For several decades, digital formative assessment tools and platforms have been promoted as promising for supporting adaptive teaching. For learning goals such as procedural fluency, this promise has been held across a range of subject-matter topics. For conceptual learning goals, however, the available digital formative assessment tools are much less widespread, and the research needed to develop them still seems to be underestimated. This paper reports on the design process of the Mastering Math Online-Check to illustrate approaches to a research-based design of targeted digital formative assessment tools that (a) are relevant, (b) are valid, and (c) provide in-depth, informative insights into students' understanding of subject-matter concepts (in this case, basic arithmetic concepts such as the meaning of multiplication and division). The described research needs and the considerations about research-informed designs refer to improving the depth of insights into students' understanding and their explanations of meanings. This conceptual (rather than empirical) paper discusses pivotal questions that must be addressed to fully leverage technological advances for the assessment of deep conceptual understanding, not only for mathematical concepts but also for other abstract concepts that rely on explanations and visual representations for the assessment of their understanding.