Debugging an Artifact, Instrumenting a Bug: Dialectics of Instrumentation and Design in Technology-Rich Learning Environments

This article explores ways of conceptualizing the design of innovative learning tools as emergent from dialectics between designers and learner-users of those tools. More specifically, I focus on the reciprocities between a designer’s objectives for student learning and a user’s situated activity in a learning environment, as these interact and co-develop in cycles of design-based research. Recent investigations of technology-supported mathematics learning conducted from an ‘instrumental’ perspective provide a powerful framework for analyzing the process through which classroom artifacts become conceptual tools, simultaneously characterizing the ways students come to both implement and understand a device in the context of a task. Similarly, design-based approaches to investigating instructional activity offer epistemological grounds for treating the process of designing artifacts to support learning as unfolding in concert with rather than concluding prior to situated student use. Drawing on each of these perspectives, I describe the design and initial implementation of a set of software artifacts intended to support students’ collaborative problem solving through locally networked handheld computers. Through detailed analyses of three classroom episodes, I report on the ways one student group’s innovative and unexpected use of these tools served as an opportunity to both examine student learning in the context of that novelty and to refine the software design. This account provides an empirical example through which to consider the potential for instrumental genesis to inform design, and for design research epistemology to broaden the scope of instrumental theory.

Word problem solving and pictorial representations: insights from an exploratory study in kindergarten

The aim of this study was to investigate how pictorial representations with different semiotic characteristics affect additive word problem solving by kindergartners. The focus of the study is on three categories of additive problems (change problems, combine problems and equalize problems) and on representational pictures with different semiotic characteristics: (a) pictures in which the problem quantities are represented in pictorial form, that is, as groups of illustrated objects (PP pictures), (b) pictures in which the quantities are represented partly in pictorial form and in symbolic form (PS pictures), and (c) pictures in which the quantities are represented in symbolic form (SS pictures). Data were collected from 63 kindergartners using a paper-and-pencil test. Results showed that the semiotic characteristics of representational pictures had a strong and significant effect on performance. Children’s performance was higher in the problems with PP pictures but declined in the problems with PS and SS pictures. However, the differences in children’s performance across the problems with different representational format varied between the problem categories and their mathematical structures. The semiotic characteristics of representational pictures had an important role in the establishment of close relations between children’s solutions in problems in different categories. Detailed analysis of children’s answers to the problems revealed a number of picture-related difficulties. Findings are discussed and directions for future research are drawn considering the methodological limitations of the study.     

Survivable network design under optimal and heuristic interdiction scenarios

We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer’s network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.     

Making Sense by Building Sense: Kindergarten Children’s Construction and Understanding of Adaptive Robot Behaviors

This study explores young children’s ability to construct and explain adaptive behaviors of a behaving artifact, an autonomous mobile robot with sensors. A central component of the behavior construction environment is the RoboGan software that supports children’s construction of spatiotemporal events with an a-temporal rule structure. Six kindergarten children participated in the study, three girls and three boys. Activities and interviews were conducted individually along five sessions that included increasingly complex construction tasks. It was found that all of the children succeeded in constructing most such behaviors, debugging their constructions in a relatively small number of cycles. An adult’s assistance in noticing relevant features of the problem was necessary for the more complex tasks that involved four complementary rules. The spatial scaffolding afforded by the RoboGan interface was well used by the children, as they consistently used partial backtracking strategies to improve their constructions, and employed modular construction strategies in the more complex tasks. The children’s explanations following their construction usually capped at one rule, or two condition-action couples, one rule short of their final constructions. With respect to tasks that involved describing a demonstrated robot’s behavior, in describing their constructions, explanations tended to be more rule-based, complex and mechanistic. These results are discussed with respect to the importance of making such physical/computational environments available to young children, and support of young children’s learning about such intelligent systems and reasoning in developmentally-advanced forms.     

A Priori knowledge contextualised and Benacerraf’s dilemma

In this article, I discuss Hawthorne’s contextualist solution to Benacerraf’s dilemma. He wants to find a satisfactory epistemology to go with realist ontology, namely with causally inaccessible mathematical and modal entities. I claim that he is unsuccessful. The contextualist theories of knowledge attributions were primarily developed as a response to the skeptical argument based on the deductive closure principle. Hawthorne uses the same strategy in his attempt to solve the epistemologist puzzle facing the proponents of mathematical and modal realism, but this problem is of a different nature than the skeptical one. The contextualist theory of knowledge attributions cannot help us with the question about the nature of mathematical and modal reality and how they can be known. I further argue that Hawthorne’s account does not say anything about a priori status of mathematical and modal knowledge. Later, Hawthorne adds to his account an implausible claim that in some contexts a gettierized belief counts as knowledge.     

