Variations on play with interactive computer simulations: balancing competing priorities

U.S. mathematics teachers face considerable pressures to keep up with pacing guides and to prepare students for standardized tests. At the same time, they are called upon to engage students in innovative exploratory activities and to incorporate new technologies into their lessons. These competing priorities pose considerable challenges. Against this backdrop, we investigated how middle-school mathematics teachers incorporated play into lessons involving interactive computer simulations (sims). The teachers used PhET sims in a variety of lessons. Following general guidelines for teaching with PhET sims, these lessons included a short period of play prior to more structured work with the sim. Our analysis of 15 mathematics lessons involving play led to the identification of four characteristics that distinguish the play phases of these lessons. Based on combinations of these characteristics, we identified three specific profiles of play, which lie at different points along a continuum of priorities from foregrounding students’ ideas to keeping pace. We discuss the implications associated with each profile of the play phase, and we begin to articulate a theory that frames teaching with play as a matter of balancing divergent and convergent modes of activity.     

On the importance of mathematical play

A number of existing theories and proposals for the meaning and characteristics of ‘play’ are considered before the authors suggest six characteristics of mathematical play, including the idea that it is not confined to childhood. Previous studies provide evidence for relating play to cognitive gain while the place of mathematical play in research activities is illustrated by describing a mathematician's approach to a number investigation from the classroom-The Six Circles. The problem-solving process for the Six Circles and observations of students solving calculator and integration problems are analysed in relation to theories of play and cognitive gain and also considered from the perspective of the students' experience. Piaget's theory for the assimilation and accommodation of new information and Davis's view of play as ‘space to support learning’ are reflected in the authors' rationale for suggesting that open questions and mathematical play provide opportunities for students to develop their own conjectures, with no threat of failure, and provide a foundation for mathematical learning. Some difficulties of implementing a ‘play’ approach in the classroom are discussed and further research questions proposed.     

Exchange rate uncertainty and employment: an algorithm describing ‘play’

The paper deals with the impacts of exchange rate uncertainty on the relationship between macroeconomic labour market variables. Under uncertainty, areas of weak reactions—so‐called ‘play’ areas—have to be considered at the macrolevel. The width of the play area is a positive function of the degree of uncertainty. When changes go beyond the play‐area suddenly strong reactions (‘spurts’) occur. These non‐linear dynamics are captured in a simplified linearized way. An algorithm describing linear play hysteresis is developed and implemented into a regression framework. As an empirical application, the exchange rate impacts on German employment are analysed considering play effects. Copyright © 2001 John Wiley & Sons, Ltd.     

不完全博弈学习过程的虚拟行动规则

本文把博弈学习虚拟行动规则的学习时间一般化,研究不完全学习过程中参与人策略选择的收敛性及效用一致性问题. 分析结果表明：当学习时间不完全时,在学习一致到达的条件下,虚拟行动规则对严格纳什均衡是吸收的; 在学习时间充分及时与虚拟行动非频繁转换的条件下,各参与人的虚拟行动具有效用一致性.     

Playing disjunctive sums is polynomial space complete

A position in a disjunctive sum of games is simply a collection of positions, one from each game: to move in a sum is to move in any one of its constituents. Sums have been studied extensively by Conway and others, and play an important rÔle in Go. It is shown that the problem of best play in a sum of trivial games is polynomial space complete. Hence it may be conjectured that there is no feasible algorithm for deriving a strategy of play in a sum from knowledge about its constituent games.     

Dynamic programming in cricket: choosing a night watchman

In cricket, when a batsman is dismissed towards the end of a day's play, he is often replaced by a lower-order batsman (a ‘night watchman’), in the hope that the remaining recognised batsmen can start their innings on the following day. A dynamic programming analysis suggests that the common practice of using a lower-order batsman is often sub-optimal. Towards the end of a day's play, when the conventional wisdom seems to be to use a night watchman, it may be best to send in the next recognised batsman in the batting order. Sending in a night watchman may be good judgement when there are several recognised batsman and several lower order batsmen still to play (say four of each). However, with smaller numbers (two of each, for example), then, with very few overs left to play, it may be better to send in a recognised batsman.     

