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.     

Bilattice logic of epistemic actions and knowledge

Baltag, Moss, and Solecki proposed an expansion of classical modal logic, called logic of epistemic actions and knowledge (EAK), in which one can reason about knowledge and change of knowledge. Kurz and Palmigiano showed how duality theory provides a flexible framework for modeling such epistemic changes, allowing one to develop dynamic epistemic logics on a weaker propositional basis than classical logic (for example an intuitionistic basis). In this paper we show how the techniques of Kurz and Palmigiano can be further extended to define and axiomatize a bilattice logic of epistemic actions and knowledge (BEAK). Our propositional basis is a modal expansion of the well-known four-valued logic of Belnap and Dunn, which is a system designed for handling inconsistent as well as potentially conflicting information. These features, we believe, make our framework particularly promising from a computer science perspective.     

Developing flexible procedural knowledge in undergraduate calculus

Mathematics experts often choose appropriate procedures to produce an efficient or elegant solution to a mathematical task. This flexible procedural knowledge distinguishes novice and expert procedural performances. This article reports on an intervention intended to aid the development of undergraduate calculus students’ flexible use of procedures. Two sections of the same course were randomly assigned to treatment and control conditions. Treatment students completed an assignment on which they resolved derivative-finding problems with alternative methods and compared the two resulting solutions. Control students were assigned a list of functions to differentiate. On the post-intervention test, treatment students were more likely to use a variety of solution methods without prompting than the control. Moreover, the set of treatment section solutions were closer to those of a group of mathematics experts. This study presents evidence that not only is flexible procedural knowledge a key skill in tertiary mathematics, it can be taught.     

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.     

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.     

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.     

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.     

The incoming statistical knowledge of undergraduate majors in a department of mathematics and statistics

Studies have shown that at the end of an introductory statistics course, students struggle with building block concepts, such as mean and standard deviation, and rely on procedural understandings of the concepts. This study aims to investigate the understandings entering freshman of a department of mathematics and statistics (including mathematics education), students who are presumably better prepared in terms of mathematics and statistics than the average university student, have of introductory statistics. This case study found that these students enter college with common statistical misunderstandings, lack of knowledge, and idiosyncratic collections of correct statistical knowledge. Moreover, they also have a wide range of beliefs about their knowledge with some of the students who believe that they have the strongest knowledge also having significant misconceptions. More attention to these statistical building blocks may be required in a university introduction statistics course.     

Examining prospective mathematics teachers' pedagogical content knowledge on fractions in terms of students' mistakes

The aim of the study is to examine prospective mathematics teachers’ pedagogical content knowledge in terms of knowledge of understanding students and knowledge of instructional strategies which are the subcomponents of pedagogical content knowledge. The participants of this research consist of 98 prospective teachers who are studying in two universities in Turkey. The participants were selected with the purposive sampling method which is one of the non-random sampling methods. Case study method, which is based on the qualitative research approach, was used. The answers given by secondary school students to fraction-related open-ended questions in the study of Soylu and Soylu were used as the data collection tool. The obtained data were analyzed via the content analysis technique. The analyses showed that the prospective mathematics teachers’ pedagogical content knowledge on fractions was not at an adequate level in identifying and correcting students’ errors. However, it was observed that the prospective teachers experienced more difficulty in the knowledge of instructional strategies compared to the knowledge of understanding students.     

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.     

Behavioral and procedural consequences of structural variation in value trees

Our experiment shows that the division of attributes in value trees can either increase or decrease the weight of an attribute. The structural variation of value trees may also change the rank of attributes. We propose that our new findings related to the splitting bias, some other phenomena appearing with attribute weighting in value trees, and the number-of-attribute-levels effect in conjoint analysis may have the same origins. One origin for these phenomena is that decision makers' responses mainly reflect the rank of attributes and not to the full extent the strength of their preferences as the value theory assumes. We call this the unadjustment phenomenon. A procedural source of biases is the normalization of attribute weights. One consequence of these two factors is that attribute weights change if attributes are divided in a value tree. We also discuss how the biases in attribute weighting could be avoided in practice.     

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).     

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.     

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.     

Recent progress in probability and statistics in China

After one year's delay due to the big earthquake in 2011,the Second Institute of Mathematical Statistics Asia Pacific Rim Meeting (IMS-APRM) was held in Tsukuba,Japan,in July of 2012.As a member of the Scientific Committee of the meeting,I was responsible to mobilize participants from China and to recommend invited speakers.My inexperience in fulfilling the task was easily compensated by the enthusiasm of my colleagues.More than 10 statisticians and probabilists from the mainland of China gave invited talks.In particular,Professor Zhi Geng of Peking University delivered a Distinguished Lecture.Recent progress in probability and statistics in China was well presented.When I was asked by the Editor in Chief,Professor K.-C.Chang,to compile a special issue for Frontiers of Mathematics in China,I once again turned to the speakers for help.     

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.