The SEXTANT software: A tool for automating the comparative analysis of mental models of dynamic systems

The comparison of mental models of dynamic systems improves our understanding of how people comprehend, interpret, and subsequently influence dynamic management tasks. Approaches to comparing mental models currently used in managerial and organizational cognition research, especially the distance-ratio and the closeness-approaches, have been criticized for not considering essential characteristics of dynamic managerial situations. This paper builds on a recent analysis method developed to compare mental models of dynamics systems, and introduces this mathematical approach to management and organizational researchers by means of the SEXTANT software. It presents the process of mental model elicitation, analysis, comparison, and interpretation. An example with four elicited mental models illustrates the software’s features to analyze and present the results. Then, the software is compared with existing software to map and compare mental models. Our conclusion is that SEXTANT marks a significant step in enabling large-scale studies about mental models of dynamic systems.     

Investigating the transformation of a secondary teacher’s knowledge of trigonometric functions

Shulman (1987) defined pedagogical content knowledge as the knowledge required to transform subject-matter knowledge into curricular material and pedagogical representations. This paper presents the results of an exploratory case study that examined a secondary teacher’s knowledge of sine and cosine values in both clinical and professional settings to discern the characteristics of mathematical schemes that facilitate their transformation into learning artifacts and experiences for students. My analysis revealed that the teacher’s knowledge of sine and cosine values consisted of uncoordinated quantitative and arithmetic schemes and that he was cognizant only of the behavioral proficiencies these schemes enable, not the mental actions and conceptual operations they entail. Based on these findings, I hypothesize that the extent to which a teacher is consciously aware of the mental activity that comprises their mathematical conceptions influences their capacity to transform their mathematical knowledge into curricular material and pedagogical representations to effectively support students’ conceptual learning.     

Between Counting and Multiplication: Low-Attaining Students’ Spatial Structuring,Enumeration and Errors in Concretely-Presented 3D Array Tasks

The move from additive to multiplicative thinking requires significant change in children’s comprehension and manipulation of numerical relationships, involves various conceptual components, and can be a slow, multistage process for some. Unit arrays are a key visuospatial representation for supporting learning, but most research focuses on 2D (rectangular) arrays, and when focusing on 3D (cuboid) arrays still frequently uses 2D representations. This article documents low-attaining children’s partially developed multiplicative thinking as they work on concretely presented 3D array tasks; it also presents a framework for microanalysis of learners’ early multiplicative thinking in array tasks. Data derives from a small but cognitively diverse set of participants, all arithmetically low-attaining and relying heavily on counting: this enabled detailed analysis of small but significant differences in their arithmetical engagement with arrays. The analytical framework combines and builds on previous structural and enumerative categorizations, and may be used with a variety of array representations.     

System dynamics and microworlds for policymakers

In the past ten years, system dynamics has become more accessible to policymakers and to the academic community. The paper reviews four major developments in the subject that have brought about this change. There have been improvements in the symbols and software used to map and model system structure. New ideas have been adopted from behavioural decision theory which help to transfer policymakers' knowledge into computer models. There have been improvements in methods of simulation analysis that enable modelers and model users to gain better insight into dynamic behaviour. Greater emphasis has been placed on small transparent models, on games and on dialogue between ‘mental models’ and computer simulations. Together these developments allow modelers to create computer-based learning environments (or microworlds) for policymakers to ‘play-with’ their knowledge of business and social systems and to debate policy and strategy change. The paper concludes with some thoughts on future research.     

Understanding the role of the teacher in emerging classroom practices: searching for patterns of participation

The relationship between acquisitionism and participationism is a challenge in research on and with teachers. This study uses a patterns-of-participation framework (PoP), which aims to develop coherent and dynamic understandings of teaching as well as to meet the conceptual and methodological problems of other approaches. The paper presents PoP theoretically, but also illustrates its empirical use. It presents a novice teacher, Anna, who often engages with mathematics and with aspects of ‘the reform’ in ways that link well with how she builds relationships with her students and positions herself in her team of teachers. However, in other situations her engagement with mathematics is overshadowed by her involvement in other practices. The study suggests that there is some potential in PoP in spite of methodological difficulties.     

Secondary Students’ Quantification of Ratio and Rate: A Framework for Reasoning about Change in Covarying Quantities

Contributing to a growing body of research addressing secondary students’ quantitative and covariational reasoning, the multiple case study reported in this article investigated secondary students’ quantification of ratio and rate. This article reports results from a study investigating students’ quantification of rate and ratio as relationships between quantities and presents the Change in Covarying Quantities Framework, which builds from Carlson, Jacobs, Coe, Larsen, and Hsu’s (2002) Covariation Framework. Each of the students in this study was consistent in terms of the quantitative operation he or she used (comparison or coordination) when quantifying both ratio and rate. Illustrating how students can engage in different quantitative operations when quantifying rate, the Change in Covarying Quantities Framework helps to explain why students classified as operating at a particular level of covariational reasoning appear to be using different mental actions. Implications of this research include recommendations for designing instructional tasks to foster students’ quantitative and covariational reasoning.     

