The validity of connected understanding interpretation of three concept mapping techniques was studied, focusing on the correspondence between mapping-intended task demands, inferred cognitive activities, and scores obtained. The concurrent and retrospective verbalizations of subjects at different levels of competency as they performed the mapping task were studied, and the directedness of the mapping tasks, the characteristics of the verbalizations, and the scores were compared. Subjects were nine chemistry students and two chemistry teachers. The three mapping techniques were: (1) construct a map from scratch; (2) fill in the nodes of a skeleton map; and (3) fill in the linking lines. The three mapping techniques provided different pictures of student knowledge. With high-directed techniques (fill-in-the-map), students' knowledge was close to the maximum criterion, but with a low-directed technique (construct-the-map), students' knowledge was revealed as partial. Low-directed tasks seemed to provide students with more opportunities to reflect their actual conceptual understanding. In addition, the magnitude of the correlation between construct-a-map and fill-in-the-lines scores was much higher than that observed in previous studies. An appendix contains a table of verbal units by subcategory, group, and assessment. (Contains 16 references.) (SLD)

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We investigated discrete-time quantum walks with an arbitary unitary coin. Here we discover that the average position x = max( x ) sin(α + γ), while the initial state is 1/ √ 2(| 0L + i | 0R ). We prove the result and get some symmetry properties of quantum walks with a U(2) coin with | 0L and | 0R as the initial state.

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