Spontaneous usage of different shortcuts based on the commutativity principle.

Based on research on expertise a person can be said to possess integrated conceptual knowledge when she/he is able to spontaneously identify task relevant information in order to solve a problem efficiently. Despite the lack of instruction or explicit cueing, the person should be able to recognize which shortcut strategy can be applied--even when the task context differs from the one in which procedural knowledge about the shortcut was originally acquired. For mental arithmetic, first signs of such adaptive flexibility should develop already in primary school. The current study introduces a paper-and-pencil-based as well as an eyetracking-based approach to unobtrusively measure how students spot and apply (known) shortcut options in mental arithmetic. We investigated the development and the relation of the spontaneous use of two strategies derived from the mathematical concept of commutativity. Children from grade 2 to grade 7 and university students solved three-addends addition problems, which are rarely used in class. Some problems allowed the use of either of two commutativity-based shortcut strategies. Results suggest that from grade three onwards both of the shortcuts were used spontaneously and application of one shortcut correlated positively with application of the other. Rate of spontaneous usage was substantial but smaller than in an instructed variant. Eyetracking data suggested similar fixation patterns for spontaneous an instructed shortcut application. The data are consistent with the development of an integrated concept of the mathematical principle so that it can be spontaneously applied in different contexts and strategies.

Hierarchical modeling of contingency-based source monitoring: a test of the probability-matching account.

According to the probability-matching account of source guessing (Spaniol & Bayen, Journal of Experimental Psychology: Learning, Memory, and Cognition 28:631-651, 2002), when people do not remember the source of an item in a source-monitoring task, they match the source-guessing probabilities to the perceived contingencies between sources and item types. In a source-monitoring experiment, half of the items presented by each of two sources were consistent with schematic expectations about this source, whereas the other half of the items were consistent with schematic expectations about the other source. Participants' source schemas were activated either at the time of encoding or just before the source-monitoring test. After test, the participants judged the contingency of the item type and source. Individual parameter estimates of source guessing were obtained via beta-multinomial processing tree modeling (beta-MPT; Smith & Batchelder, Journal of Mathematical Psychology 54:167-183, 2010). We found a significant correlation between the perceived contingency and source guessing, as well as a correlation between the deviation of the guessing bias from the true contingency and source memory when participants did not receive the schema information until retrieval. These findings support the probability-matching account.

Schema bias in source monitoring varies with encoding conditions: support for a probability-matching account.

Two experiments examined reliance on schematic knowledge in source monitoring. Based on a probability-matching account of source guessing, a schema bias will only emerge if participants do not have a representation of the source-item contingency in the study list, or if the perceived contingency is consistent with schematic expectations. Thus, the account predicts that encoding conditions that affect contingency detection also affect schema bias. In Experiment 1, the schema bias commonly found when schematic information about the sources is not provided before encoding was diminished by an intentional source-memory instruction. In Experiment 2, the depth of processing of schema-consistent and schema-inconsistent source-item pairings was manipulated. Participants consequently overestimated the occurrence of the pairing type they processed in a deep manner, and their source guessing reflected this biased contingency perception. Results support the probability-matching account of source guessing.