then the greater chance you have of getting closer to that theoretical value.' Two of the remain-ing students did not think that increasing the number of trials would affect the empirical probability,and the remainder suggested only a vague connection between the empirical and the theoreticalprobabilities when the number of trials was increased.

Five of the nine students showed some understanding of how to calculate empirical probabilityin the radio problem; however, the extent of their understanding was dependent on whether theyperceived the task as a one- or a two-dimensional problem.

While Ivan and Ellen correctly explained how to calculate the experimental probability based onvalid two-dimensional trials, the probabilities determined by the other two students were based onone-dimensional trials. For example, Christy drew a chip 10 times and recorded the results. Asked tointerpret her results, she explained,

so then you have five out of 10 times you have hip-hop so that's a 50% chance you get hip-hop. Sothen that means one out of two times, like 50% of the time, so one out of two times you would havehip-hop out of those two times you turned it on. [italics added]

It appears that Christy transformed a one-dimensional probability into a two-dimensional prob-ability; that is, she interpreted a 50% chance of getting hip-hop to mean a 50% chance of getting hip-hop both times you turned the radio on. Hanna also used one-dimensional trials, but she added thetwo probabilities to determine the probability that hip-hop would come up both times. The remainingfour students were unable to calculate the probability from the results of their simulation trials. Aliciawas one such student. She did not think that she could use her trials to determine the probability ofthe problem. When asked why, Alicia responded, 'I don't think there's a probability 'cause the timesthat I did it, it came out different.' Upon questioning, Alicia said that more hip-hop songs would needto appear before one could determine the probability of the problem. Her reasoning appeared to betied to her belief that each trial should match the target outcome. Like Alicia, Frank was unable todetermine the experimental probability of the radio problem because of his belief that simulationcannot be used to model a real-world problem. Frank's responses focused on what he thought thedeejay might or might not do. Both Frank's and Alicia's beliefs are discussed in more detail in the nextsection.

Student beliefs

Various student beliefs about simulation emerged from our analysis of the data. These beliefs aresummarized in Table 4 and have been categorized as either helpful or problematic. Helpful beliefs areconsidered to be potentially beneficial in learning because they can be linked to a normative view ofsimulation. Problematic beliefs, on the other hand, may constrain students from learning variousaspects of simulation because they incorporate misconceptions that are contrary to a normative viewof the simulation process. We examined the student beliefs that emerged in each category.

Helpful beliefe

While beliefs stated as assumptions may not always be helpful in probability problems, fourstudents in this study recognized the need for inherent assumptions that were appropriate for theradio task. Gino, Hanna, and Ivan stated explicitly and up front their assumption of a 24-hour periodfor the radio problem. Although Frank did not initially list his assumptions, he frequently referred toassumptions implied in the situation, as, for example, when he stated, 'I 'm going to assume they'restarting a program at ten in the morning. ' Later, when asked to demonstrate his simulation for the radio

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problem, Frank recognized the difficulty of being able to complete a trial because of assumptionshidden in the problem: 'Maybe the deejay likes variety; maybe he's going to go through all thedifferent types of music and then go back and play some more... there's all these different questionsbut those aren't addressed here.'

Table 4: Student beliefs

StudentDoug Beth Alicia Frank Ellen Christy Gino Hanna Ivan

Helpful beliefs•There are inherent - - - Yes - - Yes Yes Yesassumptions in asimulation model

•Probability Yes Some Some Some Yes Yes Yes Yes Yesgenerator shouldcorrespond togiven probabilities

•Experimental Yes No No Yes Yes - Some Yes Yesprobability willapproach theoreticalprobability

Problematic beliefs•Simulation cannot Some - - Yes - - Some Some Somebe used to modelreal-world problem

