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1997, Vol. 9, No.1, 39-59 Mathematics Education Research Journal A Framework for Assessing Middle School Students' Thinking in Conditional Probability and Independence James E. Tarr Graham A. Jones Middle Tennessee State University Illinois State University Based on a synthesis of research and observations of middle school students, a framework for assessing students' thinking on two constructs-eond.tional probability and independence-was formulated, refined and validated. For both constructs, four levels of thinking which reflected a continuum from subjective to numerical reasoning were established. The framework was validated from interview data with 15 students from Grades 4-8 who served as case studies. Student profiles revealed that levels of probabilistic thinking were stable across the two constructs and were consistent with levels of cognitive functioning postulated by some neo-Piagetians. The framework provides valuable benchmarks for instruction and assessment. Introduction Recent world-wide curriculum reforms in school mathematics (e.g., Australian Education Council, 1994; National Council of Teachers of Mathematics, 1989) have advocated broadening the scope of probability in the middle school mathematics curriculum. This emphasis on broader explorations of probability concepts in problem settings has established the need for further research into the probabilistic thinking of middle school students (Shaughnessy, 1992). Although there has been substantial research on middle school students' probabilistic thinking (e.g., Falk, 1983; Fischbein, Nello & Marino, 1991; Hawkins & Kapadia, 1984; Piaget & Inhelder, 1975; Shaughnessy, 1992), little of the research has focused on students' thinking in conditional probability and independence. This lack of research on students' thinking in these two concepts is a matter of concern given the fact that these concepts are increasingly being identified as important ideas in probability instruction for the middle school (Australian Education Council, 1994; Watson, 1995; Zawojewski, 1991). Accordingly, this study addresses the need for a framework which systematically describes and predicts middle school students' thinking in conditional probability and independence. The need for such a framework is critical if instructional programs and assessment targets in probability for the middle school are to be informed by research-based knowledge of students' thinking (Fennema, Carpenter & Peterson, 1989). I 40 Tarr& Jones Aims of the Research Through a synthesis of relevant research, coupled with observation and interviews with middle school children, this study set out: (a) to develop an initial framework for describing how middle school children think about conditional probability and independence; (b) to generate an assessment protocol for assessing middle school students' thinking in independent and conditional probabilistic situations; and(c) to refine and validate the framework. Definitions This study used the definition of conditional probability provided by Hogg and Tanis (1993). The background for their definition is that, in some random experiments, there is interest only in those outcomes that are elements of a subset B of the sample space S. Within this context, they define the conditional probability of an event A, given that event B has occurred, written P(A I B), as the probability of A considering as possible outcomes only those outcomes of the random experiment that are elements of B. Hogg and Tanis (1993) noted that a special case of conditional probability occurs in a random experiment carried out under "without replacement" conditions. As an example, consider a bag with one red ball, one green ball, and one blue ball, and an experiment where one ball is drawn and not replaced. The sample space immediately prior to the second draw will be a subset of the original sample space and the probability of "red," for example, will be conditional on the outcome of the first draw. If a red is picked on the first draw, the probability of "red" given the event "red" on the first draw will be O. On the other hand if a blue ball is picked on the first draw, the probability of "red" given the event "blue" on the first draw will be 0.5. "Without replacement" situations, like those described above, not only generate special cases of conditional probability, but are especially explicit because the sample space is visually reduced. Because of their explicit nature "without replacement" situations were used in this study to measure middle school students' informal knowledge of conditional probability. Hogg and Tanis (1993) also stated that two events are independent if the occurrence of one of them does not change the probability of the occurrence of the other. As an example, suppose a coin is flipped twice and the sequence of heads and tails is observed. If A is the event, "head on the first flip," and B is the event, "tails on the second flip, then P(B I A) = P(B). In other words the occurrence of A has not changed the probability of B, and A and B are independent events. Some mathematicians specify the events A and B in the above illustration as independent trials (e.g., Larsen & Marx, 1986). In this study, we make no distinction between independent events and independent trials, and we consistently use settings involving repeated trials to assess middle school students' thinking with respect to independence. In contrast with the conditional probability settings, some of these repeated trials involve independent events arising from "with replacement" situations. The use of random experiments involving repeated trials is readily observable for these students and is consistent with the settings used in this study for conditional probability. Framework for Assessing Thinking in Conditional Probability 41 Although conditional probability and independence are clearly related both mathematically and in the settings used in this study, there is no guarantee that students will take cognisance of the relationship between these two concepts. In essence this is one of the questions that this study sought to answer. Theoretical Considerations Conditional probability and independence, as defined above, are the key constructs incorporated into the initial framework for describing and predicting middle school students' thinking in probability (Table 1). The development of this framework, which was built on an earlier framework Gones, Langrall, Thornton & Mogill, 1997), is based on the assumption that'middle school students' thinking in conditional probability and independence develops over time and can be described across four levels. As is highlighted in Table I, Levell is associated with subjective thinking, Level 2 is seen to be a transitional stage between subjective and naive quantitative thinking, Level 3 involves the use of informal quantitative thinking and Level 4 incorporates numerical re~soning. Furthermore it is posited that a student's probabilistic thinking at any given time is stable over both constructs. The notion of levels of thinking within specific knowledge domains is also in concert with cognitive research which recognises developmental stages (Piaget & Inhelder, 1975) and, more particularly, with neo-Piagetian theories which postulate the existence of sub-stages or levels that recycle during stages (Biggs & Collis, 1991; Case, 1985; Fischer, 1980; Mounoud, 1986). These neo-Piagetian researchers claim that the sub-stages or levels, usually five, reflect shifts in the