수학영재들이 조립제법 일반화 과정을 유추하는 사례연구

Abstract

The synthetic division in school mathematics is introduced as a method to easily obtain the quotient and remainder in the division of polynomials, but it is limited to when the divisor is a first-order polynomial and is handled mechanically based only on calculation convenience. Accordingly, in this paper, we studied how analogical thinking is expressed in synthetic divison and generalization of synthetic division in long divison for mathematically giftedstudents. In mathematically gifted students when the divisor is a quadratic polynomial and further in the generalization of synthetic division. At first, mathematically gifted students were unable to find the quotient and remainder using long division method when the divisor polynomial was quadratic, and found the quotient and remainder using the concept of identity. However, by analyzing the principle of long division process and comparing it with synthetic division method, they were able to find the quotient and remainder when the divisor polynomial was quadratic. By analogy, synthetic division was completed. Later, in the generalization of synthetic division, the principle of synthetic division was described and a method for calculating the quotient and remainder was successfully presented. Therefore, the analogy is positive for developing mathematical thinking and fostering an investigative attitude to understand the detailed principles of synthetic division by analyzing long division process, breaking away from the simple mechanical synthetic division when the divisor is a linear polynomial and developing an attitude of inquiry that can make generalizations.