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Haptic force feedback systems are unique in their ability to dynamically render physical representations. Although haptic devices have shown promise for supporting learning, prior work mainly describes results of haptic-supported learning without identifying underlying learning mechanisms. To this end, we designed a haptic-supported learning environment and analyzed four students who used it to make connections between two different mathematical representations of sine and cosine: the unit circle, and their graph on the Cartesian plane. We highlight moments where students made connections between the representations, and identify how the haptic feedback supported these moments of insight. We use this evidence in support of a proposed theoretical and design framework for educational haptics. This framework captures four types of haptic representations, and focuses on one -- the haptic bridge -- that effectively scaffolds sense-making with multiple representations. 

Haptic technology has the potential to expand and transform the ways that students can experience a variety of science, technology, engineering, and math (STEM) topics. Designing kinesthetic haptic devices for educational applications is challenging because of the competing objectives of using low-cost components, making the device robust enough to be handled by students, and the desire to render high fidelity haptic virtual environments. In this paper, we present the evolution of a device called "Hapkit": a low cost, one-degree-of-freedom haptic kit that can be assembled by students. From 2013-2015, different versions of Hapkit were used in courses as a tool to teach haptics, physics, and control. These include a Massive Open Online Course (MOOC), two undergraduate courses, a graduate course, and a middle school class. Based on our experience using Hapkit in these educational environments, we evolved the design in terms of its structural materials, drive mechanism, and mechatronic components. Our latest design, Hapkit 3.0, includes several features that allow students to manufacture and assemble a robust and high-fidelity haptic device. First, it uses 3-D printed plastic structural material, which allows the design to be built and customized using readily available tools. Second, the design takes into account the limitations of 3-D printing, such as warping during printing and poor tolerances. This is achieved at a materials cost of approximately US $50, which makes it feasible for distribution in classroom and online education settings. The open source design is available at http://hapkit.stanford.edu.