Mathematical Inquiry and its Phase about Expansion and Reflection of the Multiplication Table

Abstract

　　The purpose of this paper is to investigate elementary school children’s mathematical inquiry and its phase about expansion and reflection of the multiplication table. Therefore we carry out the teaching unit “review and expand the multiplication table” for third graders by an 8-hour class.

　　We investigated children’s mathematical inquiry by teaching unit about reviewing and expanding the multiplication table by qualitative methods.

　　As a result of our discussions, we obtained several insights:

(1) Interpretation through expressing the multiplication table in various ways, such as expressing a figure appearing on the circumference by connecting mutually promotes the mathematical inquiry about finding property and regularity of the multiplication table.

(2) The reasoning of the analogical inference that is going to apply the relations between steps of two multiplication promotes the expansion and reflection of the multiplication table and its progressive thinking.

(3) Third grader children have difficulty about the meaning interpretation of multiplication of 0. Children are going to overcome this conflict through applying thinking to lead to a distributive law and extrapolation.