Fictitious play with incomplete learning

In this paper, we consider a case that a game is played repeatedly in an incomplete learning process where each player updates his belief only in the learning periods rather than all the stages. For fictitious play process with incomplete learning, we discuss the absorbability of Nash equilibriums and the consistency of utilities in a finite game and discuss the convergence in a 2×2 game with an identical learning-period set. The main results for incomplete learning models are that, if it is uniformly played, a strict Nash equilibrium is absorbing in a fictitious play process; a fictitious play has the property of utility consistency if it exhibits infrequent switches and players learn frequently enough; a 2×2 game with an identical learning-period set has fictitious play property that any fictitious process for the game converges to equilibrium provided that players learn frequently enough.     

Internalist Virtues and Knowledge

What role can intellectual virtues play in an account of knowledge when we interpret those virtues internalistically, i.e., as depending only on internal states of the cognizer? Though it has been argued that internalist virtues are ill suited to play any role in an account of knowledge, I will show that, on the contrary, internalist virtues can play an important role in recent accounts of knowledge developed to utilize externalist virtues. The virtue account of knowledge developed by Linda Zagzebski is intended to be supplemented by her version of the intellectual virtues which require an external success component. John Greco and Wayne Riggs both develop credit accounts of knowledge on which the abilities we use when we get credit for a true belief must be reliable. I examine the similarities between these three accounts of knowledge and demonstrate that internalist virtues fit into these accounts just as well as externalist virtues. Thus, although internalist virtues do not require a reliable connection to truth, they can still play an important role in defining the truth-requiring concept of knowledge.     

Scheduling sports competitions with a given distribution of times

In some sports competitions there is only one facility (court, etc.) and so teams or individuals are required to play at different times of day. It is desirable that these times of play be equitably distributed for each team, consistent with the constraints imposed by some contestants being unable to play at certain times for occupational reasons. This condition, treated as an additional requirement to those of a round-robin tournament, is formulated as a combinatorial design problem and certain cases are solved. In the absence of constraints imposed by particular teams, the problem is that of a balanced tournament design (BTD) and we present a new construction for BTDs with certain parameter values.     

Does observation of others affect learning in strategic environments? An experimental study

This paper presents experimental results from an analysis of two similar games, the repeated ultimatum game and the repeated best-shot game. The experiment examines whether the amount and content of information given to players affects the evolution of play in the two games. In one experimental treatment, subjects in both games observe not only their own actions and payoffs, but also those of one randomly chosen pair of players in the just-completed round of play. In the other treatment, subjects in both games observe only their own actions and payoffs. We present evidence suggesting that observation of other players' actions and payoffs may affect the evolution of play relative to the case of no observation. Received February 1996/Final version April 1998     

The play operator,the truncated variation and the generalisation of the Jordan decomposition

The play operator minimalizes the total variation on intervals, [0,T],T > 0, of functions approximating uniformly given regulated function with given accuracy and starting from a given point. In this article, we link the play operator with the so‐called truncated variation functionals, introduced recently by the second‐named author, and provide a semi‐explicit expression for the play operator in terms of these functionals. Generalisation for time‐dependent boundaries is also considered. This gives the best possible lower bounds for the total variation of the outputs of the play operator and its Jordan‐like decomposition. Copyright © 2014 John Wiley & Sons, Ltd.     

On discounted subfair primitive casino

Summary As is well known, in a subfair primitive casino a gambler with an initial fortune f, 0<f<1, desiring to reach 1 (his goal) should play boldly since there is no other strategy that can provide him with a higher utility (the probability of reaching his goal). Now suppose the game is modified by adding a discount factor which is used to motivate the gambler to recognize the time value of his goal and complete the game as quickly as is reasonably consistent with reaching his goal. Then one would intuitively suspect that again the bold play would be optimal. We will show in this paper that for certain subfair or fair primitive casinos the bold play is always optimal regardless of the discount factor; however, for some subfair or fair primitive casinos, there exist some discount factors for which the bold play is no longer optimal.This work is supported in part by NSF Grant MCS78-01336     

Dynkin Diagram Classification of λ-Minuscule Bruhat Lattices and of d-Complete Posets

d-Complete posets are defined to be posets which satisfy certain local structural conditions. These posets play or conjecturally play several roles in algebraic combinatorics related to the notions of shapes, shifted shapes, plane partitions, and hook length posets. They also play several roles in Lie theory and algebraic geometry related to -minuscule elements and Bruhat distributive lattices for simply laced general Weyl or Coxeter groups, and to -minuscule Schubert varieties. This paper presents a classification of d-complete posets which is indexed by Dynkin diagrams.     

