Recursive reasoning is a powerful tool used extensively in problem solving. For us, recursive reasoning includes iteration, sequences, difference equations, discrete dynamical systems, pattern identification, and mathematical induction; all of these can represent how things change, but in discrete jumps. Given the school mathematics curriculum's later emphasis on calculus-the mathematics of change in continuous contexts-it is surprising that the curriculum seems to neglect recursive thinking after the early grades. Research shows that recursion supports the learning of algebra among younger students, but the lack of similar research with older students is concerning. In this paper we suggest possible affordances from teaching recursive modeling, including a basic model of the spread of contagious diseases. We also discuss different ways to present these models at various points in the curriculum that might develop connections between mathematics and the real world, and support students' learning of mathematics. This leads to what we, as mathematicians, think would be interesting research questions for mathematical educators.