Indicator C.1.1. Know Relevant Mathematical Content

Well-prepared beginning teachers of mathematics understand and solve problems in more than one way, explain the meanings of key concepts, and explain the mathematical rationales underlying key procedures. For example, a well-prepared beginner for Upper Elementary Grades recognizes that simplifying 3 ÷ ⅕  indicates the question “How many fifths are in 3?” Using a visual diagram as in Figure 2.1 and considering that one whole is comprised of five fifths leads one to realize that the answer will be 3 × 5 or 15.

Figure 2.1. Fraction-bar representation of the problem 3 ÷ ⅕ .

This result can be generalized so that students recognize that dividing by any unit fraction is equivalent to multiplying by the denominator. Thus, this procedure can be built on a solid and flexible understanding of underlying mathematics. (See Chapters 4 through 7 for additional examples of the specific content for well-prepared beginners at each grade-band.)