Students’ construction and use of statistical models: a socio-critical perspective

This paper addresses how students explore, construct, validate and use statistical models when facing situations designed from a socio-critical perspective. The case study used is a statistics lesson designed by a statistics teacher and a researcher. The lesson centers on nutritional information and was implemented in a 7th-grade classroom at a public school in a Northwest Colombian city. In small groups, students gathered their own data, and subsequently organized and analyzed the data, and presented their findings to the class. The main sources of data were students’ discourse in the classroom, students’ artifacts and the researcher’s journal. The findings describe a route in which students explore, construct, use, and validate their models. The results elaborate the technological and the reflective knowledge that took place in the model building activity.     

The Cauchy–Riemann Equations: Discretization by Finite Elements,Fast Solution of the Second Variable,and a Posteriori Error Estimation

In this paper we will concentrate on the numerical solution of the Cauchy–Riemann equations. First we show that these equations bring together the finite element discretizations for the Laplace equation by standard finite elements on the one hand, and by mixed finite element methods on the other. As a consequence, methods for a posteriori error estimation for both finite element methods can derive their validity from each other. Moreover, we show that given a finite element approximation of one of the vectorfields, the missing can be accurately computed in optimal complexity.     

Mobile-phone-based classroom response systems: Students’ perceptions of engagement and learning in a large undergraduate course

Using a Classroom Response System (CRS) has been associated with positive educational outcomes, by fostering student engagement and by allowing immediate feedback to both students and instructors. This study examined a low-cost CRS (VotApedia) in a large first-year class, where students responded to questions using their mobile phones. This study explored whether the use of VotApedia retained the advantages of other CRS, overcame some of the challenges of other CRS, and whether new challenges were introduced by using VotApedia. These issues were studied within three themes: students’ perceptions of using VotApedia; the impact of VotApedia on their engagement; and the impact of VotApedia on their learning. Data were collected from an online survey, focus groups and student feedback on teaching and course content. The results indicated that using VotApedia retains the pedagogical advantages of other CRS, while overcoming some of the challenges presented by using other CRS, without introducing any new challenges.     

Students’ self-regulation of emotions in mathematics: an analysis of meta-emotional knowledge and skills

Over the past decade, the concept of self-regulated learning has broadened to include motivational, volitional, and emotional components next to (meta-)cognitive ones. In this article, we present a meta-emotion perspective as an essential component of a conceptual framework on self-regulation that fully acknowledges the role of emotions. Against this background, a study is presented that attempts to contribute to the clarification of the relevance and the functioning of students’ meta-emotional knowledge and emotional regulation skills in school-related mathematical activities. It investigates the coping strategies that 393 students of the second (age 14) and fourth (age 16) year of secondary school report to use to regulate their emotions in three different mathematical school settings (i.e., a mathematics test, a difficult mathematics homework, and a difficult mathematics lesson). More specifically, it aims (1) to document the nature and frequency of the reported coping strategies, and (2) to explore—for the three different mathematical school settings—relationships between these reported coping strategies and personal characteristics (i.e., students’ familiarity with the particular school settings, their track in secondary education, their achievement level, their age, and gender). The results indicate that students report to know and to make use of several coping strategies in school-related mathematical activities, and reveal that the use of these strategies is related to specific person-related characteristics. In conclusion, we elaborate on how schools and teachers can stimulate students to acquire appropriate strategies and skills to self-regulate their emotions.     

Between Counting and Multiplication: Low-Attaining Students’ Spatial Structuring,Enumeration and Errors in Concretely-Presented 3D Array Tasks

The move from additive to multiplicative thinking requires significant change in children’s comprehension and manipulation of numerical relationships, involves various conceptual components, and can be a slow, multistage process for some. Unit arrays are a key visuospatial representation for supporting learning, but most research focuses on 2D (rectangular) arrays, and when focusing on 3D (cuboid) arrays still frequently uses 2D representations. This article documents low-attaining children’s partially developed multiplicative thinking as they work on concretely presented 3D array tasks; it also presents a framework for microanalysis of learners’ early multiplicative thinking in array tasks. Data derives from a small but cognitively diverse set of participants, all arithmetically low-attaining and relying heavily on counting: this enabled detailed analysis of small but significant differences in their arithmetical engagement with arrays. The analytical framework combines and builds on previous structural and enumerative categorizations, and may be used with a variety of array representations.     

Secondary Students’ Quantification of Ratio and Rate: A Framework for Reasoning about Change in Covarying Quantities

Contributing to a growing body of research addressing secondary students’ quantitative and covariational reasoning, the multiple case study reported in this article investigated secondary students’ quantification of ratio and rate. This article reports results from a study investigating students’ quantification of rate and ratio as relationships between quantities and presents the Change in Covarying Quantities Framework, which builds from Carlson, Jacobs, Coe, Larsen, and Hsu’s (2002) Covariation Framework. Each of the students in this study was consistent in terms of the quantitative operation he or she used (comparison or coordination) when quantifying both ratio and rate. Illustrating how students can engage in different quantitative operations when quantifying rate, the Change in Covarying Quantities Framework helps to explain why students classified as operating at a particular level of covariational reasoning appear to be using different mental actions. Implications of this research include recommendations for designing instructional tasks to foster students’ quantitative and covariational reasoning.     

