This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using "Trees," and describing "subexpression addresses"--all help students avoid these longstanding mistakes and, more significant, cultivate true fluency in algebra. Herein, he discusses the "glue" concept, diagramming expressions as "trees," where the subexpression "lives," and tackling "strange mistakes" in simplifying fractions. All the exercises discussed force students to engage with the structure of algebraic expressions and with how that structure permits subsequent transformations. The author concludes that once his students have mastered these skills, they are delighted to have become technically proficient enough to truly understand the structure of the algebraic expressions with which they are working.