Counterexample Search in Diagram-Based Geometric Reasoning

Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object-wise, could underlie diagram-based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions.

Understanding Strategy Change: Contextual, Individual, and Metacognitive Factors

Learning, development, and response to instruction often involve changes in the strategies that learners use to solve problems. In this chapter, our focus is on mathematical problem solving in both children and adults. We offer a selective review of research on three classes of factors that may influence processes of strategy change in mathematical problem solving: contextual factors, individual factors, and metacognitive factors. Contextual factors involve information that learners encounter in the learning context, such as feedback about prior strategies and examples of alternative strategies. Individual factors involve the abilities, dispositions, and knowledge that learners bring to the learning context. Metacognitive factors involve knowledge about strategies and factors that affect the application of strategies-including perceptions of problem difficulty, confidence in the strategies one already knows, and judgments about the qualities of alternative strategies. These factors operate both independently and in combination to influence learners' behavior. Therefore, we argue that scientific progress in understanding strategy change will require comprehensive conceptual models that specify how different factors come together to explain behavior. We discuss several such models, including vulnerability-trigger models, cumulative risk models, and dynamic systems models. Research guided by such models will contribute to greater progress in understanding processes of strategy use and strategy change.

The Simulation of Situation Models Aids Analogical Transfer Between Algebra Problems

Several studies on analogical transfer to algebra word problems have demonstrated that adapting solutions learned from worked examples to nonisomorphic problems of the same type is challenging and that most instructional aids do not alleviate this difficulty. At the same time, various authors have suggested that transfer difficulties sometimes originate in students’ lack of disposition to relate algebraic formulas to the real-world situations to which they refer. We designed a noninteractive intervention encouraging students to elaborate situation models for base and target problems, and to ground algebraic formalisms in these representations. One experimental group simulated situation models by physical object manipulation, whereas another experimental group performed those simulations mentally. Both conditions outperformed a control group that did not run simulations. This intervention was more effective when the transformations posed by target problems were intrinsically more difficult to assimilate into the learned equation. Implications for the design of instructional interventions are discussed.