The Variety of Variables in Automated Real-Time Refinement

The refinement calculus is a well-established theory for deriving program code from specifications. Recent research has extended the theory to handle timing requirements, as well as functional ones, and we have developed an interactive programming tool based on these extensions. Through a number of case studies completed using the tool, this paper explains how the tool helps the programmer by supporting the many forms of variables needed in the theory. These include simple state variables as in the untimed calculus, trace variables that model the evolution of properties over time, auxiliary variables that exist only to support formal reasoning, subroutine parameters, and variables shared between parallel processes

Establishing a Content Taxonomy for the Coherent Study of Engineering in P-12 Schools

Engineering education has increasingly become an area of interest at the P-12 level, yet attempts to align engineering knowledge, skills, and habits to existing elementary and secondary educational programming have been parochial in nature (e.g., for a specific context, grade, or initiative). Consequently, a need exists to establish a coherent P-12 content framework for engineering teaching and learning, which would serve as both an epistemological foundation for the subject and a guide for the design of developmentally appropriate educational standards, performance expectations, learning progressions, and assessments. A comprehensive framework for P-12 engineering education would include a compelling rationale and vision for the inclusion of engineering as a compulsory subject, content organization for the dimensions of engineering literacy, and a plan for the realization of this vision. The absence of such a framework could yield inconsistency in authentically educating students in engineering. In response, this study was conducted to establish a taxonomy of concepts related to both engineering knowledge and practices to support the development of a P-12 curricular framework. A modified Delphi method and a series of focus groups—which included teachers, professors, industry professionals, and other relevant stakeholders—were used to reach a consensus on engineering concepts deemed appropriate for secondary study. As a result, a content taxonomy for knowledge and practices appropriate for P-12 engineering emerged through multiple rounds of refinement. This article details the efforts to develop this taxonomy, and discusses how it can be used for standards creation, curriculum development, assessment of learning, and teacher preparation