Read the following description of algebraic thinking:
What does algebraic thinking really mean? Two components of algebraic thinking, the development of mathematical thinking tools and the study of fundamental algebraic ideas, have been discussed by mathematics educators and within policy documents (e.g., NCTM, 1989, 1993, 2000; Driscoll, 1999). Mathematical thinking tools are analytical habits of mind. They include problem solving skills, representation skills, and reasoning skills. Fundamental algebraic ideas represent the content domain in which mathematical thinking tools develop. Within this framework, it is understandable why conversations and debates occur within the mathematics community regarding what mathematics should be taught and how mathematics should be taught. In reality, both components are important. One can hardly imagine thinking logically (mathematical thinking tools) with nothing to think about (algebraic ideas). On the other hand, algebra skills that are not understood or connected in logical ways by the learner remain "factoids" of information that are unlikely to increase true mathematical understanding and competence.
From Shelley Kriegler's project "Mathematics Content Programs for Teachers," UCLA Department of Mathematics, January 2000.
This passage points out two components of algebraic thinking: mathematical thinking tools and algebraic ideas. In this session, and in the sessions that follow, we will immerse ourselves in these two components of algebraic thinking. We'll use mathematical thinking tools like problem solving, reasoning, and representation skills to help us make sense of situations. We'll also take a look at algebraic ideas, including patterns, variables, and functions. Note 2
