The session I went to was about how to think about unit placement and place value for middle school students, but really applies to all students across the board. During the conversation we talked a bit about how the emphases in the Common Core curriculum are supposed to focus more on the conceptual basis for how we do things like multiplying 34x38 but on the ground in schools teachers are still using the same pedagogical tools.
The thrust of the argument is this: more often than not math teachers are teaching procedures and specific guided techniques instead of the concepts behind the mathematics. This might be so because our teachers are poorly trained or because the tests don't emphasize enough of the concepts or some other reason. Whatever it is, the problem exists and we are tasked with fixing it.
So, now for the example:
The basic problem is multiplying 34x38, as I mentioned earlier. If you are like me, you learned to set it up this way and use the procedure where you multiply the 4 and the 8, then the 3 and the 8, then the 4 and the 3, then the 3 and the 3. If you complete the steps correctly, you will get an answer of 1292.
This is all well and good, but it is such an isolated skill. With some deeper understanding, you can connect it to fractions, polynomials, and more. Let's take a closer look.
If you see the text at the top, it says "3 tens times 3 tens = 9 hundreds;" then "3 tens times 8 ones is 24 tens" and so on and so forth. Instead of using a procedure, we are thinking about the units being used. Since we know that 3x3 = 9 and tens x tens = hundreds, we can conceive of 900 in a deeper way. We still get the same, correct, answer, but now we can apply it elsewhere.
This then extends to if you had a math problem like (x + 3)(x - 2). Now I am multiplying "1 x times 1 x = 1 x^2" and "1 x times -2 ones = -2 x's." It doesn't look so good in text, but I hope you get the idea.
If we want to get some use out of the Common Core we need to understand the mathematics at a deeper level and learn methods to explain that deeper level to our students. Dr. Clay and the Math Forum will help that happen.