Visualization plays an important role in mathematics learning, but in the United States where many prospective teachers (PTs) have few if any experiences learning mathematics through visualization, mathematics teacher educators are challenged to design tasks that generate within PTs' thinking an appreciation for the role visualization plays in mathematics learning. This chapter examines the affordances of progressive visualization tasks and semiotic chaining for use in mathematics teacher preparation. To the literature on dyadic and nested forms of semiotic chaining, data analysis in this chapter contributes a new type of semiotic chaining based on Peirce's three principles of diagrammatic reasoning.