This research analyzes how external representations created by a student, Robert, helped him in building mathematical understanding over a sixteen-year period. Robert (also known as Bobby), was an original participant of the Rutgers longitudinal study where students were encouraged to work on problem-solving tasks with minimum intervention (Maher, 2005). The research demonstrates how Robert built robust counting techniques by tracing the evolvement of his problem-solving heuristics, strategies, justifications and external representations. The study also examines how Robert made connections to his earlier problem solving. In addition, the origins of Robert's ideas related to Pascal's Triangle and Pascal's Pyramid are investigated. Fifteen sessions were selected between Robert's fifth grade (February 26, 1993) and post-graduate interviews (March 27, 2009) yielding more than twenty hours of video data. Powell, Francisco, and Maher (2003) model was used for analysis where by each session was viewed, transcribed and coded for critical events to create a comprehensive narrative. The study reveals that mature combinatorial techniques were a part of Robert's counting strategies as early as middle school. Robert used binary notation to count two-colored candle arrangements and later to count the number of ways a team could win a World Series; modified exponential formulae to account for combinations for a garage door opener, arrangements for n-colored candles and n-toppings pizzas; discovered the combinations formula, C(n, 2), in his eleventh grade; and connected these solutions to Pascal's identities. In general, Robert looked for patterns in his solutions; generalized the findings; and identified structural similarities in tasks presented to him as he connected three-position garage door opener to three-colored candles arrangements, pizza with four toppings to towers four high, and directions on Pascal's Triangle to routes for a taxi on a two-dimensional grid. External representations created by Robert served as communication tools for him and provided insight into his problem solving heuristics and mathematical understanding. The research contributes to the growing body of case studies from Rutgers longitudinal study providing evidence that building of early mathematical ideas is the foundation of more advanced learning (Davis & Maher, 1997). [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]