Picturing student beliefs in statistics

Statistical skills and statistical literacy have emerged as important areas in education. While it has a rich mathematical basis, successful understanding and application of statistics incorporates other types of knowledge. In a similar way, beliefs about statistics can be described using the same framework as beliefs about mathematics, but statistical beliefs bring other aspects as well. This article describes a project for investigating student beliefs in statistics through the creation of pictures of understanding. It presents a classification of statistical concepts and attitudes which are motivated by research in statistical literacy and then shows how these can be refined to be more reliably applied in practice.     

Building a local conceptual framework for epistemic actions in a modelling environment with experiments

A local conceptual framework for the construction of mathematical knowledge in learning environments with experiments is developed. For this purpose, the mathematical modelling framework and the epistemic action model for abstraction in context are used simultaneously. In a case study, experiments of pre-service teachers with the motion of a ball on a circular billiard table are analysed within the local conceptual framework. The role of the experiments for epistemic actions of mathematical abstractions is described. In the case study, two different types of students’ approaches to the role of experiments can be distinguished.     

The mathematical knowledge and beliefs of elementary mathematics specialist-coaches

This paper reports on one aspect of a larger research project conducted in the United States that designed and implemented an elementary mathematics, specialist-coach preparation program and evaluated the effect of qualified specialist-coaches on student achievement. The paper discusses a conceptual framework for coaching in which a specialist-coach is to serve as a “more knowledgeable other” for a community of practice in a school, and ultimately to impact both the knowledge and professional practice of teachers and the school’s mathematics program as a whole. Specialist-coaches have unique opportunities and challenges in this daunting task, and the paper discusses one program designed to prepare well-respected teachers for the transition to the role and responsibilities of a specialist-coach. The reported analyses document changes in specialist-coaches’ mathematical content knowledge, mathematical knowledge for teaching, and beliefs regarding mathematics teaching and learning over the preparation program and during the specialist-coaches’ first years of service in a school. These specialist-coaches’ mathematical content knowledge grew and their beliefs became more aligned with a Making Sense perspective during the preparation program, and their changed state persisted throughout 2–3 years of service as specialist-coaches. Evidence addressing the specialist-coaches’ mathematical knowledge for teaching was mixed, but suggested that growth occurred both during the preparation program and in their first year of coaching, stabilizing in the years following.     

Common knowledge,communication, and convergence of beliefs

The aim of this paper is to examine the adequacy of the so-called expected utility theory. It is concluded that the theory is inadequate when the decision-maker is decreasingly risk averse and his initial wealth is risky.     

Model and algorithm of the conceptual scheme formation for knowledge domain in distance learning

The article describes a knowledge model, oriented to the asynchronous mode of distance learning. The formalization of the knowledge model for a given domain, the operations on the knowledge and the algorithm of the knowledge model creation are submitted. All received decisions can be realized in a program environment compatible with the SCORM standard. The described methodology, based on a generalized knowledge model, enables to develop a distance learning course mainly for the fundamental knowledge. In this paper we describe the methodology and illustrate its use through a project to develop a distance learning course for a queuing system. Moreover, a practical application is proposed based on the eQuality project.     

Fifth graders’ additive and multiplicative reasoning: Establishing connections across conceptual fields using a graph

This paper looks at 21 fifth grade students as they discuss a linear graph in the Cartesian plane. The problem presented to students depicted a graph showing distance as a function of elapsed time for a person walking at a constant rate of 5 miles/h. The question asked students to consider how many more hours, after having already walked 4 h, would be required to reach 35 miles. To answer this question, the students needed to extend the graph that was presented, either mentally or on paper, as the axes did not go up to 7 h or 35 miles. They also needed to be able to consider not only the total number of hours to reach 35 miles, but also the interval of time after 4 h. The purpose of this paper is to consider the student responses from the viewpoint of multiplicative and additive reasoning, and specifically within Vergnaud's framework of multiplicative and additive conceptual fields and scalar and functional approaches to linear relationships (Vergnaud, 1994). The analysis shows that: some student answers cannot be classified as either scalar or functional; some students combined several kinds of approaches in their explanations; and that the representation of the problem using a graph may have facilitated responses that are different from those typically found when the representation presented is a function table.     

Knowledge structure,knowledge granulation and knowledge distance in a knowledge base

One of the strengths of rough set theory is the fact that an unknown target concept can be approximately characterized by existing knowledge structures in a knowledge base. Knowledge structures in knowledge bases have two categories: complete and incomplete. In this paper, through uniformly expressing these two kinds of knowledge structures, we first address four operators on a knowledge base, which are adequate for generating new knowledge structures through using known knowledge structures. Then, an axiom definition of knowledge granulation in knowledge bases is presented, under which some existing knowledge granulations become its special forms. Finally, we introduce the concept of a knowledge distance for calculating the difference between two knowledge structures in the same knowledge base. Noting that the knowledge distance satisfies the three properties of a distance space on all knowledge structures induced by a given universe. These results will be very helpful for knowledge discovery from knowledge bases and significant for establishing a framework of granular computing in knowledge bases.     

