Using video cases to encourage participants’ engagement with research and theory: Emergent pedagogies from an online course on digital technologies for mathematical learning

In this paper we present the model behind our on-line master level module Digital Technologies for Mathematical Learning, which focuses on the teaching and learning supported by digital technologies. In our evaluation of the delivery of this module over two years, we reflect here on the emergence of two pedagogies: the online pedagogy of the tutors, ensuring that online teaching and learning is effective and the participants’ developing RiTPACK (Research Informed Technological Pedagogical Content Knowledge - our own acronym for this concept) as they start experimenting with using the digital technology in their classroom practice and linking it with the research knowledge base of the module

Building Bridges to Algebra through a Constructionist Learning Environment

Context: In the digital era, it is important to investigate the potential impact of digital technologies in education and how such tools can be successfully integrated into the mathematics classroom. Similarly to many others in the constructionism community, we have been inspired by the idea set out originally by Papert of providing students with appropriate “vehicles” for developing “Mathematical Ways of Thinking.” Problem: A crucial issue regarding the design of digital tools as vehicles is that of “transfer” or “bridging” i.e., what mathematical knowledge is transferred from students’ interactions with such tools to other activities such as when they are doing “paper-and-pencil” mathematics, undertaking traditional exam papers or in other formal and informal settings. Method: Through the lens of a framework for algebraic ways of thinking, this article analyses data gathered as part of the MiGen project from studies aiming at investigating ways to build bridges to formal algebra. Results: The analysis supports the need for and benefit of bridging activities that make the connections to algebra explicit and for frequent reflection and consolidation tasks. Implications: Task and digital environment designers should consider designing bridging activities that consolidate, support and sustain students’ mathematical ways of thinking beyond their digital experience. Constructivist content: Our more general aim is to support the implementation of digital technologies, especially constructionist learning environments, in the mathematics classroom

“Whiteboxing” the Content of a Formal Mathematical Text in a Dynamic Geometry Environment

In this article, we provide an empirical example of how digital technology; in this case, GeoGebra may assist students in uncovering—or whiteboxing—the content of a mathematical proof, in this case that of Proposition 41 from Euclid’s Elements. In the discussion of the example, we look into the impact of GeoGebra’s “dragging” functionality on students’ interactions and the possession and development of students’ proof schemes. The study and accompanying analysis illustrate that, despite the positive whiteboxing effects in relation to the mathematical content of the proposition, whiteboxing through dragging calls for caution in relation to students’ work with proof and proving—in particular, in relation to students seeing the necessity for formal proof. Moreover, caution must be paid, e.g., by teachers, so that students do not jump to conclusions and in the process develop inexpedient mathematical proof schemes upon which they may stumble in their future mathematical work

Design of teacher assistance tools in an exploratory learning environment for algebraic generalisation

The MiGen project is designing and developing an intelligent exploratory environment to support 11-14 year-old students in their learning of algebraic generalisation. Deployed within the classroom, the system also provides tools to assist teachers in monitoring students' activities and progress. This paper describes the architectural design of these Teacher Assistance tools and gives a detailed description of one such tool, focussing in particular on the research challenges faced, and the technologies and approaches chosen to implement the necessary functionalities given the context of the project

Supporting teachers’ technological pedagogical content knowledge of fractions through co-designing a virtual manipulative

This study explores the impact that co-designing a virtual manipulative, Fractions Lab, had on teachers’ professional development. Tapping into an existing community of practice of mathematics specialist teachers, the study identifies how a cooperative enquiry approach utilising workshops and school-based visits challenged 23 competent primary school teachers’ technological, pedagogical and content knowledge of fraction equivalence, addition and subtraction. Verbal and written data from the workshops alongside observations and interviews from the school visits were analysed using the technological, pedagogical and content knowledge (TPACK) framework. The findings show that the assumptions of even experienced teachers were challenged when Fractions Lab was shared as an artefact on which they were asked to co-design and subsequently interact with, using it in subsequent phases of the cooperative inquiry process. Two original aspects of successful co-design of virtual manipulatives through communities of practice are identified and offered to others: (1) careful bootstrapping of the first design iteration that gathers intelligence about the content area and the technological affordances and constraints available; and (2) involvement of highly motivated teachers who perceive themselves as agents of change in the domain area

Similarity-based grouping to support teachers on collaborative activities in exploratory learning environments

