Arithmetical thinking in children attending special schools for the intellectually disabled

This article focuses on spontaneous and progressive knowledge building in “the arithmetic of the child.” The aim is to investigate variations in the behavior patterns of eight pupils attending a school for the intellectually disabled. The study is based on the epistemology of radical constructivism and the methodology of multiple clinical interviews. Theoretical models elucidate behavior patterns and the corresponding mental structures underlying them. The individual interviews of the pupils were video recorded. The results show that the activated behavior patterns, which are responses to well-adapted contexts presented by the researcher, are compatible with findings in Swedish compulsory schools. Six of the pupils’ mental structures in the study are numerical. A substantial implication for special education is the harmonization of the content in teaching with the children's own ways of operating, which implies a triadic teaching process.     

Supervision of teachers based on adjusted arithmetic learning in special education

This article reports on 20 children's learning in arithmetic after teaching was adjusted to their conceptual development. The report covers periods from three months up to three terms in an ongoing intervention study of teachers and children in schools for the intellectually disabled and of remedial teaching in regular schools. The researcher classified each child's current counting scheme before and after each term. Recurrent supervision, aiming to facilitate the teachers’ modelling of their children's various conceptual levels and needs of learning, was conducted by the researcher. The teaching content in harmony with each child's ability was discussed with the teachers. This approach gives the teachers the opportunity to experience the children's own operational ways of solving problems. At the supervision meetings, the teachers theorized their practice together with the researcher, ending up with consistent models of the arithmetic of the child. So far, the children's and the teachers’ learning patterns are promising.     

Teaching arithmetic to low-performing, low-SES first graders

To develop their logico-mathematical foundation of number as described by Piaget (1947/1950, 1967/1971, 1971/1974), 26 low-performing, low-SES first graders were given physical-knowledge activities (e.g., Pick-Up Sticks and “bowling”) during the math hour instead of math instruction. During the second half of the school year, when they showed “readiness” for arithmetic, the children were given arithmetic games and word problems that stimulated the exchange of viewpoints. At the end of the year, the children in the experimental group were compared with a similar (low-performing, low-SES) group of 20 first graders who received traditional exercises focusing narrowly on number (e.g., counting objects, making one-to-one correspondences, and answering questions like 2 + 2). The experimental group was found to be superior both in mental arithmetic and in logical reasoning as revealed by word problems.     

Toward a student-centred process of teaching arithmetic

This article describes a way toward a student-centred process of teaching arithmetic, where the content is harmonized with the students’ conceptual levels. At school start, one classroom teacher is guided in recurrent teaching development meetings in order to develop teaching based on the students’ prerequisites and to successively learn the students’ arithmetic. The students are assessed in interviews. Two special teachers participate and their current models of each student's arithmetic are tested when assessing the students. The students’ conceptual diversity and the consequent different content in teaching are shown. Further, the special teachers’ assessments and the class teacher's opinion of the new way of teaching are reported. A wide range both of the students’ conceptual levels and of the kinds of relevant problems was found. The special teachers manage their duties well and the classroom teacher has so far been satisfied with the new teaching process.     

Using the K5Connected Cognition Diagram to analyze teachers’ communication and understanding of regions in three-dimensional space

This paper reports on a study that introduces and applies the K₅Connected Cognition Diagram as a lens to explore video data showing teachers’ interactions related to the partitioning of regions by axes in a three-dimensional geometric space. The study considers “semiotic bundles” ( Arzarello, 2006), introduces “semiotic connections,” and discusses the fundamental role each plays in developing individual understanding and communication with peers. While all teachers solved the problem posed, many failed to make or verbalize connections between the types of semiotic resources introduced during their discussions.     

