Characterizing Interaction with Visual Mathematical Representations

This paper presents a characterization of computer-based interactions by which learners can explore and investigate visual mathematical representations (VMRs). VMRs (e.g., geometric structures, graphs, and diagrams) refer to graphical representations that visually encode properties and relationships of mathematical structures and concepts. Currently, most mathematical tools provide methods by which a learner can interact with these representations. Interaction, in such cases, mediates between the VMR and the thinking, reasoning, and intentions of the learner, and is often intended to support the cognitive tasks that the learner may want to perform on or with the representation. This paper brings together a diverse set of interaction techniques and categorizes and describes them according to their common characteristics, goals, intended benefits, and features. In this way, this paper aims to provide a preliminary framework to help designers of mathematical cognitive tools in their selection and analysis of different interaction techniques as well as to foster the design of more innovative interactive mathematical tools. An effort is made to demonstrate how the different interaction techniques developed in the context of other disciplines (e.g., information visualization) can support a diverse set of mathematical tasks and activities involving VMRs.     

Extending the GGobi pipeline from R

This paper describes progress towards developing a platform for rapid prototyping of interactive data visualizations, using R, GGobi, rggobi and RGtk2. GGobi is a software tool for multivariate interactive graphics. At the core of GGobi is a data pipeline that incrementally transforms data through a series of stages into a plot and maps user interaction with the plot back to the data. The GGobi pipeline is extensible and mutable at runtime. The rggobi package, an interface from the R language to GGobi, has been augmented with a low-level interface that supports the customization of interactive data visualizations through the extension and manipulation of the GGobi pipeline. The large size of the GGobi API has motivated the use of the RGtk2 code generation system to create the low-level interface between R and GGobi. The software is demonstrated through an application to interactive network visualization.     

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.     

Epistemic management in mathematics classroom interactions: Student claims of not knowing or not understanding

An important role of classroom interaction is the processes involved in knowing or coming to know. Following a conversation analytical approach, this study investigates how students’ claims of not knowing, not remembering or not understanding are handled by mathematics teachers in whole class interactions. The study draws on video recordings of 42 mathematics lessons from 8 secondary schools in England. It is argued that claims of not knowing and claims of not remembering perform different social actions and are consequently treated differently by teachers. Claims of not knowing can challenge the assumption that knowledge can be taken-as-shared in a way that claims of not remembering do not. This contributes to the research field of mathematics classroom interaction as it nuances the epistemic management within these interactions and how this can contribute to the norms around the negotiation of meaning.     

Visualizing number sequences: Secondary preservice mathematics teachers’ constructions of figurate numbers using magnetic color cubes

This study is about preservice secondary mathematics teachers’ visualization of summation formulas modeled by magnetic color cubes representations. The theoretical framework for this research draws from studies on quantitative reasoning (Smith and Thompson, 2008, Thompson, 1995) and quantitative transformations (Schwartz, 1988). Data consist of videotaped qualitative interviews during which preservice mathematics teachers were asked to construct growing rectangles representing summation formulas. Data analysis is based on analytic induction and constant comparison methodology. Preservice teachers provided a diversity of additive and multiplicative visualizations. Results indicate that quantitative reasoning and mapping structures are fundamental constructs in establishing additive and multiplicative visualizations, hence constructing summation formulas meaningfully. Preservice teachers often had difficulties in explaining the relationships between the same-valued linear and areal quantities. They also established the rectangle condition as the essence of multiplicative visualization.     

An epistemic model of task design in dynamic geometry environment

Dynamic geometry environment (DGE) has been a catalytic agent driving a paradigm shift in the teaching and learning of school geometry in the past two decades. It opens up a pedagogical space for teachers and students to engage in mathematical explorations that niche across the experimental and the theoretical. In particular, the drag-mode in DGE has been a unique pedagogical tool that can facilitate and empower students to experiment with dynamic geometrical objects which can lead to generation of mathematical conjectures. Furthermore, the drag-mode seems to open up a new methodology and even a new discourse to acquire geometrical knowledge alternative to the traditional Euclidean deductive reasoning paradigm. This discussion paper proposes an epistemic model of techno-pedagogic mathematic task design which serves as a theoretical combined-lens to view mathematics knowledge acquisition. Three epistemic modes for techno-pedagogic mathematical task design are proposed. They are used to conceptualize design of dynamic geometry tasks capitalizing the unique drag-mode nature in DGE that opens up an explorative space for learners to acquire mathematical knowledge.     

