To teach or not to teach algorithms

Second, third, and fourth graders in 12 classes were individually interviewed to investigate the effects of teaching computational algorithms such as those of “carrying.” Some of the children had been encouraged to invent their own procedures and had not been taught any algorithms in grades 1 and 2, or in grades 1–3. Others had been taught the conventional algorithms prescribed by textbooks. The children were asked to solve multidigit addition and multiplication problems and to explain how they got their answers. It was found that those who had not been taught any algorithms produced significantly more correct answers. If children made errors, the incorrect answers of those who had not been taught any algorithms were much more reasonable than those found in the “Algorithms” classes. It was concluded that algorithms “unteach” place value and hinder children's development of number sense.     

Channels in cellular systems: To borrow or not to borrow...

This note develops an analytic model to answer the question whether it is advantageous to borrow a channel in cellular mobile communications systems. Such questions arise dynamically, whenever there is a pending call which will be blocked unless a channel is borrowed. The expected number of calls which will be blocked in the donor cells is calculated assuming that the channel is borrowed and held for the call holding duration. The resulting borrowing rule is simple — the channel should not be borrowed if this number is more than one and borrowed otherwise. If there is an option of borrowing from one of several donor groups, then the donor group which suffers the minimum number of expected blocked calls should be borrowed from. This approach provides a simple practical solution to a rather complicated problem. The results apply to any layout of cells. Channels could have been assigned to the cells via any method and not all channels may be borrowable.     

To be or not to be constructive,that is not the question

In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of mathematics, called intuitionism. Intuitionism was revolutionary in many respects but stands out – mathematically speaking – for its challenge of Hilbert’s formalist philosophy of mathematics and rejection of the law of excluded middle from the ‘classical’ logic used in mainstream mathematics. Out of intuitionism grew intuitionistic logic and the associated Brouwer–Heyting–Kolmogorov interpretation by which ‘there exists $x$’ intuitively means ‘an algorithm to compute $x$ is given’. A number of schools of constructive mathematics were developed, inspired by Brouwer’s intuitionism and invariably based on intuitionistic logic, but with varying interpretations of what constitutes an algorithm. This paper deals with the dichotomy between constructive and non-constructive mathematics, or rather the absence of such an ‘excluded middle’. In particular, we challenge the ‘binary’ view that mathematics is either constructive or not. To this end, we identify a part of classical mathematics, namely classical Nonstandard Analysis, and show it inhabits the twilight-zone between the constructive and non-constructive. Intuitively, the predicate ‘$x$ is standard’ typical of Nonstandard Analysis can be interpreted as ‘$x$ is computable’, giving rise to computable (and sometimes constructive) mathematics obtained directly from classical Nonstandard Analysis. Our results formalise Osswald’s longstanding conjecture that classical Nonstandard Analysis is locally constructive. Finally, an alternative explanation of our results is provided by Brouwer’s thesis that logic depends upon mathematics.     

To split or not to split: Capital allocation with convex risk measures

Convex risk measures were introduced by Deprez and Gerber [Deprez, O., Gerber, H.U., 1985. On convex principles of premium calculation. Insurance: Math. Econom. 4 (3), 179-189]. Here the problem of allocating risk capital to subportfolios is addressed, when convex risk measures are used. The Aumann-Shapley value is proposed as an appropriate allocation mechanism. Distortion-exponential measures are discussed extensively and explicit capital allocation formulas are obtained for the case that the risk measure belongs to this family. Finally the implications of capital allocation with a convex risk measure for the stability of portfolios are discussed. It is demonstrated that using a convex risk measure for capital allocation can produce an incentive for infinite fragmentation of portfolios.     

To match or not to match: Aspects of marital matchmaking under uncertainty

Researchers have paid scant attention to matchmaking from the perspective of a matchmaker. Therefore, our purpose is to analyze the circumstances under which a matchmaker optimally accepts or rejects individual matching assignments. We concentrate on two specific cases. In the first (second) case, our matchmaker's reservation profit is exogenous (endogenous).     

To wait or not to wait: Strategic behaviors in an observable batch-service queueing system

This paper deals with an observable batch service queueing system in which customers rationally choose whether to form a batch with another customer or not, in addition to deciding whether or not to join the queue. When choosing whether to form a batch, a customer in an incomplete batch decides on an optimal waiting time for the next customer to arrive and share the service fee. When choosing whether to join the queue, customers follow a threshold strategy, which depends on the strategy identified in the former game.     