structural complexity of students' thinking and that each level subsumes the previous one. For examp~e, Biggs and Collis (1991) distinguish five basic levels in the learning cycle: prestructural, unistructural, multistructural, relational, and extended abstract. Prestructural responses belong in the previous stage or mode of representation, extended abstract responses stretch into the next mode of representation and unistructural, multistructural and relational levels fall within the present mode. In addition, Biggs and Collis maintain that this learning cycle of five levels is consistent across different stages and is applicable to school based tasks. Each of the constructs, conditional probability and independence, is described below and is amplified through reference to research on students' probabilistic thinking. In addition, research on the two constructs is interpreted in the context of the levels of the framework. Conditional Probability In this study, students' intuitive understanding of conditional probability was measured by their ability to recognise and justify when the probability of an event was and was not changed by the occurrence of another event. Students were also expected to determine probabilities in response to the occurrence of another event. Although there has been relatively little research into middle school students' 42 Tarr & Jones Table 1 Initial Framework for Assessing Middle School Students' Thinking in Conditional Probability and Independence LEVEL i (Subjective) LEVEL 2 (Transitional) • Recognises when"certain" and "impossible" events arise in replacement and non-replacement situations. . .Generally uses subjective reasoning in considering the conditional probability of any event in a "with" or "without" replacement situation. • Recognises that the probabilities of some events change in a "without replacement" situation. Recognition is incomplete, however, and is usually confined to events that have previously occurred. • May revert to subjective judgments or use inappropriate quantitative measures. • Unaware that two events mayor may not influence each other. • Holds a pervasive belief that they can control the outcome of an event. • Uses subjective reasoning which precludes any meaningful focus on the independence or dependence of events. ·Shows some recognition as to whether consecutive events are related or unrelated. • Frequently uses a "representativeness" strategy, either a positive or negative recency orientation. • May also revert to subjective reasoning. CONDITIONAL PROBABILITY • Keeps track of the complete composition of the sample space in judging the relatedness of two events in both "with" and "without" replacement situations. • Recognises that the probabilities of all events change in a "without replacement" situation, and that none change in a "with replacement" situation. • Can quantify, albeit imprecisely, changing probabilities in a "without replacement" situation. • Assigns numerical probabilities in "with" and "without" replacement situations. • Uses numerical reasoning to compare the probabilities of events before and after each trial in "with" and "without" replacement situations. INDEPENDENCE • Recognises when the outcome of the first event does or does not influence the outcome of the second event. In "with replacement" situations, sees the sample space as restored. .Can differentiate, albeit imprecisely, independent and dependent events in "with" and "without" replacement situations. • May revert to the use of a representativeness strategy. • Distinguishes dependent and independent events in "with" and "without" replacement situations; using numerical probabilities to justify their reasoning. CONDITIONAL· PROBABILITY INDEPENDENCE thinking in conditional probability situations, two studies (Fischbein & Gazit, 1984; Jones, Langrall, Thornton & Mogill, 1997) are seminal to the development of the Framework for Assessing Thinking in Conditional Probability 43 framework for this research. These two studies also incorporate some earlier research on conditional" probability reported by Piaget and Inhelder (1975), Borovcnik and Bentz (1991), and Shaughnessy (1992). Fischbein and Gazit (1984) carried out a teaching experiment involving 285 students from Grades 5, 6 and 7. They found that, when students were asked to determine conditional probabilities in "with" and "without" replacement situations, the performance of sixth and seventh graders dropped dramatically for "without replacement" tasks when compared with "with replacement" tasks. The percentage of correct responses for sixth graders fell from 63% to 43% and for seventh graders from 89% to 71%. The fifth graders response rate was only 24% for the "with replacement" tasks and remained approximately the same for the "without replacement" tasks. Based on their analysis, Fischbein and Gazit identified two misconceptions in students' thinking in conditional probability: (a) they did not realise that the sample space had changed in a "without replacement" situation; and (b) they found the probability of an event in a "without replacement" situation by comparing the number of favourable outcomes for the event before and after the first trial rather than making comparisons with the total number of outcomes. In a related study with Grade 3 children over a period of one year, Jones, Langrall, Thornton and Mogill (1997) identified four levels of thinking in conditional probability. Children at Levell (Subjective) could not list the complete set of outcomes in either a "with" or "without" replacement situation and did not recognise that the probability of any event changed in a "without replacement" situation. Their reasoning was almost always subjective. By Level 2 (Transitional), children were beginning to recognise that the conditional probability of some but not all events changed in a "without replacement" situation. This recognition was generally restricted to the event that had occurred in the previous trial. Level 3 (Informal Quantitative) thinkers recognised that the probability of all events changed in a "without replacement" situation and had begun to use numbers in an informal way to determine conditional probabilities. Students at Level 4 (Numerical) were able to find valid numerical measures for changing probabilities in "without replacement" tasks, but the measures they used were ratios rather than fractions. Jones, Langrall, Thornton and Mogill noted that, in their study, none of the students reached Level 4 even after instruction. With respect to conditional probability, our framework has built on all of these findings. In particular, in describing students' probabilistic thinking from Levels 1 to 4 we have used similar descriptors to those identified by Jones, Langrall, Thornton and Mogill (1997) and have amplified these descriptors with the findings of Fischbein and Gazit (1984). Independence In this study, students' intuitive understanding of independence was measured by their ability to recognise and justify when the occurrence of one event had no influence on the occurrence of the other. More particularly, students had to recognise that, when the probability of an event was not changed by the occurrence of another event, the two events were independent. A key study in the area of 44 Tarr& Jones independence was carried out by Fischbein, Nello and Marino (1991) with 618 students in Grades 4-8. In this study, the researchers asked students to determine which event was more likely, obtaining three "heads" by tossing one coin three times or by tossing