Constructivism,Factoring, and Beliefs

The errors made by remedial intermediate algebra students in factoring polynomials were analyzed in the light of student definitions of factoring. Certain belief sets about factoring are found to logically entail many of the errors made. The vocabulary of epistemic semantics is used to describe how beliefs are changed. It is suggested that belief-based teaching can be successful in teaching factoring.     

On the Coherence of Wittgensteinian Constructivism

Michael Dummett presents a modus tollens argument against a Wittgensteinian conception of meaning. In a series of papers, Dummett claims that Wittgensteinian considerations entail strict finitism. However, by a “sorites argument”, Dummett argues that strict finitism is incoherent and therefore questions these Wittgensteinian considerations.In this paper, I will argue that Dummett’s sorites argument fails to undermine strict finitism. I will claim that the argument is based on two questionable assumptions regarding some strict finitist sets of natural numbers. It will be shown that strict finitism entails none of these assumptions. Hence, the argument does not prove that the view is internally incoherent, and consequently, Dummett fails to undermine the Wittgensteinian conception of meaning.     

Beyond the dichotomies

The present paper starts from the assumption that a semiotic approach might provide a fresh start in reconciling prevailing dichotomies in educational reflection and research. Especially as regards the popular dichotomies of individual against social learning and constructive against receptive learning. Three exemplary sections will illustrate the salient features of a semiotic approach: sign process seen as mediating means; sign processes as creating networks of objects, signs and interpretants; and the metaphor of map and territory as a relation of sign and activity. Throughout the paper it is tried to capitalize from the tension between the semiotic approaches of Peirce and Vygotskij. Der vorliegende Beitrag gründet sich auf die Annahme, dass ein semiotischer Ansatz möglicherweise dem Versuch zu neuem Erfolg verhelfen kann, herrschende Dicho-tomien in der pädagogischen und didaktischen forschung zu überwinden. Dies betrifft besounders die beliebte Gegenüber-stellung von sozial und individuall und von konstruktiv und rezeptiv beim Lernen. In drei exemplarischen Abschnitten wird die Fruchtbarkeit einer semiotischen Perspektive demonstriert: Zeichenprozess als Mittel und Vermittlung. Zeichenprozesse als Vernetzung von Objekten, Zeichen und Interpretanten; und die Beziehung von Landkarte und Territorium als Metapher für die Beziehung von Zeichen und Tätigkeit. Die Spannung zwischen den semiotischen Perspektiven von Peirce und Vygotskij bildet dabei einen permanenten Bezugspunkt.     

Berkeley and Beyond

This article is excerpted from a book by the author, to be published in February by Basic Books, Inc. The volume, The Conflict of Generations, represents an effort to view student movements in various countries from historical and sociological perspectives. This article is taken from Chapter Nine: The Berkeley Student Movement, in which the author traces the history of the Berkeley rebellion, its results, and what he believes to be its consequences for the nation.     

Beyond the Chu-construction

Starting from symmetric monoidal closed (= autonomous) categories, Po-Hsiang Chu showed how to construct new *-autonomous categories, i.e., autonomous categories that are self-dual by virtue of having a dualizing object. Recently, Michael Barr extended this to the nonsymmetric, but closed, case, utilizing monads and modules between them. Since these notions are well-understood for bicategories, we introduce a notion of cyclic *-autonomy for these that implies closedness and, moreover, is inherited when forming bicategories of monads and of interpolads. Since the initial step of Barr's construction also carries over to the bicategorical setting, we recover his main result as an easy corollary. Furthermore, the Chu-construction at this level may be viewed as a procedure for turning the endo-1-cells of a closed bicategory into the objects of a new closed bicategory, and hence conceptually is similar to constructing bicategories of monads and of interpolads.     

