The role of model comparison in young learners’ reasoning with statistical models and modeling

The goal of this study is to explore the role of model comparison, which is a key activity of young learners’ informal reasoning, with statistical models and modeling in the context of informal statistical inference. We suggest a framework to describe this reasoning (the RISM framework), and offer an illustrative case study of two-sixth graders showcasing its utility. In particular, we illustrate the benefit of untangling the informal modeling process into three separate, though not independent, modeling processes: modeling a conjecture, modeling data, and comparing them by means of a comparison model. This case study shows the possible progression of a comparison model, and its potential role as a catalyst for the development of the other two modeling processes. Finally, an expansion of our initial framework is discussed, highlighting the centrality of model comparisons.     

Dimensions of students’ views of themselves as learners of mathematics

Students’ views of themselves as learners of mathematics are a decisive parameter for their engagement and success in school. We are interested in students’ experiences with mathematics encompassing cognitive, emotional and motivational aspects. In particular, we focus on capturing the structural properties of affect related to mathematics. Participants in our study were 1,436 randomized chosen students of secondary schools from overall Finland. In the Finnish upper secondary school, there are two different syllabi for mathematics: the general and the advanced one. Schools were invited to organize the survey by one of their year 2 general syllabus courses and one of their year 2 advanced syllabus courses in grade 11. By means of factor analysis, we obtained seven dimensions in which students’ hold beliefs and emotions about mathematics partly intertwined with their motivational orientations. These dimensions are described by reliable scales, which allow outlining an average image of Finnish students’ views of themselves as learners of mathematics. Moreover, we analyzed relations between the seven dimensions and what kind of structure they generate. Thereby, a core of three high correlating dimensions could be identified, yielding different accentuations with regard to course choice.     

The stability of learners’ choices for real-life situations to be used in mathematics

One of the efforts to improve and enhance the performance and achievement in mathematics of learners is the incorporation of life-related contexts in mathematics teaching and assessments. These contexts are normally, with good reasons, decided upon by curriculum makers, textbook authors, teachers and constructors of examinations and tests. However, little or no consideration is given to whether students prefer and find these real-life situations interesting. There is also a dearth of studies dealing explicitly with the real-life situations learners prefer to deal with in mathematics. This issue was investigated and data on students’ choices for contextual issues to be used in mathematics were collected at two time periods. The results indicate that learners’ preferences for contextual situations to be used in mathematics remained fairly stable. It is concluded that real-life issues that learners highly prefer are not normally included in the school mathematics curriculum and that there is a need for a multidisciplinary approach to develop mathematical activities which take into account the expressed preferences of learners.     

Warfield’s Duality and Malcev’s Matrix

In this work, we investigate relations between Malcev’s matrices of a torsion-free group G of finite rank and Malcev’s matrices of groups Hom(R,G) and Hom(G,R), where G is a locally free group and R is a torsion-free group of rank 1.     

Bergmann’s Dilemma and Internalism’s Escape

Michael Bergmann has argued that internalist accounts of justification face an insoluble dilemma. This paper begins with an explanation of Bergmann??s dilemma. Next, I review some recent attempts to answer the dilemma, which I argue are insufficient to overcome it. The solution I propose presents an internalist account of justification through direct acquaintance. My thesis is that direct acquaintance can provide subjective epistemic assurance without falling prey to the quagmire of difficulties that Bergmann alleges all internalist accounts of justification cannot surmount.     

Hall’s and Kőnig’s theorem in graphs and hypergraphs

We investigate the relation between Hall’s theorem and K?nig’s theorem in graphs and hypergraphs. In particular, we characterize the graphs satisfying a deficiency version of Hall’s theorem, thereby showing that this class strictly contains all K?nig–Egerváry graphs. Furthermore, we give a generalization of Hall’s theorem to normal hypergraphs.     

On generalized Stokes’ and Brinkman’s equations with a pressure- and shear-dependent viscosity and drag coefficient

We study generalizations of the Darcy, Forchheimer, Brinkman and Stokes problem in which the viscosity and the drag coefficient depend on the shear rate and the pressure. We focus on existence of weak solutions to the problem, with the chief aim to capture as wide a group of viscosities and drag coefficients as mathematically feasible and to provide a theory that holds under minimal, not very restrictive conditions. Even in the case of generalized Stokes system, the established result answers a question on existence of weak solutions that has been open so far.     

Dedekind’s and Frege’s views on logic

A contextual and comparative analysis shows that Dedekind and Frege do not understand the terms “logic” and “arithmetic” in the same way. More specifically the meaning and the scope of the corresponding concepts are essentially different for them. Consequently Dedekind and Frege have different conceptions of the relationship between arithmetic and logic.     

Auditing Shaked and Shanthikumar’s ‘excess wealth’

A notion called “excess wealth” was introduced by Shaked and Shanthikumar around 1998 (Probab. Eng. Inf. Sci. 12:1–23, 1998). Subsequent to this, much has been written on it, mostly by Shaked and his colleagues; see Sordo (Insur. Math. Econ. 45(3):466–469, 2009) for a recent review. These works have appeared in the literatures of reliability theory and stochastic orderings. Since the term excess wealth connotes a measure of income inequality—much like its dual, poverty—it should have had an impact in economics and the econometric literature. This, it appears is not the case, at least to the extent that it should be. The purpose of this paper is to investigate the above disconnect by looking at the notion of excess wealth more carefully, but keeping in mind the angle of economics and income. Our conclusion is that an alternative definition of excess wealth better encapsulates what one means by a colloquial use of the term. Our motivation for being attracted to this topic arises from two angles. The first is that the stochastics of diagnostic and threat detection tests, in which we have an interest, has a strong bearing on indices of concentration like the Lorenz Curve, the Gini index, and the entropy. Thus the notion of excess wealth, which conveys a sense of income concentration should also be relevant to diagnostics. The second motivation is to honor Moshe Shaked, a prolific researcher and a friend of the first author, by developing a paper based on an idea that is co-attributed to him.     