•Representativeness No Yes • Yes No Yes No Yes No No

•Target outcomes Yes - Yes — - - - - -should appear inthe first trial

To some degree, all students believed that the probability generator should correspond to theprobabilities stated in the problem context. Six of the students either explicitly or tacitly believed thatthe probabilities stated in the problem should be matched to a probability generator. Christy exempli-fied this belief: 'Because it's like 60% of 100, and you know it's like six and four out of 10. So it's allrelative, and so by taking out a chip it's like saying that you have a 10% chance... it's all proportional. 'The other three students also held this belief to some degree, but less strongly. For example, in thepizza problem, Alicia seemed to believe that a probability generator was acceptable as long as theoutcome for a pizza with meat was more likely than one without meat, regardless of the fact that theprobabilities did not match the given problem. Even when the interviewer asked, 'Could I pick othernumbers of chips and do the same thing? Could I pick eight chips to represent the meat and six chipsto represent the green and that would be okay?' Alicia answered, 'yeah. ' Interestingly, Alicia seemedto overcome this belief while working on the radio problem. In that problem she was able to constructa valid model, one in which the outcomes and probabilities were in correspondence. Frank and Bethwere the other two students who had difficulty consistently modelling probability situations. In bothcases this difficulty appeared to be caused by competing beliefs. For example, Frank believed that itwas not really possible to simulate a real phenomenon like ordering pizza using a probability genera-tor. We will discuss this belief in more detail in the section dealing with problematic beliefs.

Six of the nine students exhibited a belief that the empirical probability will approach the theoreti-cal probability as the number of trials increases. Frank and Ivan actually used the term 'theoretical

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probability,' but Ellen's response was more typical: 'It would probably get more accurate I wouldimagine. Like your probability was 20% or something and then you did it again, it would probably bestill around 20% but then it might be an even closer number. ' Two of the students, Beth and Alicia, didnot hold this belief, and the remaining student did not respond to the probe. In an overall sense, thisresult is heartening because such a belief could be a powerful mechanism when a teacher is attempt-ing to have students construct key elements of the simulation process.

Problematic beließ

Five of the nine students, albeit to varying degrees, expressed the belief that a simulation cannotor should not be used to model a real-world problem (see Table 4). Frank was explicit in stating thatyou cannot simulate a real-world phenomenon. For example, when asked whether the simulationcould be used to determine the probability in the pizza problem, Frank replied, 'I would think nobecause it's one thing to have theoretical probability which is what he has determined here and it'sanother thing to actually go through with it.' Later in the interview Frank was even more emphatic:'It's one thing to just draw chips out of a bag. It's one [another] thing to actually having people doingit.' Frank appeared to hold this belief more strongly than the other four students. For them, the belief

. was more an outward manifestation of their actions than an articulated disposition. For example,when asked to simulate the radio problem, Doug said, 'just turn on the radio at 10 and 2:30 and see ifthey're on [the hip-hop songs] and all that... just listen for the whole time for four hours and listenhow many times you hear hip-hop and figure out what the percent is and everything like that.' LikeGino, Hanna, and Ivan, when probed further, Doug did attempt to design a simulation using colouredchips.

Four of the nine students in this study demonstrated a belief in representativeness (Kahneman& Tversky, 1972) under the conditions operating in the pizza problem. More specifically, these fourstudents believed that the outcomes of the simulation should always approximate the given prob-abilities. Beth's response illustrates this belief. When asked if the outcomes of the pizza problemcould be used to determine the probability that the next two phone orders are for pizza with meat, sheresponded, 'I wouldn't think so because now that I look at this, it doesn't seem like it would be veryaccurate because it said the red chips represented an order for pizza with meat, and they said 60% oftheir phone orders had meat and 40% didn't and it seems that the results pose the opposite.' Ginorevealed a similar disposition towards the pizza problem. When probed, he conceded that in real lifesuch results could be possible. However, when asked again about using the outcomes in the problem,he replied, 'I wouldn't be able to [use the results] because the red chip only came out 22 times and thegreen chip came out 28 times out of 50.' These responses typically reflect a belief in representative-ness; that is, students expect the short-term probabilities to always reflect the probabilities of theparent population (Kahneman & Tversky, 1972).

