Modeling the dynamics of classroom education using teaching games

Modeling the dynamics of classroom education using teaching games Michael C. Frank mcfrank@stanford.edu Department of Psychology Stanford University Abstract The parallel between teaching and communication emerges in recent formal work using probabilistic models. Shafto and Goodman (2008) described a model of teaching a single student, in which teacher and learner reason recur- sively about one another’s intentions. In that model, the teacher’s goal is to choose an example or set of examples to allow a student to acquire a concept (say, a rectangle in the coordinate plane). The student then reasons about the examples that the teacher has chosen, given that the teacher has chosen them with the student in mind (e.g. in order to maximize what is learned). This recursive form—teacher chooses examples to maximize student learning, student con- siders teacher’s choice—leads to much stronger inferences than those possible in scenarios where the teacher does not reason about the student and vice versa (Shafto, Goodman, & Frank, 2012). Though some models of this type employ very deep recursion—learners reasoning about speakers, who in turn reason about learners, on down until until asymptotic con- vergence (J¨ager, 2010)—greater depth does not necessarily provide a good fit to human behavior. In a recent model of pragmatic communication, Frank and Goodman (2012) de- veloped a version of this framework that described speakers as choosing their words to convey maximal information to a naive listener (one who interprets words literally). This two- level framing nevertheless led to inferences beyond the lit- eral. The spirit of that framework guides the current work. Teachers are modeled as speakers who consider the beliefs of a group of students when choosing what example to present, but students are not assumed to reason about the teachers’ motivation (though adding such recursion could be of interest in future investigations). The fundamental unit of the analysis here is a teaching game. A teacher attempts to guide a group of students to discover an extremely simple concept. For the concept in this paper, we choose the weight of a biased coin (the parameter of a Bernoulli distribution). Though a single coin weight is an extremely simple case study, many interesting dynamics become clearer because of this simplicity. In a more complex scenario, the dynamics of optimal teaching can be arbitrarily complex, and patterns are much harder to discern. I return to this issue in the general discussion. The challenge for teachers is that they can only convey the target concept by the use of examples. What examples should a teacher choose in order to convey a particular coin weight? If students are perfectly unbiased, teachers’ examples should follow the overall distribution of the coin. So for a fair coin they should provide e.g., 0, 1, 0, 1. But what if they know that What makes a good teacher, and how should we structure classrooms to promote effective teaching? This paper inves- tigates the idea that a good teacher is a good communicator, using models of optimal pragmatic communication to explore the dynamics of classroom learning. The proposed model de- scribes teaching as choosing examples of a concept to present to students in order to maximize their information gain. Under this model, the key challenge for the teacher is communicat- ing to a heterogenous audience. A number of results emerge naturally, including decreases in performance with increases in class size and increases in performance based on tailoring instruction to groups of students based on prior knowledge or ability. Keywords: Teaching; education; communication; pragmatics; Bayesian modeling. Introduction Many questions about classroom education are both foun- dational and highly controversial. For example, do larger classes produce worse learning outcomes for students (Glass & Smith, 1979; Slavin, 1989)? Does separating students by ability produce better learning (Slavin, 1987; Tomlinson, 1999)? Is it better to spend class time testing students’ initial performance in order to customize curriculum, or is it better to standardize materials (Fuchs & Fuchs, 1986)? While there is a vast empirical literature that attempts to answer these questions (and many others), relatively little extant work fo- cuses on providing quantitative theory that would provide a priori predictions about these questions. The goal of the cur- rent work is to take a first step in this direction. My guiding inspiration comes from considering what qual- ities make a good teacher. Intuitively, a good teacher is a good communicator, choosing explanations and examples that allow students to learn effectively. The model I describe here explores this parallel between teaching and communi- cation, examining how variability in both the size and the knowledgeability of teachers’ audience affects communica- tion strategies. This paper presents ideal observer analysis, similar to those used in perception, where models provide a baseline from which to assess human performance (Geisler, 2003; Frank, 2013). A model of classroom education could provide a tool for reasoning about the causal factors involved in student out- comes. Further down the road, to the extent its predictions match human behavior they could be a guide for future re- search or even—in combination with a model of the cost of interventions—to promote effective decision-making. And to the extent that its predictions mismatch data, this mismatch would be a prompt for revision of the model.