Describing collective creative acts in a mathematical problem-solving environment

In this article, I report on a study of the nature of collective creativity in mathematics learning settings. Based on an extensive review of the literature on creativity, the study was initially framed by a number of themes to encompass a variety of visions of creativity. These themes were used in the analysis of data collected in a Grade 6 mathematics learning environment. The themes were then refined and (re)developed to distill four creative acts to describe the experience of creativity with(in) the collective. These collective creative acts are: summing forces, expanding possibilities, divergent thinking, and assembling things in new ways. Here, I present the findings of the study, then, I speculate on some of the logical implications of the four collective creative acts for teaching and learning.     

Survivable network design under optimal and heuristic interdiction scenarios

We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer’s network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.     

Optimistic problem-solving activity: enacting confidence,persistence, and perseverance

Optimism supports creative mathematical problem-solving. To elaborate its nature, empirical data were analyzed to identify relationships between optimism and more commonly researched constructs, confidence, and persistence. To do so, theoretical links between these constructs were first explored. Theoretically, confidence and persistence were found to be mutually exclusive personal characteristics possessed by optimistic students. Then, five elementary school students were purposefully selected from a broader longitudinal video-stimulated interview study of the role of optimism in collaborative problem-solving to find whether all combinations of confidence and persistence existed. Activity of students possessing different combinations of confidence and persistence was analyzed to determine whether there were differences in their problem-solving activity. Perseverance emerged as a third mutually exclusive characteristic within optimism. By distinguishing between persistence and perseverance, the crucial nature of perseverance in creative mathematical thinking was illuminated. These findings should inform teachers, teacher educators, and researchers interested in building optimism to increase problem-solving capacity.     

Collective Mathematical Understanding as Improvisation

This article explores the phenomenon of mathematical understanding, and offers a response to the question raised by Martin (2001) at the Annual Meeting of the Psychology of Mathematics Education Group (North American Chapter) about the possibility for and nature of collective mathematical understanding. In referring to collective mathematical understanding, we point to the kinds of learning and understanding we may see occurring when a group of learners work together on a piece of mathematics. We characterize the growth of collective mathematical understanding as a creative and emergent improvisational process and illustrate how this can be observed in action. In doing this, we demonstrate how a collective perspective on mathematical understanding can more fully explain its growth. We also discuss how considering the growth of mathematical understanding as a collective process has implications for classroom practice and in particular for the setting of mathematical tasks.     

学英语（英文）

<正>There are 7 questions in total,presenting various different question types.While you attempt to resolve the problems,remember to be creative.During accomplishing these flexible mathematical exercises,you can inspire your mathematical thinking.1.An alligator population in a nature preserve in the Everglades decreases by 60alligators over     

Cosmological natural selection and the purpose of the universe

The cosmological natural selection (CNS) hypothesis holds that the fundamental constants of nature have been fine‐tuned by an evolutionary process in which universes produce daughter universes via the formation of black holes. Here, we formulate the CNS hypothesis using standard mathematical tools of evolutionary biology. Specifically, we capture the dynamics of CNS using Price's equation, and we capture the adaptive purpose of the universe using an optimization program. We establish mathematical correspondences between the dynamics and optimization formalisms, confirming that CNS acts according to a formal design objective, with successive generations of universes appearing designed to produce black holes. © 2013 Wiley Periodicals, Inc. Complexity 18: 48–56, 2013     

A property of creative sets

A new approach to the study of creative sets using the notion of a table is offered. Making use of tables conforming to recursively enumerable sets, novel properties of creative sets are established. Harrington's theorem on the definability of creative sets in the lattice of recursively enumerable sets is proved, and we reprove Lachlan's theorem which states that one of the factors in a direct product of creative sets is again creative. Supported by RFFR grant No. 93-01-16014. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 294–307, May–June, 1996.     

Advancing Mathematical Activity: A Practice-Oriented View of Advanced Mathematical Thinking

The purpose of this article is to contribute to the dialogue about the notion of advanced mathematical thinking by offering an alternative characterization for this idea, namely advancing mathematical activity. We use the term advancing (versus advanced) because we emphasize the progression and evolution of students' reasoning in relation to their previous activity. We also use the term activity, rather than thinking. This shift in language reflects our characterization of progression in mathematical thinking as acts of participation in a variety of different socially or culturally situated mathematical practices. For these practices, we emphasize the changing nature of students' mathematical activity and frame the process of progression in terms of multiple layers of horizontal and vertical mathematizing.     

Monoids for Which Condition (P) Acts are Projective

A characterisation of monoids for which all right S-acts satisfying conditions (P) are projective is given. We also give a new characterisation of those monoids for which all cyclic right S-acts satisfying condition (P) are projective, similar in nature to recent work by Kilp [6]. In addition we give a sufficient condition for all right S-acts that satisfy condition (P) to be strongly flat and show that the indecomposable acts that satisfy condition (P) are the locally cyclic acts.     

Robin-type boundary control problems for the nonlinear Boussinesq type equations

In this paper, we study a linear and a nonlinear boundary control problems arising from viscous flows. The equations are of nonlinear Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the temperature and the salinity. The essential difficulties are due to the nonlinear nature of a part of the boundary conditions and to the nature of the equations: time-dependent, coupled and nonlinear. The existence and the conditions of the uniqueness of the solution, for the variational problem, are studied. The control is of linear or nonlinear Robin-type and acts on a part of the boundary during a time T. The cost function measures the distance between the observed and the computed vorticity. The existence of an optimal control in the admissible set of states and controls is proved. A first order necessary conditions of optimality are obtained.     

