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In this paper, the multidigit arithmetic-related strategy adaptivity, strategy flexibility and solution accuracy of Danish compulsory school students is examined. Participants, 749 grade three, 731 grade six and 818 grade eight, were drawn from twenty demographically different schools. Drawing on a tri-phase assessment tool, each student completed a series of tasks designed to elicit shortcut strategies. First, students solved each task by means of their preferred strategy; those using shortcut strategies were construed as adaptive for that task. Second, students solved the same tasks by means of whatever alternative strategies they had available; those offering at least two strategies were construed as flexible for that task. Third, for each task, students were asked to indicate which of their strategies they believed was optimal. Across all grades, students were more flexible than adaptive. Overall, sixth graders exhibited higher levels of flexibility than third graders and marginally lower levels than eighth graders. Sixth graders exhibited higher levels of adaptivity than those in either grade three or grade eight. Students’ accuracy, which improved with maturation, was influenced positively by both adaptivity and flexibility, with flexibility having the greatest influence in grade three and adaptivity in grade six. The findings raise further questions concerning, inter alia, culture’s influence on students’ strategy choices and the interaction of adaptivity, flexibility and maturity on accuracy.

We consider the new statements of problems in cryptography and steganography, which depend on the features of cloud systems. We also analyze the aspects of developing and implementing crypto- and steganosystems for cloud computing.

Replication studies play a critical role in scientific accumulation of knowledge, yet replication studies in mathematics education are rare. In this study, the authors replicated Thanheiser’s (Educational Studies in Mathematics 75:241–251, 2010) study of prospective elementary teachers’ conceptions of multidigit number and examined the main claim that most elementary pre-service teachers think about digits incorrectly at least some of the time. Results indicated no statistically significant difference in the distribution of conceptions between the original and replication samples and, moreover, no statistically significant differences in the distribution of sub-conceptions among prospective teachers with the most common conception. These results suggest confidence is warranted both in the generality of the main claim and in the utility of the conceptions framework for describing prospective elementary teachers’ conceptions of multidigit number. The report further contributes a framework for replication of mathematics education research adapted from the field of psychology.

Dutch mathematics programs emphasize mental addition and subtraction in the lower grades. For two-digit numbers up to 100, instruction focuses on \"counting by tens from any number\" (N10), a strategy that is difficult to learn. Therefore, many children prefer as an easier alternative \"decomposition\" in tens (1010) and units. Instead of the use of arithmetic blocks (BL), the hundredsquare (HU) was introduced in the 1980s because of a (supposed) better modeling function for teaching N10. In a field study with several schools, (a) we compared the strategies N10 and 1010 on procedural effectiveness and error types, and (b) we assessed the influence of the support conditions BL versus HU on the acquisition of mental strategies (we had also a control condition NO with no extra materials or models). Results confirmed the greater effectiveness of N10 but also the preference of many weaker children for 1010. Support for BL or HU had differential effects on mental strategies. Differences are discussed in terms of cognitive psychology: the role of declarative knowledge and the relation between conceptual and procedural knowledge. New Dutch proposals for the 1990s emphasize teaching both strategies N10 and 1010 to enhance the flexibility of students' mental arithmetic.

In an article that appeared in the Arithmetic Teacher , Madell (1985) described findings from a private school in New York City in which children were not taught any algorithms until the end of the third grade. Without algorithms, the children devised their own ways of solving computation problems. Madell's observation of the children's thinking led him to conclude that “children not only can but should create their own computational algorithms” (p. 20) and that “children can and should do their own thinking” (p. 22). The purpose of the present article is to reiterate Madell's call for reform, with supporting evidence from a public school near Birmingham, Alabama.