One of the current limitations in digital educational experiences is the lack of touch. Touch is a critical component in the learning process and in creating inclusive educational experiences for sensorially diverse learners. From haptic devices to tangible user interfaces (TUI), a growing body of research is investigating ways to bring touch back into the digital world, yet many focus on a specific dimension (e.g. haptic feedback or kinesthetic manipulation) of touch. Learning, however, is a multi-dimensional touch experience - it is about moving and being moved. This work presents the Action Quad - a novel haptic-TUI design for teaching geometry (specifically quadrilaterals). The Action Quad is a multi-point-of-contact, reconfigurable tool that synergizes the affordances of both kinesthetic interaction and haptic feedback into a single form factor. We present findings from an initial user study (N = 11) investigating how sighted-hearing individuals approach, interact, and experience the Action Quad, and we present a case study with an individual with blindness. We share key takeaways from the design process and participant feedback on interactions with this novel haptic-TUI device, sharing design insights on an emerging area of research that could support a new class of educational learning tools rooted in touch.