Recent educational surveys show that students have difficulty applying math knowledge to practical problem solving. The goal of Phase I was to develop knowledge elicitation techniques to diagnose where these deficiencies lie. Phase I compared the problem solving strategies and knowledge of math practitioners to those of middle schoolers. The main difference between the two was that math students tried to solve problems exclusively through formulas, while practitioners used real world knowledge to develop a model of the problem domain and select an appropriate problem solving technique. This suggests that practitioners are effective not because they have more mathematical knowledge, but because they are better able to bring real-world knowledge and reasoning skills to practical problems. These findings gave rise to an exploratory technique called """"""""teaching math without numbers"""""""" that was pilot tested in a Reston middle school. Additionally, a proof-of-concept software tool was developed to help elicit and store problem solving knowledge of math practitioners and students so that comparisons could be made. Phase II would extend the Phase I work by: 1) further developing the knowledge elicitation approach to diagnosing students' math skills; 2) developing and testing instructional techniques; 3) developing software tools to help students and teachers diagnose skills.
