Shared communication in building mathematical ideas: A longitudinal study

• Students worked on combinatorics problems in middle school and revisited them in high school.
• Their work helped them understand the isomorphic relationship between the combinatorics problems.
• Ultimately, they were able to generate Pascal's Identity using standard mathematical notation.

Students make sense of mathematical ideas using a variety of representations including physical models, pictures, diagrams, spoken words, and mathematical symbols. As students’ understanding of mathematical ideas becomes more general and abstract, there is a need to express these ideas using mathematical notation. This paper describes students’ movement from model building and personal notations to elegant use of mathematical symbols that show their understanding of advanced counting ideas. Specifically, this paper shows how earlier ideas from investigations of specific combinatorics problems (questions about making pizzas with different toppings and using cubes to build towers) are retrieved and built upon using the formal mathematical register to explain the meaning of Pascal's Identity, the addition rule of Pascal's Triangle. This analysis also shows the power of shared communication in mathematical problem solving.