Laurie H. Rubel

Brooklyn College of the City University of New York

Journal of Statistics Education Volume 15, Number 2 (2007), http://jse.amstat.org/v15n2/rubel.html

Copyright © 2007 by Laurie H. Rubel all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.

**Keywords:**availability heuristic, combinatorial thinking, middle school, high
school

Teams Item: Now consider a group of 10 Dutchmen (authors’s note: the Dutchman is the mascot of the school at which the study was undertaken) who want to form a 2-person team. Also consider a group of 10 Dutchmen who want to form an 8-person team.

- There are more 2-person teams
- There are more 8-person teams
- There are the same number of 2-person and 8-person teams

The number of 2-person teams in the above item is equal to the number of 8-person teams. However, on similarly constructed tasks, adults typically indicate that there are more 2-person teams (Shaughnessy, 1981; Tversky and Kahneman, 1973). Their reasoning for this response can be explained by the “availability heuristic,” by which one estimates the probability of an event according to the ease with which instances of the event can be conceived (Tversky and Kahneman, 1973). Two-person teams are considered more available than eight-person teams for two reasons. First of all, ten people can be split into five distinct two-person teams at one time, whereas ten people can form only one eight-person team at a time. Secondly, any specific eight-person team has a greater overlap with the other eight-person teams. Median values of college students’ estimations of the number of distinct teams of size n out of a total of ten people were found to decrease with larger values of n (Tversky and Kahneman, 1973). Specifically, the median value of the subjects’ estimations of the number of 2-person teams was greater than sixty, while the median value of the estimations of the number of 8-person teams was less than twenty.

More recently, Fischbein and Schnarch (1997) posed a question similar to the Teams Item to a convenience sample of undergraduate students and students in 5th, 7th, 9th and 11th grades, and found, surprisingly, that higher proportions of the older students responded that there are more two-person teams. In their study, 10% of 5th graders, 20% of 7th graders, 65% of 9th graders, 85% of 11th graders, and 72% of undergraduate education students indicated that there are more two-person teams. Fischbein and Schnarch explained this surprising result in terms of the development of the cognitive schema of combinatorial reasoning. In other words, since older students possess more developed combinatorial reasoning skills, they can better conceptualize the number of combinations of two objects out of ten, which leads them toward the availability heuristic. At the same time, though, 55% of 5th graders and 40% of 7th graders in Fischbein and Schnarch’s study did not respond to this particular item, in contrast with the nearly complete response rates of the older students. The goal of this article is to further investigate Fischbein and Schnarch’s results by presenting the Teams Item to a new sample of students and by including an analysis of students’ justifications of their responses.

For example, consider the comparison of two jars of marbles, one containing six black marbles out of a total of eight marbles, and the other containing three black marbles out of a total of four. When school students say that it is more likely to pick a black marble out of the first jar, a typical justification is that there are more black marbles in that jar. In other words, they have compared the two jars in terms of one quality, the number of black marbles. They then use the More A More B intuitive rule to reason that since the first jar has more black marbles, it will also have a greater likelihood to yield a black marble.

The Teams Item is an additional example of a comparative task, in which information is provided about one quality of the two objects, the size of each team. If a student indicates erroneously that there will be more eight-person teams, this could be interpreted as a result of the More A More B scheme. The eight-person teams have more members, so seemingly there ought to be more of them.

If the two objects or systems are equal in terms of one quality A, students often indicate that they will be equal in terms of a second quality B, which Tirosh and Stavy term Same A Same B. For example, when comparing the likelihood that a two-child family is comprised of a son and a daughter with the likelihood that a four-child family is comprised of two sons and two daughters, a typical response is that these events are equally likely since the ratios of the number of boys to the number of girls in each family are equal. In other words, since the two families are the same in terms of one quality, the ratio of girls to boys, they should seemingly be the same in terms of a second quality, their likelihoods. In the case of the Teams item, one could reason that since we are interested in forming two-person teams out of a group of ten people, and eight-person teams out of a group of ten people, the constant ten people in each scenario is an implication that there will be the same number of 2-person and 8-person teams.

