The purpose of this study was to describe elementary preservice teachers' difficulties with understanding algebraic generalizations that were set in an authentic context. Fifty-eight preservice teachers enrolled in an elementary mathematics methods course participated in the study. These students explored and practiced with authentic, hands-on materials called "object boxes," then created sets of their own object box materials. Each algebra rules object box contained materials to illustrate and describe four different algebraic generalizations, or "rules." The variables "n" and "z" were used in each of the generalizations. For each generalization, there was a set of objects attached to a piece of mat board that showed three cases of the generalization for different values of "n." Two sets of cards accompanied these objects, giving word problems, defining variables, stating equations, and explaining the algebraic generalizations. Students matched word problems to the object sets, defined variables and checked their work, then wrote algebraic generalizations for the object sets and used the reverse sides of the equation cards to check their work. Projects were graded with a rubric. Students were then surveyed about their difficulties. Results of the analysis showed that students were able to make an assortment of authentic materials in a variety of contexts and enjoyed the creative aspects of the project, but found the algebraic content challenging. The most common mathematical difficulties were being able to define the variable, and identify the pattern. Examples of effective student materials are provided. (Contains 4 tables and 16 figures.) [This paper was presented at the: (1) Annual Conference of the New York State Association of Teacher Educators (NYSATE) (Saratoga Springs, New York, April 28) [Conference presentation title: Preservice Elementary Candidate Assessment of Algebraic Understanding for Accreditation]; and (2) Annual Conference of the Association of Mathematics Teachers of New York State (AMTNYS) (Saratoga Springs, New York, October 26-28) [Conference presentation title: Using Hands-On Materials to Write Algebraic Generalizations (Grades 5-8)].]