Corey Brady

Fostering computational literacy in science classrooms

An agent-based approach to integrating computing in secondary-school science courses.

All Roads Lead to Computing: Making, Participatory Simulations, and Social Computing as Pathways to Computer Science

Computer science (CS) is becoming an increasingly diverse domain. This paper reports on an initiative designed to introduce underrepresented populations to computing using an eclectic, multifaceted approach. As part of a yearlong computing course, students engage in Maker activities, participatory simulations, and computing projects that foreground the social and collaborative aspects of CS. Collectively, these activities are designed to introduce learners to the growing diversity of what CS looks like in the 21st century. This paper lays out the practical and theoretical motivations for the Computational Thinking for Girls (CT4G) project, specifically highlighting the use of Making through physical and social computing as ways to engage students in CS. A snapshot of one activity from the program is provided—Wearing the Web—in which students use open-hardware programmable badges to explore the underlying structure and technology that enables the Internet. Data from the first year of the CT4G program are presented to show the positive effects that this diverse introduction to CS is having on the students with respect to their attitudes toward CS.

Technological Supports for Mathematical Thinking and Learning: Co-action and Designing to Democratize Access to Powerful Ideas

The enterprise of understanding and supporting processes of mathematical cognition is both epistemologically deep and politically urgent. We cannot ignore that new technology-mediated learning environments have the potential to democratize access to powerful ideas. The importance of technology in this respect is bound up with the essential nature of mathematical objects as symbolic entities that can only be expressed and conjured up through the mediation of representations. A key question for the design of technology-enhanced learning environments is whether the cognitive tools—material and digital-symbolic—that have been developed in recent decades might offer learners access to modes of activity with disciplinary structures that have historically been achievable only by ‘maestros’ of the discipline. In this article we elaborate the construct of “co-action” as a means of describing humans’ mathematical interactions with the support of such tools.

Extending the Reach of the Models and Modelling Perspective: A Course-Sized Research Site

For over 30 years, researchers have engaged in inquiry within the Models and Modelling Perspective (MMP), taking as a fundamental principle that learners’ ideas develop in coherent conceptual systems called models. Under appropriate conditions, such as in Model Eliciting Activities (MEAs), this research has shown how learners’ models can grow through rapid cycles of development toward solutions involving creative mathematics. These externalized models, and other thought-revealing artifacts, can become rich objects for reflection by learners, for formative assessment by teachers, and for analysis of idea-development by researchers. This chapter describes a new research effort to expand the reach of this MMP tradition, engaging questions about the interconnected models and modelling processes of students and teachers at larger, course-length scales.

The Student Experience of Model Development Activties: Going Beyond Correctness to Meet a Client’s Needs

For today’s students to be successful tomorrow in the world beyond school, their mathematics education must go beyond assembling a fixed toolkit of skills and procedures. They will need to be able to adapt and apply the material they learn in school in subtle ways, in order to produce tactful and appropriate solutions to problems under constraints and uncertainty. In this chapter, we describe a genre of activities that provide this kind of experience to learners in classrooms settings. We consider the student experience of these activities, and we show how uncertainty about how to apply the knowledge they have learned opens up space for students to use mathematical ways of thinking to interpret the world. We then show how the presence of a concrete Client provides a means for students to continually test whether their emerging solutions are responsive to the human needs and perspectives of this Client, as well as sensitive to other people in the problem who are affected by their solution.

Designing Powerful Environments to Examine and Support Teacher Competencies for Models and Modelling: Original Text – Chapter 8

The purpose of this chapter is to highlight how work focusing on student‐level modelling and idea development can also serve as a powerful context to investigate parallel and interacting modelling processes at the teacher level. We assert that expertise in teaching is partly reflected in how teachers interpret and respond to classroom situations—in what they see and recognize, as well as in what they do. To establish a setting for extended studies of teacher‐level modelling, different research groups have found it desirable to move beyond single‐activity implementations in one of several ways. We begin our description of this work by identifying several values or beliefs about the nature of mathematical thinking and learning and corresponding instructional practices, which comprise key teacher competencies for the teaching and learning of modelling that have been emphasized by these projects. Then, we describe three different extensions from previous reports of our ongoing research, each aimed at examining and supporting students’ and teachers’ models and modelling over long time‐scales, such as the level of an entire course. Our results suggest that, in course‐sized studies of student development, significant powerful changes can also occur in teacher‐level competencies for the teaching and learning of modelling. One reason for this teacher‐level development is that in our student‐level modelling activities, students naturally express important aspects of their thinking in forms that can be observed directly by both teachers and researchers. Insights about the nature of students’ thinking have proven to be a powerful impetus to encourage important aspects of teacher development. Furthermore, course‐sized research sites enable us to directly observe significant teacher competencies that cannot be seen in studies focusing on single modelling activities.

