'Children learn mathematics when they are engaged in productive struggle.' I agree with this statement because children will learn that it is a part of the process of doing mathematics and they will embrace the struggle when they reach a solution. As a teacher, I feel that we should know our children ability to do mathematics well enough, because they should not be given a problem out of their reach and yet not be given a problem that is straight forward.
Both constructivist and sociocultural theories describe how students learn mathematics and emphasize on the learner building connections among existing and new ideas.
The more ideas used and the more connections made, the better one will understand.
I strongly agree that in the continuum of understanding, we would want children to fall into relational understanding of knowing what to do and why rather than the instrumental understanding of doing something without understanding.
Mathematics should be learned through 'doing'. As teachers, we should provide ample tools for children to learn mathematical concept and as mentioned by Piaget, he says that children learn best through hands-on experiences, when they play with concrete materials.