We discuss teaching and learning situations that surfaced when computer programming and mathematics were brought together in a course where students write computer code to explore mathematics problems. Combining programming and mathematics creates a rich ecosystem which, on top of traditional mathematics activities (writing solutions, proofs, etc.), offers simulation and experimentation, invites discussions about structure, requires logic and testing strategies, and handles mathematics objects with an added feeling of "reality." Focusing on novice and inexperienced programmers, we look for answers to the practice-oriented question, "How do students reason through their difficulties when using programming to explore a mathematics problem?" Following literature review and methodology, we build the "programming model," which we use to study students' experiences as they approach a mathematical problem by writing computer code. Our research is based on analyzing students' in-class work and class notes, author's observations of students working on their computers, and his interactions with students in class and elsewhere. In the four case studies that we present we touch upon students' difficulties in working with complex conditional statements and recurrence relations. As well, we discuss cases where resolving a programming issue demands posing and answering mathematical questions.