Role of Interaction in Enhancing the Epistemic Utility of 3D Mathematical Visualizations

Many epistemic activities, such as spatial reasoning, sense-making, problem solving, and learning, are information-based. In the context of epistemic activities involving mathematical information, learners often use interactive 3D mathematical visualizations (MVs). However, performing such activities is not always easy. Although it is generally accepted that making these visualizations interactive can improve their utility, it is still not clear what role interaction plays in such activities. Interacting with MVs can be viewed as performing low-level epistemic actions on them. In this paper, an epistemic action signifies an external action that modifies a given MV in a way that renders learners’ mental processing of the visualization easier, faster, and more reliable. Several, combined epistemic actions then, when performed together, support broader, higher-level epistemic activities. The purpose of this paper is to examine the role that interaction plays in supporting learners to perform epistemic activities, specifically spatial reasoning involving 3D MVs. In particular, this research investigates how the provision of multiple interactions affects the utility of 3D MVs and what the usage patterns of these interactions are. To this end, an empirical study requiring learners to perform spatial reasoning tasks with 3D lattice structures was conducted. The study compared one experimental group with two control groups. The experimental group worked with a visualization tool which provided participants with multiple ways of interacting with the 3D lattices. One control group worked with a second version of the visualization tool which only provided one interaction. Another control group worked with 3D physical models of the visualized lattices. The results of the study indicate that providing learners with multiple interactions can significantly affect and improve performance of spatial reasoning with 3D MVs. Among other findings and conclusions, this research suggests that one of the central roles of interaction is allowing learners to perform low-level epistemic actions on MVs in order to carry out higher-level cognitive and epistemic activities. The results of this study have implications for how other 3D mathematical visualization tools should be designed.     

Designing for Mathematical Abstraction

Our focus is on the design of systems (pedagogical, technical, social) that encourage mathematical abstraction, a process we refer to as designing for abstraction. In this paper, we draw on detailed design experiments from our research on children’s understanding about chance and distribution to re-present this work as a case study in designing for abstraction. Through the case study, we elaborate a number of design heuristics that we claim are also identifiable in the broader literature on designing for mathematical abstraction. Our previous work on the micro-evolution of mathematical knowledge indicated that new mathematical abstractions are routinely forged in activity with available tools and representations, coordinated with relatively na?ve unstructured knowledge. In this paper, we identify the role of design in steering the micro-evolution of knowledge towards the focus of the designer’s aspirations. A significant finding from the current analysis is the identification of a heuristic in designing for abstraction that requires the intentional blurring of the key mathematical concepts with the tools whose use might foster the construction of that abstraction. It is commonly recognized that meaningful design constructs emerge from careful analysis of children’s activity in relation to the designer’s own framework for mathematical abstraction. The case study in this paper emphasizes the insufficiency of such a model for the relationship between epistemology and design. In fact, the case study characterises the dialectic relationship between epistemological analysis and design, in which the theoretical foundations of designing for abstraction and for the micro-evolution of mathematical knowledge can co-emerge.     

Product life cycle analysis: a goal programming approach

This paper considers the role of Life-Cycle Costing (LCC), Time-Based Competition (TBC), and other relevant goals at the three planning-horizon levels of strategic, intermediate, and tactical with respect to time and cost of products to market in a concurrent engineering environment. A conceptual framework has been developed in which the general areas of LCC and TBC are delineated. Based on the conceptual framework, the specific components of LCC and TBC at each level of planning are developed and presented. Subsequently, a Preemptive Goal Programming (PGP) model is developed. This model uses the components of LCC, TBC, and other relevant goals as the decision variables. Then, a case study of a company that has developed LCC and TBC is considered. The PGP model is then applied to the company's operations by using the actual data obtained from the company in order to capture the essence of the conceptual framework. This example explains and validates the PGP model in practice and provides managerial implications of the solution. Finally, a set of conclusions and an assessment of the results are presented.     

