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Historically, what people can learn is co-determined by the representational infrastructure for knowledge building. When Latin was the required medium of knowledge building, few could engage in scholarly activities; without the change to the vernacular, nearly universal access to higher education would not be possible. The highly compact, abstract, and opaque symbolism of mathematics presents similar barriers to the necessary democratization of access to important mathematics.
Over the course of a program of research lasting more than 20 years and involving contributors from institutions throughout the United States and worldwide (Hegedus & Roschelle, 2013), the representationally innovative design of SimCalc Mathworlds® has provided affordances for novel and effective approaches to teaching important algebraic and calculus-related ideas. When integrated with appropriate curricular workbooks, teacher professional development, and other instructional factors, dynamic representation has enabled diverse populations to learn more advanced mathematics. Research has included both design research as well as large-scale experiments involving hundreds of teachers and thousands of students; overall, the approach also has an unusually strong base of empirical support. We focus on lasting, essential design contributions of this body of work with a special emphasis on the dialectic relationship between affordances of technology and curricular progressions.