Robot Manipulations: A Synergy of Visualization, Computation and Action for Spatial Instruction

This article considers the use of a learning environment, RoboCell, where manipulations of objects are performed by robot operations specified through the learner's application of mathematical and spatial reasoning. A curriculum is proposed relating to robot kinematics and point-to-point motion, rotation of objects, and robotic assembly of spatial puzzles. Various instructional methods are supported by the RoboCell robot system, such as interactive demonstrations, modeling, computer simulations and robot operations, providing diverse activities in spatial perception, mental rotation and visualization. Pre-course and post-course tests in two middle schools and a high school indicated significant student progress in the tasks related to the categories of spatial ability which were practiced in the course. This revised version was published online in July 2006 with corrections to the Cover Date.     

Reflected acting in mathematical learning processes

Acting and thinking are strongly interconnected activities. This paper proposes an approach to mathematical concepts from the angle of hands-on acting. In the process of learning, special emphasis is put on the reflection of the own actions, enabling learners to act consciously. An illustration is presented in the area number representation and extensions of number fields. Using didactical materials, processes of mathematical acting are stimulated and reflected. Mathematical concepts are jointly developed with the learners, trying to address shortcomings from own experiences. This is accompanied by reflection processes that make conscious to learners the rationale of mathematical approaches and the creation of mathematical concepts. Teaching mathematics following this approach does intent to contribute to the development of decision-making and responsibility capabilities of learners.     

Embodied Semiotic Activities and Their Role in the Construction of Mathematical Meaning of Motion Graphs

This paper examines the relation between bodily actions, artifact-mediated activities, and semiotic processes that students experience while producing and interpreting graphs of two-dimensional motion in the plane. We designed a technology-based setting that enabled students to engage in embodied semiotic activities and experience two modes of interaction: 2D freehand motion and 2D synthesized motion, designed by the composition of single variable function graphs. Our theoretical framework combines two perspectives: the embodied approach to the nature of mathematical thinking and the Vygotskian notion of semiotic mediation. The article describes in detail the actions, gestures, graph drawings, and verbal discourse of one pair of high school students and analyzes the social semiotic processes they experienced. Our analysis shows how the computerized artifacts and the students’ gestures served as means of semiotic mediation. Specifically, they supported the interpretation and the production of motion graphs; they mediated the transition between an individual’s meaning of mathematical signs and culturally accepted mathematical meaning; and they enable linking bodily actions with formal signs.     

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.     

Explorative and dynamic visualization of data in virtual reality

Summary  A software system has been developed for the study of dynamic glyph visualizations in the context of Visual Data Mining in Virtual Reality. The system uses parallel processing to calculate data visualizations in real-time, with real-time interaction and dynamic changes to the view. The system allows morphing between different visualizations, the use of dynamic features like “vibrations” and “rotations” of thousands of objects individually, and dynamic visualization, where the influence of any variable of a dataset with a “reasonable” distribution, can be shown as a dynamic development. It appears that these facilities for dynamic data visualization have a very promising potential, but their optimal use will depend on further developments in the context of their individual practical application.     

Types of reasoning required in university exams in mathematics

Empirical research shows that students often use reasoning founded on copying algorithms or recalling facts (imitative reasoning) when solving mathematical tasks. Research also indicate that a focus on this type of reasoning might weaken the students’ understanding of the underlying mathematical concepts. It is therefore important to study the types of reasoning students have to perform in order to solve exam tasks and pass exams. The purpose of this study is to examine what types of reasoning students taking introductory calculus courses are required to perform. Tasks from 16 exams produced at four different Swedish universities were analyzed and sorted into task classes. The analysis resulted in several examples of tasks demanding different types of mathematical reasoning. The results also show that about 70% of the tasks were solvable by imitative reasoning and that 15 of the exams could be passed using only imitative reasoning.     

High school prospective teachers’ problem-solving reasoning that involves the coordinated use of digital technologies

The aim of this study was to characterize and discuss ways of reasoning that prospective high school mathematics teachers develop and exhibit in a problem-solving scenario that involves the coordinated use of digital technologies. A conceptual framework that includes Virtual Learning Spaces (VLS) and Resources, Activities, Support and Evaluation (RASE) essentials is used to introduce and support a problem-solving approach to structure learners’ problem-solving activities that encouraged them to share ideas, discuss and extend mathematical discussions beyond formal settings. Main results indicated that prospective high school teachers relied on a set of tool affordances (dragging objects, looking and exploring object’s loci, using sliders, quantifying and visualizing mathematical relations, etc.) to formulate, explore and identify properties or relations to share, discuss and support mathematical conjectures. In this context, the participants recognized and valued the importance of using several tools to both dynamically represent and explore mathematical tasks and to share and constantly refine their mathematical ideas and problem-solving approaches.     