three coins simultaneously. Thirty-eight percent of fourth and fifth graders and 30% of junior high students, with no prior instruction in probability, responded that the probabilities were not equal. By a ratio of nearly 2:1, students at each grade level believed the probability of obtaining three heads, by tossmg a single coin three times, was higher. Based on follow-up interviews, ~ Fischbein, Nello and Marino found that students harboured a pervasive belief that the outcomes of a coin toss can be controlled by the individual. They concluded that such a belief is incompatible with the notion of independence, given the probability of obtaining a head on each trial remains constant at 0.5. Similar misconceptions were evident in the National Assessment of Educational Progress in Mathematics (Brown, Carpenter, Kouba, Lindquist, Silver &. Swafford, 1988) which asked students to state the most likely outcome on the next toss of a fair coin which had landed "TITT" on four successive trials. Results indicated that only 47% of the seventh graders selected the correct alternative-heads and tails are equally likely. Slightly higher achievement was obtained in a study of 2930 British students aged 11 to 16 years (Green, 1983). In this study a fair coin was flipped four times, each time landing heads up. When asked to name the most likely outcome of the fifth toss, 75% of all students answered correctly, including 67% of 11-12 year olds. In a third study using this same item (Konold, Pollatsek, Well, Lohmeier & Lipson, 1~93), only 70% of the undergraduates in a remedial mathematics course responded correctly. Moreover in extensions to this item, Konold et al. asked the students to state which of the following sequences was most likely and which was least likely to occur when a fair coin was tossed five times: (a) HHHTI; (b) THHTH; (c) THTIT; (d) HTHTH; and (e) all four sequences are equally likely. In the most likely case, approximately 61% of the undergraduates enrolled in the remedial course responded correctly but only about 35% responded correctly in the case of the least likely sequence. Clearly, a substantial number of students who demonstrated some understanding of independence in the most likely case abandoned this thinking in the least likely case. Konold et al.concluded that a conflict existed between the belief that a coin has an equal chance of coming up heads or tails and that roughly half heads and half tails are expected in a sample of coin flips. Two categories of misconceptions were identified in these· studies. The most common misconception (gambler's fallacy or negative recency effect) was the tendency to believe that, after a run of tails, heads should be more likely to come up. A far less common but related misconception (positive recency effect) was the tendency to believe that, after a run of tails, a tail is more likely to come up. Both of these misconceptions illustrate a judgmental heuristic known as "representativeness"-the belief that a sample or even a single outcome should reflect the parent population (Kahneman & Tversky, 1972). Such a belief is powerful and is generally in conflict with the concept of independence. Thus, even when students seemingly exhibit an understanding of the concept of independence, the representativeness heuristic may still prevail. Framework for Assessing Thinking in Conditional Probability 45 In relation to independence, our framework attempts to capture the change in thinking from subjective judgments and beliefs about control, to thinking that is based on numerical judgments and resolves conflicts between beliefs about distribution expectations and event probabilities. The framework descriptors also try to recognise growth in thinking that results from the ability to make connections between conditional probability and independence. Methodology Subjects Students in Grades four and five in an ~lementary school in Bloomington, Illinois, and students in Grades six through eight in a junior high school in Eureka, Illinois, formed the population for this study~ Fifteen students, three from each grade level, served as case studies in this investigation. The students represented the full range of abilities in each grade, with one being randomly chosen from the top third, one from the middle third and one from the lowest third in mathematics achievement. None of these students had been exposed to instruction in probability. The Validation Process In formulating the framework for this study, four levels of probabilistic thinking were posited and initial descriptions were generated for each of these levels over two constructs, conditional probability and independence (Table 1). Adopting the validation process used by Jones, Thornton and Putt (1994) and Jones, Langrall, Thornton and Mogill (1997), this study sought to: (a) refine the initial descriptions of the four levels of probabilistic thinking; (b) examine the profiles and stability of case-study students' thinking over the two constructs; and (c) illuminate the distinguishing characteristics of each thinking level within the framework. Qualitative analysis was used to address all three parts of the validation. Instrumentation and Data Collection The framework (Table 1) guided the design of the structured interview protocol used in this study. The interview protocol, based on the framework, was administered by the first author to each of the students who, served as case studies. All interviews were conducted within a two-week period and were audiotaped for subsequent analysis. The interview protocol comprised 14 tasks, of which 8 assessed thinking in conditional probability and 6 assessed thinking in independence. The assessment tasks in conditional probability focused on probability situations involving "with" and "without" replacement conditions, and the tasks on independence assessed whether students could recognise and justify independent or dependent trials generated in various probability settings. Selected items from the interview protocol are presented in Table 2. In generating the items for the interview protocol the researchers used problem contexts such as gumball machines, locker 46 Tarr& Jones Table 2 Selected Probability Tasks from the Assessment Protocol CONDITIONAL PROBABILITY CPl Suppose you were to reach into the bag of Halloween candy and draw out a ,candy bar. What kind of candy bar do you have the most chance of drawing? Why? A candy bar is drawn and not replaced. Suppose you ate the candy bar and then reached into the bag again. What kind of candy bar do you have the most chance of drawing? Why? Has your chance of drawing a Snickers bar changed or is it the same chance as before? Explain. CP2 Suppose your teacher is going to have a drawing to determine who gets to go first to lunch today. If your colour is drawn, then you get to go to lunch first. Allow the student to observe that the cup contains 4 blue, 3 green, 2 red, 1 yellow chips. What colour do you want to be? Why? Suppose you are assigned green and your friend is assigned yellow. I'm going to reach in and draw a chip. What colour do you predict will be drawn? Why? A blue chip is drawn and not replaced. Suppose your teacher has another drawing on the following day. Has your chance of winning changed or is it the same as before? Has your friend's chance of winning changed or is it the same as bef~re? Why? CP3 Your class is going to elect a president and vice president. There are five people rurining. Show the INDEPENDENCE INOI Allow the student to examine a die. I'm going to roll the die. If I gave you one chance to pick, do you think you could predict what number will come up? Why? Roll the die. I'm going to roll the die again. Do you think the chance of rolling a (result of first roll)has changed? Why? Suppose I wanted ·to roll a 3. How many rolls would it take to guarantee that a 3 would come up? Why? IN02 Allow the student to examine a chip that is coloured red on one side, white on the other. Suppose I flipped the chip five times and kept track of the result. Which of the following sequences is the most likely result of five flips? a) ••• 00 b) 0.