To ask or not to ask,that is the question

Applicants for credit have to provide information for the risk assessment process. In the current conditions of a saturated consumer lending market, and hence falling “take” rates, can such information be used to assess the probability of a customer accepting the offer?     

To bat or not to bat: An examination of match outcomes in day-night limited overs cricket

The tradition of tossing a coin to decide who bats first in a cricket match introduces a randomly assigned advantage to one team that is unique in sporting contests. The potential importance of the toss rule in determining cricket match results has been the subject of some investigation, which is further advanced in this paper that utilizes a data set relating to the increasingly popular, but contentious, day-night form of limited overs cricket as played at international level. We employ logit regression models to examine the effects of winning the toss and choice of batting order on the likelihood of a match victory, while controlling for home advantage and (relative) team quality. Our findings suggest that winning the toss and batting first increases the probability of winning whereas winning the toss and bowling first does not.     

The two-machine open shop problem: To fit or not to fit, that is the question

We consider the open shop scheduling problem with two machines. Each job consists of two operations, and it is prescribed that the first (second) operation has to be executed by the first (second) machine. The order in which the two operations are scheduled is not fixed, but their execution intervals cannot overlap. We are interested in the question whether, for two given values D₁ and D₂, there exists a feasible schedule such that the first and second machine process all jobs during the intervals [0,D₁] and [0,D₂], respectively.We formulate four simple conditions on D₁ and D₂, which can be verified in linear time. These conditions are necessary and sufficient for the existence of a feasible schedule. The proof of sufficiency is algorithmical, and yields a feasible schedule in linear time. Furthermore, we show that there are at most two non-dominated points (D₁,D₂) for which there exists a feasible schedule.     

To stop or not to stop: a time-constrained trip covering location problem on a tree network

Annals of Operations Research - Location of new stations/stops in public transportation networks has attracted much interest from both the point of views of theory and applications. In this paper...     

To change or not to change: adapting mathematical knowledge for teaching (MKT) measures for use in Korea

This paper examines challenges in adapting mathematical knowledge for teaching (MKT) measures developed in the United States for use in Korea. After an initial analysis of candidate issues regarding the “fit” of items to the Korean context—whether items were familiar, authentic, and realistic as characterized by Delaney et al. (J Math Teach Educ 11:171–197, 2008)—we adapted and administered an instrument developed by the Learning Mathematics for Teaching project with 93 Korean teachers and conducted follow-up interviews with nine teachers. Based on analysis of this data, we conducted a second round of revision and then administered the revised test to 101 Korean teachers. Results showed that small modifications that were made to increase fit often increased teachers’ performance on the items as expected, but the impact of changes was at times difficult to interpret. For several items, modifications introduced unanticipated validity issues. The paper discusses dynamics that arise in making changes to MKT items—in particular, the tension in modifying items to increase the fit to specific educational contexts while maintaining validity.     

Bruno de Finetti and imprecision: Imprecise probability does not exist!

We review several of de Finetti’s fundamental contributions where these have played and continue to play an important role in the development of imprecise probability research. Also, we discuss de Finetti’s few, but mostly critical remarks about the prospects for a theory of imprecise probabilities, given the limited development of imprecise probability theory as that was known to him.     

Maximal arcs in Desarguesian planes of odd order do not exist

Forq an odd prime power, and 1<n<q, the Desarguesian planePG(2,q) does not contain an(nq–q+n,n)-arc.Supported by Italian M.U.R.S.T. (Research Group onStrutture geometriche, combinatoria, loro applicazioni) and G.N.S.A.G.A. of C.N.R.     

Decision Analysis and Real Options: A Discrete Time Approach to Real Option Valuation

In this paper we seek to enhance the real options methodology developed by Copeland and Antikarov (2001) with traditional decision analysis tools to propose a discrete time method that allows the problem to be specified and solved with off the shelf decision analysis software. This method uses dynamic programming with an innovative algorithm to model the project’s stochastic process and real options with decision trees. The method is computationally intense, but simpler and more intuitive than traditional methods, thus allowing for greater flexibility in the modeling of the problem.     

To be or to become: how dynamic geometry changes discourse

In this article, we investigate the impact of the introduction of a dynamic geometry environment on mathematical thinking by identifying changes in discourse engendered by its introduction in a high school geometry class. Our focus is on the teacher, and we find significant differences between static and dynamic geometry in terms of the ways in which the teacher talks about geometric objects, makes use of visual artifacts and models geometric reasoning. Even though these changes have major implications for the geometry being studied, they are made only very implicitly in the classroom.