In 2009, the National Research Council published a report entitled, “Mathematics Learning in Early Childhood: Paths toward Excellence and Equity.” The report’s purpose was providing information to adults who were actively involved in the lives of young children about ways they could engage them in play-based math activities and thus boost math achievement in later years. Several chapters of the report present the describe teaching-learning paths or learning trajectories. These paths are composed of multiple stepping stones — hierarchical progressions children typically follow when learning mathematics. For example, stepping stones for the ability to count to 100, include counting to 10, and then, counting to 20.  By leading a child’s thinking up a particular trajectory, an adult can help the child continually increase math knowledge and skill fluency. Teaching math along these trajectories also promotes enjoyment, satisfaction, and confidence. Enjoyment is present when children, on their own, discover next steps along a trajectory. Satisfaction takes root when they discover math makes sense and is useful. Confidence is boosted as children connect new ideas with what they already know.
 The NRC report focuses several of its chapters on teaching-learning paths. Professor Douglas Clements, one of the committee members charged with writing the report have studied learning trajectories in mathematics for many years. He and his wife Dr. Julie Sarama have published widely on a learning trajectories approach in early mathematics and are now professors at the University of Denver. A New York Times article – “Studying Young Minds, and How to Teach Them” (Benedict Carey; 12/21/09) does a good job of interpreting the work of Clements and Sarama for the general public. http://www.nytimes.com/2009/12/21/health/research/21brain.html