· 0 • 000 c) o. d) • 0 • 0 • e) All five sequences are equally likely. Suppose I flipped the chip five times and kept track of the results. Which of the above sequences is the least likely result of five flips? IND3 student five cards: Beth, Steve, Maria, Rick, and YOU. All five students are considered to have an equal chance of winning. After school the results are announced. Is it more likely the class president will be a boy or a girl? Why? Is it more likely your name will be read or more likely it won't be read? Reveal the name of the president. Is it more likely the vice president will be a boy or a girl? Why? Compared to last time, has the chance that your name will be read changed or is it the same as before? Why? CP4 Place the "One Away" game board in front of the student. (see IND4) There is a secret three-digit number. It is not 2-8-5, but each digit is one away from 2-8-5. That is, the first digit can be 1 or 3, etc. If I gave you one chance to pick, do you think you could win? Is it more likely you would win, more likely you would lose, or is it the same chance? Allow the student to pick the first digit. (Regardless of their choice) You are correct. Has your chance of winning changed? Allow the student to pick the second digit. (Regardless of their choice) You are correct. Has your chance of winning changed? Why? I'm going to spin it with the needle on the line. Do you think it's more likely for it to land on red, on yellow, or is it the same chance? Why? Spin the needle. I'm going to spin it again. Do you think it's more likely for it to land on red, on yellow, or is it the same chance? Why? (Repeat again) Suppose I wanted it to land on red. Should I start the needle on red, on yellow, on the line, or does it not make a difference? Why? lN04 174 L L :s:: 396 Place the "One Away" game board in front of the student. What is your chance· of picking the first digit correctly? Allow the student to pick the first digit. (Regardless of their choice) You are correct. Has your chance of correctly picking the second digit changed? Allow the student to pick the second digit. (Regardless of their choice) You are correct. Has your chance of correctly picking the third changed? Why? Framework for Assessing Thinking in Conditional Probability 47 combination numbers, Halloween candy drawings, first to lunch drawings, and games involving coin flips and die rolls, all of which were very much part of these students' real world. The design of each of the conditional probability and independence items enabled the researchers to explore students' thinking across all four levels of the framework. In each case-study interview, tasks on conditional probability and tasks on independence were· administered in an alternating sequence. Items were structured so that students would respond orally, but they were also encouraged to use probability materials to demonstrate and explain their respons~s. , A double-coding procedure described by Miles and Huberman (1984) was used to code the interview protocols. Initially, both researchers independently coded each item. of each student's interview assessment. Using the framework criteria (Table I), items were coded according to the level of thinking exhibited by the student. These codings were then used to determine the dominant (modal) probabilistic thinking level for each student on both conditional probability and independence. Agreement between the two researchers was achieved on the coding of 28 out of 30 levels, that is, a reliability of 93%. Variations were then clarified until consensus was reached on each student's dominant level of thinking on both conditional probability and independence. During the coding process described above, both researchers used a groundedtheory approach (Bogdan & Biklen, 1992) to discern the key thinking patterns exhibited by students at each level of the framework and across both constructs. These patterns were used to refine the framework descriptors and to generate summaries that illuminated the descriptors. Data Analysis and Results: Validating the Framework The results of the validation process are presented in three parts. The first part describes refinements made to. the framework following the collection of data. In the second part, the profiles and stability of students' thinking across the two constructs of the revised framework are examined. Finally, based on an analysis of students' thinking, summaries and exemplars are presented to illuminate the four levels of probabilistic thinking in the revised framework. Refinements to the Framework The refined framework (Table 3) was developed following the analysis of the case-study data. Although only the first descriptor under Level I, Independence (Table 1) was rejected in refining the initial framework (Table 3), several additional descriptors were added to the refined framework to enhance the initial descriptors for conditional probability and independence. These additional descriptors have been italicised in Table 3. The descriptor, "Unaware whether two events influence each other or not" (see Table 1: Levell, Independence, first descriptor) was found to be inappropriate and was omitted from the refined framework. This action was taken because neither of the case studies at Level 1 exhibited. ambivalence concerning the influence of consecutive events on each other. On the contrary, both case study students maintained that consecutive events were always dependent; that is, they rejected the concept of independence. In order to underscore Level 1 students' 48 Tarr& Jones Table 3 Refined Framework for Assessing Middle School Students' Thinking in Conditional Probability and Independence LEVEL 1 (Subjective) CONDITIONAL PROBABILITY -Recognises when "certain" and "impossible" events arise in replacement and nonreplacement situations. -Generally uses subjective reasoning in considering the conditional probability of any event in a "with" or "without" replacement situation. -Ignores given numerical information in formulating predictions. -Predisposition to consider that consecutive events are always related. INDEPENDENCE - Pervasive belief that they can control the outcome of an event. - Uses subjective reasoning which precludes any meaningful focus on the independence. -Exhibits unwarranted confidence in predicting successive outcomes. CONDITIONAL PROBABILITY INDEPENDENCE - Recognises that the probabilities of all events change in a "without "replacement" situation, and that none change in a "with replacement" situation. -Keeps track of the complete composition of the sample space in judging the relatedness of two events in both "with" and "without" replacement situations. -Can quantify, albeit imprecisely, changing probabilities in a "without replacement" situation. - Recognises when the outcome of the first event does or does not influence the outcome of the second event. In "with replacement" situations, sees the sample space as restored. -Can differentiate, albeit imprecisely, independent