Is this a coincidence? The role of examples in fostering a need for proof

It is widely known that students often treat examples that satisfy a certain universal statement as sufficient for showing that the statement is true without recognizing the conventional need for a general proof. Our study focuses on special cases in which examples satisfy certain universal statements, either true or false in a special type of mathematical task, which we term “Is this a coincidence?”. In each task of this type, a geometrical example was chosen carefully in a way that appears to illustrate a more general and potentially surprising phenomenon, which can be seen as a conjecture. In this paper, we articulate some design principles underlying the choice of examples for this type of task, and examine how such tasks may trigger a need for proof. Our findings point to two different kinds of ways of dealing with the task. One is characterized by a doubtful disposition regarding the generality of the observed phenomenon. The other kind of response was overconfidence in the conjecture even when it was false. In both cases, a need for “proof” was evoked; however, this need did not necessarily lead to a valid proof. We used this type of task with two different groups: capable high school students and experienced secondary mathematics teachers. The findings were similar in both groups.     

What Is Entitlement?

In his seminal paper, “Content Preservation,” Tyler Burge defends an original account of testimonial knowledge. The originality of the account is due, in part, to the fact that it is cast within a novel epistemic framework. The central feature of that framework is the introduction of the concept of entitlement, which is alleged to be a distinctive type of positive epistemic support or warrant. Entitlement and justification, according to Burge, are sub-species of warrant. Justification is the internalist form of warrant, but entitlement is epistemically externalist. My focus in this paper is Burge’s conception of entitlement, and there are three primary issues that I wish to address. What is the relationship between entitlement and the more traditional concept of justification? In what sense is entitlement epistemically externalist? Has Burge introduced a new epistemic concept or merely coined a new term for a familiar epistemic concept?     

Is Personal Identity Analysable?

Trenton Merricks has argued that given endurantism personal identity is unanalysable in terms of psychological continuity, while Anthony Brueckner has argued against this claim. This article shows that neither philosopher has made a compelling case and also shows what it would take to settle the issue either way. It is then argued that whether personal identity is analysable or not may not be of crucial importance to those wanting to defend a psychological continuity approach to personal identity.     

WHEN am I ever going to use this?

<正>SCIENCE Bottle-nosed dolphins and killer whales are the best Gknown species ofthe dolphin family.Akiller whale,which can grow to a length of 9 meters,is 4 meters longer than a bottle-nosed dolphin.1.What does x represent in the figure?2.Write an expression to represent 4 meters longer than a dolphin.     

What Is Variable Bandwidth?

We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator where p > 0 is a strictly positive function. Denote by the orthogonal projection of A_p corresponding to the spectrum of A_p in ; the range of this projection is the space of functions of variable bandwidth with spectral set in Λ. We will develop the basic theory of these function spaces. First, we derive (nonuniform) sampling theorems; second, we prove necessary density conditions in the style of Landau. Roughly, for a spectrum the main results say that, in a neighborhood of , a function of variable bandwidth behaves like a band‐limited function with local bandwidth . Although the formulation of the results is deceptively similar to the corresponding results for classical band‐limited functions, the methods of proof are much more involved. On the one hand, we use the oscillation method from sampling theory and frame‐theoretic methods; on the other hand, we need the precise spectral theory of Sturm‐Liouville operators and the scattering theory of one‐dimensional Schrödinger operators. © 2017 Wiley Periodicals, Inc.     

Is simulation in health different?

It is often stated that health simulation is quite different and even that it is more difficult than in other sectors. But is simulation in health really different to simulation elsewhere? In this paper we explore this question through a survey of simulation modellers and academics. We elicit their opinions across a range of factors concerning the difficulties of health modelling against modelling in other domains. The analysis considers the responses of the whole group of respondents and the sub-group of respondents who have experience both in and outside of health modelling. The results show that overall, there is a perception that health modelling is different and that it is more difficult across a range of factors. The implications for simulation research and practice in health are discussed.     

How Fast Is the Bandit?

Abstract

In this article we investigate the rate of convergence of the so-called two-armed bandit algorithm. The behavior of the algorithm turns out to be highly non standard: no central limit theorem, possible occurrence of two different rates of convergence with positive probability.     

If this is our mathematics, what are our stories?

This paper sets out to examine how narrative modes of thinking play a part in the claiming of mathematical territories as our own, in navigating mathematical landscapes and in conversing with the mathematical beings that inhabit them. We begin by exploring what constitutes the narrative mode, drawing principally on four characteristics identified by Bruner and considering how these characteristics manifest themselves in the activities of mathematicians. Using these characteristics, we then analyse a number of examples from our work with expressive technologies; we seek to identify the narrative in the interactions of the learners with different computational microworlds. By reflecting on the learners’ stories, we highlight how particular features, common across the microworlds—motion, colour, sound and the like—provided the basis for both the physical and psychological grounding of the behaviour of the mathematically constrained computational objects. In this way, students constructed and used narratives that involved situating mathematical activities in familiar contexts, whilst simultaneously expressing these activities in ways which—at least potentially—transcend the particularities of the story told.

Is mathematics “Out there”?

The Opinion column offers mathematicians the apportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-in-chief, Chandler Davis.