Quantum teleportation and quantum epistemic semantics

Quantum information gives rise to some puzzling epistemic problems that can be interestingly investigated from a logical point of view. A characteristic example is represented by teleportation phenomena, where knowledge and actions of observers (epistemic agents) play a relevant role. By abstracting from teleportation, we propose a simplified semantics for a language that consists of two parts:

1. the quantum computational sub-language, whose sentences α represent pieces of quantum information (which are supposed to be stored by some quantum systems)
2. the classical epistemic sub-language, whose atomic sentences have the following forms: agent a has a probabilistic information about the sentence α; agent a knows the sentence α.

Interestingly enough, some conceptual difficulties of standard epistemic logics can be avoided in this framework.     

Coordinating visualizations of polysemous action: values added for grounding proportion

We contribute to research on visualization as an epistemic learning tool by inquiring into the didactical potential of having students visualize one phenomenon in accord with two different partial meanings of the same concept. 22 Grade 4–6 students participated in a design study that investigated the emergence of proportional-equivalence notions from mediated perceptuomotor schemas. Working as individuals or pairs in tutorial clinical interviews, students solved non-symbolic interaction problems that utilized remote-sensing technology. Next, they used symbolic artifacts interpolated into the problem space as semiotic means to objectify in mathematical register a variety of both additive and multiplicative solution strategies. Finally, they reflected on tensions between these competing visualizations of the space. Micro-ethnographic analyses of episodes from three paradigmatic case studies suggest that students reconciled semiotic conflicts by generating heuristic logico-mathematical inferences that integrated competing meanings into cohesive conceptual networks. These inferences hinged on revisualizing additive elements multiplicatively. Implications are drawn for rethinking didactical design for proportions.     

Learning mathematics in a computer algebra environment: obstacles are opportunities

Using computer algebra is not as easy it may seem. Students often encounter obstacles while working in a computer algebra environment. In this paper, global and local obstacles are distinguished, and obstacles from both categories are identified. The theory of instrumentation provides a framework for interpreting an obstacle as an unbalance of the conceptual and technical aspects of an instrumentation scheme. It is argued that making the obstacles explicit and trying to overcome them leads to conceptual development. Therefore, obstacles are opportunities for learning.     

Four (algorithms) in one (bag): an integrative framework of knowledge for teaching the standard algorithms of the basic arithmetic operations

In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic operations with rational numbers. In order to evaluate the potential applicability of the framework to task design, it was used for the design of mathematical learning tasks for teachers. The article includes examples of the tasks, their theoretical analysis, and empirical evidence of the sensitivity of the tasks to variations in teachers’ knowledge of the subject. This evidence is based on a study of 46 primary school teachers. The article concludes with remarks on the applicability of the framework to research and practice, highlighting its potential to encourage teaching the four algorithms with an emphasis on conceptual understanding.     

Stability of Timoshenko systems with thermal coupling on the bending moment

The Timoshenko system is a distinguished coupled pair of differential equations arising in mathematical elasticity. In the case of constant coefficients, if a damping is added in only one of its equations, it is well‐known that exponential stability holds if and only if the wave speeds of both equations are equal. In the present paper we study both non‐homogeneous and homogeneous thermoelastic problems where the model's coefficients are non‐constant and constants, respectively. Our main stability results are proved by means of a unified approach that combines local estimates of the resolvent equation in the semigroup framework with a recent control‐observability analysis for static systems. Therefore, our results complement all those on the linear case provided in [22], by extending the methodology employed in [4] to the case of Timoshenko systems with thermal coupling on the bending moment.     

Complex Listening: Supporting Students to Listen As Mathematical Sense-makers

Participating in reform-oriented mathematical discussion calls on teachers and students to listen to one another in new and different ways. However, listening is an understudied dimension of teaching and learning mathematics. In this analysis, we draw on a sociocultural perspective and a conceptual framing of three types of listening—evaluative, interpretive, and hermeneutic (Davis, 1996, 1997)—in order to interpret the listening interactions in a fourth-grade classroom. Using interaction analysis (Jordan & Henderson, 1995) to pay close attention to how participants responded to one another during a carefully selected lesson segment, findings reveal that these students listened in complex ways with explicit support from their teacher. From this revelatory case, we offer a framework for understanding the teacher’s role in supporting complex listening.     