Comparing German and Taiwanese secondary school students’ knowledge in solving mathematical modelling tasks requiring their assumptions

Mathematization is critical in providing students with challenges for solving modelling tasks. Inadequate assumptions in a modelling task lead to an inadequate situational model, and to an inadequate mathematical model for the problem situation. However, the role of assumptions in solving modelling problems has been investigated only rarely. In this study, we intentionally designed two types of assumptions in two modelling tasks, namely, one task that requires non-numerical assumptions only and another that requires both non-numerical and numerical assumptions. Moreover, conceptual knowledge and procedural knowledge are also two factors influencing students’ modelling performance. However, current studies comparing modelling performance between Western and non-Western students do not consider the differences in students’ knowledge. This gap in research intrigued us and prompted us to investigate whether Taiwanese students can still perform better than German students if students’ mathematical knowledge in solving modelling tasks is differentiated. The results of our study showed that the Taiwanese students had significantly higher mathematical knowledge than did the German students with regard to either conceptual knowledge or procedural knowledge. However, if students of both countries were on the same level of mathematical knowledge, the German students were found to have higher modelling performance compared to the Taiwanese students in solving the same modelling tasks, whether such tasks required non-numerical assumptions only, or both non-numerical and numerical assumptions. This study provides evidence that making assumptions is a strength of German students compared to Taiwanese students. Our findings imply that Western mathematics education may be more effective in improving students’ ability to solve holistic modelling problems.     

Omnipresence, Multipresence and Ubiquity: Kinds of Generality in and Around Mathematics and Logics

A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: ‘omnipresent’ and ‘multipresent’ theories, and ‘ubiquitous’ notions that form dependent parts, or moments, of theories. The category of ‘facets’ is also introduced, primarily to assess the roles of diagrams and notations in these two disciplines. Various consequences are explored, starting with means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. The influence of theories already in place upon theory-building is emphasised. The roles in both mathematics and logics of set theory, abstract algebras, metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted in algebraic logic and in mathematical logic. Finally, the issue of monism versus pluralism in these two disciplines is rehearsed, and some suggestions are made about the special character of mathematical and logical knowledge, and also the differences between them. Since the article is basically an exercise in historiography, historical examples and case studies are described or noted throughout.     

A validation study of the use of mathematical knowledge for teaching measures in Ireland

Researchers who study mathematical knowledge for teaching (MKT) are interested in how teachers deploy their mathematical knowledge in the classroom to enhance instruction and student learning. However, little data exists on how teachers’ scores on the US-developed measures relate to classroom instruction in other countries. This article documents a validation study of Irish teachers’ scores on measures of MKT that were adapted for use in Ireland. A validity argument is made identifying elemental, structural and ecological assumptions. The argument is evaluated using qualitative and quantitative data to analyse inferences related to the three assumptions. The data confirmed the elemental assumption but confirming the structural and ecological assumptions was more difficult. Only a weak association was found between teachers’ MKT scores and the mathematical quality of instruction. Possible reasons for this are outlined and challenges in validating the use of measures are identified.     

Distribution systems design with two-level routing considerations

In this study, we formulate and analyze a strategic design model for three-echelon distribution systems with two-level routing considerations. The key design decisions considered are: the number and locations of distribution centers (DC’s), which big clients (clients with larger demand) should be included in the first level routing (the routing between plants and DC’s), the first-level routing between plants, DC’s and big clients, and the second-level routing between DC’s and other clients not included in the first-level routing. A hybrid genetic algorithm embedded with a routing heuristic is developed to efficiently find near-optimal solutions. The quality of the solution to a series of small test problems is evaluated—by comparison with the optimal solution solved using LINGO 9.0. In test problems for which exact solutions are available, the heuristic solution is within 1% of optimal. At last, the model is applied to design a national finished goods distribution system for a Taiwan label-stock manufacturer. Through the case study, we find that the inclusion of big clients in the first-level routing in the analysis leads to a better network design in terms of total logistic costs.     

