"The Common Core Standards in Mathematics stress the importance of conceptual understanding as a key component of mathematical expertise. Alas, in my experience, many math teachers do not understanding."
Being helped to generalize from one’s specific knowledge is key to genuine understanding.
What is Conceptual Understanding?
knowing that multiplying two negative numbers yields a positive result is not the same thing as understanding why it is true.
…knowledge of procedures is no guarantee of conceptual understanding
"students demonstrate understanding of –
1) which mathematical ideas are key, and why they are important
2) which ideas are useful in a particular context for problem solving
3) why and how key ideas aid in problem solving, by reminding us of the systematic nature of mathematics (and the need to work on a higher logical plane in problem solving situations)
4) how an idea or procedure is mathematically defensible – why we and they are justified in using it
5) how to flexibly adapt previous experience to new transfer problems."
"Instruction for it (conceptual understanding) has to be different than the learning of basic skills and facts." - Dan Willingham