Development of Counting Skills: Role of Spontaneous Focusing on Numerosity and Subitizing-Based Enumeration

Children differ in how much they spontaneously pay attention to quantitative aspects of their natural environment. We studied how this spontaneous tendency to focus on numerosity (SFON) is related to subitizing-based enumeration and verbal and object counting skills. In this exploratory study, children were tested individually at the age of 4-5 years on these skills. Results showed 2 primary relationships in children's number skills development. Performance in a number sequence production task, which is closely related to ordinal number sequence without reference to cardinality, is directly associated with SFON. Second, the association of SFON and object counting skills, which require relating cardinal and ordinal aspects of number, is mediated by subitizing-based enumeration. This suggests that there are multiple pathways to enumeration skills during development.     

Development of Counting Skills: Role of Spontaneous Focusing on Numerosity and Subitizing-Based Enumeration

Children differ in how much they spontaneously pay attention to quantitative aspects of their natural environment. We studied how this spontaneous tendency to focus on numerosity (SFON) is related to subitizing-based enumeration and verbal and object counting skills. In this exploratory study, children were tested individually at the age of 4–5 years on these skills. Results showed 2 primary relationships in children's number skills development. Performance in a number sequence production task, which is closely related to ordinal number sequence without reference to cardinality, is directly associated with SFON. Second, the association of SFON and object counting skills, which require relating cardinal and ordinal aspects of number, is mediated by subitizing-based enumeration. This suggests that there are multiple pathways to enumeration skills during development.     

Counting metric spaces

If κ is a cardinal number, then any class of mutually non-homeomorphic metric spaces of size κ must be a set whose cardinality cannot exceed 2 ^κ . Our main result is a vivid construction of 2 ^κ mutually non-homeomorphic complete and both path connected and locally path connected metric spaces of size κ for each cardinal number κ from continuum up. Additionally we also deal with counting problems concerning countable metric spaces and Euclidean spaces.     

A Direct Combinatorial Proof of a Positivity Result

Objects lying in four different boxes are rearranged in such a way that the number of objects in each box stays the same. Askey, Ismail, and Koornwinder proved that the cardinality of the set of rearrangements for which the number of objects changing boxes is even exceeds the cardinality of the set of rearrangements for which that number is odd. We give a simple counting proof of this fact.     

Cardinality bounds via covers by compact sets

Acta Mathematica Hungarica - We establish results concerning covers of spaces by compact and related sets. Several cardinality bounds follow as corollaries. Introducing the cardinal invariant...     

Mary Wynne Warner (1932-1998)

Mary Warner, as she was mainly known in the mathematical world,died in April 1998. At a time when few women mathematiciansreached the top in their profession, she succeeded in doingso through her ability and determination. Her research contributionswere commemorated at a recent international conference on fuzzytopology, the field in which she was one of the pioneers andrecognized as one of the leading figures for the past thirtyyears. She was also an outstanding teacher. But to understandher achievements properly it is necessary to know somethingof her life.     

“Can you help me count these pennies?”: Surfacing preschoolers’ understandings of counting

ABSTRACT

Capturing the breadth and variety of children’s understanding is critical if studies of children’s mathematical thinking are to inform policy and practice in early childhood education. This article presents an investigation of young children’s counting. Detailed coding and analyses of assessment interviews with 476 preschoolers revealed understandings that would be overlooked by solely assessing the accuracy of their responses. In particular, many children demonstrated understandings of counting principles on a challenging task that were not captured by other, simpler tasks. We conclude that common approaches to capturing young children’s mathematical understanding are likely underestimating their capabilities. This study contributes to researchers’ understanding of what making sense of counting looks and sounds like for preschool age children (3–5 years), the development and relations among counting principles (one-to-one, cardinal, and patterns of the number sequence), and the affordances of challenging, open-ended tasks. We close by considering the implications of recognizing and building from what children know and can do for researchers, practitioners, and policymakers.     