Finally, two students, Doug and Alicia, held the problematic belief that the simulation was flawedunless the target outcome in the problem appeared in the first trial. In essence, they seemed to ignorethe possibility that the other outcomes in the sample might occur. For example, in carrying out the firsttrial of the radio problem, Doug commented, 'Blue is alternative and then we put it back so there's thesame chance of alternative. And then red is hip-hop. So that didn't work. I'm completely confused onthis one.' Alicia reflected a similar belief. When the first chip that she drew represented a countrysong rather than a hip-hop song, she said, 'Let's see ... there's a problem already that they're goingto play a country song.' While this belief seems trivial, both of these students held it sufficientlystrongly that they were not able to proceed with their simulation when the first trial did not producethe target outcome.

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Discussion

Probability simulation is an integral part of Advanced Placement Statistics (APS), a program thathas shown a 200% growth in student numbers across the United States since its inception in 1997.However, there is almost no research on the kind of knowledge of probability simulation that studentsbring to such a program. This study began the process of addressing this void by documenting thenature and scope of nine high school students' reasoning and beliefs when they encountered two-dimensional tasks in probability simulation. The findings of the study, which are discussed below,relate to key aspects of probability simulation: modelling the probability context, defining a two-dimensional trial, and calculating empirical probability. In addition, we discuss how the students'beliefs played out while they were engaged in the simulation tasks.

With respect to modelling probability contexts, six of the students were able to recognize anappropriate probability generator and eight were able to construct an appropriate generator. Thisresult is consistent with the findings of Benson and Jones (1999), who report that all but one of theirparticipants used appropriate strategies when identifying probability generators. Moreover, like thestudents in the Benson and Jones study, most of the participants in our study used a strategy basedon establishing a one-to-one correspondence between context and generator outcomes and theirprobabilities. In fact, the widespread use of correspondence among students in this study suggeststhat the concept of correspondence should be a key part of a teacher's learning trajectory (Simon,1995) when planning instruction in simulation.

With respect to dimensionality of trials, only one of the nine students, Ivan, recognized theinappropriateness of the one-dimensional outcomes in the pizza problem. Furthermore, only Ivan andone other student were able to clearly define a two-dimensional trial for the radio problem. Students'problems with two-dimensional tasks in this study manifested themselves in several ways. Somestudents immediately transformed a two-dimensional problem into a one-dimensional problem; othersshowed evidence of intending to generate a two-dimensional trial but were diverted by lack of under-standing of the nature of two-dimensional sample space or by beliefs about the need for the targetevent to occur on the first trial. The process of determining the dimensionality of a trial also connectsnaturally to the process of calculating the empirical probability of an event. Hence it is not surprisingthat only the two students who defined dimensionality correctly were able to explain how to calculatethe empirical probability of the target event in the two-dimensional radio problem. In most other caseswhere students calculated an empirical probability, it was usually based on one-dimensional data.Although earlier research has attested to the difficulty of two-dimensional probability tasks (Benson& Jones, 1999; Fischbein & Schnarch, 1997; Green, 1983), this study provided further insights intostudents' difficulties with two-dimensional problems by examining their reasoning in the context of asimulation. Given these findings, teachers need to be aware of the complexities of two-dimensionalsimulations and the difficulties that students have in representing a two-dimensional trial.

This research also documents the varied beliefs, both helpful and problematic, that studentsbring to a simulation task. The fact that all students, at least to some extent, believed that the probabil-ity generator should correspond to the given probabilities was both predictable (Benson & Jones,1999) and heartening. It was also reassuring that two-thirds of the participants held a reasonablycoherent belief about the empirical probability approaching the theoretical probability when the numberof trials increased. Teachers may well be able to capitalize on both these beliefs by linking instructionto what some students already believe.