Fractional commensurate, composition, and adding schemes: Learning trajectories of Jason and Laura: Grade 5

A case study of two 5th-Grade children, Jason and Laura, is presented who participated in the teaching experiment, Children’s Construction of the Rational Numbers of Arithmetic. The case study begins on the 29th of November of their 5th-Grade in school and ends on the 5th of April of the same school year. Two basic problems were of interest in the case study. The first was to provide an analysis of the concepts and operations that are involved in the construction of three fractional schemes: a commensurate fractional scheme, a fractional composition scheme, and a fractional adding scheme. The second was to provide an analysis of the contribution of interactive mathematical activity in the construction of these schemes. The phrase, “commensurate factional scheme” refers to the concepts and operations that are involved in transforming a given fraction into another fraction that are both measures of an identical quantity. Likewise, “fractional composition scheme” refers to the concepts and operations that are involved in finding how much, say, 1/3 of 1/4 of a quantity is of the whole quantity, and “fractional adding scheme” refers to the concepts and operations involved in finding how much, say, 1/3 of a quantity joined to 1/4 of a quantity is of the whole quantity. Critical protocols were abstracted from the teaching episodes with the two children that illustrate what is meant by the schemes, changes in the children’s concepts and operations, and the interactive mathematical activity that was involved. The body of the case study consists of an on-going analysis of the children’s interactive mathematical activity and changes in that activity. The last section of the case study consists of an analysis of the constitutive aspects of the children’s constructive activity, including the role of social interaction and nonverbal interactions of the children with each other and with the computer software we used in teaching the children.     

The Cauchy problems for evolutionary pseudo-differential equations over p-adic field and the wavelet theory

We solve the Cauchy problems for p-adic linear and semi-linear evolutionary pseudo-differential equations (the time-variable t∈R and the space-variable ). Among the equations under consideration there are the heat type equation and the Schrödinger type equations (linear and nonlinear). To solve these problems, we develop the “variable separation method” (an analog of the classical Fourier method) which reduces solving evolutionary pseudo-differential equations to solving ordinary differential equations with respect to real variable t. The problem of stabilization for solutions of the Cauchy problems as t→∞ is also studied. These results give significant advance in the theory of p-adic pseudo-differential equations and can be used in applications.     

On contracts for VMI program with continuous review (r, Q) policy

Based on continuous review (r, Q) policy, this paper deals with contracts for vendor managed inventory (VMI) program in a system comprising a single vendor and a single retailer. Two business scenarios that are popular in VMI program are “vendor with ownership” and “retailer with ownership”. Taking the system performance in centralized control as benchmark, we define a contract “perfect” if the contract can enable the system to be coordinated and can guarantee the program to be trusted. A revenue sharing contract is designed for vendor with ownership, and a franchising contract is designed for retailer with ownership. Without consideration of order policy and related costs at the vendor site, it is shown that one contract can perform satisfactorily and the other one is a perfect contract. With consideration of order policy and related costs at the vendor site, it is shown that one contract can perform satisfactorily and the performance of the other one depends on system parameters.     

Shapley mappings and the cumulative value for n-person games with fuzzy coalitions

In this paper we prove existence and uniqueness of the so-called Shapley mapping, which is a solution concept for a class of n-person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions. The Shapley mapping, when it exists, associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency, the symmetry, and the null-player conditions. It determines a “cumulative value” that is the “sum” of all coalitional allocations for whose computation we provide an explicit formula.     

Rationals and decimals as required in the school curriculum: Part 1: Rationals as measurement

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment, carried out jointly with his wife, Nadine, in her classroom at the École Jules Michelet, was to teach all of the material on rational and decimal numbers required by the national program with a carefully structured, tightly woven and interdependent sequence of “situations.” This article describes and discusses the first portion of that experiment.     

A degree sum condition for graphs to be prism hamiltonian

Win, in 1975, and Jackson and Wormald, in 1990, found the best sufficient conditions on the degree sum of a graph to guarantee the properties of “having a k-tree” and “having a k-walk”, respectively. The property of “being prism hamiltonian” is an intermediate property between “having a 2-tree” and “having a 2-walk”. Thus, it is natural to ask what is the best degree sum condition for graphs to be prism hamiltonian. As an answer to this problem, in this paper, we show that a connected graph G of order n with σ₃(G)≥n is prism hamiltonian. The degree sum condition “σ₃(G)≥n” is best possible.     

