Understanding and the CCSSM

Every semester, at this time, I introduce my students to the idea of understanding maths. It's not an easy task especially for those graduate students who may not have experienced learning elementary school maths for many years. Even with some of my undergrads the process of having to understand something they have known by rote, "off by heart",  for some times seems like a pointless task.  But, we have to understand the maths we are planning to teach so we go through the topic carefully to make sure everyone is on board.

In every field of study or human endeavor there are giants; people whose ideas, or deeds, have stood the test of time and are as fresh and relevant today as they were back then. One  such person in math education is Richard Skemp.

In 1976, Richard Skemp, a British educator published a definitive article in the journal, Mathematics Teaching, in which he described two types of understanding of mathematics; instrumental understanding; "rules without reason", "rote learning", "pure memorization",  and relational understanding; or understanding the what, why, how, when, connected with,  and so on; the type of understanding to be found in the CCSSM. Having found that my students have a hard time remembering the terms I now call them fragile understanding and robust understanding which always seems somewhat onomatopoeic.   

The wonderful thing about robust understanding is that it not only makes problem solving so much easier, it makes the retention and recall of all those illuminating but tedious  math facts children have to learn infinitely more efficient and effective. If, for example you can relate the multiplication facts to the addition facts they suddenly have a new structure that gives them sense and aids in meaningful recall. 6 x 5 for example is 5 x 5 plus another 5. Then if you can count by multiples of  5 the mental structures you are creating makes forgetting almost impossible.

We all learn that Pi is  3.14 and just about everyone knows that it goes on forever without establishing a repeating pattern. Every high school hallway is adorned with it on pi-day on March 14. (July 22 in the UK).  But ask anyone what it means and very few can tell you that it's a ratio between the diameter and circumference of a circle. Every circle is just over 3 times further around the outside than across the center.

This is what the CCSSM are designed to do.