Subitising activity relative to units construction: a case study

Subitising, a quick apprehension of the numerosity of a small set of items, has been found to change from an individual's reliance on perceptual to conceptual processes. In this study, we utilised a constructivist teaching experiment methodology to investigate how the subitising activity of one preschool student, Amy, related to her construction of prenumerical units. Subitising and counting tasks were designed to assess and perturb Amy's thinking relative to her construction of units, and to observe changes in Amy's activity associated with the different tasks. Findings indicate that as Amy's subitising activity changed from perceptual to conceptual, she constructed subitised motor units and subitised figurative units. Implications of this study suggest that the construction of subitised units may support young children's later development of arithmetic units.     

Constructions and applications of rigid spaces,I

The two major results proved are: (1) The category TOP of topological spaces contains a complete nonreflective subcategory. (2) Under the assumption (2^m)⁺ < 2^2m, for each infinite cardinal number m there exists a Hausdorff space of cardinality m, in which the identity map is the only nonconstant continuous self-map. The first result is proved as a consequence of another result which answers a question of Herrlich concerning strongly rigid spaces; it is then used to settle in the negative a conjecture concerning the characterization of reflective subcategories in TOP. In addition, several interesting spaces are constructed.     

Universality among graphs omitting a complete bipartite graph

For cardinals λ,κ,θ we consider the class of graphs of cardinality λ which has no subgraph which is (κ,θ)-complete bipartite graph. The question is whether in such a class there is a universal one under (weak) embedding. We solve this problem completely under GCH. Under various assumptions mostly related to cardinal arithmetic we prove non-existence of universals for this problem. We also look at combinatorial properties useful for those problems concerning κ-dense families.     

Making sense of instruction on fractions when a student lacks necessary fractional schemes: The case of Tim

This paper critically examines the discrepancies among the pre-requisite fractional concepts assumed by a curricular unit on operations with fractions, the teacher's assumptions about those concepts and a particular student's understanding of fractions. The paper focuses on the case of one student (Tim) in the teacher's 6th grade class who was interviewed by one of the authors once a week during the teaching of the unit. The teaching materials and the teacher's instruction were based on the assumption that students understood the concept of a unit fraction as being one of several equal parts of a given whole. The teacher neither emphasized the need for equal parts nor the part-to-whole relation. The teacher's reasonable assumptions about her students’ understanding of fractions were severely challenged by the cognitive constructs that Tim exhibited during his first two interviews. When she viewed tapes of the class instruction and the interviews with Tim she realized Tim lacked essential constructs to make sense of her instruction. She subsequently made adjustments in her instruction, making effective use of more appropriate representations based on tasks from the unit that we modified and used with Tim in our interviews. These adjustments helped Tim to construct partitioning operations and an appropriate unit fractional scheme. This study illustrates the importance of coming to understand a student's mathematical activity in terms of possible conceptual schemes and modifying instructional strategies to build on those schemes. The coordinated design of the research study facilitated these instructional modifications.     

Fuzzy幂群的基数定理

文（１）提出了幂群的概念，给出了幂群中各元素是等势的基数定理，文（２）提出了Ｆｕｚｚｙ幂群的概念，但没研究其中各元素的基数问题，本文深入研究这一问题，得到了由Ｄ．Ｄｕｂｏｉｓ等在文（３）中提出的和由李洪兴等在文（４）中提出的两种Ｆｕｚｚｙ集基数形式下的Ｆｕｚｚｙ幂群的基数定理，并给出了Ｆｕｚｚｙ幂群中与基数有关的若干结果。     

Some universal properties of the category of clones

In [6], O. C. García and W. Taylor asked if the breadth of the lattice of interpretability types of varieties is uncountable. The present paper solves the problem by two different constructions. Both of them show that any cardinal number is the cardinality of an antichain in the named lattice and that the existence of a proper class antichain is equivalent to the negation of Vopěnka's principle. The first construction gives in a way a minimal solution of the problem, whereas the second one gives stronger results about the category of clones. Received November 12, 1999; accepted in final form June 19, 2001.     

