METHODOLOGY
Setting
This study is a part of a larger three-year, research project, Informal Mathematics Learning (IML), a partnership program between the Robert B. Davis Institute for Learning (RBDIL) and an economically depressed urban school district in the USA. The case study was conducted with middle school aged students (11 to 13 years). A primary goal was to investigate how mathematical ideas and ways of reasoning by students were developed over time in an informal, after school environment. The content strand reported here is algebra. Ariel and other members of his cohort, were introduced to functions by engaging in Guess My Rule activities The algebra strand began at the end of the second year of the program. Ariel participated in six group algebra sessions over 3 weeks and one interview about his experiences in the algebra domain 15 months later. Each session lasted 60- to 80 minutes. At all of the sessions, the student participants worked on open-ended problem solving tasks that challenged them to build the rule that described the function. Task For this report, we focus on a task that asks students to determine how many light green Cuisenaire rods (Figure 1) would be needed to build a ladder with different number of rungs. The shortest ladder has only one rung and can be built with 5 light green Cuisenaire rods. A two-rung ladder would be modelled using 8 light green rods. It was of interest to see if students could provide a general solution to the problem. The problem was presented as follows: The Ladders Problem: Build a rod model to represent a 3-rung ladder. How many rods did you use? How many rods would you need to build a ladder with 10 rungs? How could you represent the number of rods needed if you were to build a ladder with any number of rungs? Justify your solution.