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.     

Neural correlates of counting large numerosity

The mastery of counting numerosities larger than those correctly estimated by infants or non-human species is an important foundation for the development of higher level calculation skills. The cognitive processes involved in counting are related to spatial attention, language, and number processing. However, the respective involvement of language- and/or visuo-spatial-based brain systems during counting is still under debate. In the present functional magnetic resonance imaging study, we asked 27 right-handed participants to perform an enumeration task on visual arrays of bars that varied in numerosity. Each enumeration condition was contrasted to a color-detection condition that was numerically and spatially matched to the counting condition. The results showed a behavioral discontinuity in response time slopes between large (6–10) and small (1–5) numerosities during enumeration, suggesting that during large enumeration, participants engaged counting processes. Comparing brain regional activity during the enumeration of large numerosity to the enumeration of smaller numerosity, we found increased activation in the bilateral fronto-parietal attentional network, the inferior parietal gyri/intraparietal sulci, and the left ventral premotor and left inferior temporal areas. These results indicated that in adults who master enumeration, counting more than five items requires the strong involvement of spatial attention and eye movements, as well as numerical magnitude processes. Counting large numerosity also recruited verbal working memory areas, subtending a subvocal articulatory code and a visual representation of numbers.     

Kindergartners’ Spontaneous Focusing on Numerosity in Relation to Their Number-Related Utterances During Numerical Picture Book Reading

This study investigated the relationship between kindergartners’ Spontaneous Focusing on Numerosity (SFON) and their number-related utterances during numerical picture book reading. Forty-eight 4- to 5-year-olds were individually interviewed via a SFON Imitation Task and a numerical picture book reading activity. We expected differences in the frequency of number-related utterances during picture book reading between children with a higher SFON score, providing more number-related utterances, and children with a lower SFON score. Our results showed large inter-individual differences in both kindergartners’ SFON and the frequency of their number-related utterances during picture book reading, yet SFON was not related to the frequency of number-related utterances. This unexpected result is discussed in terms of its scientific, methodological, and educational implications.     

Das Kardinalzahlkonzept

This case-study investigates different aspects of the concept of cardinality of an eighteen-year-old student with mental retardation. At the age of six she could not relate number words, finger and objects in counting. These errors still persist in the classroom situation. This investigation shows that nevertheless her concept of cardinality is fairly highly developed. She knows that in counting she must match number words and objects one to one, the number word sequence she uses is stable, and her insight into the irrelevance of order of enumeration when counting, which she finds by trial, is a sign of the robustness of her cardinal concept. She also understands the relationships of equivalence and order of sets, and she solves arithmetical problems by counting on or down, which means that she understands the number words as cardinal and at the same time as sequence numbers. Errors occur in complex situations, where several components have to be considered. But her concept of cardinality is also incomplete: she has special difficulties concerning counting out objects bundled in tens. The same problems occur when she uses multidigit numbers: she does not see a ten-unit as composed of ten single unit items, that is to say, she replaces the hierarchic structure of the number sequence by a concatenated one. These difficulties must be interpreted as a consequence of her special weakness concerning synthetic thinking and simultaneous performing, as similar patterns can be seen in her spatial perception and in her speech. In the syntactic structure of her utterances, too, the combination of simple entities to complicated units is replaced by a mere concatenation. This means that due to brain dysfunction her behavior is determined by a particular pattern which repeatedly appears intrapersonally, and which is characteristic of some mentally retarded persons though not of all of them. Evidently mathematical thinking is also not a determined system, but a variable one. Mentally retarded students may therefore have great difficulties concerning some areas and at the same time make better progress in others. In particular, difficulties in counting objects are no obstacle to knowledge of cardinality.     

Teaching young children decomposition strategies to solve addition problems: An experimental study

The ability to count has traditionally been considered an important milestone in children's development of number sense. However, using counting (e.g., counting on, counting all) strategies to solve addition problems is not the best way for children to achieve their full mathematical potential and to prepare them to develop more complex and advanced computational skills. In this experimental study, we demonstrated that it was possible to teach children aged 5-6 to use decomposition strategy and thus reduced their reliance on counting to solve addition problems. The study further showed that children’ ability to adopt efficient strategies was related to their systematic knowledge of the part-part-whole relationship of the numbers 1-10.     

Stories about Groups and Sequences

The main theme of this article is that counting orbitsof an infinite permutation group on finite subsets or tuplesis very closely related to combinatorial enumeration; this pointof view ties together various disparate ``stories'. Among theseare reconstruction problems, the relation between connected andarbitrary graphs, the enumeration of N-free posets, and someof the combinatorics of Stirling numbers.     

The research and progress of the enumeration of lattice paths

The enumeration of lattice paths is an important counting model in enumerative combinatorics. Because it can provide powerful methods and technical support in the study of discrete structural objects in different disciplines, it has attracted much attention and is a hot research field. In this paper, we summarize two kinds of the lattice path counting models that are single lattice paths and family of nonintersecting lattice paths and their applications in terms of the change of dimensions, steps, constrained conditions, the positions of starting and end points, and so on. (1) The progress of classical lattice path such as Dyck lattice is introduced. (2) A method to study the enumeration of lattice paths problem by generating function is introduced. (3) Some methods of studying the enumeration of lattice paths problem by matrix are introduced. (4) The family of lattice paths problem and some counting methods are introduced. (5) Some applications of family of lattice paths in symmetric function theory are introduced, and a related open problem is proposed.     

Counting the number of independent sets in chordal graphs

We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of independent sets of a fixed size. With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant time per output. On the other hand, we prove that the following problems for a chordal graph are #P-complete: (1) counting the number of maximal independent sets; (2) counting the number of minimum maximal independent sets. With similar ideas, we also show that finding a minimum weighted maximal independent set in a chordal graph is NP-hard, and even hard to approximate.     

