Glossary

The learning axes and bridging tools design framework is grounded in research on neo-Piagetian developmental theory (Robbie Case), phenomenological philosophy (Martin Heidegger, Maurice Merleau–Ponty), pedagogy and design (Hans Fruedenthal, Karen Fuson, Ernst von Glasersfeld, Seymour Papert, Uri Wilensky), artificial intelligence (Marvin Minsky), cognitive science (Gilles Fauconnier & Mark Turner), and creativity (George Steiner). To read more about the framework, see Abrahamson (2004), Abrahamson (2006), Abrahamson and Cendak (2006), and Abrahamson and Wilensky (2007).

Apprehending Zone
Apprehending Zone diagramThe apprehending zone is an emergent design-research model of student learning in social contexts. The AZ model informs implementation of constructivist philosophy in the form of objects, activities, and facilitation emphases. The AZ model foregrounds the complementarity of personal and interpersonal aspects of classroom learning processes. Student learning is interpreted as a personal multi-modal process of linking up situations and representations (see central bubble) as modeled by the teacher and peers in the classroom shared problem-solving space (arrows going in and out of the central bubble). Through this process, designed tools (bottom bubble) that initially have little if any meaning for the student are deployed in the spatial–temporal classroom space to be interwoven (linked, coordinated) into a concept-specific semiosis (central bubble). Thus, through participation in classroom activities the student has developed residual conceptual structures with sufficient overlap with those of the designer (top bubble), so as to be mathematically accurate enough.

Bridging Tool
A bridging tool is a spatial–numerical mathematical artifact specifically designed to elicit students' cognitive resources pertaining to a mathematical concept, such that the students can coordinate successfully between situated and symbolic aspects of this concept (as well as mathematical procedures and vocabulary). The bridging tool is an "ambiguous" object: different activity contexts make salient in the bridging tool properties that elicit different cognitive resources. Yet, unlike the famous duck–rabbit ambiguous figure, in bridging tools both interpretations pertain to the target content: students are guided to experience cognitive conflict between the interpretations, to reflect on this conflict, and resolve it. By virtue of articulating the reconciliation of the cognitive conflict, the student constructs core issues of the target concept. The design of bridging tools is based on the analysis of mathematical representations as conceptual composites.

Conceptual Composite
'Conceptual composite' is a cognitive-studies characterization of mathematical representations, such as diagrams. As part of a domain analysis that intentionally focuses on the artifacts used in mathematical practice, the representations are treated as implicitly binding/blending/superimposing two (or more) historical idea elements. Students often need help to see these idea elements within the mathematical representation. The bridging tool is the designer's attempt to unravel the conceptual composite so as to foster ontogenetic capitulation of the phylogenetic construct (the students are to reinvent the conceptual composite).

Designed Selection Constraint

Embodied Spatial Articulation
Embodied spatial articulation is an individual's design-facilitated negotiation between personal and cultural resources pertaining to the visuo–spatiality of mathematical situations and representations. The personal resources are proto-mathematical action-based images, and the cultural resources are the appropriate seeing-in-using of classroom spatial–numerical artifacts. Embodied spatial articulation, I conjecture, underpins human interacting with epistemic artifacts historically, developmentally, in the designer's workshop, and in classroom space–time. [read more]

Learning Axis
A learning axis is a metaphoric line extending between two idea elements, one at each of the poles of the axis, which pertain to the study of a mathematical concept. These idea elements are difficult to see in conceptual composites yet are "distilled" by a bridging tool. In interventions, students engage bridging tools in problem-solving and construction activities designed so that the students dwell along the axis. Data analysis of students' engagement with the bridging tools tracks the students' alternation between the poles and whether the student apprehends both poles and works between the poles on resolving the conflict they engender.

Learning Issue
A learning issue is an embodiment of a learning axis within a particular bridging tool. When we say a student is "learning concept X," we mean that the student is mastering a set of learning issues. Different designs for concept X usually (perhaps necessarily) have overlap in learning issues, but they also have distinctive learning issues, too. Students move from difficulty to understanding with the set of design-specific learning issues. This process can be articulated fully in terms of the cognitive links the students are forming amoung personal konwledge and the classroom referents. The learning issues enable education researchers to describe learning in terms of the tools that are introduced to the students and the activities around these tools (and how these activities are facilitated). Thus, the learning issues also enable clear communication between researchers and practitioners and serve as foci of teacher attention, classroom discussion, and formative assessment. Importantly, the learning issues effectively frame data analysis in terms of students' interaction with the artifacts.

Platonic (Adjusted) Combinatorial Space
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Stratified Learning Zone
A stratified learning zone is an undesirable classroom dynamic that may emerge in the implementation of collaborative construction projects. In the SLZ, labor is divided and associated with individual students such that they become "type cast" into a particular task that may or may not foster the development of mathematical knowledge. More formally, the SLZ is a design-engendered non-continuousness hierarchy of students' potential learning trajectories along problem-solving skill sets, each delimited in its conceptual scope, and all simultaneously occurring within a classroom. In comparison, the term continuous learning zone depicts a space wherein students can each embark from a core problem, sustain engagement in working on this problem, and build a set of skills wherein each accomplishment suggests, contextualizes, and supports the exploration and learning of the successive skill, so that a solution path is learned as a meaningful continuum. [read more]