Representational decisions when learning population dynamics with an instructional simulation

DEMIST is a multi-representational simulation environment that supports understanding of the representations and concepts of population dynamics. We report on a study with 18 subjects with little prior knowledge that explored if DEMIST could support their learning and asked what decisions learners would make about how to use the many representations that DEMIST provides. Analysis revealed that using DEMIST for one hour significantly improved learners' understanding of population dynamics though their knowledge of the relation between representations remained weak. It showed that learners used many of DEMIST's features. For example, they investigated the majority of the representational space, used dynalinking to explore the relation between representations and had preferences for representations with different computational properties. It also revealed that decisions made by designers impacted upon what is intended to be a free discovery environment

Mathematical practice, crowdsourcing, and social machines

The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a "social machine", a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent Computer Mathematics, CICM 2013, July 2013 Bath, U

Validation of Solutions of Construction Problems in Dynamic Geometry Environments

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the drag test criterion may permit constructions created using measurement tools to be considered valid; however, these constructions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is necessarily problematic, we examine what would constitute the analogue of the straightedge-and-compass criterion in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the compatibility criterion , a new but not necessarily superior criterion to the drag test criterion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42932/1/10758_2004_Article_6999.pd

The TA Framework: Designing Real-time Teaching Augmentation for K-12 Classrooms

Recently, the HCI community has seen increased interest in the design of teaching augmentation (TA): tools that extend and complement teachers' pedagogical abilities during ongoing classroom activities. Examples of TA systems are emerging across multiple disciplines, taking various forms: e.g., ambient displays, wearables, or learning analytics dashboards. However, these diverse examples have not been analyzed together to derive more fundamental insights into the design of teaching augmentation. Addressing this opportunity, we broadly synthesize existing cases to propose the TA framework. Our framework specifies a rich design space in five dimensions, to support the design and analysis of teaching augmentation. We contextualize the framework using existing designs cases, to surface underlying design trade-offs: for example, balancing actionability of presented information with teachers' needs for professional autonomy, or balancing unobtrusiveness with informativeness in the design of TA systems. Applying the TA framework, we identify opportunities for future research and design.Comment: to be published in Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems, 17 pages, 10 figure

Theories of Mathematics Education: A Global Survey of Theoretical Frameworks/Trends in Mathematics Education Research

In this article we survey the history of research on theories in mathematics education. We also briefly examine the origins of this line of inquiry, the contribution of Hans-Georg Steiner, the activities of various international topics groups and current discussions of theories in mathematics education research. We conclude by outlining current positions and questions addressed by mathematics education researchers in the research forum on theories at the 2005 PME meeting in Melbourne, Australia

Proof Planning with Multiple Strategies

. Humans have different problem solving strategies at their disposal and they can flexibly employ several strategies when solving a complex problem, whereas previous theorem proving and planning systems typically employ a single strategy or a hard coded combination of a few strategies. We introduce multi-strategy proof planning that allows for combining a number of strategies and for switching flexibly between strategies in a proof planning process. Thereby proof planning becomes more robust since it does not necessarily fail if one problem solving mechanism fails. Rather it can reason about preference of strategies and about failures. Moreover, our strategies provide a means for structuring the vast amount of knowledge such that the planner can cope with the otherwise overwhelming knowledge in mathematics. 1 Introduction The choice of an appropriate problem solving strategy is a crucial human skill and is typically guided by some meta-level reasoning. Trained mathematicia..