Evolution of the magnetic field in plasma in a neighborhood of zero points

We consider a mathematical model of magnetic reconnection in a neighborhood of singular points in the framework of 3D magnetohydrodynamics. In the case of linear dependence of the magnetic field and the velocity on coordinates, we present self-similar solutions of the system of MHD equations. For an arbitrary initial magnetic configuration, we state a mixed problem, which is solved numerically. We describe a semi-implicit algorithm for the solution of the difference problem and present the results of several numerical experiments.     

What Is Entitlement?

In his seminal paper, “Content Preservation,” Tyler Burge defends an original account of testimonial knowledge. The originality of the account is due, in part, to the fact that it is cast within a novel epistemic framework. The central feature of that framework is the introduction of the concept of entitlement, which is alleged to be a distinctive type of positive epistemic support or warrant. Entitlement and justification, according to Burge, are sub-species of warrant. Justification is the internalist form of warrant, but entitlement is epistemically externalist. My focus in this paper is Burge’s conception of entitlement, and there are three primary issues that I wish to address. What is the relationship between entitlement and the more traditional concept of justification? In what sense is entitlement epistemically externalist? Has Burge introduced a new epistemic concept or merely coined a new term for a familiar epistemic concept?     

Remarks on strict efficiency in scalar and vector optimization

The aim of this paper is to study optimality conditions for strict local minima to constrained mathematical problems governed by scalar and vectorial mappings. Unlike other papers in literature dealing with strict efficiency, we work here with mappings defined on infinite dimensional normed vector spaces. Firstly, we (mainly) consider the case of nonsmooth scalar mappings and we use the Fréchet and Mordukhovich subdifferentials in order to provide optimality conditions. Secondly, we present some methods to reduce the study of strict vectorial minima to the case of strict scalar minima by means of some scalarization techniques. In this vectorial framework we treat separately the case where the ordering cone has non-empty interior and the case where it has empty interior.     

The Value of Lesser Goods: The Epistemic Value of Entitlement

The notion of entitlement plays an important role in some influential epistemologies. Often the epistemological motive for introducing the concept is to accommodate certain externalist intuitions within an internalist framework or, conversely, to incorporate internalist traits into an otherwise externalist position. In this paper two prominent philosophers will be used as examples: Tyler Burge as a representative of the first option and Fred Dretske as one of the second. However, even on the assumption that the concept of entitlement is sufficiently clarified, accomplishing these results is easier said than done – especially if we also want to ascribe positive epistemic value to entitlement. It will be shown that the epistemic value of entitlement is either granted at the expense of the epistemic value of justification or the value ends up below the level of value at which the epistemologists employing the concept of entitlement are aiming.     

Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework

The article contributes to the ongoing discussion on ways to deal with the diversity of theories in mathematics education research. It introduces and systematizes a collection of case studies using different strategies and methods for networking theoretical approaches which all frame (qualitative) empirical research. The term ‘networking strategies’ is used to conceptualize those connecting strategies, which aim at reducing the number of unconnected theoretical approaches while respecting their specificity. The article starts with some clarifications on the character and role of theories in general, before proposing first steps towards a conceptual framework for networking strategies. Their application by different methods as well as their contribution to the development of theories in mathematics education are discussed with respect to the case studies in the ZDM-issue.     