and dependent events in "with" and "without" replacement situations. -May revert to the use of a representativeness strategy. LEVEL 2 (Transitional) - Recognises that the probabilities of some events change in a "without replacement" situation. Recognition is incomplete, however, and is usually confined to events that have previously occurred. -Inappropriate use of numbers in determining conditional probabilities. For example, when the sample space contains two outcomes, always assumes that the two outcomes are equally likely. -Representativeness acts as a confounding effect when making decisions about conditional probability. - May revert to subjective judgments. -Shows some recognition as to whether consecutive events are related or unrelated. - Frequently uses a "representativeness" strategy, either a positive or negative recency orientation. - May also revert to subjective reasoning. - Assigns numerical probabilities in "with" and "without" replacement situations. - Uses numerical reasoning to compare the probabilities of events before and after each trial in "with" and "without" replacement situations. -States the necessary conditions under which two events are related. - Distinguishes dependent and independent events in "with" and "without" replacement situations, using numerical probabilities to justify their reasoning. -Observes outcomes ofsuccessive trials but rejects a representativeness strategy. -Reluctance or refusal to predict outcomes when events are equally likely. Framework for Assessing Thinking in Conditional Probability 49 preoccupation with dependence, an additional descriptor, "Predisposition to consider that consecutive events are always related," was included (see Table 3: Levell, Independence, first descriptor). A somewhat related addition was necessitated by Levell students' tendency to ignore quantitative information inherent in the sample space and to display unwarranted confidence in predicting successive outcomes in replacement situations (see Table 3: Levell, Independence, fourth descriptor). By way of contrast, both students exhibiting level 4 thinking were clearly reluctant to predict outcomes when events were equally likely, and often refused to pick altogether unless the odds were in their favour (see Table 3: Levell, Independence, fourth descriptor). Two other additions to the framework (see Table 3: Level 2, Conditional Probability, third descriptor; and Level 4, Independence, second descriptor) were incorporated to build a more comprehensive picture of students' thinking with respect to the representativeness strategy. When these additional descriptors were cOlnbined with the original descriptors pertaining to representativeness they resulted in a more ordered description of student's use of the representativeness strategy across the four thinking levels. The remaining additions (see Table 3: Levell, Conditional Probability, third descriptor; Level 2, Conditional Probability, second descriptor; and Level 4, Conditional Probability, third descriptor) were seen to add greater precision and amplification to the original descriptions and are illustrated in the next section through the thinking of Alicia (Levell), Brian (Level 2) and Denise (Level 4). Profiles and Stability Across the Two Constructs From a validation perspective, the stability of students' thinking across the two constructs is of prime importance. In particular, if the framework is to be used in curriculum development and assessment it is essential that descriptors provide parallel benchmarks at each level across both constructs, conditional probability and independence. Profiles of the students' thinking levels (Figure 1) revealed strong internal consistency in that thinking patterns across the two constructs were at the same level for 11 of the 15 students. In the case of the four, exceptions, it was noteworthy that none of the differences were more than one level apart and that two of the differences were associated with higher thinking levels on independence and two with higher levels on conditional probability. These observations support the "stability" hypothesis for the framework and this position is further endorsed by a Pearson correlation coefficient of r = 0.83, indicating 69% of shared variance between thinking levels for independence and conditional probability. Notwithstanding the constraints associated with a sample of 15 students, the individual profiles yield further findings. Given the fact that the two students who exhibited Level 1 thinking were in Grades 4 and 5 and one of the students who exhibited Level 4 thinking was in Grade 8, it is compelling to conclude that thinking becomes more sophisticated with increased age. However, such a conclusion may be oversimplifying the situation because within each grade there was a diversity of probabilistic thinking (see Figure 1). Moreover, two fourth-grade 50 Tarr& Jones students exhibited Level 3 thinking while one eighth-grade student was at Level 2. These extremes and the diversity of probabilistic thinking within grades appear to be just as explicit as differences between grades. Conditional Probability f:A Jana CI} ~ Grade 4 Mary 4 CI} ~ ~ ._ ~ I i5 0 ~ CP 0 IND Ashley CI} Q) :> Grade 5 ~ 4 Q) :> i5 I CP Q) ._ ~ ._ ~ I I 0 Grade 7 gf2 CI} Q) CP 0 IND 4 i5 3 bJ> ] 2 c .- I ~ CP 0 IND CI} 4 Q) :> 3 Q) :> 3 ~ CP IND ~ IND gf2 ~ .- 1 0 CP Denise Sergio gf2 I 0 Q) ~ 4 ~ .- IND 4 CI} ~ 3 CP Kurt CI} ~I gf2 ~ IND ~ 2 Q) Grade 8 0 CP 4 Melissa ~ ~ .3bJ> 3 I 0 ~ .- J Vanessa 4 ~ 2 CI} Q) .3 3 IND IND 4 gf2 0 CP Daniel CI} ~I .3bJ> 3 i§ 0 IND 4 Brian Q) CP ~ CP CI} ~ ~ 3 ....:l 0 IND Kiko Q) 1 CP 4 CI} gf2 CI} 3 gf2 ~ 0 IND Christina 4 .3 i§ CP gf2 IND Q) Grade 6 ~ 4 Neil CI} ._ ~ I .3 3 ~ 0 bJ>2 c 3 ~ gf2 3 ~ Brad CI} 3 ~ CI} bJ>2 c I 4 ~ 3 ~ gf2 Independence Carlos 4 ~ 3 l1li ~ CP IND I 0 CP IND Figure 1. Probabilistic thinking patterns: Profiles of case-study students. Framework for Assessing Thinking in Conditional Probability 51 Analysis of the Probabilistic Thinking at Each Level The responses and thinking of each of the students who served as case studies were analysed, and summaries and exemplars were produced to illuminate the thinking patterns outlined in the refined' framework (Table 3). Although all students at a particular level on a construct tended to generate similar probabilistic thinking and misconceptions, we have, for ease of reading, used case-study students who were at the same level on both constructs. Level 1 Summary. In interpreting probability situations, students exhibiting Levell thinking tend to rely on subjective judgments, ignore relevant quantitative information and generally believe that they can control the outcome of an event. Given' this void of quantitative referents, Level 1 students form conditional probability judgments by constructing their own reality, either searching for patterns which do not exist or by imposing their own system of regularity. They predict outcomes with unwarranted confidence and often use their own recent experiences in playing games (availability heuristic-see Shaughnessy (1992)) to predict or estimate the chance of an event. Level 1 students tend to believe that previous outcomes always influence future outcomes and they essentially deny the existence of independence. In essence, Levell students' proclivity to use subjective judgments, to impose regularity or to rely on readily available experiences involving chance precludes any meaningful focus on independence and conditional probability. Alicia, a fifth-grade student, demonstrated a reliance