Familiar to the students as a teacher at this school, I visited each of the twelve represented mathematics classes and delivered an informal presentation about their participation this particular study. I gave students an opportunity to ask questions about the study and they could opt out of participation at any time. The students completed the Probability Inventory, a written questionnaire, during their regular mathematics class period. Each item on the Probability Inventory prompted students for an explanation or justification of the given answer. The Task Item, as stated at the beginning of this article, was one of the ten items. I categorized the responses to each item on the Probability Inventory in two phases, first according to students’ answers, and then in terms of justification type. The age levels of this sample match those of Fischbein and Schnarch’s sample, as does the written format of the task. In contrast with Fischbein and Schnarch’s study, in this case, students were also asked to explain their answer. 33 students were selected to participate in clinical interviews using the tasks from the Probability Inventory. Data from these interviews are used in this article to instantiate or further clarify justification categories.

Grade 5 (n=36) | Grade 7 (n=45) |
Grade 9 (n=50) | Grade 11 (n=42) | Total (n=173) | |

More 2- person teams | 61% (22) | 60% (27) |
58% (29) | 31% (13) | 53% (91) |

More 8- person teams | 8% (3) | 16% (7) |
16% (8) | 36% (15) | 19% (33) |

Same- number of 2- person and 8-person teams (correct answer) | 17% (6) | 22% (10) |
18% (9) | 19% (8) | 19% (33) |

Other or no- answer | 14% (5) | 2% (1) |
8% (4) | 14% (6) | 9% (16) |

Table 1 contains the distribution of responses to the Teams item. About 19% of the students at each grade level gave the correct answer. About half of all of the students answered that there are more 2-person teams, and another 19% answered that there are more 8-person teams. While we might expect that the correct response rate would be higher among the older students, the correct response rates remain roughly stable across the grade levels. However, if we categorize students’ responses according to their justifications, a different picture emerges, as explained below.

Table 2 further specifies the results by including students’ method of justification along with their response type. In this section, I present student reasoning leading to the “more 2-person teams” response, continue with student reasoning for the “more 8-person teams” response, and conclude with student reasoning for the response “same number of teams.”

Grade 5 (n=36) | Grade 7 (n=45) |
Grade 9 (n=50) | Grade 11 (n=42) | Total (n=173) | |

MORE 2-PERSON TEAMS | 61% (22) | 60% (27) |
58% (29) | 31% (13) | 53% (91) |

Availability | 50% (11) | 37% (10) |
41% (12) | 62% (8) | 45% (41) |

Partition interpretation | 36% (8) | 52% (14) |
48% (14) | 23% (3) | 43% (39) |

Other or no justification | 14% (3) | 11% (3) |
10% (3) | 15% (2) | 12% (11) |

MORE 8-PERSON TEAMS: More A More B | 8% (3) | 16% (7) |
16% (8) | 36% (15) | 19% (33) |

SAME NUMBER- of 2-PERSON and 8-PERSON TEAMS (correct answer) (correct answer) | 17% (6) | 22% (10) |
18% (9) | 19% (8) | 19% (33) |

Counting | 17% (1) | 0% (0) |
0% (0) | 25% (2) | |

Inclusion/exclusion | 17% (1) | 10% (1) |
44% (4) | 75% (6) | |

Same A Same B | 34% (2) | 50% (5) |
33% (3) | 0% (0) | |

Other justification | 34% (2) | 40% (4) |
22% (2) | 0% (0) | |

Other or no- answer | 14% (5) | 2% (1) |
8% (4) | 14% (6) | 9% (16) |

Availability (41 of 91 students). Students using this justification indicated that since two is a smaller number than eight, there are more possible combinations. For example, as 11th grader indicated, “There are less 8-person teams. It’s obvious…2 is a lot less than 8 out of 10. There are a lot more combinations.” Seven of these students counted approximately forty-five possible two-person teams, and then said that this must be more than the number of eight-person teams.

Alternate interpretation of the task: partitioning (39 of 91 students). To explain this form of
student reasoning, let us consider a new question: Suppose there are ten people in a room. Scenario
A requires that these people divide themselves, at one time, into two-person teams. This scenario
is a partition interpretation of the Teams item, and is fundamentally different from counting the
number of unique ways to choose two people out of ten. The number of ways to complete Scenario A
is given by 10!/(2!)^{5}, or 113,400. Scenario B, on the other hand, requires that these
people divide themselves into eight-person teams. Since one cannot partition ten people strictly
into groups of size eight, we are effectively counting the number of ways to choose eight people
out of ten, which is forty-five. So, we see, using a partition interpretation, the number of
two-person teams is greater than the number of eight-person teams. While none of the students gave
the complete argument outlined above, some students wrote that ten people could be split into five
two-person teams as opposed to only one eight-person team, indicating a partition interpretation of
the task. For example, five 5th graders responded that there are more 2-person teams and justified
this answer by writing “5>1.” An eleventh grader’s response further explains this reasoning: “In
10 people one can create 5 teams made up of two people and in the same group only one 8-person team
can be made.”