Dragging in a Dynamic Geometry Environment Through the Lens of Variation

What makes Dynamic Geometry Environment (DGE) a powerful mathematical knowledge acquisition microworld is its ability to visually make explicit the implicit dynamism of thinking about mathematical geometrical concepts. One of DGE’s powers is to equip us with the ability to retain the background of a geometrical configuration while we can selectively bring to the fore dynamically those parts of the whole configuration that interest us. That is, we can visually study the variation of an aspect of a DGE figure while keeping other aspects constant, hence anticipating the emergence of invariant patterns. The aim of this paper is to expound the epistemic value of variation of the Dragging tool in DGE in mathematical discovery. Functions of variation (contrast, separation, generalization, fusion) proposed in Marton’s theory of learning and awareness will be used as a framework to develop a discernment structure which can act as a lens to organize and interpret dragging explorations in DGE. Such a lens focuses very strongly on mathematical aspects of dragging in DGE and is used to re-interpret known dragging modalities (e.g., Arzarello et al.) in a potentially more mathematically-relevant way. The exposition will centre about a specific geometrical problem in which two dragging trajectories are mapped out, consequently resulting in a DGE theorem and a visual theorem. In doing so, a new spectral dragging strategy will be introduced that literally allows one to see the drag mode in action. A model for the lens of variation in the form of a discernment nest structure is proposed as a meta-tool to interpret dragging experiences or as a meta-language to relate different dragging analyses which consequently might give rise to pedagogical and epistemological implications.     

Self-Inquisitiveness: the Structure and Role of an Epistemic Virtue

The motivating virtue account claims that inquisitiveness or curiosity is the motivating epistemic virtue. In the case of self-knowledge, self-inquisitiveness, intrinsic and instrumental, is the motivating epistemic virtue that mobilizes other virtues, skills, and epistemic character virtues, needed to achieve such knowledge. Its proper object is substantial self-knowledge, knowledge of one’s dispositions and causal powers that has historically played a central role in philosophy, and is now, under various names, investigated by psychologists. It has been, until recently, comparatively neglected within analytical epistemology of self-knowledge. Self-inquisitiveness thus instantiates the general paradigm of curiosity-inquisitiveness that organizes and motivates other epistemic virtues (virtues-abilities and character virtues). And it is perhaps responsible for intrinsic value of self-knowledge.     

Paradox,programming, and learning probability: A case study in a connected mathematics framework

Formal methods abound in the teaching of probability and statistics. In the Connected Probability project, we explore ways for learners to develop their intuitive conceptions of core probabilistic concepts. This article presents a case study of a learner engaged with a probability paradox. Through engaging with this paradoxical problem, she develops stronger intuitions about notions of randomness and distribution and the connections between them. The case illustrates a Connected Mathematics approach: that primary obstacles to learning probability are conceptual and epistemological; that engagement with paradox can be a powerful means of motivating learners to overcome these obstacles; that overcoming these obstacles involves learners making mathematics—not learning a “received” mathematics and that, through programming computational models, learners can more powerfully express and refine their mathematical understandings.     

Design of electronic assembly lines: An analytical framework and its application

The design of component assembly lines in Printed Circuit Board (PCB) manufacturing environments is a challenging problem faced by many firms in the electronics industry. The main design approaches to such component assembly lines are the Mini-Line, Flexible Flow Line, and Hybrid Line designs. In this paper, we discuss the operational trade-offs associated with these design alternatives and present a mathematical programming framework that captures relevant system design issues. Each of the design alternatives can be viewed as a special case of the stated mathematical programming model. We develop effective algorithms to solve these mathematical programs. We have used the framework in a specific PCB manufacturing environment to advise managers on the best configuration of their lines. The models were used as sensitivity analysis tools. The results of our computational experiments, combined with qualitative comparisons of different design approaches developed by a crossfunctional team (engineers, manufacturing and product managers), have led to the development of a set of managerial guidelines for the selection of the design plan for component assembly lines in the studied environment.     

Numerical simulation of discontinuous waves propagating over a dry bed

A numerical algorithm is proposed for simulating the propagation of discontinuous waves over a dry bed governed by the shallow water equations in the first approximation. The algorithm is based on a modified conservation law of total momentum that takes into account the concentrated momentum loss associated with the formation of local eddy structures within the framework of the long-wave approximation. The modified conservation law involves a heuristic parameter that is chosen so as to agree with laboratory experiments. Numerical results are presented for the formation, propagation, and transformation of a discontinuous wave arising in a complete or partial (in the planned case) collapse of a dam over a bed with a horizontal or sloping bottom or a bottom with a local obstacle in the tailwater area.