## Multiplying Fractions with Paper Squares

In the last decade or so there has been quite a push back regarding discovery math. I have always found this to be very perplexing. How can it be denied that we are not born knowing mathematical concepts other than a very base form of quantity (Dehaene, 2011)? At some point, we all need to discover these mathematical ideas for ourselves. The question is really what is the best method for discovering these mathematical ideas. We could discover them through someone showing us (direct instruction), or it could be us figuring them out for ourselves through an experience, or it could be some combination of the two, but regardless at some point, we all need to discover these concepts and begin growing our own personal conceptual images for these ideas.

I am most interested in what types of activities contribute to the richest conceptual understanding and what are the most engaging and interesting. It is this combination that will cause students to excel and …