MULTIPLICATION WITH RATIONAL NUMBERS: TEACHING PROSPECTIVE K-8 TEACHERS

Diane S. Azim

Eastern Washington University

dazim@ewu.edu

The majority of current K-8 teacher education candidates have not experienced Standards-based math instruction, emphasizing number and operation sense, in their K-12 schooling. They must, therefore, construct meaning for multiplication with positive rational numbers during their teacher education coursework. Meaning about multiplication can be constructed through multiple perspectives (the Conceptual Field theory). In work and research by the author with prospective K-8 teachers over 6 years, four perspectives on -- or conceptual dimensions within -- multiplication have required special instructional attention for the majority of teacher education candidates:

1. Understanding and Modeling the Types of Relationships Modeled by Multiplication: modeling quantitative relationships such as of 1 or 1 quantities of , which includes understanding what these relationships mean and having a method for quantitatively determining them;

2. Connecting Non-Whole Rational Number and Whole Number Multiplication Meanings: connecting the types of relationships described above with multiplication -- i.e., with whole number meanings for multiplication); interpreting multiplication expressions in terms of the relationships involved -- using two interpretations (by commuting the two factors); translating multiplication expressions into real world situations through translating the understood relationships into situations within real world contexts;

3. Interpreting Referents: understanding the referents involved in the factors and products of multiplication situations, particularly understanding the numerical and referent meaning of the product in non-whole positive rational number situations -- countering such misconceptions as x as representing pizza multiplied by pizza or interpreting the product of x as , rather than as ;

4. Understanding Multiplication as an Invariant Operation: recognizing real world situations that could be modeled by whole number and non-whole number positive rational number multiplication as multiplication situations and being able to describe multiplication as one invariant operation with one invariant meaning across whole and non-whole positive rational numbers.