On Andrunakievich’s chain and Koethe’s problem

In 1969 Andrunakievich asked whether one gets a ring without nonzero nil left ideals from an arbitrary ring R by factoring out the ideal A(R) which is the sum of all nil left ideals of R. Recently, it was shown that this problem is equivalent to Koethe’s problem. In this context one may consider the chain of ideals, which starts with A ₁(R) = A(R) ⊆ A ₂(R), where A ₂(R)/A ₁(R) = A(R/A ₁(R)), and extends by repeating this process. We study the properties of this chain and show that, assuming a negative solution of Koethe’s problem, this chain can terminate at any given ordinal number.     

Gauss’ lemma and valuation theory

Gauss’ lemma is not only critically important in showing that polynomial rings over unique factorization domains retain unique factorization; it unifies valuation theory. It figures centrally in Krull’s classical construction of valued fields with pre-described value groups, and plays a crucial role in our new short proof of the Ohm-Ja?ard-Kaplansky theorem on Bezout domains with given lattice-ordered abelian groups. Furthermore, Eisenstein’s criterion on the irreducibility of polynomials as well as Chao’s beautiful extension of Eisenstein’s criterion over arbitrary domains, in particular over Dedekind domains, are also obvious consequences of Gauss’ lemma. We conclude with a new result which provides a Gauss’ lemma for Hermite rings.     

Normality and Montel’s Theorem

In this article, we prove a normality criterion for a family of meromorphic functions having zeros with some multiplicity which involves sharing of a holomorphic function by the members of the family. Our result generalizes Montel’s normality test in a certain sense.     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008     

Random and algebraic permutations’ statistics

An algebraic permutation $\hat{A}\in S(N=n^{m})$ is the permutation of the N points of the finite torus ? _n ^m , realized by a linear operator A∈SL(m,? _n ). The statistical properties of algebraic permutations are quite different from those of random permutations of N points. For instance, the period length T(A) grows superexponentially with N for some (random) permutations A of N elements, whereas $T(\hat{A})$ is bounded by a power of N for algebraic permutations  $\hat{A}$ . The paper also contains a strange mean asymptotics formula for the number of points of the finite projective line P¹(? _n ) in terms of the zeta function.     

Kobayashi’s and Teichmuller’s Metrics and Bers Complex Manifold Structure on Circle Diffeomorphisms

Given a modulus of continuity ω,we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk.We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms.Using these distortion properties,we give the Bers complex manifold structure on the Teichm(u| ")ller space TC^1+H as the union of over all0 <α≤1,which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure.Furthermore,we prove that with the Bers complex manifold structure on TC^1+H ,Kobayashi’s metric and Teichmuller’s metric coincide.     

A link between Wilson’s and d’Alembert’s functional equations

If \({f, g : G \to \mathbb{C}}\), f ≠ 0, is a solution of Wilson’s functional equation on a group G, then g is a d’Alembert function.     

Brownian motion with killing and reflection and the ‘‘hot–spots’’ problem

We investigate the hot–spots property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the hot–spots property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other.Supported in part by NSF Grant # 9700585-DMSSupported in part by NSF Grant # 0203961-DMS     

Teachers’ metacognitive and heuristic approaches to word problem solving: analysis and impact on students’ beliefs and performance

We conducted a 7-month video-based study in two sixth-grade classrooms focusing on teachers’ metacognitive and heuristic approaches to problem solving. All problem-solving lessons were analysed regarding the extent to which teachers implemented a metacognitive model and addressed a set of eight heuristics. We observed clear differences between both teachers’ instructional approaches. Besides, we examined teachers’ and students’ beliefs about the degree to which metacognitive and heuristic skills were addressed in their classrooms and observed that participants’ beliefs were overall in line with our observations of teachers’ instructional approaches. In addition, we investigated how students’ problem-solving skills developed as a result of teachers’ instructional approaches. A positive relationship between students’ spontaneous application of heuristics to solve non-routine word problems and teachers’ references to these skills in their problem-solving lessons was found. However, this increase in the application of heuristics did not result in students’ better performance on these non-routine word problems.     

Bartnik’s Mass and Hamilton’s Modified Ricci Flow

We provide estimates on the Bartnik mass of constant mean curvature surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new notion we call the asphericity mass. The asphericity mass is defined by applying Hamilton’s modified Ricci flow and depends only upon the restricted metric of the surface and not on its mean curvature. The theorem is proven by studying a class of asymptotically flat Riemannian manifolds foliated by surfaces satisfying Hamilton’s modified Ricci flow with prescribed scalar curvature. Such manifolds were first constructed by the first author in her dissertation conducted under the supervision of M. T. Wang. We make a further study of this class of manifolds which we denote Ham₃, bounding the ADM masses of such manifolds and analyzing the rigid case when the Hawking mass of the inner surface of the manifold agrees with its ADM mass.