Entwicklung wirkungsvoller Umgebungen zur Untersuchung von Lehrerkompetenzen zum mathematischen Modellieren

In diesem Kapitel wird verdeutlicht, wie Modellierungsprozesse von Lernenden fur die Lehrerausbildung zum Lehren von mathematischer Modellierung als ein wirkungsvolles Instrument eingesetzt werden konnen. Wir behaupten, dass sich die Expertise des Unterrichtens zum Teil darin widerspiegelt, wie Lehrkrafte Unterrichtssituationen interpretieren und darauf reagieren – sowohl darin, was sie sehen und erkennen, als auch darin, was sie tun. Wir beginnen unsere Ausfuhrungen mit der Identifikation verschiedener Beliefs in Bezug auf mathematisches Denken und Lernen sowie den entsprechenden Unterrichtsmethoden, die die Schlusselkompetenzen der Lehrkrafte zum Lehren und Lernen von Modellieren umfassen, die bei diesem Projekt in den Vordergrund gestellt wurden. Unsere Ergebnisse legen nahe, dass – bei Studien zur Entwicklung von Modellierungskompetenz bei Lernenden uber die Dauer eines ganzen Kurs hinweg – signifikante wirkungsvolle Veranderungen auch bei den Kompetenzen auf Lehrerebene fur das Lehren und Lernen von Modellieren stattfinden konnen. Ein Grund fur diese Entwicklung auf Lehrerebene ist, dass die Lernenden bei den Modellierungsaktivitaten wichtige Aspekten ihres Denkens auf eine Weise ausdrucken, die direkt sowohl von Lehrenden als auch von Forschenden beobachtet werden konnen. Einblicke in die Denkweisen der Lernenden haben sich als starke Impulse zur Forderung von Lehrerkompetenzen zum Lehren mathematischer Modellierung erwiesen.

Schülererfahrungen mit Aktivitäten zur Modellentwicklung: Über die Korrektheit hinausgehen, um die Bedürfnisse eines Kunden zu erfüllen

Damit die Schuler durch den heutigen Unterricht auch in der Welt auserhalb der Schule erfolgreich sein konnen, muss ihre mathematische Ausbildung uber einen festgelegten Satz an Fertigkeiten und Routinen hinausgehen. Stattdessen mussen sie in der Lage sein, den in der Schule erlernten Stoff auf subtile Weise anzupassen und anzuwenden, um mit Feingefuhl und Geschicklichkeit bei der Losung von Problemen, die gekennzeichnet sind durch Einschrankungen und Unklarheiten, reagieren zu konnen. In diesem Kapitel beschreiben wir Aktivitaten, die den Lernenden diese Art der Erfahrung ermoglichen. Wir betrachten die Erfahrungen der Schuler mit diesen Aktivitaten und wir zeigen, wie die Ungewissheit uber die Art der Anwendung des erlernten Wissens einen Raum fur die Verwendung von mathematischen Denkweisen zur Interpretation der Welt eroffnet. Wir zeigen dann, wie die Prasenz eines konkreten Kunden fur die Schuler die Moglichkeit bietet, Losungen zu formulieren, die auf die menschlichen Bedurfnisse und die Sichtweise dieses Kunden sowie anderer Menschen, die von der Losung betroffen sind, eingehen.

Dynamic hyperbolic geometry: building intuition and understanding mediated by a Euclidean model

ABSTRACT This paper explores a deep transformation in mathematical epistemology and its consequences for teaching and learning. With the advent of non-Euclidean geometries, direct, iconic correspondences between physical space and the deductive structures of mathematical inquiry were broken. For non-Euclidean ideas even to become thinkable the mathematical community needed to accumulate over twenty centuries of reflection and effort: a precious instance of distributed intelligence at the cultural level. In geometry education after this crisis, relations between intuitions and geometrical reasoning must be established philosophically, rather than taken for granted. One approach seeks intuitive supports only for Euclidean explorations, viewing non-Euclidean inquiry as fundamentally non-intuitive in nature. We argue for moving beyond such an impoverished approach, using dynamic geometry environments to develop new intuitions even in the extremely challenging setting of hyperbolic geometry. Our efforts reverse the typical direction, using formal structures as a source for a new family of intuitions that emerge from exploring a digital model of hyperbolic geometry. This digital model is elaborated within a Euclidean dynamic geometry environment, enabling a conceptual dance that re-configures Euclidean knowledge as a support for building intuitions in hyperbolic space—intuitions based not directly on physical experience but on analogies extending Euclidean concepts.

Mathematical Reflections: The Design Potential of Ensemble Performance

This paper describes learning in new forms of mathematical activity that involve multiple, moving bodies. It reports on initial findings from interviews with participants from various disciplines, who were asked to find and make sense of emergent mathematics in performances from the opening ceremony of the 2016 Rio Olympic Games. Findings from these interviews will inform future design studies of experimental mathematics teaching and learning.