CHILDREN INVENTING SIGNS FOR MENTAL NUMBER STRATEGIES

The research study described here was conducted with a small group of 5- and 6-year-old children in a 35-pupil rural school in the Langdale Valley (Lake District, UK) over a period of 14 weeks. It considers the theoretical implications of children inventing their own signs for their counting actions instead of using the culturally inherited signs of conventional arithmetic. It also raises questions concerning ‘transformational addition’ in which ordinals (‘position numbers’) are distinguished from cardinals (‘size numbers’), and suggests that the invented signs may have a consistency and validity comparable with the signs of conventional arithmetic.     

Learning beginning algebra in a computer-intensive environment

We present a design research on learning beginning algebra in an environment where spreadsheets were available at all times but the decision about using them or not, and how, in any particular situation was left to the students. Students’ activity is analyzed in Kieran’s framework of generational, transformational and global/meta-level activity, and compared to the designers’ intentions. We do this by focusing on the activity of one student in four sessions spread over several months and discussing the activity of 51 additional students in view of the analysis of the focus student. We show that the environment enables a number of different entries into algebra and as such supports students in becoming autonomous learners of algebra, and in making the shift from arithmetic to algebra via generational and global/meta-level activity before dealing with the more technical transformational activities.     

A secure and efficient entropy coding based on arithmetic coding

A novel security arithmetic coding scheme based on nonlinear dynamic filter (NDF) with changeable coefficients is proposed in this paper. The NDF is employed to generate the pseudorandom number generator (NDF-PRNG) and its coefficients are derived from the plaintext for higher security. During the encryption process, the mapping interval in each iteration of arithmetic coding (AC) is decided by both the plaintext and the initial values of NDF, and the data compression is also achieved with entropy optimality simultaneously. And this modification of arithmetic coding methodology which also provides security is easy to be expanded into the most international image and video standards as the last entropy coding stage without changing the existing framework. Theoretic analysis and numerical simulations both on static and adaptive model show that the proposed encryption algorithm satisfies highly security without loss of compression efficiency respect to a standard AC or computation burden.     

A Symbolic Dance: The Interplay Between Movement,Notation, and Mathematics on a Journey Toward Solving Equations

This article analyzes the use of the software Grid Algebra with a mixed ability class of 21 nine-to-ten-year-old students who worked with complex formal notation involving all four arithmetic operations. Unlike many other models to support learning, Grid Algebra has formal notation ever present and allows students to “look through” that notation and interpret it either in terms of physical journeys on a grid or in terms of mathematical operations. A dynamic fluidity was found between the formal notation, imagery of movements on a grid, and the process of mathematical operations. This fluidity is interpreted as a “dance” between these three. The significant way in which this dynamic took place reflects the scaffolding and fading offered by the software, which was crucial to the students’ fluency with formal notation well beyond what has been reported from students of that age.     

The teacher’s role in guiding children’s mathematical ideas toward meeting lesson objectives

This paper investigates the classroom interactive pattern, in which the teacher aims to introduce new mathematical content to children by focusing on their mathematical thinking. First, by drawing on the results of studies on the features of social interaction patterns in mathematics classrooms, we develop a framework that we call a “guided focusing pattern,” composed of four phases. Next, we use this framework and Sfard’s (J Res Math Educ 31(3):296–327, 2000) theory of focal analysis to examine the social interaction occurring in a series of mathematics lessons conducted by an experienced teacher. In the ten consecutive lessons that we analyzed, the guided focusing pattern was salient; the teacher introduced key mathematical content to children while offering support and guidance in a variety of forms within each phase and when transitioning to the next phase. On the basis of the results, we highlight the teacher’s key instructional actions that facilitate the pattern of progressing through the mathematical content as closely linked to and guided by her lesson objectives.     

A multi-criteria framework as a decision support system for urban growth management applications: Central city redevelopment

The paper introduces a decision support system based on a multi-criteria framework that integrates several socio-economic, physical planning and urban design processes recently used in the U.S. for growth management of central city redevelopment. It formalizes and operationalizes the concept of the ‘critical coalition’, i.e. user groups representative of the community's interests who share common perceived values, objectives and problems within an urban setting. This critical coalition provides the decision model with the socio-economic qualitative input to establish representative priorities and importance weights, and to provide the political base for the implementation of policy actions. The framework is applied to establish locational suitability of development or redevelopment actions within a central city. This application is illustrated with examples from a pilot study to establish the suitability of residential infill within the central business district of the City of Columbus, Ohio.     

Normalising geometrical figures: dynamic manipulation and construction of meanings for ratio and proportion

Enlarging-shrinking geometrical figures by 13 year-olds is studied during the implementation of proportional geometric tasks in the classroom. Students worked in groups of two using ‘Turtleworlds’, a piece of geometrical construction software which combines symbolic notation, through a programming language, with dynamic manipulation of geometrical objects by dragging on sliders representing variable values. In this paper we study the students’ normalising activity, as they use this kind of dynamic manipulation to modify ‘buggy’ geometrical figures while developing meanings for ratio and proportion. We describe students’ normative actions in terms of four distinct Dynamic Manipulation Schemes (Reconnaissance, Correlation, Testing, Verification). We discuss the potential of dragging for mathematical insight in this particular computational environment, as well as the purposeful nature of the task which sets up possibilities for students to appreciate the utility of proportional relationships.     