An application of Pólya's theory of counting to an enumeration problem arising in quadratic form theory

In this paper we apply Pólya's theory of counting to compute the number of isomorphism types of reduced Witt rings of fields with a fixed finite number of orderings. The problem is first transformed into a graph theoretical enumeration problem involving unlabeled rooted trees with certain numbers assigned to the vertices.     

A Logarithmic Connection for Circular Permutation Enumeration

A general theorem is obtained for the enumeration of permutations equivalent under cyclic rotation. This result gives the generating function as the logarithm of a determinant which arises in the enumeration of a related linear permutation enumeration. Applications of this theorem are given to a number of classical enumerative problems.     

Multidimensional permanents in enumeration problems

Permanents are an effective tool for solving many combinatorial problems about enumeration. The respective theory is well developed and has numerous applications. In this paper the problem of counting the number of distinct 1-perfect binary codes is reduced to computing a generalized permanent of a particularly constructed multidimensional matrix.     

ZFC的模型的上限序数的一个性质

S1 引言 Forcing方法假设存在ZFC的一个可数可传的模型M。记满足αM的最小序数α为,显然M中一切序数所成的集合即。由于M是ZFC的模型,故应具有某些性质。本文证明了它满足关系,故为ε数或1级关键数,进而证明了是H级关键数(H为任意自然数)。文中的记号等引用。     

Arithmetical Semigroups Related to Trees and Polyhedra,II — Maps on Surfaces

This paper extends recent investigations by Arnold Knopfmacher and John Knopfmacher [10] of asymptotic enumeration questions concerning finite graphs, trees and polyhedra. The present emphasis is on the counting of non‐isomorphic maps of not necessarily connected finite graphs on arbitrary surfaces. A significant aid towards this goal is provided by an extended abstract prime number theorem, based partly on more delicate tools of analysis due to W. K. Hayman [8].     

Stories about groups and sequences

The main theme of this article is that counting orbits of an infinite permutation group on finite subsets or tuples is very closely related to combinatorial enumeration; this point of view ties together various disparate stories. Among these are reconstruction problems, the relation between connected and arbitrary graphs, the enumeration of N-free posets, and some of the combinatorics of Stirling numbers.Dedicated to Hanfried Lenz on the occasion of his 80th birthday.     

A scheduling algorithm for open pit mines

An open pit (opencast) mine can be described by a three-dimensionalarray of blocks, each of which is assigned a number of valuesdefining its characteristics. Scheduling an open pit consistsin finding a sequence in which the blocks should be removedfrom the mine in order to maximize the total discounted profitfrom the mine subject to a variety of technical and economicconstraints. This paper proposes to model the mine-schedulingproblem as one of sequential optimization, and develops an algorithmfor its solution. To overcome the difficulty caused by an extremelylarge number of states in the problem at hand, we consider atechnique which is related to dynamic programming but avoidsthe complete enumeration of the state space. Our algorithm isa combination of this technique with powerful heuristics derivedfrom the specific properties of open pit mining.     

Part-whole number knowledge in preschool children

Ability to reflect on a number as an object of thought, and to isolate its constituent parts, is basic to a deep knowledge of arithmetic, as well as much practical and applied mathematical problem solving. Part-whole reasoning and counting are closely related in children’s numerical development. The mathematical behavior of young children in part-whole problem settings was examined by using a dynamic problem situation, in which a small set of items was partitioned such that one of the subsets remained perceptually inaccessible. Issues addressed include the problem solving strategies successful children used, adaptations children make in response to successive administrations of the task over time, and characterizations of children’s mathematical thinking based on their responses to the task.     

Counting labelled trees with given indegree sequence

For a labelled tree on the vertex set [n]:={1,2,…,n}, define the direction of each edge ij to be i→j if i<j. The indegree sequence of T can be considered as a partition λ?n−1. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on [n] with indegree sequence corresponding to a partition λ. In this paper we give two proofs of Cotterill's conjecture: one is “semi-combinatorial” based on induction, the other is a bijective proof.     

Aggregation of ordinal information

Our interest is with the fusion of information which has an ordinal structure. Information fusion in this environment requires the availability of ordinal aggregation operations. Basic ordinal operations are first introduced. Next we investigate conjunctive and disjunction aggregations of ordinal information. The idea of a pseudo-log in the ordinal environment is presented. We discuss the introduction of a zero like point on an ordinal scale along with the related ideas of bipolarity (positive and negative values) and uni-norm aggregation operators. We introduce mean like aggregation operators as well weighted averages on a ordinal scale. The problem of selecting between ordinal models is considered.     

Enumeration problems for classes of self-similar graphs

We describe a general construction principle for a class of self-similar graphs. For various enumeration problems, we show that this construction leads to polynomial systems of recurrences and provide methods to solve these recurrences asymptotically. This is shown for different examples involving classical self-similar graphs such as the Sierpiński graphs. The enumeration problems we investigate include counting independent subsets, matchings and connected subsets.     

Complete voting systems with two classes of voters: weightedness and counting

We investigate voting systems with two classes of voters, for which there is a hierarchy giving each member of the stronger class more influence or important than each member of the weaker class. We deduce for voting systems one important counting fact that allows determining how many of them are for a given number of voters. In fact, the number of these systems follows a Fibonacci sequence with a smooth polynomial variation on the number of voters. On the other hand, we classify by means of some parameters which of these systems are weighted. This result allows us to state an asymptotic conjecture which is opposed to what occurs for symmetric games.