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.     

Numerosities of labelled sets: a new way of counting

The notions of “labelled set” and “numerosity” are introduced to generalize the counting process of finite sets. The resulting numbers, called numerosities, are then used to develop nonstandard analysis. The existence of a numerosity function is equivalent to the existence of a selective ultrafilter, hence it is independent of the axioms of ZFC.     

Overcoming intuitive interference in mathematics: insights from behavioral, brain imaging and intervention studies

It is well known that many students encounter difficulties when solving problems in mathematics. Research indicates that some of these difficulties may stem from intuitive interference with formal/logical reasoning. Our research aims at deepening the understanding of these difficulties and their underlying reasoning mechanisms to help students overcome them. For this purpose we carried out behavioral, brain imaging and intervention studies focusing on a previously demonstrated obstacle in mathematics education. The literature reports that many students believe that shapes with a larger area must have a larger perimeter. We measured the accuracy of responses, reaction time, and neural correlates (by fMRI) while participants compared the perimeters of geometrical shapes in two conditions: (1) congruent, in which correct response was in line with intuitive reasoning (larger area–larger perimeter) and (2) incongruent, in which the correct answer was counterintuitive. In the incongruent condition, accuracy dropped and reaction time for correct responses was longer than in the congruent condition. The congruent condition activated bilateral parietal brain areas, known to be involved in the comparison of quantities, while correctly answering the incongruent condition activated bilateral prefrontal areas known for their executive control over other brain regions. The intervention, during which students’ attention was drawn to the relevant variable, increased accuracy in the incongruent condition, while reaction times increased in both congruent and incongruent conditions. The findings of the three studies point to the importance of control mechanisms in overcoming intuitive interference in mathematics. Overall, it appears that adding a cognitive neuroscience perspective to mathematics education research can contribute to a better understanding of students’ difficulties and reasoning processes. Such information is important for educational research and practice.     

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.     

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.     

Quantitative Estimation: One, Two, or Three Abilities?

The mathematics education literature refers to 3 types of quantitative estimation skill: numerosity, measurement, and computational estimation. The psychometric literature includes a confusing array of tests intended to define quantitative estimation. This study examined relations among tests for numerosity, measurement, and computational estimation, and recognized tests for numerical facility and quantitative reasoning using principal components analysis. 2 components were identified. The first component aligned computational estimation with numerical facility and general quantitative reasoning. The second component included the tests of numerosity and measurement estimation. It was suggested that this second component might be related to spatial ability. Implications for mathematics education and assessment are discussed.     

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.     

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.     

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.     

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.     

关于K圈标号图的计数

本文研究含K个圈的标号图的计数问题,得出了有n个标定顶点且有K个交于一点的圈的连通图的计数公式,并得到了双圈连通标号图的计数公式,从而解决了K-2时连通图的计数问题。     

Enumeration of vacuously transitive relations

The concept of vacuously transitive relation is defined and under an appropriate isomorphism, the equivalence classes of such relations are enumerated by use of the power group enumeration theorem [3]. This enumeration is shown to be combinatorially equivalent to a counting series derived by Harary and Prins [4] for certain kinds of bicolored graphs. Finally, it is shown that the main result can be extended to cover two additional cases of interest.     

Dynamic Enumeration of All Mixed Cells

The polyhedral homotopy method, which has been known as a powerful numerical method for computing all isolated zeros of a polynomial system, requires all mixed cells of the support of the system to construct a family of homotopy functions. The mixed cells are reformulated in terms of a linear inequality system with an additional combinatorial condition. An enumeration tree is constructed among a family of linear inequality systems induced from it such that every mixed cell corresponds to a unique feasible leaf node, and the depth-first search is applied to the enumeration tree for finding all the feasible leaf nodes. How to construct such an enumeration tree is crucial in computational efficiency. This paper proposes a dynamic construction of an enumeration tree, which branches each parent node into its child nodes so that the number of feasible child nodes is expected to be small; hence we can prune many subtrees which do not contain any mixed cell. Numerical results exhibit that the proposed dynamic construction of an enumeration tree works very efficiently for large scale polynomial systems; for example, it generated all mixed cells of the cyclic-15 problem for the first time in less than 16 hours.     

Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function

Using traditional educational research methods, it is difficult to assess students’ understanding of mathematical concepts, even though qualitative methods such as task observation and interviews provide some useful information. It has now become possible to use functional magnetic resonance imaging (fMRI) to observe brain activity whilst students think about mathematics, although much of this work has concentrated on number. In this study, we used fMRI to examine brain activity whilst ten university students translated between graphical and algebraic formats of both linear and quadratic mathematical functions. Consistent with previous studies on the representation of number, this task elicited activity in the intra-parietal sulcus, as well as in the inferior frontal gyrus. We also analysed qualitative data on participants’ introspection of strategies employed when reasoning about function. Expert participants focused more on key properties of functions when translating between formats than did novices. Implications for the teaching and learning of functions are discussed, including the relationship of function properties to difficulties in conversion from algebraic to graphical representation systems and vice versa, the desirability of teachers focusing attention on function properties, and the importance of integrating graphical and algebraic function instruction.     

Bias expansion of spatial statistics and approximation of differenced lattice point counts

Investigations of spatial statistics, computed from lattice data in the plane, can lead to a special lattice point counting problem. The statistical goal is to expand the asymptotic expectation or large-sample bias of certain spatial covariance estimators, where this bias typically depends on the shape of a spatial sampling region. In particular, such bias expansions often require approximating a difference between two lattice point counts, where the counts correspond to a set of increasing domain (i.e., the sampling region) and an intersection of this set with a vector translate of itself. Non-trivially, the approximation error needs to be of smaller order than the spatial region’s perimeter length. For all convex regions in 2-dimensional Euclidean space and certain unions of convex sets, we show that a difference in areas can approximate a difference in lattice point counts to this required accuracy, even though area can poorly measure the lattice point count of any single set involved in the difference. When investigating large-sample properties of spatial estimators, this approximation result facilitates direct calculation of limiting bias, because, unlike counts, differences in areas are often tractable to compute even with non-rectangular regions. We illustrate the counting approximations with two statistical examples.     

Wavelet Analysis of Big Data Contaminated by Large Noise in an fMRI Study of Neuroplasticity

Functional magnetic resonance imaging (fMRI) allows researchers to analyze brain activity on a voxel level, but using this ability is complicated by dealing with Big Data and large noise. A traditional remedy is averaging over large parts of brain in combination with more advanced technical innovations in reducing fMRI noise. In this paper a novel statistical approach, based on a wavelet analysis of standard fMRI data, is proposed and its application to an fMRI study of neuron plasticity of 24 healthy adults is presented. The aim of that study was to recognize changes in connectivity between left and right motor cortices (the neuroplasticity) after button clicking training sessions. A conventional method of the data analysis, based on averaging images, has implied that for the group of 24 participants the connectivity increased after the training. The proposed wavelet analysis suggests to analyze pathways between left and right hemispheres on a voxel-to-voxel level and for each participant via estimation of corresponding cross-correlations. This immediately necessitates statistical analysis of large-p-small-n correlation matrices contaminated by large noise. Furthermore, distributions that we are dealing in the analysis are neither Gaussian nor sub-Gaussian but sub-exponential. The paper explains how the problem may be solved and presents results of a dynamic analysis of the ability of a human brain to reorganize itself for 24 healthy adults. Results show that the ability of a brain to reorganize itself varies widely even among healthy individuals, and this observation is important for our understanding of a human brain and treatment of brain diseases.     

Counting and enumeration complexity with application to multicriteria scheduling

In this paper we tackle an important point of combinatorial optimisation: that of complexity theory when dealing with the counting or enumeration of optimal solutions. Complexity theory has been initially designed for decision problems and evolved over the years, for instance, to tackle particular features in optimisation problems. It has also evolved, more or less recently, towards the complexity of counting and enumeration problems and several complexity classes, which we review in this paper, have emerged in the literature. This kind of problems makes sense, notably, in the case of multicriteria optimisation where the aim is often to enumerate the set of the so-called Pareto optima. In the second part of this paper we review the complexity of multicriteria scheduling problems in the light of the previous complexity results. This paper appeared in 4OR 3(1), 1–21, 2005.     

Counting and enumeration complexity with application to multicriteria scheduling

In this paper we tackle an important point of combinatorial optimisation: that of complexity theory when dealing with the counting or enumeration of optimal solutions. Complexity theory has been initially designed for decision problems and evolved over the years, for instance, to tackle particular features in optimisation problems. It has also evolved, more or less recently, towards the complexity of counting and enumeration problems and several complexity classes, which we review in this paper, have emerged in the literature. This kind of problems makes sense, notably, in the case of multicriteria optimisation where the aim is often to enumerate the set of the so-called Pareto optima. In the second part of this paper we review the complexity of multicriteria scheduling problems in the light of the previous complexity results.Received: November 2004 / Received version: March 2005MSC classification: 90B40, 90C29, 68Q15