How should multiplication be defined in the elementary school mathematics program? How do children conceptualize mutiplicative situations prior to classroom instruction in multiplication? Answers to these quest ions appear to be based on opinion rather than on the results of research.

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### Margaret A. Hervey and Bonnie H. Litwiller

Mathematics is often referred to as a study of relations and patterns. Many elementary teachers would agree with this idea, but they may have difficulty in finding materials that will provide opportunities for their students to discover patterns. The addition table may be used as an effective vehicle for the discovery of some number patterns. While searching for patterns, the students are engaging in purposeful practice, which results in a “practice-discovery” activity.

### Margaret A. Hervey and Bonnie H. Litwiller

Number patterns have interested mathematicians from the time of the ancient Greeks. Some of this interest centered around the relations between number expressed geometrically and algebraically. This idea is basic to the study of polygonal numbers. Polygonal numbers, which are sometimes called figurate numbers, include triangular numbers, square numbers. pentagonal numbers, hexagonal numbers, and so on. For example, if a number of objects can be arranged in the form of a regular triangle, that number is considered to be a triangular number. Figure I shows a geometrical pattern for the first five triangular numbers.

### Margaret A. Hervey and Bonnie H. Litwiller

Have you ever considered a graphical representation of multiples of whole numbers? Could such a representation be used to reinforce and extend the ideas of prime numbers and composite numbers? These questions were posed in our elementary methods classes. In considering these questions the students constructed graphs. As the students constructed their graphs they discovered that they had a representation that summarized many ideas of arithmetic.