Classroom algebra instruction and the learning structure of algebra

Abstract

This research consists of a qualitative study of the behavior and discourse of Native American students who participated in two consecutive, intensive, beginning level-algebra classes at a western state university. The records of students' test responses and of their discourse related to the specific mathematical content of the course were analyzed in order to infer the types of mathematical meanings that the students were capable of assigning to the algebraic topics under study. Transcripts of classroom lectures were analyzed to determine the levels of mathematical conceptualization of the subject matter that was presented to the students. This information was used to assess the effectiveness of the particular classroom learning environment and to provide recommendations for the development of a model of effective classroom learning. A group of students in the first class were found to utilize a type of classificatory recognition system that allowed them to correctly manipulate algebraic symbols through rote memorization or by referencing master examples. The students in the second class were more capable of producing mathematically correct work than the first group of students but were not able to develop an appropriate level of mathematical conceptualization for ideas associated with algebraic equations. A model of classroom algebra learning was developed that postulates the following guidelines for effective classroom learning and instruction. It is necessary to (1) identify the levels of mathematical conceptualization of the intended subject matter, (2) provide an appropriate instructional match between the existing levels of the students' mathematical knowledge and the objects of study, and (3) use guided instructional activities that promote the active participation of the students in the learning environment. A model of participatory learning and a mathematical analysis of the learning structure of algebra were developed to address the implementation of the above guidelines.