on subjective reasoning that exemplifies Level 1 thinking. When asked to predict what colour would be drawn in First to Lunch (Table 2, CP2), Alicia immediately chose red because, "it ~s my favourite colour." Her response typically ignored the numerical composition of the cup which contained fewer red chips than either blue or green. The distinctive feature of Alicia's conditional thinking was her lack of awareness of the role that numbers play in making probability judgments-to her, numbers were simply not relevant. With respect to the concept of independence, Alicia sought to impose her own system of regularity when interpreting random experiments. After subsequent rolls of 2,3, and 4 (Table 2, INO 1) she believed that the chance of rolling a 6 had increased, saying, "it's gone 2, 3, 4, so it's laying a little pattern." Her response is indicative of the common belief among Levell students that previous results influence outcomes in "with replacement" situations. Typically she also exhibited unwarranted confidence in predicting outcomes. When predicting the initial roll of a die (Table 2, INO 1), Alicia was emphatic that the result would be a 6 and vividly recalled how she had rolled consecutive 6's when playing a game in second grade. Although she appeared to use an availability heuristic (Shaughnessy, 1992) in recalling an earlier success in rolling 6's, her strategy may well be subsumed under the more general rubric of "imposing regularity where randomness prevails." These observations demonstrate that Alicia was not receptive to the existence of independence. Level 2 Summary. Students who demonstrate Level 2 probabilistic thinking are in transition between subjective and informal quantitative thinking. They sometimes make appropriate use of quantitative information in making 52 Tarr& Jones conditional probability judgments, but are easily distracted by irrelevant features. For example, they tend to put too much faith in the distribution of previous outcomes when forming predictions; as a result, they often use a representativeness heuristic (Shaughnessy, 1992), incorporating either a positive or negative recency orientation. When considering conditional probabilities, some Level 2 students may also start from a subjective perspective or may even revert to subjective judgments when outcomes do not occur as they expected. Others faced with a probability situation containing two outcomes are prone to assuming that the two outcomes are equally likely. Their vulnerability to irrelevant features or imprecise reasoning during this transitional stage seems to explain the vacillations they exhibit when faced with tasks which' involve conditional probability and independence. Even when they do use informal quantitative reasoning, their thinking is limited. They are able to recognise that the probabilities of some events change in "without replacement" situations, but recognition is incomplete and is usually confined to events that have previously occurred. Level 2 thinking was exemplified by Brad, a fifth grader, and by Brian, a seventh grader. At times, both students were able to use quantitative information appropriately, but they were also prone to subjective judgments and imprecise quantitative reasoning. Although Brad and Brian chose blue in the drawing to be first to lunch (Table 2, CP2), Brian had some reservations, "Red is my favourite colour, but I'll go with blue because there's more of them." In the same conditional probability task, after a blue chip was drawn and not replaced, both thought the probability that green would be drawn had increased because there was an equal number of blue and green chips. Brad explained, "You already have one blue out so that makes three blues, three greens, and that's the same amount." However, typical of Level 2 students, their conditional thinking was not complete as each maintained in the same problem that the probability that yellow would be drawn was "still the same," because "there's still only one yellow in the cup." A further illustration of Brian's imprecision in using quantitative information occurred when the sample space contained two outcomes. He consistently assumed that both events were equally likely as the following exemplars show. When asked the probability of selecting the three-digit number in one pick (Table 2, CP4, IND4), Brian said, "There's a 50-50 chance I could or could not." After learning that he had correctly selected the first digit, Brian still maintained that his chance of winning versus losing was "still 50-50" even though it had increased from 1/8 to 1/4. Both students exhibited some recognition of independent events but were easily distracted by the outcome distribution of successive trials. When asked to predict if a spinner was more likely to land·on red or yellow (Table 2, IND 3), Brad said, "It's the same chance it'll be either one of those." After the spinner landed on red, Brad maintained this stance. However, when it stopped on red a second time, Brad predicted it was now more likely to land on red, "Because the last time you spinned it, it was red so it might be red this time." Neil, a sixth grader, who had a dominant thinking level of 3, actually shifted his thinking in the opposite direction to Brad on this task. He initially predicted that the needle would land on red because it started on red. After the needle stopped on yellow he abandoned his subjective reasoning in favour of a more quantitative argument saying, "half ofthe circle is still red and half of it is still yellow." Framework for Assessing Thinking in Conditional Probability 53 The pervasiveness of a representativeness strategy confounds Level two students' ability to make proper judgments in conditional probability and independence. In selecting the most likely outcome of five flips of a chip (Table 2, IND2), Brian chose B "because it looks like the one that's most common-it's mixed." When asked which result was the least likely result of five flips, Brian chose C "because there are more whites than red, and if anything it would probably come up even-well, it can't come up even, but pretty close to even." Level 3 Summary. Students exhibiting Level 3 thinking are aware of the role that quantities play in forming conditional probability judgments. Although these students do not usually assign precise numerical p~obabilities they often use relative frequencies, ratios or some form otodds in an appropriate way to determine conditional probabilities after each trial of an experiment "with" or "without" replacement. They keep track of the complete composition of the sample space and usually recognise that the conditional probabilities of all events change in "without replacement" situations. Level 3 students' predisposition to monitor the sample space composition also enables them to recognise independent events in "without replacement" situations; however they sometimes revert to representativeness strategies (Shaughnessy, 1992) in dealing with a series of independent trials. Levei 3 thinking was exemplified by Carlos, a fourth grader, and Christina, a sixth grader. Both students kept track of the composition of the sample space before and after each trial and consequently recognised that the conditional probabilities of all events changed in "without replacement" situations. For example, after a Milky Way was drawn from a bag of Halloween candy and not replaced (Table 2, CPl), Carlos explained that the chance of being selected had increased for both Butterfingers and Snickers. He said, "Butterfingers is tied with the most