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. Attention then turns to focusing on the whole number, the integer and the rational number systems. Models of the operations in each system are discussed and studied. Set theory is used to introduce the set models of whole number operations, and then the whole number models are used to build some of the integer models.
Students again see that in the structure of mathematics, new ideas build on previous ideas.
In addition to the models of the operations, students investigate properties of the number systems and elementary number theory. Manipulatives add to the understanding of, for example, the commutative property of integer addition. The course concludes with an understanding of fractions and the rational numbers and the operations of addition and subtraction in this system. Mathematics for Elementary Teachers II begins with a review of the rational numbers and concludes the treatment of them by studying the operations of multiplication and division. The real number system, with decimals and irrational numbers is then considered to conclude the study of number systems.