The effect of motion analysis activities in a video-based laboratory in students’ understanding of position,velocity and frames of reference

In this article, we present the results of a qualitative research project on the effect of motion analysis activities in a Video-Based Laboratory (VBL) on students’ understanding of position, velocity and frames of reference. The participants in our research were 48 pre-service teachers enrolled in Education Departments with no previous strong science or mathematics background. VBLs are presented in the context of a category of technology tools that allow the analysis of motion events for educational purposes followed by a review of literature regarding the use of VBL in science education. The qualitative methodology that we undertook is presented and justified and the results of our study are stated, escorted by selected and detailed dialogue extracts from our interviews with the participating students. The outcomes of our research are discussed and found to be positive and some implications for further research are stated.     

Characteristics of good mathematics teaching in Singapore grade 8 classrooms: a juxtaposition of teachers’ practice and students’ perception

This paper examines the instructional approaches of three competent grade 8 mathematics teachers. It also examines their students’ perception of the lessons they taught as well as characteristics of good lessons. The findings of teachers’ practice and students’ perception are juxtaposed to elicit characteristics of good teaching in Singapore grade 8 classrooms. With limitation, the findings of the paper suggests that good mathematics teaching in Singapore schools centres around building understanding and is teacher-centred but student focused. Some characteristic features of good lessons are that their instructional cycles have specific instructional objectives such that subsequent cycles incrementally build on the knowledge. The examples used in such lessons are carefully selected and vary in complexity from low to high. Teachers actively monitor their student’s understanding during seatwork, by moving from desk to desk guiding those with difficulties and selecting appropriate student work for subsequent whole-class review and discussion. Finally, during such lessons teachers reinforce their students’ understanding of knowledge expounded during whole-class demonstration by detailed review of student work done in class or as homework.     

Toward Homological Characterization of Semirings: Serre’s Conjecture and Bass’s Perfectness in a Semiring Context

Among other results on homological characterization of semirings, we prove that the classes of projective and free right (left) semimodules over the polynomial semiring R[x₁, x₂,..., x_n] over an additively regular division semiring R coincide iff R is a field. Also it is shown that an additively regular commutative semiring R is perfect (in H. Basss sense) iff R is a perfect ring.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived July 27, 2003; accepted in final form April 2, 2004.     

The Method of Scientific Discovery in Peirce’s Philosophy: Deduction, Induction, and Abduction

In this paper we will show Peirce’s distinction between deduction, induction and abduction. The aim of the paper is to show how Peirce changed his views on the subject, from an understanding of deduction, induction and hypotheses as types of reasoning to understanding them as stages of inquiry very tightly connected. In order to get a better understanding of Peirce’s originality on this, we show Peirce’s distinctions between qualitative and quantitative induction and between theorematical and corollarial deduction, passing then to the distinction between mathematics and logic. In the end, we propose a sketch of a comparison between Peirce and Whitehead concerning the two thinkers’ view of mathematics, hoping that this could point to further inquiries.     

Dynamic conformance and design quality in a supply chain: an assessment of contracts’ coordinating power

We consider a two-echelon supply chain involving one manufacturer and one supplier who collaborate on improving both design and conformance quality. Design quality is supposed to increase product desirability, and therefore market demand, while conformance quality should reduce the proportion of defective items, and therefore increase the manufacturer’s sales revenue. We investigate how the supply chain parties allocate effort between design and conformance quality under both cooperative and non-cooperative settings in an intertemporal framework. Furthermore, we evaluate wholesale price contracts and revenue-sharing contracts in terms of their performance and coordination power. We show that although a revenue-sharing contract enables the manufacturer to effectively involve the supplier in quality improvement, neither contract type allows for perfect coordination resulting in the quality that can be achieved by a cooperative supply chain. We thus suggest a reward-based extension to the revenue-sharing contract, to ensure system-wide optimal quality performance. Importantly, we find that the supplier would be better off adopting a reward-based revenue sharing contract and refusing a standard revenue-sharing contract, while the opposite would be true for the manufacturer.     