Almost fifth powers in arithmetic progression

We prove that the product of k consecutive terms of a primitive arithmetic progression is never a perfect fifth power when 3?k?54. We also provide a more precise statement, concerning the case where the product is an “almost” fifth power. Our theorems yield considerable improvements and extensions, in the fifth power case, of recent results due to Gy?ry, Hajdu and Pintér. While the earlier results have been proved by classical (mainly algebraic number theoretical) methods, our proofs are based upon a new tool: we apply genus 2 curves and the Chabauty method (both the classical and the elliptic verison).     

Generalized Zernike or disc polynomials: An application in quantum optics

An application of the “generalized Zernike or disc polynomials”, recently introduced in the literature, is shown, resorting to the Lie algebra based investigation of the dynamics of quantum systems driven by two-mode interaction Hamiltonians. Further properties of the associated “disc functions” are deduced. Also, a generalization of the disc polynomials towards the Hahn polynomials is suggested.     

Rationals and decimals as required in the school curriculum: Part 2: From rationals to decimals

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment, carried out jointly with his wife, Nadine, in her classroom at the École Jules Michelet, was to teach all of the material on rational and decimal numbers required by the national program with a carefully structured, tightly woven and interdependent sequence of “situations.” This article describes and discusses the second portion of that experiment.     

An adaptive multiphase approach for large unconditional and conditional p-median problems

A multiphase approach that incorporates demand points aggregation, Variable Neighbourhood Search (VNS) and an exact method is proposed for the solution of large-scale unconditional and conditional p-median problems. The method consists of four phases. In the first phase several aggregated problems are solved with a “Local Search with Shaking” procedure to generate promising facility sites which are then used to solve a reduced problem in Phase 2 using VNS or an exact method. The new solution is then fed into an iterative learning process which tackles the aggregated problem (Phase 3). Phase 4 is a post optimisation phase applied to the original (disaggregated) problem. For the p-median problem, the method is tested on three types of datasets which consist of up to 89,600 demand points. The first two datasets are the BIRCH and the TSP datasets whereas the third is our newly geometrically constructed dataset that has guaranteed optimal solutions. The computational experiments show that the proposed approach produces very competitive results. The proposed approach is also adapted to cater for the conditional p-median problem with interesting results.     

Arithmetic partial differential equations, I

We develop an arithmetic analogue of linear partial differential equations in two independent “space-time” variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us to “flow” integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves carry certain canonical “arithmetic flows” that are arithmetic analogues of the convection, heat, and wave equations, respectively. The same is true for the additive and the multiplicative group.     

Arithmetic partial differential equations, II

We continue the study of arithmetic partial differential equations initiated in [7] by classifying “arithmetic convection equations” on modular curves, and by describing their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions (Mori, 1995 [13], Buium, 2003 [4]) of the same modular forms; in this sense, our arithmetic convection equations can be seen as “unifying” the two types of expansions. The theory can be generalized to one of “arithmetic heat equations” on modular curves, but we prove that they do not carry “arithmetic wave equations.” Finally, we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation.     

On p-norm linear discrimination

We consider a p-norm linear discrimination model that generalizes the model of Bennett and Mangasarian (1992) and reduces to a linear programming problem with p-order cone constraints. The proposed approach for handling linear programming problems with p-order cone constraints is based on reformulation of p-order cone optimization problems as second order cone programming (SOCP) problems when p is rational. Since such reformulations typically lead to SOCP problems with large numbers of second order cones, an “economical” representation that minimizes the number of second order cones is proposed. A case study illustrating the developed model on several popular data sets is conducted.     

A regularization method for solving the Cauchy problem for the Helmholtz equation

In this paper, we investigate a Cauchy problem associated with Helmholtz-type equation in an infinite “strip”. This problem is well known to be severely ill-posed. The optimal error bound for the problem with only nonhomogeneous Neumann data is deduced, which is independent of the selected regularization methods. A framework of a modified Tikhonov regularization in conjunction with the Morozov’s discrepancy principle is proposed, it may be useful to the other linear ill-posed problems and helpful for the other regularization methods. Some sharp error estimates between the exact solutions and their regularization approximation are given. Numerical tests are also provided to show that the modified Tikhonov method works well.     

Optimal and better transport plans

We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value ∞. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided that {c=∞} is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a “better” notion of optimality called robust optimality.