Constructivism is part of a distinguished intellectual history, including the work of Jean Piaget and John Dewey, and learning from that history is important. However, several distinguishing contributions of constructivism set it apart. Students can construct knowledge, even from lectures, if they listen to, and think about, what is appropriate for them. However, a difference exists between talking at and talking with students, both in giving students opportunities to invent mathematics and in encouraging positive beliefs about learning mathematics. One overarching goal should be that students become autonomous and self-motivated in their learning.
Constructivist views emphasize understanding and building on students' thinking.
This idea implies that as teachers we need to learn from other teachers and researchers, and study our own students, to better understand how children think. Instead, we need to study how children think about the particular mathematical topics we teach, and we need to work to understand our students' thinking at a level deeper than everyday communication. Both teachers and students become more responsible for learning. People create and recreate mathematics. Constructivism emphasizes active, living systems, not passive empirical data.