A model of students’ combinatorial thinking

Combinatorial topics have become increasingly prevalent in K-12 and undergraduate curricula, yet research on combinatorics education indicates that students face difficulties when solving counting problems. The research community has not yet addressed students’ ways of thinking at a level that facilitates deeper understanding of how students conceptualize counting problems. To this end, a model of students’ combinatorial thinking was empirically and theoretically developed; it represents a conceptual analysis of students’ thinking related to counting and has been refined through analyzing students’ counting activity. In this paper, the model is presented, and relationships between formulas/expressions, counting processes, and sets of outcomes are elaborated. Additionally, the usefulness and potential explanatory power of the model are demonstrated through examining data both from a study the author conducted, and from existing literature on combinatorics education.     

On large half-factorial sets in elementary<Emphasis Type="Italic">p</Emphasis>-groups: Maximal cardinality and structural characterization

Half-factoriality is a central concept in the theory of non-unique factorization, with applications for instance in algebraic number theory. A subsetG ₀ of an abelian group is called half-factorial if the block monoid overG ₀, which is the monoid of all zero-sum sequences of elements ofG ₀, is a half-factorial monoid. In this paper we study half-factorial sets with large cardinality in elementaryp-groups. First, we determine the maximal cardinality of such half-factorial sets, and generalize a result which has been only known for groups of even rank. Second, we characterize the structure of all half-factorial sets with large cardinality (in a sense made precise in the paper). Both results have a direct application in the study of some counting functions related to factorization properties of algebraic integers. This work was supported by the Austrian Science Fund FWF (Project P16770-N12) and by the Austrian-French Program ‘Amadeus 2003–2004’.     

A climbing girl's reflections about angles

The main research question in this paper is whether a climbing discourse can be a resource for a school-geometry discourse. The text is based on a 12-year old girl's story from an exciting climbing trip during her summer holiday. The girl uncovers some of her knowledge that had been invisible to her; she is guided to see some relations between her climbing and her understanding of angles. In the beginning, this girl believes her story does not concern angles at all. The tools for uncovering angles in her story are based on different levels of visibility and objects of the climbing discourse combined with different conceptions of space. The girl develops her consciousness about angles as natural elements in her climbing activity and she is guided to see the angle as an object of her climbing discourse.     

Reflexive abelian groups and measurable cardinals and full MAD families

Answering problem (DG) of [1], [2], we show that there is a reflexive group of cardinality equal to the first measurable cardinal.     

Possible size of an ultrapower of \omega

Let be the first infinite ordinal (or the set of all natural numbers) with the usual order . In § 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of , whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In § 2 we construct several -Archimedean ultrapowers of under some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exist a -Archimedean ultrapower of for some uncountable cardinal . This answers a question in [8], modulo the assumption of measurability. Received: 19 November 1996     

Showing by Avowing

Dorit Bar-On aims to account for the distinctive security of avowals by appealing to expression. She officially commits herself only to a negative characterization of expression, contending that expressive behavior is not epistemically based in self-judgments. I argue that her account of avowals, if it relies exclusively on this negative account of expression, can't achieve the explanatory depth she claims for it. Bar-On does explore the possibility that expression is a kind of perception-enabling showing. If she endorsed this positive account, her argument would re-gain an explanatory advantage over its rivals. But extending this account to linguistic expressive behavior would bring Bar-On very close to constitutive accounts of first-person authority.     

Binary words avoided by the Thue-Morse sequence

This paper deals with one problem concerning avoidable words. Namely, the set of words over a two-letter alphabet avoided by the Thue-Morse sequence is described. This set is a fully invariant ideal of a two-generated free semigroup, and we find its aminimal generating set.