and it was behind ... Snickers is one away from the top and it was two away from the top." A similar approach, for which we have coined the term "competition strategy/' was utilised by Christina in predicting the results of the class election (Table 2, CP3). After the name of the president was revealed, she compared her chance of being named vice president to her prior chance of being named president as follows: "It [vice president] is better-now there's three people against me instead of four." By monitoring the composition of the sample space, both students were able to determine whether two outcomes were independent. In replacement situations, they recognised when the sample space had been restored and linked this phenomena to the concept of independence. For example, after a spinner had twice landed on yellow (Table 2, IND 3), Christina stated that it was equally likely for red or yellow; she said, "It's the same chance .. '. because half of the circle is still red and half of it is still yellow (authors' italics)." Similar thinking was generated by Melissa, an eighth grader, when she considered whether probabilities had changed with successive rolls of the die (Table 2, IND 1). After the first roll yielded a I, she stated that the chance of rolling a 1 had not changed: "That number hasn't been eliminated ... so it just starts all over again." 54 Tarr& Jones Neither student at Level 3 assigned numerical probabilities in forming probability judgments. This limitation in making precise probability judgments may have produced some reversion to a representativeness strategy for both students. For example, when asked to determine the most likely result of five flips of a chip (Table 2, IND 2), Carlos initially said, "1 think they're all equally likely." However, when pressed to compare the relative likelihood of [A] and [B], Carlos hedged and said, "No, not [A], three times in a row. Maybe not." Only after comparing all four sequences did Carlos reaffirm his position: "all have the exact same chance." Level 4 Summary. Students who demonstrate Level 4 thinking use numerical reasoning to interpret probability situations. They are keenly aware of the composition of the sample space, recognise its importance in determining conditional probabilities, and can assign numerical probabilities spontaneously and with explanation. Using numerical thinking, Level 4 students consistently distinguish between independent and dependent events and can even identify the conditions under which two events are related. Their reliance on numerical reasoning enables them to hold strong convictions when making conditional probability judgments. In "without replacement" probability settings, Level 4 students can determine the minimum number of picks before an event is certain, and in "with replacement" situations, they realise that there is no maximum number of trials that will produce a certain event. With respect to replacement situations they are aware of the distribution of outcomes in an on-going series of trials but they reject a representativeness strategy in favour of using more precise conditional probability measures. As further instance of their strong numericallybased convictions, these students are reluctant to make predictions when all outcomes are equally likely. However, they point out when an outcome occurs with the odds, and acknowledge the possibility that events can occur against the odds. Level 4 thinking was demonstrated by Daniel, a sixth grader, and Denise, an eighth grader. Both students consistently and pervasively used numerical reasoning to form probability judgments. For example, as the following interaction shows, Denise was able to use her numerical precision to focus on and interrelate multiple tasks involving conditional probability and independence. When asked if she could correctly select the three-digit number in one pick (Table 2, CP4, IND4), Denise said it was unlikely because there were eight possibilities and only one correct combination. Then, after learning that the first digit was 1, she said, "Now I have a better chance [of guessing the three-digit number] because I got to eliminate four choices-396, 394, 376, 374-so now there's only four for me to choose from." Moreover even though she correctly selected the first digit, Denise indicated that her chances of picking the second digit had not changed. She explained, "If you got the first one wrong, you'd still have the same chance of getting the second one right or wrong." Upon learning that the second digit was 7 she replied, "1 have a better chance now because now I know that it's not 194 or 196; it's either 174 or 176." She reiterated that her chances of correctly picking the next digit had not changed and stated, "It's the same because it doesn't matter what you get the first two times." Her ability to keep everything in perspective during this exchange is a tribute to her strong sense of numerical precision. Framework for Assessing Thinking in Conditional Probability 55 The use of numerical reasoning was also evident when Denise interpreted activities involving independent events. When asked if a spinner was more likely to land on red or yellow (Table 2, IND3) she replied, "It's the same because each side is equal ... there's the same amount of red as there is yellow." In seven successive trials the spinner landed alternately on red and yellow. Denise noted the pattern but rejected a representativeness strategy. She articulated, "It went red, yellow, red, yellow, red, yellow, but even though it's going in a pattern, there's still the same chance because both sides are equal." With her adherence to numerical reasoning, Denise could even identify the conditions under which two events are related or unrelated. For example, after a candy bar was drawn and not replaced (Table 2, CP1), she said the probability of selecting a Snickers bar changes "unless you'put it back in." . Daniel also realised that probabilities do not change in "with replacement" situations. He was reluctant to predict the outcome in the roll of a die (Table 2, IND1) because the chance was the same for each trial. Daniel indicated that if he had to predict he would choose 4 but added, "If I do happen to get it, it would be defying the odds." After a 4 was rolled, he said the chance of rolling a 4 had not changed because "all the numbers are still there ... it's still the same because it's still one out of six." By realising that a non-changing sample space in "with replacement" situations is tantamount to probabilities remaining constant, he was able to distinguish independent and dependent events. When asked how many rolls would be required to guarantee that a 1 would come up Daniel said, "There is no real maximum number of hitting that I." Discussion There is a growing consensus that knowledge of learners is an important component of teacher knowledge (Shulman, 1986). Accordingly, the lack of research on middle school students' thinking in conditional probability and independence is problematic given the emerging emphasis on these probability topics in the middle school curriculum. This research sought to develop and validate a framework for describing and predicting students' thinking in conditional probability and independence that might inform instructional programs in probability. The initial framework, based largely on previous research, identified four levels of probabilistic thinking which ranged from subjective judgments to numerical reasoning. Following analysis of case-study data, a refined framework was developed and profiles of case-study students' thinking levels for each construct were generated. These profiles of students' thinking revealed a general pattern of growth in probabilistic thinking with age; however, broad ranges of probabilistic thinking existed within each grade level. The profiles also suggest that levels of probabilistic thinking, as described in the refined framework, appeared to be stable across both conditional probability and independence. Such stability is important from a validation perspective because it implies that the framework not only presents a coherent picture of middle school students' thinking in conditional probability and independence but also provides parallel benchmarks for the two constructs which can be used in instruction and assessment. 