Using Schemas to Develop Algebraic Thinking
This article describes how students generalize and formalize patterns using student developed schemas including subtracting out and building up. The article includes samples of student work including students explanations for each schema. This article could supplement readings from section II and IV.
Steele, Diana F. (2005). Using Schemas to Develop Algebraic Thinking. Mathematics: Teaching in the Middle School 11(1), pgs. 40-46.