As students begin to be conscious of their thinking and how to explain themselves, the course moves to the foundational topics of set theory and logic. These topics provide the background knowledge for the development of later topics, such as models of whole number addition. Also, they expose the student to the formal structure of mathematics as an axiomatic system. While not going into the depth of an entire formal course in either of the topics, the course gives students a sense of the underlying structure and unity of mathematics. Once the foundations have been laid, the course moves into a treatment of numeration systems.
Students investigate numeration systems in general, looking at place-value and bases.
An example of an activity that students do in the course, relates to this topic. The activity suggests a base five system, but students are encouraged to be creative as they work in small groups. Each group must submit a written description of their system, including an evaluation of its strengths and weaknesses. In addition, each group must present its system to the rest of the class and answer any questions posed. As students deal with numeration systems other than the base ten place-value system with which they are familiar, they gain an appreciation for the difficulty and questions children experience.