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Numbo: a study in cognition and recognition

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... Despite the fact that games based on four-operations are very common, the limited number of scientific studies have been conducted in this regard. Defays [2], [3] used artificial intelligence search methods to solve these games. Hutton [4] developed a simple but ineffective functional program for solving four-operations problems. ...
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Combination problems are one of the most important issues of probability theory. The four-operations combination problem underlies the basis of some competition programs broadcasted in many national channels. In these competition programs, the competitors are expected to reach the target number by using six numbers and four basic arithmetic operators. The numbers are used at most once, the operators can be used any desired number to reach the target number. In this problem, all four-operations combinations include the operation blocks consisting of two numbers and an operator. Therefore, the fouroperations combination problem is solved by developing a "Type-2 Tree Structure" which is a new approach to accurately model the operation blocks. The performance of the proposed method for the four-operations combination problem is examined by a simulation study. Also, the statistics from experimental results are given in this study.
... Other relevant models include those devised by Lenat and Brown (1984) and by Fajtlowicz (1988Fajtlowicz ( , 1989 to model mathematicians' conjecturing and those that Siegler and his colleagues (Siegler & Jenkins, 1989) are developing with regard to children's strategy choice in arithmetic. Defays' (1988Defays' ( , 1990 computer program Numbo is probably the most relevant to the present study. It plays a game involving the construction of a given number (the "target") from a given set of five other numbers (the "bricks"), using the arithmetical operations of addition, multiplication, and subtraction. ...
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Selected groups of 44 academic ''pure'' mathematicians, 44 accountants, 44 psychology students, and 44 English students were given Levine's (1982) computational estimation task, which involved mentally estimating the products and quotients of 20 multiplication and division problems and describing their strategies. The mathematicians were the most accurate estimators, and the English students the least accurate, with psychology students and accountants obtaining similar scores intermediate between the other groups. All groups demonstrated an impressively versatile use of appropriate strategies. The mathematicians and accountants used significantly larger numbers of appropriate strategies than the other groups and strongly resembled one another in this respect, despite the significantly greater accuracy of the mathematicians. All the non-mathematician groups used significantly larger numbers of inappropriate strategies than did the mathematicians. We discuss (1) the implications for cognitive psychology of the great variability of strategy use in an apparently simple task; and (2) the relationship between people's mathematical knowledge and experience and their estimation accuracy and strategy variability.
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