Counting (3 of 33 students). One 5th grader and two 11th graders counted the number of combinations of 2-person teams and the number of 8-person teams.

Inclusion/exclusion (12 of 33 students). One 5th grader, one 7th grader, four 9th graders, and six 11th graders reasoned that including two people on a team is equivalent to excluding the other eight people from the team. For example, a 5th grader wrote, “For two-person teams, there will be eight people who aren’t. 1-8 | 9-10. You don’t know which is the team.” Similarly, a 9th grader wrote, “Choosing a 2-person team is the same as choosing eight people to not put on the team.”

“Same A Same B” (10 of 33 students). Two 5th grader, five 7th graders, and three 9th graders answered the question correctly but explained their answer using Same A Same B reasoning (Tirosh and Stavy, 1999). In other words, the same number of people in total makes the same number of teams, whatever their size. For example, a 5th grader wrote, “because the same amount of people do the same amount of teams.” A 7th grader wrote, “You have the same number of people for both.” The Same A Same B leads to a correct answer with the parameters of the Teams Item as stated. However, this reasoning would lead to an incorrect answer if comparing the number of possible 2-person and 3-person teams from the pool of ten people.

The task was difficult for most of the students in this sample, as only 33 of the 173 students answered this question correctly, and even fewer students provided a correct justification to that response. More of the older students used the mathematically precise inclusion/exclusion approach, while more of the younger students used the intuitive Same A Same B approach. About half of the students responded that there are more two-person teams, as the availability heuristic would dictate. However, fewer than half of those students actually justified their answer using availability reasoning. This calls into question the widespread acceptance of the availability heuristic as the dominant solution strategy to this task. Almost as many students interpreted this question as a partition question, in which case there are many more two-person teams. The partition interpretation of the task seems to be a new finding and warrants attention in future research.

This study also has instructional implications that extend from its methodology. Students were asked to answer a question and to justify their answer. This enabled the analysis to include descriptions of student reasoning to mathematically correct as well as incorrect responses. As a result, insights were gained, both into the types of errors students make, as well as the methods students use to arrive at correct answers. Statistics educators can benefit from a better understanding of students’ reasoning on this specific task, but more broadly, statistics educators could use this, or a similarly constructed, item and its categorization of responses as a formative assessment tool. The actual process of asking for justifications and then paying attention to students’ reasoning leading to correct and incorrect answers has significance for classroom assessment as well as future statistics education research.

Fischbein, E. and Schnarch, D. (1997). “The Evolution with Age of Probabilistic, Intuitively Based
Misconceptions,” *Journal for Research in Mathematics Education* 28, 96-105.

Jones, G. A. (Ed.) (2005). Exploring Probability in School: Challenges for Teaching and Learning. New York: Springer Verlag.

Rubel, L.H. (2002). “Probabilistic Misconceptions: Middle and High School Students’ Mechanisms for Judgments under Uncertainty.” Unpublished PhD dissertation. Teachers College, Columbia University.

Shaughnessy, J.M. (1981). “Misconceptions of Probability: From Systematic Errors to Systematic
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* Reston, Virginia: NCTM.

Shaughnessy, J.M. (1992). “Research in Probability and Statistics: Reflections and Directions.” In
D. Grouws (Ed.), *Handbook of Research on MathematicsTeaching and Learning.* Reston, VA: National
Council of Teachers of Mathematics.

Tirosh, D. and R. Stavy. (1999). “Intuitive Rules: A Way to Explain and Predict Students’
Reasoning,” *Educational Studies in Mathematics* 38: 51-66.

Tirosh, D. and R. Stavy. (2000). *How Students (Mis-) understand Science and Mathematics*,
New York: Teachers College Press.

Tversky, A. and Kahneman, D. (1973). “Availability: A Heuristic for Judging Frequency and
Probability,” *Cognitive Psychology* 5, 207-232.

Laurie H. Rubel

School of Education

Brooklyn College of the City University of New York

2900 Bedford Avenue

Brooklyn, NY 11210

U.S.A.
*LRubel@brooklyn.cuny.edu*

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