While manipulatives have played an important role in children’s mathematics development for decades, employing tangible objects together with digital systems in the classroom has been rarely explored yet. In a transdisciplinary research project with computer scientists, mathematics educators and a textbook publisher, we investigate the potentials of using tangible user interfaces for algebra learning and develop as well as evaluate a scalable system for different use cases. In this paper, we present design implications for tangible user interfaces for algebra learning that were derived from a comprehensive field study in a grade 9 classroom and an expert study with textbook authors, who also are teachers. Furthermore, we present and discuss the resulting system design.

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Physical models for equation solving typically lack feedback regarding their appropriate use. Such feedback is possible in virtual environments and could be implemented in hybrid models. Based on an epistemological analysis, this article presents a touch gesture as a way for users to signal they want to divide both sides of an equation and a design for feedback on the use of this so-called 'division gesture'. The design is investigated by contrasting a case study, in which students used an app with the division gesture, with a preparatory study where students had to perform corresponding actions on physical manipulatives. This investigation revealed insight into feedback functions, steps of understanding dividing with this touch gesture and, furthermore, showed problems that students have with the boundary case where the dividend is 0. The study informs possible improvements of the design of the division gesture and of the overall learning environment. The results are reflected on, in order to illuminate known problems of learning how to solve linear equations, and theorized to contribute to the wider discussion around the design of digital and physical manipulatives, in particular the design of modes of interaction enabled by new technologies. Keywords Division. Touchscreen. Touch gesture. Feedback. Solving linear equations. Digital manipulatives For many students, solving equations is a challenge, because of the changing role of the equal sign from an operation sign to a relational one (Kieran 1981, 2014; Alibali et al. 2007). Several (physical and virtual) manipulative models have been introduced,