There is a concept I find very hopeful in contemporary mathematics education, the concept of Learning Trajectory (LT), whose simplest definition is derived from (Clements and Sarama, 2009): LT has a mathematical goal, a developmental progression, or a learning path to reach it, and the instruction that helps to move along that path.(p.17). The very idea of an LT in mathematics recalls the best, in my opinion, American pedagogical approach of the Texan Discovery Method created by Prof. R.L.Moore from Chicago School of mathematics of early nineteen hundreds. Each course of calculus, topology or any other mathematical subject designed through that method was constructed along a learning trajectory specified by the collection of problems and axioms, e.g Calculus 1 in 142 problems and axioms.
The main strength and virtue of learning trajectories lies in the possibility to design instruction both in agreement with the mathematical development of concepts and with student learning pathways. They offer to the instructor a long range view upon mathematics involved and upon student learning, which can be utilized for cyclical 9from semester to semester) design of instruction. The art of design learning trajectories lies in creation of appropriate cognitive distance between two consecutive problems along the trajectory – they must be challenging but possible for students to deal with. The previous post On Understanding at https://cunymathblog.commons.gc.cuny.edu/2012/09/18/on-understanding-in-mathematics/ includes an excellent performance task from DOE NYC Mathematics Task Force, The Fir Tree designed along the learning trajectory with those characteristics. Its LT design made it an excellent measurement of understanding for the concept of a Variable.
At the same time teaching along a hypothetical learning trajectory gives a teacher sense of direction and increases awareness of learning process. So, with the introduction of Common Core Mathematics standards which employ, in its design, the learning trajectory approach in the service of understanding, there is hope and chance for success in Math Ed.
However, the teaching-research in the community colleges of the Bronx has suggested that student success, depends not only on its cognitive dimension, but in equal measure on affective and self-regulatory dimensions. Our colleagues in the Bronx CC pursue the path of mathematical creativity as the source of motivation for to enjoyment and to learning mathematics (described on the previous post On Creativity @ https://cunymathblog.commons.gc.cuny.edu/2012/09/27/on-creativity-in-mathematics/ ). The question of facilitation of the “self-regulatory practices” that is of “ how to study” for our college freshman, in for example, remedial algebra is still a mystery to me.