This paper describes a computer-based tool that helps teachers group their students for collaborative activities in the classroom, the challenge being to organise groups of students based on their recent work so that their collaboration results in meaningful interactions. Students first work on an exploratory task individually, and then the computer suggests possible groupings of students to the teacher. The complexity of the tasks is such that teachers would require too long a time to create meaningful groups. The paper describes the design of the tool, the algorithms and metrics used for generating the groups, the evaluation of the tool, and the pedagogical context in which the tool was designed

Learning from the pandemic: Capitalising on opportunities and overcoming challenges for mathematics teaching and learning practices with and through technology

This new working group (WG) was created to discuss the theoretical and methodological challenges faced by the mathematics education field when the prevailing boundaries of the classroom shifted; alongside the changed nature of the classroom interactions between the humans (teachers and students) and the chosen technologies. Starting with the assumption that technology resources are being used, the WG explored the nature of these tools and their affordances for the mathematical teaching and learning. The work was framed by the following three pedagogic activities, which are proving to be particularly challenging: introducing and developing understanding of new mathematical topics; managing interaction and communication in mathematics; and assessing mathematics, both formatively and summatively. Three case studies of teachers’ practices were presented to initiate discussions with respect to these challenges and to highlight some existing theoretical and methodological frames

Establishment of methodical basics and initial analysis for the physiological characterization of Roseobacter clade bacteria

Die Roseobacter-Gruppe gehört zu den dominanten Bakteriengruppen mariner Habitate. Über ihre Physiologie und zugehörige Genetik ist wenig bekannt. Im ersten Abschnitt der vorliegenden Arbeit wurden deshalb die methodischen Grundlagen für eine genetische Analyse von Mitgliedern der Roseobacter-Gruppe gelegt. Es wurden Transformationsprotokolle mittels Elektroporation und Transduktion etabliert. Stabil replizierbare Plasmide wurden identifiziert und ein Sauerstoff-unabhängiges Reportergensystem erfolgreich getestet. Im zweiten Teil wurde das Sekretom verschiedener Stämme der Roseobacter-Gruppe funktionell mittels Enzymnachweisverfahren und strukturell mittels 2D-Gelelektrophorese basierten Proteomics bestimmt. Eine Adaption des Sekretoms an sich ändernde Nährstoffbedingungen wurde beobachtet. Schließlich wurde die anaerobe Genexpression von Dinoroseobacter shibae umfassend für denitrifizierende und Arginin-fermentierende Bedingungen in Abhängigkeit von Licht mittels DNA-Array Analysen erfasst. Eine generelle anaerobe Induktion von über siebzig Genen ohne großen Einfluss von Nitrat, Arginin oder Licht wurde beobachtet. Zusammengefasst wurden im Rahmen dieser Arbeit wesentliche methodische und intellektuelle Grundlagen für eine zukünftige systembiologische Charakterisierung der Roseobacter-Gruppe gelegt.The Roseobacter clade belongs to the dominant groups of bacteria in marine habitats. Little is know about their physiology and its corresponding genetics. Therefore, in the first section of the present work, a methodological background was established which enables the genetic analysis of members of the Roseobacter clade. Transformation protocols using electroporation and transduction were established. After identification of stable replicating plasmids, an oxygen-independent reporter gene system was developed. In the second part, the secretome of different strains of the Roseobacter group were determined. Functional analysis were performed via enzyme assays while structural analysis were carried out via 2D-gel electrophoresis based Proteomics. An adaptation of the secretome towards changing nutrient supplies was observed. Finally, the anaerobic gene expression of Dinoroseobacter shibae both under denitrifying as well as arginine-fermentative conditions according to light was comprised extensively via DNA-array analysis. A general anaerobic induction of more than sixty genes without any influence of nitrate, arginine or light was observed. Summarised, essential methodological and intellectual basics for future characterization of the Roseobacter-group via system biology were created

Towards a definition of "mathematical digital competency"

Due to the embeddedness of digital technologies in mathematics education of today, we often see examples of students simultaneously using their mathematical competencies and digital competencies. In relevant literature, however, these are not seen as a connected whole. Based on reviewing existing competency frameworks, both mathematical and digital, and by exploring an empirical example, this article addresses the question of how to think about and understand a combined Bmathematical digital competency^ (MDC). In doing so, the article relies on the two theoretical frameworks of the instrumental approach and conceptual fields to Bbridge^ the mathematical and digital competency descriptions