The culture of education

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Cognition In The Wild

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The Cambridge Handbook Of The Learning Sciences

This article has two purposes: firstly to introduce this special issue on scaffolding and dialogic teaching in mathematics education and secondly to review the recent literature on these topics as well as the articles in this special issue. First we define and characterise scaffolding and dialogic teaching and provide a brief historical overview of the scaffolding metaphor. Then we present a review study of the recent scaffolding literature in mathematics education (2010–2015) based on 21 publications that fulfilled our criteria and 14 articles in this special issue that have scaffolding as a central focus. This is complemented with a brief review of the recent literature on dialogic teaching. We critically discuss some of the issues emerging from these reviews and provide some recommendations. We argue that scaffolding has the potential to be a useful integrative concept within mathematics education, especially when taking advantage of the insights from the dialogic teaching literature.

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Scaffolding and dialogic teaching in mathematics education: introduction and review

A rising epistemological paradigm in the cognitive sciences—embodied cognition—has been stimulating innovative approaches, among educational researchers, to the design and analysis of STEM teaching and learning. The paradigm promotes theorizations of cognitive activity as grounded, or even constituted, in goal-oriented multimodal sensorimotor phenomenology. Conceptual learning, per these theories, could emanate from, or be triggered by, experiences of enacting or witnessing particular movement forms, even before these movements are explicitly signified as illustrating target content. Putting these theories to practice, new types of learning environments are being explored that utilize interactive technologies to initially foster student enactment of conceptually oriented movement forms and only then formalize these gestures and actions in disciplinary formats and language. In turn, new research instruments, such as multimodal learning analytics, now enable researchers to aggregate, integrate, model, and represent students’ physical movements, eye-gaze paths, and verbal–gestural utterance so as to track and evaluate emerging conceptual capacity. We—a cohort of cognitive scientists and design-based researchers of embodied mathematics—survey a set of empirically validated frameworks and principles for enhancing mathematics teaching and learning as dialogic multimodal activity, and we synthetize a set of principles for educational practice.

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The Future of Embodied Design for Mathematics Teaching and Learning

Forty 4th and 9th grade students participated individually in tutorial interviews centered on a problem-solving activity designed for learning basic algebra mechanics through diagrammatic modeling of an engaging narrative about a buccaneering giant burying and unearthing her treasure on a desert island. Participants were randomly assigned to experimental (Discovery) and control (No-Discovery) conditions. Mixed-method analyses revealed greater learning gains for Discovery participants. Elaborating on a heuristic activity architecture for technology-based guided-discovery learning (Chase and Abrahamson 2015), we reveal a network of interrelated inferential constraints that learners iteratively calibrate as they each refine and reflect on their evolving models. We track the emergence of these constraints by analyzing annotated transcriptions of two case-study student sessions and argue for their constituting role in conceptual development.

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Searching for buried treasure: uncovering discovery in discovery-based learning

Scaffolding is the asymmetrical social co-enactment of natural or cultural practice, wherein a more able agent implements or performs for a novice elements of a challenging activity. What the novice may not learn, however, is how the expert’s co-enactments support the activity. Granted, in many cultural practices novices need not understand underlying process. But where process is content, such as mathematics, scaffolding is liable to undermine tenets of reform-oriented pedagogy. We point to tensions between traditional conceptualizations of scaffolding and discovery-based pedagogical methodology for mathematics education. Focusing on co-enactment as a critical feature of scaffolding activities, we introduce “reverse scaffolding”, wherein experts enact for novices only what they know to do rather than what they do not know to do. We demonstrate our approach by discussing a novel technological learning activity, Giant Steps for Algebra, wherein students construct models of realistic narratives. We argue for the method’s potential via reporting on findings from mixed-methods analyses of a quasi-experimental implementation with 40 students.

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Reverse-scaffolding algebra: empirical evaluation of design architecture

In the course of developing an experimental algebra unit, the researchers noted variability in their design's instructional potential across a set of implementation media. In an effort to explain this variability, we revisited the classical theoretical construct of transparency. Transparency is the perceptual and conceptual accessibility of the mechanism, logic, and application of a tool. Corroborating earlier literature, it appears that participants saw only what they had built---transparency is a subjective achievement of a learner rather than an inherent feature of a device. Our first design prompted students with an algebraic proposition, for example "3x+2=4x-1". The two equivalent expressions were to be interpreted by students as alternative quantifications of a single linear spatial interval; namely, the path that a giant took on two separate occasions to bury and recover treasure. Problem solving required manually adjusting the modeling media to coordinate two types of equivalence: (a) the total length represented by each expression; and (b) the length represented by the variable and known units. The researchers found that successful coordination was predicated on the subjective transparency of the models' perceptual. Therefore, in redesigning the activity as a computer-based application we will have learners first construct tools and only then automatize them.

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Rethinking transparency: constructing meaning in a physical and digital design for algebra

Can reflective practice help an interdisciplinary team collaborate? When a new team begins negotiating their working process toward achieving project deliverables, members implicitly bring diverse professional practices to the table. Once minimal common ground has been established, members specify what they each need from the other in order to implement their respective expertise. This process of imposing mutual constraints results in adjusted workflow protocols that may modify participants’ regular course of action yet are vital for facilitating the collaboration. Yet, we argue, this discursive process of negotiating collaboration protocols in interdisciplinary projects may result in more than just surface reconfiguration of local practices. The negotiation may yield an articulated reification of implicit know-how in the form of new theoretical constructs bearing potential impact beyond the local context of the project. We support the argument by presenting and analyzing archived records gathered from an...

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Interfacing practices: domain theory emerges via collaborative reflection

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Kiera Chase

Papers

Searching for buried treasure: uncovering discovery in discovery-based learning

Reverse-scaffolding algebra: empirical evaluation of design architecture

Rethinking transparency: constructing meaning in a physical and digital design for algebra

Interfacing practices: domain theory emerges via collaborative reflection

Reflective Practice: International and Multidisciplinary Perspectives