Adapting Japanese Lesson Study to enhance the teaching and learning of geometry and spatial reasoning in early years classrooms: a case study

Increased efforts are needed to meet the demand for high quality mathematics in early years classrooms. Despite the foundational role of geometry and spatial reasoning for later mathematics success, the strand receives inadequate instructional time and is limited to concepts of static geometry. Moreover, early years teachers typically lack both content knowledge and confidence in teaching geometry and spatial reasoning. We describe our attempt to deal with these issues through a research initiative known as the Math for Young Children project. The project integrates effective features of both design research and Japanese Lesson Study and is designed to support teachers in developing content knowledge and new approaches for teaching geometry and spatial reasoning. Central to our Professional Development model is the integration of four adaptations to the Japanese Lesson Study model: (1) teachers engaging in the mathematics, (2) teachers designing and conducting task-based clinical interviews, (3) teachers and researchers co-designing and carrying out exploratory lessons and activities, and (4) the creation of resources for other educators. We present our methods and the results of our adaptations through a case study of one Professional Learning Team. Our results suggest that the adaptations were effective in: (1) supporting teachers’ content knowledge of and comfort level with geometry and spatial reasoning, (2) increasing teachers’ perceptions of young children’s mathematical competencies, (3) increasing teachers’ awareness and commitment for the inclusion of high quality geometry and spatial reasoning as a critical component of early years mathematics, and (4) the creation of innovative resources for other educators. We conclude with theoretical considerations and implications of our results.     

Using interactive algebra software to support a discourse community

In this work we studied the impact of using NuCalc, an interactive computer algebra software, on the development of a discourse community in a college level mathematics class. Qualitative and quantitative data were collected over the course of 3 weeks of instruction. We examined the influence of the software on: group interactions; the mathematical investigations of learners; and the teacher’s interactions with students. Data points to four distinct ways in which the presence of NuCalc positively impacted the learning community we studied: (1) it served as a tool for extending students’ mathematical thinking, (2) it motivated students’ engagement in group discourse, (3) it became a tool for mediating discourse, (4) it became a catalyst for refining the culture of classroom, shifting the patterns of interactions between the teacher and learners.     

A characterization of dynamic reasoning: Reasoning with time as parameter

Students incorporate and use the implicit and explicit parameter time to support their mathematical reasoning and deepen their understandings as they participate in a differential equations class during instruction on solutions to systems of differential equations. Therefore, dynamic reasoning is defined as developing and using conceptualizations about time as a parameter that implicitly or explicitly coordinates with other quantities to understand and solve problems. Students participate in the following types of mathematical activity related to dynamic reasoning: making time an explicit quantity, using the metaphor of time as “unidimensional space”, using time to reason both quantitatively and qualitatively, using three-dimensional visualization of time related functions, fusing context and representation of time related functions, and using the fictive motion metaphor for function. The purpose of this article is to present a characterization of dynamic reasoning and promote more explicit attention to this type of reasoning by teachers in K-16 mathematics in order to improve student understanding in time related areas of mathematics.     

“Your truth isn’t the Truth”: Data activities and informal inferential reasoning

This paper investigates data activities in an afterschool setting, offering a deeper understanding of the social nature of students’ informal inferences by investigating how informal inferences are negotiated in group interactions, influenced by social norms, and how statistical concepts come into play in learners’ informal inferential reasoning (IIR). Analyses take up a multi-sited orientation to investigate how youth used quantitative and contextual resources during a research activity to make meaning of data and negotiate emergent social tensions. Findings show how data activities that are part of informal inferential reasoning, such as collection, interpretation, generalization, inference, and representation unfolded as social, political, and personal. Implications call for designs for learning that better support working with data and understanding real-world phenomena and sociopolitical issues in ways that leverage youths’ experiences, enabling them to take part in social action as critical community actors.     

Toward a situation model in a cognitive architecture

The ability to coherently represent information that is situationally relevant is vitally important to perform any complex task, especially when that task involves coordinating with team members. This paper introduces an approach to dynamically represent situation information within the ACT-R cognitive architecture in the context of a synthetic teammate project. The situation model represents the synthetic teammate’s mental model of the objects, events, actions, and relationships encountered in a complex task simulation. The situation model grounds textual information from the language analysis component into knowledge usable by the agent-environment interaction component. The situation model is a key component of the synthetic teammate as it provides the primary interface between arguably distinct cognitive processes modeled within the synthetic teammate (e.g., language processing and interactions with the task environment). This work has provided some evidence that reasoning about complex situations requires more than simple mental representations and requires mental processes involving multiple steps. Additionally, the work has revealed an initial method for reasoning across the various dimensions of situations. One purpose of the research is to demonstrate that this approach to implementing a situation model provides a robust capability to handle tasks in which an agent must construct a mental model from textual information, reason about complex relationships between objects, events, and actions in its environment, and appropriately communicate with task participants using natural language. In this paper we describe an approach for modeling situationally relevant information, provide a detailed example, discuss challenges faced, and present research plans for the situation model.