Inquiry-based learning in mathematics and science: a comparative baseline study of teachers’ beliefs and practices across 12 European countries

In the European educational context, reports by expert groups have identified the necessity of a renewed pedagogy in schools to overcome deficits in science and mathematics teaching and to raise the standards of scientific and mathematical literacy. Inquiry-based learning (IBL) is considered the method of choice. However, it remains open to what extent IBL is actually used in day-to-day teaching. In the study presented here we elaborate—from the perspective of teachers—the current status of IBL in day-to-day teaching. Further, we explore what problems teachers anticipate when implementing IBL. In order to gain insight into the wide spectrum of practices in mathematics and science teaching in relation to IBL, a baseline study using teacher questionnaires was carried out in the 12 participating countries. We present selected results from this study that for the first time provides an overview of teachers’ beliefs and their reports on the current use of IBL practices in a European context. The results facilitate a cross-cultural comparison on the potentials and challenges of implementing IBL from the perspective of practicing teachers. Furthermore, the study reveals considerable differences between the teaching of mathematics and science subjects. The findings of the baseline study can serve as a reference line against which the impact of interventions to improve the quality of teaching and learning can be evaluated.     

Correction to: ‘The UCT, the Milnor Sequence, and a Canonical Decomposition of the Kasparov Groups’

In this note we correct a mistake in K-Theory 10 (1996), 49–72. In that paper we asserted that under bootstrap hypotheses the short exact sequence

Telling and illustrating stories of parity: a classroom-based design experiment on young children’s use of narrative in mathematics

This paper examines ways to engage young children in constructing and interpreting narratives to develop their understanding of parity. It reports on a teaching intervention that was developed over three research cycles of a classroom-based design experiment, and focuses on the last of these cycles. The teaching intervention set out to investigate how young children (5–6-year-olds) can be supported to draw on narrative in their explanations of whether a whole number less than 20 is odd or even. Evidence of the effectiveness of the intervention is provided through comparison of children’s performance on pre- and post-tests in the form of semi-structured individual interviews. Also, authentic examples are provided of how children utilised their power of ‘imagining and expressing’ to tell stories of whether a whole number is odd or even, using either a counting, partitive, or quotitive model for division. Implications for research and practice are discussed in light of these findings.     

Dickson’s Lemma,Higman’s Theorem and Beyond: A survey of some basic results in order theory

We provide proofs for the fact that certain orders have no infinite descending chains and no infinite antichains.     

On the stability of discrete tripole,quadrupole, Thomson’ vortex triangle and square in a two-layer/homogeneous rotating fluid

A two-layer quasigeostrophic model is considered in the f-plane approximation. The stability of a discrete axisymmetric vortex structure is analyzed for the case when the structure consists of a central vortex of arbitrary intensity Γ and two/three identical peripheral vortices. The identical vortices, each having a unit intensity, are uniformly distributed over a circle of radius R in a single layer. The central vortex lies either in the same or in another layer. The problem has three parameters (R, Γ, α), where α is the difference between layer thicknesses. A limiting case of a homogeneous fluid is also considered.A limiting case of a homogeneous fluid is also considered.The theory of stability of steady-state motions of dynamic systems with a continuous symmetry group G is applied. The two definitions of stability used in the study are Routh stability and G-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex multipole, and the G-stability is the stability of a three-parameter invariant set O _G , formed by the orbits of a continuous family of steady-state rotations of a multipole. The problem of Routh stability is reduced to the problem of stability of a family of equilibria of a Hamiltonian system. The quadratic part of the Hamiltonian and the eigenvalues of the linearization matrix are studied analytically.The cases of zero total intensity of a tripole and a quadrupole are studied separately. Also, the Routh stability of a Thomson vortex triangle and square was proved at all possible values of problem parameters. The results of theoretical analysis are sustained by numerical calculations of vortex trajectories.     

Building teachers’ expertise in understanding, assessing and developing children’s mathematical thinking: the power of task-based, one-to-one assessment interviews

In this paper, we outline the benefits to teachers’ expertise of the use of research-based, one-to-one assessment interviews in mathematics. Drawing upon our research and professional development work with teachers and students in primary and middle years in Australia and the research of others, we argue that the use of the interviews builds teacher expertise through enhancing teachers’ knowledge of individual and group understanding of mathematics, and also provides an understanding of typical learning paths in various mathematical domains. The use of such interviews also provides a model for teachers’ interactions and discussions with children, building both their pedagogical content knowledge and their subject matter knowledge.     

An extension of Krugman’s core–periphery model to the case of a continuous domain: Existence and uniqueness of solutions of a system of nonlinear integral equations in spatial economics

We consider a model that is an extension of Krugman’s core–periphery model to the case of a bounded closed domain included in a Euclidean space. We can describe the relation of the density of workers, the density of nominal wages, and the density of real wages by the system of nonlinear integral equations of the model. If we obtain a solution of the system under the condition that the density of workers is given, then the solution is called a short-run equilibrium. In this paper we prove that this model has a short-run equilibrium, and we obtain a sufficient condition for its uniqueness. Moreover we obtain upper and lower estimates for short-run equilibria, and we construct a useful iteration scheme to numerically obtain short-run equilibria.