A fundamental part of the undergraduate statistics and data science curriculum is the year-long, calculus-based probability and statistics course sequence. Within this sequence, concepts of probability are traditionally taught in a vacuum, without regard to how they will eventually be utilized in statistical inference. We thus squander an opportunity to reinforce these concepts by illustrating how they arise in inferential contexts. Furthermore, in the traditional approach, one ends up teaching inferential concepts like estimation at only one point during the year. Because of a lack of reinforcement, conceptual understanding is less likely to be retained. In this presentation, we describe a new approach to mathematical statistics that we will be piloting at CMU in which we tackle concepts repeatedly using a distribution-based framework. For instance, after teaching probability basics, we concentrate on the normal distribution, using it to illustrate concepts of estimation and hypothesis testing, etc., then spiral back to the binomial distribution, etc. In addition to describing course structure, we will also describe how we will assess the long-term efficacy of our spiral approach.