RESUMO
A diagnostic judgment of a teacher can be seen as an inference from manifest observable evidence on a student's behavior to his or her latent traits. This can be described by a Bayesian model of inference: The teacher starts from a set of assumptions on the student (hypotheses), with subjective probabilities for each hypothesis (priors). Subsequently, he or she uses observed evidence (students' responses to tasks) and knowledge on conditional probabilities of this evidence (likelihoods) to revise these assumptions. Many systematic deviations from this model (biases, e.g., base-rate neglect, inverse fallacy) are reported in the literature on Bayesian reasoning. In a teacher's situation, the information (hypotheses, priors, likelihoods) is usually not explicitly represented numerically (as in most research on Bayesian reasoning) but only by qualitative estimations in the mind of the teacher. In our study, we ask to which extent individuals (approximately) apply a rational Bayesian strategy or resort to other biased strategies of processing information for their diagnostic judgments. We explicitly pose this question with respect to nonnumerical settings. To investigate this question, we developed a scenario that visually displays all relevant information (hypotheses, priors, likelihoods) in a graphically displayed hypothesis space (called "hypothegon")-without recurring to numerical representations or mathematical procedures. Forty-two preservice teachers were asked to judge the plausibility of different misconceptions of six students based on their responses to decimal comparison tasks (e.g., 3.39 > 3.4). Applying a Bayesian classification procedure, we identified three updating strategies: a Bayesian update strategy (BUS, processing all probabilities), a combined evidence strategy (CES, ignoring the prior probabilities but including all likelihoods), and a single evidence strategy (SES, only using the likelihood of the most probable hypothesis). In study 1, an instruction on the relevance of using all probabilities (priors and likelihoods) only weakly increased the processing of more information. In study 2, we found strong evidence that a visual explication of the prior-likelihood interaction led to an increase in processing the interaction of all relevant information. These results show that the phenomena found in general research on Bayesian reasoning in numerical settings extend to diagnostic judgments in nonnumerical settings.
RESUMO
Productive Failure (PF) facilitates students' conceptual knowledge by delaying instruction until after problem solving. While PF is well investigated in middle and high school students, little is known about its effectiveness in younger students. Studies in younger samples, which implemented delayed instruction designs similar to those used in PF studies, showed mixed results. However, these studies did not implement two core design components of PF: (1) contrasting and comparing student-generated solutions and the canonical solution during the instructional phase (contrasting activity), and (2) students' collaboration in small groups during the initial problem-solving phase. Both components can be expected to contribute to the effectiveness of PF. In a quasi-experimental study with 228 fifth graders, we implemented the first component (contrasting activity) with all students to establish whether under this condition, problem solving prior to instruction would be more effective for younger students' conceptual knowledge acquisition than direct instruction (i.e., problem solving after instruction). Further, we experimentally tested the effect of the second component (collaborative vs. individual problem solving) on students' conceptual knowledge and the number of solution ideas generated during initial problem solving. We found no empirical support for either of our hypotheses. To explore the extent to which students' collaboration actually achieved its potential and relates to students' conceptual knowledge and solution ideas in PF, we conducted analyses of collaborative processes. Our study adds to the mixed results regarding the superiority of problem solving prior to instruction for young students, thus opening the discussion about age-related prerequisites as boundary conditions for PF.
