Educational software applications (apps) on multi-touch, mobile devices provide a promising space to help learners work toward long-term educational goals, like learning with understanding (Bransford, Brown, & Cocking, 2000). Such goals are particularly relevant in supporting a learner's efforts to become more mathematically literate. Yet, a number of current apps do not appear to be living up to this potential. As such, this study drew upon the theoretical framework of Learning Science and the conceptual framework of TPACK theory (Mishra & Koehler, 2006) to define curricular characteristics that ideally support primary children's potential to learn early algebra concepts with understanding, through multi-touch, mobile, iOS mathematics education apps. Using qualitative content analysis these characteristics, then, were compared to the curricular characteristics of three authentic (i.e., real-world) apps in order to describe the general extent to which the two sets of characteristics aligned. This study found the authentic apps did not align with the majority of curricular characteristics that ideally support learning with understanding. Additionally, a number of qualitative findings emerged from the study that may be used to inform future app design. These ideas include themes related to the kinds of characteristics the authentic apps tended to align with or not, and suggested adaptations to a number of contemporary theories and models related to pedagogical content knowledge and its application toward the goal of learning with understanding. These findings have direct implications for the theory and practice of app design, and suggest revisions to the way in which the field of instructional design, historically, has been approached. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]