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 and Wilensky (2007). 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.

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. 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 among personal knowledge 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. [**read more**]

*‘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). [**read more**]