一个创新网络动态性的仿真模型

提出了一个关于创新网络动态性研究的仿真模型,模型较为全面地考察了网络的形成过程和不同的角色在创新和网络中的作用,研究结论指出:网络的形成是由于创新主体为适应创新的复杂性而寻求资源互补效应的动态相互作用的结果,中小型企业的个体在网络演化的早期和中期起了重要的作用,合作体的作用主要发挥在中后期.     

Notes on Lipschitz estimates for the stop and play operator in plasticity

We present a generalisation of existing Lipschitz estimates for the stop and play operator for an arbitrary convex and closed characteristic, which contains the origin, in a separable Hilbert space. We are especially concerned with the dependence of stop and play on different scalar products.     

Generalized double involutive geometry of the triangle

We study a generalized geometry of the triangle, based on the idea of letting two arbitrary points play the role that the centroid and the orthocenter play classically. We thus generalize some of the classical results and constructions and also prove some new results in ordinary metric geometry.     

Zones of cooperation in demographic prisoner's dilemma

The emergence of cooperation in prisoner's dilemma (PD) games is generally assumed to require repeated play (and strategies such as Tit for Tat, involving memory of previous interactions) or features (“tags”) permitting cooperators and defectors to distinguish one another. In the demographic PD, neither assumption is made: Agents with finite vision move to random sites on a lattice and play a fixed culturally-inherited zero-memory strategy of cooperate (C) or defect (D) against neighbors. Agents are indistinguishable to one another—they are “tagless.” Positive payoffs accrue to agents playing C against C, or D against C. Negative payoffs accrue to agents playing C against D, or D against D. Payoffs accumulate. If accumulated payoffs exceed some threshold, agents clone offspring of the same strategy onto neighboring sites and continue play. If accumulated payoffs are negative, agents die and are removed. Spatial zones of cooperation emerge. © 1998 John Wiley & Sons, Inc.     

页岩气区块优选的模糊动态多属性决策模型研究

页岩气作为目前最现实的可替代能源倍受各国政府和企业的高度关注。针对信息不确定下页岩气区块优选的动态多属性决策难题,在初步估测不同阶段地质资源禀赋、基础设施建设投资以及环境保护要求等不确定信息下,以优选具有商业化价值开发的页岩气目标区块为重点研究对象,从解决不同区块的投资优先次序入手,运用多属性决策、模糊优化决策和动态决策优化的方法理论,通过剖析页岩气区块优选决策过程及其复杂性特征,将页岩气区块投资优选和排序问题进行形式化描述,提出一种探索解决这种模糊动态多属性决策问题的方法,并应用到具体的页岩气区块优选问题中。本项研究不仅有助于深化不确定条件下动态多属性决策理论的研究,还为解决页岩气区块优选这一投资决策难题提供新的思路和方法。     

Prohibiting repetitions makes playing games substantially harder

If one prohibits repetition of moves (i.e., moving to a position the player has already moved to before) in combinatorial games, a won game cannot be successfully played using a strategy in the narrow sense only. In general, one has to regard the actual development of the play arbitrarily far back and process it in an arbitrarily complex way. The problem of best play in such a game whose graph of positions is explicitly given is, indeed, polynomial space complete.     

Fictitious play in stochastic games

In this paper we examine an extension of the fictitious play process for bimatrix games to stochastic games. We show that the fictitious play process does not necessarily converge, not even in the 2 × 2 × 2 case with a unique equilibrium in stationary strategies. Here 2 × 2 × 2 stands for 2 players, 2 states, 2 actions for each player in each state.