Figure 1: Rosa and Maria reason from their data
Noel Enyedy, University of California, Berkeley
This section examines ways in which activities can be designed to prompt and support epistemic discourse. In particular, it looks at the potential for discussions in which students evaluate competing ways of reasoning about probability. The potential value for these types of discussions is illustrated by an example where the lack of consideration of conflicting epistemologies may have influenced and in some cases hindered students' conceptual development.
There are three well-defined ways of reasoning about probability and uncertainty. First, there is Subjective probability, which deals with the degree of belief or confidence in an uncertain statement or event. For example, one might say there is a greater probability that Los Angeles will get a football expansion team than Monterey. In this case one is expressing their respective degree of confidence in a one-time event. Second, there is Frequentist probability, which assigns probabilities based on the long-term behavior of random events. For example, one might calculate the probability of rolling a seven on two dice by rolling the dice, recording the results, and extrapolating from this empirical data. Third, there is Classicist probability, which assigns probabilities based on the quantification of all the equally likely possible outcomes that can happen (i.e., the outcome space). In this case the probability of rolling a seven on two dice is calculated by enumerating the thirty-six possible equally likely outcomes for two dice and counting the number of ways you can roll a seven (i.e., 6 out of 36).
Each of these ways of probabilistic reasoning has a long history and strikingly different epistemological commitments. Subjectivist probability considers knowledge itself to be uncertain. Frequentist probability grows out of the epistemological tradition of empiricism, which argued that we don't really know things, but we only know about things. From this perspective perception plays a critical role in how we come to know things. Knowledge is built through induction and hypothesis testing. Classicist probability, on the other hand, emerged from the rationalist tradition, which argued that knowledge is a rediscovery of universal truths. From this perspective, knowledge is constructed deductively from a priori truths and not one's empirically verifiable perceptions.
There is evidence that students, particularly young students, intuitively reason about probability from the Subjectivist perspective . From this perspective students see each trial of an experiment, not as one of many possible outcomes, but as the single outcome of the experiment .
However, from the perspective of some mathematicians as well as the mathematical standards for K-12 mathematics education the subjectivist perspective is a misuse use of the mathematical register because the term probability is restricted to refer to the long term trends and not single events. The national mathematical framework calls for students in the sixth through eighth grade to both, "develop and evaluate inferences, prediction, and arguments that are based on data" and also to "compute simple probabilities using appropriate methods, such as, lists, tree diagrams or area models," . That is, the NCTM framework calls for students to be able to reason both from the Frequentist and Classicist point of view.
The Probability Inquiry Environment (PIE) was designed as a three week probability curriculum for seventh grade students that marries these two perspectives. The PIE curriculum consists of a number of computer-based and hands-on collaborative inquiry activities that are designed to help students build from their existing understanding of probability. In each PIE activity students were asked to determine if a game of chance was "fair". PIE provided a simulation of each game and a set of tools to help the students analyze the empirical data they generated. From the students' growing understanding of probability based on these long-term trends of their empirical data (i.e., Frequentist probability), PIE builds towards a way of reasoning about probability based on the more abstract concept of the outcome space (i.e., Classicist Probability).
In PIE both the Frequentist and Classicist ways of reasoning are anchored by representations. Frequentist reasoning practices require representations of aggregated data such as bar charts, fractions, and percentages. This is necessary because it is difficult to perceive and reliably reason about long term trends without quantifying and representing the data in some manner first. Classicist reasoning practices, on the other hand, require representations of the outcome space such as ordered lists, tables, probability trees and area models. Like empirical data, to reason successfully from the outcome space it must first be represented. PIE was successful in helping students develop ways of reasoning around both types of conceptual representations. But for some students there was an unanticipated resistance to move from Frequentist to Classicist ways of reasoning that may be rooted not in their conceptual understanding but their epistemological beliefs.
Figure 1 shows two girls using their empirical data to inductively determine that the game they are playing is fair. In their talk they reference both that the number of points the two teams are scoring is close and that the two teams are winning about the same number of games. Figure 2, on the other hand, shows a student, in interaction with the teacher, coming to the same inference about the same game deductively, based on the idea that both teams have the same number of opportunities to score.
Figure 1: Rosa and Maria reason from their data
These two ways of reasoning are then juxtaposed in a whole class discussion. In this debate many students begin to adopt the classicist perspective of deducing fairness based on the quantification of the outcome space. However, some students explicitly reject that it is a valid way of reasoning saying, "I wouldn't really know if its fair, but I wouldn't know if it is unfair either. It would be hard to tell until I played it." While by the end of the PIE curriculum most students had appropriated the Classicist perspective, a minority of students still held that they could not make predictions about future random events without first collecting empirical data first.
It is possible to speculate from this example that some of the difficulty that students have in successfully reasoning about probability is epistemological and not necessarily conceptual or development, as has been assumed . If this is true, then an essential part of the students' learning trajectories is to critically examine these different reasoning practices. Comparing and evaluating these two reasoning practices, each anchored by a different set of conceptual representations, would help bring the students' own epistemologies to the forefront of the discussion.
Unfortunately, because we had not anticipated this particular difficulty we did not design activities that juxtaposed the two perspectives to encourage and support the debate. However, it seems that a useful design principle can be learned from our experience in the classroom. That is, it is useful to consider different ways of representing (and labeling) distinct forms of knowledge or reasoning practices so that they can be visually contrasted. This comparison of ways of reasoning and the types of conceptual representations that anchor these reasoning practices would allow students to engage in a debate about the respective merits of each. This in turn would help the students to address their own epistemological perspective and how it is related to different forms of mathematical reasoning.