“Visual Math: The Function Web Book” by Yerushalmy, M., Katriel, H., and Shternberg, B. (2002). Published by CET, Ramat Aviv, Israel. www.cet.ac.il/math/function/english

This article is a review of the above web (or interaction) book. The review considers the contribution of this book with respect to the authors’ own stated aspirations to support the teaching and learning of functions. These somewhat theoretical conclusions are hopefully given enhanced meaning through discussion with a trainee teacher. I conclude that the design of the web book has much to offer anyone working in this field. Teachers as well will find it a wonderful resource though they may need to work quite hard at motivating children’s work and interpreting the many rich tasks to make them more accessible to “ordinary kids”.     

On mappings whose sum is monotone

Proposition 1 of this article points at a gap in the proof of Kuhlmann’s characterization of algebraically maximal valued fields [2]. The author [1] showed how to fill this gap. His arguments involved tools of mathematical logic and infinite combinatorics. Proposition 2 of this article provides us with a simple proof of the key fact of Kuhlmann’s characterization.     

Pivoting in Linear Complementarity: Two Polynomial-Time Cases

We study the behavior of simple principal pivoting methods for the P-matrix linear complementarity problem (P-LCP). We solve an open problem of Morris by showing that Murty’s least-index pivot rule (under any fixed index order) leads to a quadratic number of iterations on Morris’s highly cyclic P-LCP examples. We then show that on K-matrix LCP instances, all pivot rules require only a linear number of iterations. As the main tool, we employ unique-sink orientations of cubes, a useful combinatorial abstraction of the P-LCP.     

Assessing elemental validity: the transfer and use of mathematical knowledge for teaching measures in Ghana

This paper reports on a validation study that investigates the utility of US-developed mathematical knowledge for teaching measures in Ghana. Using three teachers as cases this study examines the relationship between teachers’ mathematical knowledge for teaching responses and their reasoning about their responses. Preliminary findings indicate that although the measures could be used in Ghana with adaptation to determine teachers with high mathematical knowledge, the validity of the findings are influenced by other issues such as the cultural incongruence of the item contexts.     

A changing trajectory of proof learning in the geometry inquiry classroom

The purpose of this study is to explore how students changes through learning to construct mathematical proofs in an inquiry-based middle school geometry class in Korea. Although proof has long been considered as one of the most important aspects of mathematics education, it is well-known that it is one of the most difficult areas of school mathematics for students. The geometry inquiry classroom (GIC) is an experimental class designed to teach geometry, especially focusing on teaching proof, based on students’ own inquiry. Based on a 2-year participant observation in the GIC, this study was conducted to address the following research question: how has students’ practice of mathematical proof been changed through their participation in the GIC? The in-depth analysis of the classroom discourse identified three stages through which the students’ practice of mathematical proof was transformed in the GIC: ‘emergent understanding of proof’, ‘proof learning as a goal-oriented activity’, ‘experiencing proof as the practice of mathematics’. The study found that as learning evolved through these stages, so the mathematics teacher’s role shifted from being an instructor to a mediator of communication. Most importantly, this research showed that the GIC has created a learning environment where students develop their competence in constructing meaningful mathematical proof and grow to be ‘a human who proves’, ultimately ‘a person who playfully engages with mathematics’.     

On the Integration of Digital Technologies into Mathematics Classrooms

Trouche’s [Third Computer Algebra in Mathematics Education Symposiums, Reims, France, June 2003] presentation at the Third Computer Algebra in Mathematics Education Symposium focused on the notions of instrumental genesis and of orchestration: the former concerning the mutual transformation of learner and artefact in the course of constructing knowledge with technology; the latter concerning the problem of integrating technology into classroom practice. At the Symposium, there was considerable discussion of the idea of situated abstraction, which the current authors have been developing over the last decade. In this paper, we summarise the theory of instrumental genesis and attempt to link it with situated abstraction. We then seek to broaden Trouche’s discussion of orchestration to elaborate the role of artefacts in the process, and describe how the notion of situated abstraction could be used to make sense of the evolving mathematical knowledge of a community as well as an individual. We conclude by elaborating the ways in which technological artefacts can provide shared means of mathematical expression, and discuss the need to recognise the diversity of student’s emergent meanings for mathematics, and the legitimacy of mathematical expression that may be initially divergent from institutionalised mathematics.     

