Mathematical connections established in the teaching of functions.

This study explores the types of mathematical connections established in the classroom in the teaching of functions. An extended model for mathematical connections (different representations (DR), procedural (PC), if-then (I-T), part-whole connections (PWC), feature/property (F/P), analogies, and instruction-oriented connections (IOC)) is used as the analytical framework. The context for the study is classroom observations of two secondary mathematics teachers teaching functions to Grade 9 students. The strength of teachers' mathematical knowledge for teaching (MKT) the concept of function is different: one has stronger MKT, while the other has weaker MKT. A total of 485 connections are identified in a sample of 24 ninth-grade lessons observed (12 lessons per teacher). The teacher with stronger MKT produces far more connections (f  = 317) than the teacher with weaker MKT (f  = 168), and she mostly establishes I-T, DR, PWC and F/P type of connections. The teacher with weaker MKT frequently makes procedural types of connections. This 'connections gap' may reflect differences in the teachers' MKT and in their beliefs about the teaching and learning of mathematics. The study also documents some of the important internal connections within functions based on the observed lessons, and an additional IOC has emerged from the data. [ABSTRACT FROM AUTHOR]