Studying the role of human agency in school mathematics

Mathematical discourse is often described as abstract and devoid of human presence, yet many school curricula espouse an aim to develop active, creative mathematical problem posers and solvers. The project The Evolution of the Discourse of School Mathematics(EDSM) developed an analytic scheme to investigate the nature of school mathematics discourse through the lens of high-stakes examinations in England. Following an overview of the scheme, this article ‘zooms in’ on the development of the sub-component addressing the question of how the origin of mathematical knowledge is construed, allowing investigation of the potential for students to see a role for themselves as active, creative agents in mathematical practices. Analytical tools operationalising this component are presented and their application illustrated. Results of analysis of examinations over a period of three decades suggest some increase in human agency, though some other aspects characteristic of higher-level mathematics may have reduced.     

On uniform acts over semigroups

The paper is devoted to the investigation of uniform acts over semigroups perceived as an overclass of subdirectly irreducible acts. We establish conditions for a uniform act to be subdirectly irreducible. In particular, we prove that uniform acts with two zeros are subdirectly irreducible. Ultimately we investigate monoids which are uniform as right acts over themselves and we describe regular monoids with this property.     

Optimal routing of vehicles with communication capabilities in disasters

Major emergencies and disasters such as acts of terrorism, acts of nature, or human-caused accidents may lead to disruptions in traffic flow. Minimizing the negative effects of such disruptions is critical for a nation’s economy and security. A decision support system that is capable of gathering (real-time) information about the traffic conditions following a disaster and utilizing this information to generate alternative routes for vehicles would benefit the government, industry, and the public. For this purpose, we develop a mathematical programming model to minimize the delay for vehicles with communication capabilities following a disaster. Most commercial trucks and public buses utilize QUALCOMM as a communication tool. We also develop a prediction model for vehicles that do not have any communication capabilities. Although the problem is inherently integer we developed a linear program to reduce the computational burden caused by the large size of the problem. An algorithm is proposed to update the parameters of the linear program based on a duality analysis in order to obtain better results. A monotonic speed–density relationship is embedded in the model to capture high traffic congestion that occurs after a disaster. The model and the algorithm are tested using a simulated disaster scenario. The results indicate that the proposed model improves system performance measures such as mobility and average speed.     

Sequentially Injective Hull of Acts Over Idempotent Semigroups

In this paper using the notion of a sequentially dense monomorphism we consider sequential injectivity (s-injectivity) for acts over a semigroup S. Among other things we describe the s-injective hull of acts over an idempotent semigroup S. We also give some classes of idempotent semigroups for the acts over which the notions of injectivity and s-injectivity coincide, and hence get the injective hull of acts over these classes of semigroups.     

The Effect of Nature of Science Metacognitive Prompts on Science Students’ Content and Nature of Science Knowledge,Metacognition, and Self‐Regulatory Efficacy

The purpose of the present explanatory mixed‐method design is to examine the effectiveness of a developmental intervention, Embedded Metacognitive Prompts based on Nature of Science (EMPNOS) to teach the nature of science using metacognitive prompts embedded in an inquiry unit. Eighty‐three (N = 83) eighth‐grade students from four classrooms were randomly assigned to an experimental and a comparison group. All participants were asked to respond to a number of tests (content and nature of science knowledge) and surveys (metacognition and self‐regulatory efficacy). Participants were also interviewed. It was hypothesized that the experimental group would outperform the comparison group in all measures. Partial support for the hypotheses was found. Specifically, results showed significant gains in content knowledge and nature of science knowledge of the experimental group over the comparison group. Qualitative findings revealed that students in the comparison group reported scientific thinking in similar terms as the scientific method, while the experimental group reported that scientists were creative and had to explain events using evidence, which is more closely aligned to the aspects of the nature of science. EMPNOS may have implications as a useful classroom tool in guiding students to check their thinking for alignment to the nature of science.     

SEPARABILITY CONDITIONS IN ACTS OVER MONOIDS

We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and strongly subact separable (and hence weakly subact separable); all acts over a commutative idempotent monoid are residually finite and strongly subact separable; and all acts over a Clifford monoid are strongly subact separable.

     

L－逆左系对幺半群的刻画

正则左Ｓ-系是ｖｏｎ Ｎｅｕｍａｎｎ正则半群的自然推广，逆左Ｓ-系是逆半群的自然推广．作为左逆半群的自然推广，本文引入了Ｌ-逆左系的概念，并用来刻画了几类幺半群，如左逆幺半群，逆幺半群，ａｄｅｑｕａｔｅ幺半群等．     

Injective Hulls of Acts over Left Zero Semigroups

In this paper, using the notion of "Cauchy sequences", we study the injectivity of acts over left zero semigroups, and give some equivalent conditions to it. We see that Baer criterion is true if an identity is adjoined to the semigroup. Further, we find injective hulls of acts over left zero semigroups. We hope that the technique used here will be useful to study injectivity of acts over some other more general semigroups.     

Thermodynamics of action and organization in a system

Complexity in nature is astounding yet the explanation lies in the fundamental laws of physics. The Second Law of Thermodynamics and the Principle of Least Action are the two theories of science that have always stood the test of time. In this article, we use these fundamental principles as tools to understand how and why things happen. In order to achieve that, it is of absolute necessity to define things precisely yet preserving their applicability in a broader sense. We try to develop precise, mathematically rigorous definitions of the commonly used terms in this context, such as action, organization, system, process, etc., and in parallel argue the behavior of the system from the first principles. This article, thus, acts as a mathematical framework for more discipline‐specific theories. © 2015 Wiley Periodicals, Inc. Complexity 21: 307–317, 2016