Beyond internal OR groups

The paper reviews the prospects of OR applied by dispersed practitioners, small providers of OR services, academics and OR consultants, in other words OR applied by those not located inside internal OR groups. Many in the UK have regarded the activities of OR groups as the defining practice of our profession. The Survival and Success of OR Groups (SSOR) study indicates that within a fairly stable population of OR practitioners the number of internal OR groups is in decline while the number of external, consulting groups is growing. There are good reasons in terms of the diffusion of innovation for assuming that this trend will continue. Against that possibility, the OR profession in the UK needs to reassess the defining aspects of OR and the implications for the activities of the profession in the future. Using the language of strategy, the profession needs to identify its core competencies, define how they are distinctive, consider how they can be developed and maintained, and determine through which channels they can be delivered. One of the key battlegrounds is within the management consultancies where the competing approaches of management science and organisational development vie for supremacy.     

Pure Cognitivism and Beyond

The article begins with Jonathan Dancy’s attempt to refute the Humean Theory of Motivation. It first spells out Dancy’s argument for his alternative position, the view he labels ‘Pure Cognitivism’, according to which what motivate are always beliefs, never desires. The article next argues that Dancy’s argument for his position is flawed. On the one hand, it is not true that desire always comes with motivation in the agent; on the other, even if this was the case, it would still not follow that desire is identical with the state of being motivated. When this negative work is done, the article turns to some positive, albeit admittedly tentative remarks about what sort of cognitivist theory of motivation one should endorse. The aim at this point is not to present a brand new theory, but rather to sketch an alternative that stems from what Dancy himself says and is in line with many of his endorsed commitments in other areas. In this way, by moving beyond Pure Cognitivism, the paper sketches a different, but still ‘Dancyesque’ theory of motivation.     

Beyond odious and evil

In a recent post on the Seqfan list the third author proposed conjectures concerning the summatory function of odious numbers (i.e., of numbers whose sum of binary digits is odd), and of evil numbers (i.e., of numbers whose sum of binary digits is even). We prove these conjectures here. We will also study the sequences of “generalized” odious and evil numbers, and their iterations, giving in particular a characterization of the sequences of usual odious and evil numbers in terms of functional equations satisfied by their compositions.     

Beyond the Trudinger-Moser supremum

Let Ω be a bounded, smooth domain in ${\mathbb{R}^2}$ . We consider the functional $$I(u) = \int_\Omega e^{u^2}\,dx$$ in the supercritical Trudinger-Moser regime, i.e. for ${\int_\Omega |\nabla u|^2dx > 4\pi}$ . More precisely, we are looking for critical points of I(u) in the class of functions ${u \in H_0^1 (\Omega )}$ such that ${\int_\Omega |\nabla u|^2 \, dx = 4\, \pi \, k\, (1+\alpha)}$ , for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with ${\int_\Omega |\nabla u|^2dx = 4\pi(1 + \alpha)}$ for any bounded domain Ω, 2-peak critical points with ${\int_\Omega |\nabla u|^2dx = 8\pi(1 + \alpha)}$ for non-simply connected domains Ω, and k-peak critical points with ${\int_\Omega |\nabla u|^2 dx = 4k \pi(1 + \alpha)}$ if Ω is an annulus.     

Brooks' Theorem and Beyond

We collect some of our favorite proofs of Brooks' Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon–Tarsi orientations, since analogs of Brooks' Theorem hold in each context. We conclude with two conjectures along the lines of Brooks' Theorem that are much stronger, the Borodin–Kostochka Conjecture and Reed's Conjecture.     

Beyond elementary catastrophe theory

We give a brief discussion of the relations between elementary catastrophe theory, general catastrophe theory, singularity theory, bifurcation theory, and topological dynamics. This is intended to clarify the status, and potential applicability, of “catastrophe theory,” a phrase used by different authors and at different times with different meanings. Catastrophe theory has often been criticized for (supposed) applicability only to gradient systems of differential equations; but properly speaking this criticism can apply only to the elementary version of the theory (where it is in any case wrong). Roughly speaking, elementary catastrophe theory deals with the singularities of real-valued functions, general catastrophe theory with singularities of flows. Between these lies singularity theory, which deals with vector-valued functions. All relate strongly to bifurcation theory and topological dynamics. The issue is more subtle than it appears to be, and we describe an example where elementary catastrophe theory has been used to solve a long-standing problem about nongradient flows: degenerate Hopf bifurcation.