Algebraic thinking can be organized into two major components:
1. The development of mathematical thinking tools and
2. The study of fundamental algebraic ideas.
Mathematical thinking tools include:
Analytical habits of mind: Problem solving skills, reasoning skills, and representation skills.
Fundamental algebraic ideas represent a domain in which mathematical thinking tools can develop.
They are the content for study.
What I am interested in discovering and discussing with others is this topic of algebra content. What do we consider central to the study of algebra? How do we determine when this is learned sufficiently? Can the content be learned satisfactorily if the accompanying algebra thinking tools are poorly understood by the student? Conversely, can the algebraic thinking tools be acquired and refined without delving deeply into all algebra topics.
Examples I am considering:
Do 8th graders need to know, for example, how to determine the slope of a line perpendicular to another line with known slope?
Do 8th graders need to know how to multiply binomials even if there is no practical use other than preparing their skills for future math. In this sense, is learning binomial multiplication a valid example of "algebraic thinking tools"?