Non-Inferential Moral Knowledge

In a series of recent papers, Walter Sinnott-Armstrong has developed a novel argument against moral intuitionism. I suggest a defense on behalf of the intuitionist against Sinnott-Armstrong’s objections. Rather than focus on the main premises of his argument, I instead examine the way in which Sinnott-Armstrong construes the intuitionistic position. I claim that Sinnott-Armstrong’s understanding of intuitionism is mistaken. In particular, I argue that Sinnott-Armstrong mischaracterizes non-inferentiality as it figures in intuitionism. To the extent that Sinnott-Armstrong’s account of intuitionism has been adopted by others uncritically, intuitionists have cause for concern. I develop an alternative, and more accurate, reading of what is non-inferential about intuitionistic moral knowledge. In light of this alternative reading, certain elements of Sinnott-Armstrong’s case against intuitionism are significantly weakened. But perhaps more importantly, this paper helps clarify what circumspect intuitionists mean when they claim that some moral knowledge is non-inferential.     

Game Design as an Interactive Learning Environment for Fostering Students'' and Teachers'' Mathematical Inquiry

Many learning environments, computer-based or not, have been developed for either students or teachers alone to engage them in mathematical inquiry. While some headway has been made in both directions, few efforts have concentrated on creating learning environments that bring both teachers and students together in their teaching and learning. In the following paper, we propose game design as such a learning environment for students and teachers to build on and challenge their existing understandings of mathematics, engage in relevant and meaningful learning contexts, and develop connections among their mathematical ideas and their real world contexts. To examine the potential of this approach, we conducted and analyzed two studies: Study I focused on a team of four elementary school students designing games to teach fractions to younger students, Study II focused on teams of pre-service teachers engaged in the same task. We analyzed the various games designed by the different teams to understand how teachers and students conceptualize the task of creating virtual game learning environment for others, in which ways they integrate their understanding of fractions and develop notions about students' thinking in fractions, and how conceptual design tools can provide a common platform to develop meaningful fraction contexts. In our analysis, we found that most teachers and students, when left to their own devices, create instructional games to teach fractions that incorporate little of their knowledge. We found that when we provided teachers and students with conceptual design tools such as game screens and design directives that facilitated an integration of content and game context, the games as well as teachers' and students' thinking increased in their sophistication. In the discussion, we elaborate on how the design activities helped to integrate rarely used informal knowledge of students and teachers, how the conceptual design tools improved the instructional design process, and how students and teachers benefit in their mathematical inquiry from each others' perspectives. In the outlook, we discuss features for computational design learning environments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Global Dynamics of a Novel Delayed Logistic Equation Arising from Cell Biology

The delayed logistic equation (also known as Hutchinson’s equation or Wright’s equation) was originally introduced to explain oscillatory phenomena in ecological dynamics. While it motivated the development of a large number of mathematical tools in the study of nonlinear delay differential equations, it also received criticism from modellers because of the lack of a mechanistic biological derivation and interpretation. Here, we propose a new delayed logistic equation, which has clear biological underpinning coming from cell population modelling. This nonlinear differential equation includes terms with discrete and distributed delays. The global dynamics is completely described, and it is proven that all feasible non-trivial solutions converge to the positive equilibrium. The main tools of the proof rely on persistence theory, comparison principles and an $$L^2$$-perturbation technique. Using local invariant manifolds, a unique heteroclinic orbit is constructed that connects the unstable zero and the stable positive equilibrium, and we show that these three complete orbits constitute the global attractor of the system. Despite global attractivity, the dynamics is not trivial as we can observe long-lasting transient oscillatory patterns of various shapes. We also discuss the biological implications of these findings and their relations to other logistic-type models of growth with delays.     

The Complex Process of Converting Tools into Mathematical Instruments: The Case of Calculators

Transforming any tool into a mathematical instrument for students involves a complex ‘instrumentation’ process and does not necessarily lead to better mathematical understanding. Analysis of the constraints and potential of the artefact are necessary in order to point out the mathematical knowledge involved in using a calculator. Results of this analysis have an influence on the design of problem situations. Observations of students using graphic and symbolic calculators were analysed and categorised into profiles, illustrating that transforming the calculator into an efficient mathematical instrument varies from student to student, a factor which has to be included in the teaching process. This revised version was published online in August 2006 with corrections to the Cover Date.