#### Bookmarks (1)

Unit 5: Seeing Structure and Generalizing (MP7 and MP8)

# 5.1.1 Generalization

Generalizations are the lifeblood of mathematics."

Mason, et al., 2011

By examining examples such as:

a2 × a3 = (a × a) × (a × a × a) = a5
a3 × a4 = (a × a × a) × (a × a × a × a) = a7 and so on

One can conclude that:

am × an = am+n

…thus generalizing to all cases for a specific domain for the base “a” and the exponents “m” and “n.”

In mathematics, generalization can be both a process and a product. When one looks at specific instances, notices a pattern, and uses inductive reasoning to conjecture a statement about all such patterns, one is generalizing. The symbolic, verbal, or visual representation of the pattern in your conjecture might be called a generalization.

When a student notices that the sum of an even and an odd integer always results in an odd integer, that student is generalizing. Generalizations such as this allow students to think about computations independently of the particular numbers that are used. Without this, and many other generalizations made in mathematics from the early grades, all work in mathematics would be cumbersome and inefficient.

Generalizing is the process of "seeing through the particular" by not dwelling in the particularities but rather stressing relationships… whenever we stress some features we consequently ignore others, and this is how generalizing comes about."

Mason, et al., 2011 In your Metacognitive Journal, reflect on how the structure and generalization mathematical practices are inextricably linked.  