基于全局熵值法模型的我国区域创新能力动态评价与分析

本文在对FP&S和H&M的创新能力分析框架进行修正的基础上,从创新环境、创新资源、创新成果和创新品牌四个维度建立基于企业层面的区域创新能力分析框架,以此为理论依据建立了基于企业层面的区域创新能力评价指标体系,并引入全局的思想构建全局熵值法模型,在此基础上对京津冀地区、东北地区、长三角地区和南部沿海地区2009~2012年企业层面的创新能力进行了动态评价与分析,研究结果对于认识我国各经济区的企业创新能力提供了重要的参考依据。     

Bilattice logic of epistemic actions and knowledge

Baltag, Moss, and Solecki proposed an expansion of classical modal logic, called logic of epistemic actions and knowledge (EAK), in which one can reason about knowledge and change of knowledge. Kurz and Palmigiano showed how duality theory provides a flexible framework for modeling such epistemic changes, allowing one to develop dynamic epistemic logics on a weaker propositional basis than classical logic (for example an intuitionistic basis). In this paper we show how the techniques of Kurz and Palmigiano can be further extended to define and axiomatize a bilattice logic of epistemic actions and knowledge (BEAK). Our propositional basis is a modal expansion of the well-known four-valued logic of Belnap and Dunn, which is a system designed for handling inconsistent as well as potentially conflicting information. These features, we believe, make our framework particularly promising from a computer science perspective.     

Building on mathematical events in the classroom

Mathematical events from classrooms were used as stimuli to encourage mathematical discussion in two groups of mathematics teachers at the secondary level. Each event was accompanied by an analysis of mathematics that would be useful to the teacher in such a situation. The Situations, mathematical events and analyses, were used originally to create a framework describing the Mathematical Proficiency for Teaching at the Secondary Level, and then they were used with both Prospective and Practicing teachers to validate the framework. Teachers involved in the validation research claimed that the process was instructional. The process is explained, and teachers’ quotes provide evidence that the experience provoked changes in teachers’ understanding of mathematics. This process, which builds on mathematical events from the classroom, holds potential as a professional development experience that helps teachers expand their expertise in teaching mathematics.     

Flow auctions

Flow goods (like electricity) are sold through auctions in a dynamic framework. An important design question is the frequency of such auctions. We use a simple dynamic auction model in continuous time to answer this question. We focus on the relationship between the persistency of bidders’ valuations and the optimal choice of frequency. If the seller focuses on the equilibrium in which bidders follow a repeated static Nash strategy, then the frequency of auctions should typically increase when persistency declines. However, accounting for the fact that bidders can follow different equilibria that are collusive in nature, the comparative statics are reversed, forcing the seller to reduce the frequency when bidders’ valuations are less persistent. The argument builds on the fact that high frequency auctions are more conducive to collusion among bidders.     

DAG reversal is NP-complete

Runs of numerical computer programs can be visualized as directed acyclic graphs (DAGs). We consider the problem of restoring the intermediate values computed by such a program (the vertices in the DAG) in reverse order for a given upper bound on the available memory. The minimization of the associated computational cost in terms of the number of performed arithmetic operations is shown to be NP-complete. The reversal of the data-flow finds application, for example, in the efficient evaluation of adjoint numerical programs. We derive special cases of numerical programs that require the intermediate values exactly in reverse order, thus establishing the NP-completeness of the optimal adjoint computation problem. Last but not least we review some state-of-the-art approaches to efficient data-flow reversal taken by existing software tools for automatic differentiation.     

Four (algorithms) in one (bag): an integrative framework of knowledge for teaching the standard algorithms of the basic arithmetic operations

In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic operations with rational numbers. In order to evaluate the potential applicability of the framework to task design, it was used for the design of mathematical learning tasks for teachers. The article includes examples of the tasks, their theoretical analysis, and empirical evidence of the sensitivity of the tasks to variations in teachers’ knowledge of the subject. This evidence is based on a study of 46 primary school teachers. The article concludes with remarks on the applicability of the framework to research and practice, highlighting its potential to encourage teaching the four algorithms with an emphasis on conceptual understanding.     

Perspectives of approximate dynamic programming

Approximate dynamic programming has evolved, initially independently, within operations research, computer science and the engineering controls community, all searching for practical tools for solving sequential stochastic optimization problems. More so than other communities, operations research continued to develop the theory behind the basic model introduced by Bellman with discrete states and actions, even while authors as early as Bellman himself recognized its limits due to the “curse of dimensionality” inherent in discrete state spaces. In response to these limitations, subcommunities in computer science, control theory and operations research have developed a variety of methods for solving different classes of stochastic, dynamic optimization problems, creating the appearance of a jungle of competing approaches. In this article, we show that there is actually a common theme to these strategies, and underpinning the entire field remains the fundamental algorithmic strategies of value and policy iteration that were first introduced in the 1950’s and 60’s.