Table of Contents

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

Time to Reflect

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

 

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Time to Extend

Refer to the publication, Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School (Carpenter, et al., 2003) and on the included disc, view the video of Susie as she tries to justify a + b - b = a. Susie is a second grade student who has been learning in a Cognitively Guided Instruction environment.

This video has value for all grade levels because it illustrates how even young children can reason about computation and make and justify important generalizations. After watching, consider the following:

  • In the example ½ + 11 - 11, does Susie work from left to right in her calculations?
  • What can you infer about her understanding of the structure of the number system?
  • What property is she using?
  • What does she seem to understand about the use of a variable as an indicated element of an infinite set?
  • How does she use generalization in her work?