Although four students held helpful beliefs about the need for assumptions in performing asimulation, it was somewhat disturbing to find that these four students, as well as one other, believed,albeit to different degrees, that simulation cannot be used to model a real-world probability problem.The latter belief may have been merely a healthy scepticism for four of the students, but for onestudent, Frank, it was strongly held. The results of this study demonstrate that the role of assump-tions, as well as the feasibility of simulating a contextual situation, needs to be discussed carefully

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with students.Other problematic beliefs were predictable. Two students, Doug and Alicia, believed that the

target outcome (in this case, two hip-hop songs) had to appear in the first trial in order for thesimulation to work. This belief may be related to Konold's outcome approach (1991). According toKonoid, someone who reasons using an outcome approach interprets a probabilistic situation jn anon-probabilistic way. That is, they predict the outcome of a single trial instead of considering theprobability of the event. Four students also revealed beliefs associated with representativeness; thatis, they expected the short-term probabilities to always reflect the probabilities of the parent popula-tion. Given the well-documented literature on both these heuristics (Batanero & Serrano, 1999; Konoid,1991 ; Konoid et al., 1993 ; Kahneman & Tversky, 1972), it is not surprising that they re-emerged in thecontext of a probability simulation.

This study has documented research-based knowledge on the reasoning and beliefs potentialAdvanced Placement Statistics students bring to probability simulations. Future research in the formof teaching experiments (Cobb, 2000) is needed to evaluate the viability of using this research-basedknowledge to inform instruction in regular classroom situations. More specifically, knowledge aboutstudents' thinking in relation to probability simulations can be used to build instructional sequencesor hypothetical learning trajectories (Simon, 1995). In building on this study, teaching experimentssuch as those proposed have the potential to generate sustainable instructional programs in prob-ability simulation that build on students' prior knowledge and recognize the nature of their beliefs.

Notes1 In this study we use the term two-dimensional to refer to probability activities or simulations that involve

performing two random experiments or performing one random experiment twice. In a two-dimensionalprobability activity or simulation, we obtain one outcome from each random experiment; hence we aredealing with pairs of outcomes, in this case ordered pairs. Although less complex, the two-dimensionalproblems in our study essentially belong to a class of problems that mathematicians refer to as the jointprobability density function of two random variables (Hogg & Tanis, 1997). By way of contrast, a one-dimensional probability situation would involve just one random experiment and a sample space thatcomprised single outcomes. English (1993) also uses the term 'two-dimensional' when referring to combi-natorial problems that exhibit similar structure to the problems in this study.

2 Advanced Placement courses in subjects such as calculus have existed in the United States since the 1960s.Students normally take an Advanced Placement course at their high school, but to gain advanced credit at aUS university or college, they must pass a nationally assessed examination. Both the curriculum and theexamination come under the auspices of the US College Board. Advanced Placement Statistics has beenavailable only since 1997 and includes topics such as displaying data, density curves and the normaldistribution, correlation and regression, sampling and the design of experiments, probability, simulation,random variables, the binomial and geometric distributions, sampling distributions, and inferential statisticsincluding confidence levels and tests of significance.

3 In the US high school mathematics curriculum, a one-year course in advanced algebra typically follows asequence of one-year courses in each of algebra and geometry. Advanced algebra usually includes topicssuch as linear, quadratic, exponential, logarithmic, and rational functions; conic sections; arithmetic andgeometric sequences and series; and the binomial theorem.

4 While the chance of getting 22 successes in a binomial distribution with 50 trials and p = 0.6 is less than 2%,we wanted to provide data that would reflect the limits that students might obtain in classroom or small-group simulations with a relatively small number of trials. We also wanted to see the maximum extent towhich high 'Advanced Algebra' students would be susceptible to representativeness (Kahneman & Tversky,1972).

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probability simulation: what meaning does it have for high school students?

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