Diagnostics and score statistics in regression

In a generalized linear model, we have a linear predictor. We extend to a nonlinear one and propose a unified method to establish diagnostic procedures for such models with nonlinear links. Applications of the procedures to various useful models are given with examples.Research supported by National Science Council of the Republic of China (NSC-25009).     

Examining the feasibility of using a modelling tool to assess resilience across a health-care system and assist with decisions concerning service reconfiguration

Changes in medical practice, demographic shifts and financial pressures are all examples of factors that may contribute to demand for periodic changes in the configuration of health services. When reconfiguring a service, health planners often take into account projected demand for services, patient access criteria and budgetary constraints (among other things), but typically give little consideration regarding its resilience to deliver services during and after external disruptions to its capability to deliver. In this paper we discuss a study conducted in response to a direct request from the National Health Service (NHS) Resilience Project within the Department of Health to explore the feasibility of assessing resilience across local services within the NHS and developing a computer software tool to assess resilience of different service reconfigurations. We give an account of the modelling process used, including the analytical framework we developed using both optimisation and heuristic methods, and an illustrative example of usage of a prototype software tool. We also highlight the key lessons that emerged during this project, which may be helpful to OR analysts working on similar projects regarding resilience in the public sector.     

Bipartitional polynomials and their applications in combinatorics and statistics

Bipartitional polynomials are multivariable polynomials Y_mn=Y_mn(cy₀₁,cy₁₀,cy₁₁,…,cy_mn), c^k≡c_k, defined by a sum over all partitions of the bipartite number (mn). Recurrence relations, generating functions and some basic properties of these polynomials are given. Applications in Combinatorics and Statistics are briefly indicated.     

Expert and experiential knowledge in planning

The issue of expert and experiential knowledge has received increasing attention in conjunction with the contemporary surge in interest in post-positivist approaches in planning and policy-making. So far, few concrete methods have been put forward on how to evaluate these two types of knowledge in specific policy-making situations and thereby achieve their balanced use. This paper presents two approaches that can be used to derive expert and experiential knowledge. It describes their application to two case studies, one in Sweden and one in Italy, where expert and experiential knowledge were generated and used in urban development plans. It also proposes a method for evaluating and integrating the two types of knowledge.     

Examining the balance between efficiency and resilience in closed-loop supply chains

Central European Journal of Operations Research - We study a hybrid system where the demand of customers can be satisfied by both manufacturing new products and remanufacturing used products. To...     

The impact of using multiple modalities on students’ acquisition of fractional knowledge: An international study in embodied mathematics across semiotic cultures

Principled by the Embodied, Situated, and Distributed Cognition paradigm, the study investigated the impact of using a research-based curriculum that employs multiple modalities on the performance of grade 5 students on 3 subscales: concept of unit, fraction equivalence, and fraction comparison. The sample included five schools randomly selected from a population of 14 schools in Lebanon. Eighteen 5th grade classrooms were randomly assigned to experimental (using multimodal curriculum) and control (using a monomodal curriculum) groups. Three data sources were used to collect quantitative and qualitative data: tests, interviews, and classroom observations. Quantitative data were analyzed using two methods: reliability and MANOVA. Results of the quantitative data show that students taught using the multimodal curriculum outperformed their counterparts who were instructed using a monomodal curriculum on the three aforementioned subscales (at an alpha level = .001). Additionally, fine-grained analysis using the semiotic bundle model revealed different semiotic systems across experimental and control groups. The study findings support the multimodal approach to teaching fractions as it facilitates students’ conceptual understanding.     

Classification and statistics of finite index subgroups in free products

The principal theme of this paper is the enumeration of finite index subgroups Δ in a free product Γ of finite groups under various restrictions on the isomorphism type of Δ. In particular, we completely resolve the realization, asymptotic, and distribution problems for free products Γ of cyclic groups of prime order (prior to this work, these questions were wide open even in the case of the classical modular group). This complex of problems, usually referred to as Poincaré-Klein Problem, originally arose around 1880 out of the work of Klein and Poincaré on automorphic functions and related number theory, but has also grown roots in geometric function theory and, more recently, in the theory of subgroup growth. Ideas and techniques from the theory of generalized permutation representations (an enumerative theory of wreath product representations recently developed by the first named author) play a fundamental role here. Other tools come from analytic number theory, combinatorics, and probability theory.     

Some set partition statistics in non-crossing partitions and generating functions

In this paper we shall give the generating functions for the enumeration of non-crossing partitions according to some set partition statistics explicitly, which are based on whether a block is singleton or not and is inner or outer. Using weighted Motzkin paths, we find the continued fraction form of the generating functions. There are bijections between non-crossing partitions, Dyck paths and non-nesting partitions, hence we can find applications in the enumeration of Dyck paths and non-nesting partitions. We shall also study the integral representation of the enumerating polynomials for our statistics. As an application of integral representation, we shall give some remarks on the enumeration of inner singletons in non-crossing partitions, which is equivalent to one of udu's at high level in Dyck paths investigated in [Y. Sun, The statistic “number of udu's” in Dyck paths, Discrete Math. 284 (2004) 177-186].