How we use what we learn in Math: An integrative account of the development of commutativity.

One of the crucial issues in mathematics development is how children acquire mathematical concepts and procedures. Most researchers now agree that this knowledge develops iteratively (e.g., Resnick, 1992). However, little is known about how well this knowledge is integrated into a more abstract concept and how children come to spontaneously apply such concepts. Expertise research suggests that spontaneously spot and use a principle whenever it applies requires well-integrated conceptual and procedural knowledge. Here, we report a method allowing to asses procedural and conceptual knowledge about the commutative principle in an unobtrusive manner. In two different tasks, procedural and conceptual knowledge of second and third graders as well as adult students were assessed independently and without any hint concerning commutativity. Results show that, even though second graders according to our measures already possessed procedural and conceptual knowledge about commutativity, the knowledge assessed in these two tasks was unrelated. An integrated relation between the two measures first emerged with some of the third graders and was further strengthened for adult students

Gedächtnispsychologische Untersuchungen eines Rechenkünstlers

Investigations of a mental calculation expert's memory. Summary. In this article investigations of a mental calculation expert are described, which are concerned with various components of his working memory as well as with long-term remembering of numbers. The most important results are: \ud 1) Not the frequency of memorizing, but the quality of processing determines long-term remembering. \ud 2) The expert either does not instruct himself to forget or does not utilize cues to forget. In spite of that there was no evidence for proactive inhibition. \ud 3) Retroactive inhibition was shown when active processing was impossible. \ud 4) In Sternberg's paradigm the expert processes more digits than consonants per sec. Only with respect to digits the expert processes faster than other subjects. \ud 5) The memory span for digits is large, for letters average. The digit span is not determined by a limited number of chunks but by the number of digits which can be processed within a limited space of time. This time seems to be larger than the one of other subjects. \ud 6) The symbolic distance effect for digits and letters was shown to be valid for all subjects. Only the expert's comparison of numbers is less time-consuming than the comparison of other subjects. \ud The results are discussed with respect to his computational abilities which he has demonstrated independent of these investigations (Bredenkamp, 1989)

The role of the proactive interference in mnemonic techniques

The success of many mnemonic techniques, such as the method of loci, is based on the use of specific well-known anchors, which are mentally combined with to-be-learned items and subsequently facilitate their retrieval. In our studies we intended to answer the question of whether the repeated application of the method of loci may result in proactive interference effects, as might be expected due to the applied association of items with the same loci each time the method is used. To this end, we manipulated list similarity in a typical proactive interference design and compared the method of loci with the link method and the rehearsal method, which do not involve the use of a specified set of anchors. Our results replicate those from other studies, which have shown that the use of a mnemonic technique leads to superior recall of list items compared to a simple rehearsal strategy. We were further able to show that the repeated learning of items from different categories results in moderate practice effects over three list-learning trials, whereas this effect is superimposed by an effect of proactive interference if different lists are composed of items from the same category. However, this effect of proactive interference was not increased for the method of loci, and we discuss this finding with regard to its practical implications

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

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