Effects of teaching statistical laws on reasoning about problems

farm and can apply these rules to ordinary events. It seems that learner's errors in statistical reasoning are better understood as an inability to code events in terms that make cantact with statistical rules rathex than as ignorance of the statistical rules themselves. In fact, Holland et al. state that the strongest evidence for the highly abstract existence of these rules is "the fact that the teaching of statistical rules in an abstract way, as is normally done in statistics courses, for example, has an effect on the way people reason about problems in everyday life" (p. 256). Implications for Instruction The theory about inferential rules outlined in this chapter suggests implications for injecting new rules into human informaton processing systems in school settings. Essentially, the type of inferential rule Holland et d. call Law of Large Numbers Heuristics is part of the formal array of scientific reasoning tools employed by statisticians and scientists. Chapter 1 advanced a normative premise fbr teaching reasoning and improving human intellectual competence through principles and tools fbr thinking arising from the scientific tradition of inquiry. The formal array of scientific reasoning tools is embedded in that scientific tradition and, in large measure, is already part of school curricula. However, the present view that such tools may also include inferential rules that are relatively independent of specific domains of knowledge and are applicable in everyday contexts is not a commonly held view in school settings. The promise of teaching abstract inferential rules to improve problem solving is highlighted by Holland et d. as follows: We have spoken about systems as if they had to go it alone, learning everything by the sweat of their brows or the hum of their central prmescm. Fortunately, that is not the case. New rules can be suggested to systems by e x t d agents such as teachers and programmers . The most hportant thing to say about inserted rules is that they will tend to act in some impartant aspects like induced rules. That is to say, they will enter into competition for the right to represent the environment along with the rest of the rules in the system's possession. (p. 96). Since statistical reasoning involves abstract, domain-independent inferential rules, it may be possible to improve students' abilities to reason about a wide range of events by formally teaching the abstract statistical rules themselves. This is, in some ways, made easier because students may already possess intuitive versions of some statistical rules that they have induced by reasoning about problems where uncertainty is prevalent. They may have acquired the means for encoding infbrmation in problems in terms of statistical rules. However, learned abilities to encode and apply such abstract rules vary. Application can be improved by instruction in encoding the information contained in problems in appropriate ways so that it makes contact with the rules. There are two main issues to be considered in planning instruction. First, newlyintroduced rules compete with old rules. This issue, highly relevant to instuction, is often missed by teachers who teach as if students were tabJa ram. According to Holland et al. : education should not be thought of as replacing the rules that people use for understandine the world but mthex as introducing new rules that enter into competition wit.the old ones. People reliably &st& the new rules in the diredion of the old ones, or igncne them altogether except in the highly specific domains in which they were taught. @. 206) Therefore, it is not appropriate to assume that any learner's mind is a blank slate. Instructional techniques must take students' existing beliefs (Clement, 1982) and prior knowledge, including related heuristics, into accowlt. For example, students commonly learn one set of rules for school science and another for the real world (West & Pines, 1985). Second, the newly acquited rules will not compete successfully with older induced rules if the application requires encoding that the system has not yet mastered. Knowledge about how students of various ages and capabilities en& statistical infarmaton in problems from various domains is useful information for instructional planning. Nisbett et al. (1983) have argued that people call on statistical heuristics in adQess'i everyday problems according to (a) the degree to which features of a problem can be enaxled in terms of statistical rules (i.e., when sample space and sampling processes are clear, when the events can be coded in common units); (b) the presence of chance factors or random variation in the problem; and (c) the presence of events in the problem that are "culturally" associated with random variations (e.g., gambling). Based on this analysis, three problem "domains" of statistical reasoning were identified by Fong et al.: (a) Probabilistic problems are those in which random variation in the sample data was made clear in the problem. (b) Objective problems are those that desaibe sample data in some objective metric (e.g., pounds as opposed to "W"), but there is no explicit cue about randomness of the sample. And, (c) Subjective problems are those in which data describing a sample is not reliably measurable (e.g., 'funny"). Fong a al. argue that these three are domains of everyday problems where statistical heuristics apply. The two instructionally significant issues-newly-inbroduced rules compete with old rules and newly acquired rules will not compete successfully with older induced rules if the application requites encoding that the system has not yet maskd-need to be considered in teaching abstract inferential rules such as statistical heuristics. If students readily use statistical rules only in a domain where they can easily encode the variability of events such as the Probabilistic domain, they need to be shown how to e n d events in terms that make contact with the rules when events are not so easily encoded. This might be done with examples from the Objective and Subjzziw domains. Second, if students have strong prim intuitions-pe-established, erroneous statistical or other heuristics that they apply to problems where uncertainty is prevalent-Mcmal a y h e s that incorporate demOIlSttations may overcome and defeat those prior intuitions (see Siegler, 1983; McCloskey & Kaiser, 1984). Holland et d. are specific in their suggestions for the types of demonstrations that teachers should undertake to defeat ppeexisting ermrs in reasoning: What is needed is demonstrations where students are required to make predictions about the likely outcomes of events, then to articulate the bases of their predictions, then to explore the possibility that other rules might exist that would do a better job of prediction. (p. 228) Teachers' knowledge of the other, competing heuristics or typical reasoning errors and intuitions is critical. In addition teachers need to become informed about common "default conditions" (Peakins et d) that are often triggered in the cognitions of makes-sense epistemologists who encounter problem solving activities. Knowledge of students' existing rules and misconceptions is encompassed in a recent model of teaching called the cognitivemediationa1 model where ssudent cognitions are seen as the cause for learning. Winne and Marx (1987) refer to two models of teaching tbat carrespond to process-product and cognitive-mediatima1 paradigmi The first of these is the perfcnmanc6based model of teaching effeictiveness. It was predicated on the view that a simple cause-andeffect chain exists between what teachers do and how well students learn. It meant that teaching could be analyzed as a set of discrete skills or behavioun Specific skiUs/behavioilts used by teachers would increase student learning, and a linking together of skills/behaviours would result in effective teaching. The authors argue that the most critical problem with the Pepfarmanwbased model is that "it does not include a formal description of how students actually leam from teaching" (Winne & Manr, 1987, p. 269). The missing causal link between teaching and learning is addressed by the second of the two models, the cognitive mediational model. It adds two dimensions to the perf~nnance-based model. One is a cognitive dimension focusing on the kinds of cognitions which students use in trying to learn. The other is a mediational dimension focusing on the need for students to inteapret teacher behavim in the ambiguous context of classrooms. The cognitive-mediational model shifts the cause-effect relationship for learning from teacher behaviours as causes to students' cognitim as causes. It follows that knowing how learners reason about problems becomes prerequisite understanding far improving such reasoning. Studies of Statistical Reasoning The following eight studies represent the corpus of research on how the inferential rule called the law of large numbers heuristics is applied in everyday problem solving. To investigate whether people normally reason more statistically about objective, easyta-code events than about subjective, hard-to-de events, Jepson et d(1983) tested subjects with two types of problems. In the fitst type of problem, the behaviour to be considered by the problem solver was an achievement or some other outcome, such as number of friends, that could be assessed by objective means. It was hypothesized that the objective nature of the information in the problem would allow easier identification of statistical propeaties such as variability and randomness. In the second type of problem, the behaviour to be consided was only subjectively assessable, for example, friendliness. Subjects were observed to be much more likely to use the law of large numbers and the repsion principle with the first type of problem. The investigators concluded that inductive reasoning will be guided by statistical inferential rules when peop