56 Tarr& Jones Notwithstanding the apparent stability discussedaboye, some caution should be used in claiming that the levels of probabilistic thinking across both conditional probability and independence are highly stable for.1al'geuum bers of students. Earlier research (Jones, Langrall, Thornton & Mogill,1997) revealed that the internal consistency of young children's probabjIi§tiCthinking across four constructs was more erratic than the probabilistic thihl<it1gexhi1:>ited by students in this study. Such instability is also consistent withth~fin.qit1gs of Dawson and Rowell (1995) who concluded that "during the process~fconcept development there will be periods of transition within which leamerswilluot have a full and stable belief in anyone explanation" (p. 90). . .... .......> In addition to the value these framework descri~~()~§hay¢jn instruction and assessment, they also extend and illuminate the resear~~;lit.efflture on students' thinking in conditional probability and independ~J.'l~~~\':fJ1~iian.alysis of data revealed that students exhibiting Levell thinking we~i80J.'l~i~t~n.t1y restricted to subjective judgments when interpreting tasks incg~~jti~:J.'l~lprobability and independence. In essence, they appeared to demonstrati?;tli~sl1~J.'acteristicsof what Biggs and Collis (1991) describe as the prestructurallever~fi~'el~arningcycle; that is, they engage in the task but are easily distracted by featittest!iatareirrelevant to probabilistic thinking. Note how Alicia typically ignorecixeleyant quantitative information and, instead, fixated on an irrelevant featureoftl'ietClsk-for example, her favourite colour. Students exhibiting Level 2 thinking were charact~risediby their limited quantitative reasoning in exploring probability tasks. At linles,they were able to make appropriate-albeit primitive-use of numbers to form j'\ldgments, but at other times, they made inappropriate and imprecise useo! •.quantities. Such characteristics serve to illustrate Level 2 as a period of transition in which there is a naive attempt to quantify probabilities. At this level, sh.J.q.ellts' thinking is indicative of what Biggs and Collis (1991) term the unistructurallevel in the sense that the learner engages in the task in a relevant way butQI'llY. one aspect is pursued. For example, Brian recognised that the probabilities of blue and green had changed (Table 2, CP2), but maintained that the probability of yellow had not changed. In essence, he focused on just one comparison, betweellthe target colour and the event that had just occurred, rather than making severalcomparisons with . the changed sample space including those relevant to the event"yellow." Students demonstrating Level 3 thinking effectively used quantitative reasoning when interpreting tasks based on the two constructs. ·They kept track of the composition of the sample space to determine conditional probabilities in "with" and "without" replacement situations and to decide whether two events were dependent or independent. Students at Level 3 seemed to exhibit characteristics of the multistructurallevel of the learning cycle (Biggs & Collis, 1991) in that they focused on more than one relevant feature of a task. For example, in contrast to the Level 2 thinkers who focused on only one aspect of the learning task, note how Carlos, in a similar task (see Table 2, CP1), monitored the changing composition of the sample space for all events. The ability to consider several events simultaneously and to use more than one strategy are features of Level 3 thinkers which distinguish them from Level 2 thinkers. Framework for Assessing Thinking in Conditional Probability 57 Students exhibiting Level 4 thinking use numerical reasoning to interpret probability situations. They assign numerical probabilities to determine conditional probabilities in "with" and "without" replacement situations and to decide whether two events are related or unrelated. Level 4 students appeared to exhibit thinking at what Biggs and Collis (1991) term the relational level of the learning cycle; that is, they integrate numerous relevant aspects of the task in a comprehensive and coherent manner. Denise's management of the dependent and independent features in the "One Away" problem (See Table 2, CP4, IND4) is a poignant illustration of the ability of students at Level 4 to integrate multiple aspects of the learning task. For example~ after learning that she had correctly picked the first digit, she realised that although the number of equally,.likely outcomes in the sample space for the three-digit combination had decreased from 8 to 4, ,the sample space for the second digit had been maintained. As a consequence, Denise was able to articulate that her chances of winning the game had increased but her chance of picking the second digit had not changed. Consistent with relational level thinkers, Denise was not only able to monitor, within the same context, different sample spaces and their associated probabilities, but she was also, able to integrate and isolate elements of the problem when the need arose. The framework formulated and validated in this study has implications for instruction and assessment. It offers comprehensive and internally consistent benchmarks for conditional probability and independence which teachers can use: (a) to identify key elements and tasks in an instructional program; (b) to provide the initial assessments of student's thinking that will inform instruction; and (c) provide on-going assessment once instruction has begun. Moreover, the framework can be used to establish targets and tasks for state and national assessment. 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Framework for Assessing Thinking in Conditional Probability 59 Romberg, T. A, & Carpenter, T. P. (1986). Research on teaching and learning mathematics: Two disciplines of scientific inquiry. In M. C. Wittrock (Ed.), Handbook of research on teaching (3rd ed., pp. 850-873). New York: Macmillan. Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and directions. In D. A. Grouws (Ed.), Handbook ofresearch on mathematics teaching and learning (pp. 465. 494). New York: Macmillan. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2),4-14. . Watson, J. (1995). Conditional probability: Its place in the mathematics curriculum. Mathematics Teacher, 88, 12-17. Zawojewski, J. S. (1991). Curriculum and evaluation standards for school mathematics, Addenda series, Grades 5-8: Dealing with data and chance. Reston, VA: Author. Authors James E. Tarr, Middle Tennessee State University, Department of Mathematical Sciences, . Email: [email protected] Murfreesboro, Tennessee 37132, USA Graham A Jones, Illinois State University, 4520 Mathematics Department, Normal, Illinois 61790-4520, USA. Email: [email protected]