Conclusion: Learning Trajectories Going Forward: A Foundation for Coherence in the Instructional Core

Piaget’s foundational research established that children’s points of view do not mirror those of adults, nor are they merely imperfect or partial copies of adults’ ideas. In creating the origins of constructivism, Piaget offered three key constructs that still influence much of the work on learning trajectories: genetic epistemology, schemes, and stages.

With genetic epistemology, Piaget (1970) recognized that it is not only what one asserts that constitutes one’s knowledge, but instead it is how and why one comes to believe an idea that determines the idea’s worth. From Piaget’s theories, the learning trajectories researcher is compelled to ask the questions: How did a particular idea originally evolve? What outstanding problems did it resolve? And what kinds of opportunities to make sense of the world did it engender? Asking these questions of a “big idea” in mathematics learning gives one an opportunity to consider a mathematical idea from the perspective of the child—not just what resources and beliefs a child brings to the situation, but what the new idea permits a child to do that she or he might want to do, and how this changes our own understanding of more authorized expert knowledge. Confrey (1995, 1998) referred to this challenge as negotiating the “voice-perspective” dialectic, arguing that carefully listening to the “voice” of a learner interacts with the “perspective” of the listener (the expert—a teacher or researcher), and triggers changes in the listener’s own perspective, or the listener’s understanding of his or her own knowledge. Genetic epistemology helps the researcher to consider the underlying “problematic” and “create a need for the idea” (Confrey, 1991).