Mathematics
There are 5 elements of math proficiency
National Academies report Adding It Up (2001b, p. 107)
Conceptual Understanding
Comprehension of mathematical concepts, operations, and relationships
Procedural Fluency
Skill in choosing the best method to do operations and in doing them correctly
Strategic Competence
Ability to figure out how to turn problems into a solvable form and then solve them
Adaptive Reasoning
Ability to think logically and explain arguments
Productive Disposition
Tendency to see math as useful and worth learning
Traditionally, math classes have focused almost exclusively on procedural fluency, but all five elements depend on each other to create real proficiency.
THEREFORE
Math education requires more time spent on discussion, practice with feedback, and understanding key ideas and strategies.
AND
Those discussions need to focus on the interconnectedness of mathematical concepts and how mathematical argument works.