Building Algebra One Giant Step at a Time: Toward a Reverse-Scaffolding Pedagogical Approach for Fostering Subjective Transparency Through Engineering Levels of Interaction With a Technological Learning Environment

The project brings together and studies the intersection of three ‘big’ ideas from educational research and practice. Firstly, this project is built on the constructivist assumption that meaningful learning occurs when students interact with educational materials that occasion problem solving and reflection. Secondly, this project elaborates on the theoretical construct of transparency by tracking its subjective development. Finally, the project reexamines the notion of scaffolding – the socio-cultural idea that novices receive expert intervention in the form of supports until mastery is achieved – specifically scaffolding that enables an authentic discovery- based learning experience. This work occurs within the context of early algebra. The story of learning algebra in schools is often told as the challenge of progressing from arithmetic to algebra. A main character in this story is the “=” sign or, rather, students’ evolving meanings for this sign (Herscovics & Linchevski, 1996). The balance metaphor is undoubtedly the most common visualization of algebraic propositions. Still, students’ persistent difficulty in transitioning from arithmetic to algebra suggests that the balance metaphor may not be the ideal method for building transparency for a relational understanding of equations (Jones, Inglis, Gilmore, & Evans, 2013). Therefore, this study investigates an alternative approach. Using a technological-enabled constructivist learning activity, Giant Steps for Algebra, students construct models of realistic narratives. As they build a virtual model of a problem situation, students discover technical principles for assuring the model’s fidelity to the situation. These construction heuristics, are precisely the conceptual foundations of algebra, and the activity’s situated intermediary learning objectives (SILOs). To enable to gradual development of transparency for Giant Steps, at each interaction level, the student discovers a SILO, and then the technology takes over by automatizing that SILO, thus freeing the student for further discovery. This activity architecture is called reverse scaffolding, because the tools relieve learners from performing what they know to do, as opposed to what they do not know to do. In a quasi-experimental evaluation study (Grades 4 & 9; n=40), reverse-scaffolding students outperformed baseline students, for whom the technical features were pre-automatized upfront, on measures of transparency for the SILOs. I thus conclude that discovery-based learning activities are advantageous, and that reverse-scaffolding technological activities can level the gradual development of subjective transparency