Richard R. Skemp - Relational Understanding and Instrumental Understanding

3 things that made me stop during my read-through:

1.     “There is this important difference, that one side at least cannot refuse to play.” (pg. 4)
       - a wonderful analogy describing the compulsory teacher-student bond that is forged over the years

2.     “But his relational understanding, by knowing not only what method worked but why, would have enabled him to relate the method to the problem, and possibly to adapt the method to new problems.” (pg. 9)

       - possibly the best statement supporting relational understanding from the entire article

3.     “There is more to learn – the connections as well as the separate rules – but the result, once learnt, is more lasting.” (pg. 9)

      - I've seen too many students try to 'memorize' math, it doesn't survive the test of time

      My stance is that relational understanding is superior to instrumental understanding.  Why? Much of our world consists of quick fixes and people demanding (near) immediate gratification, with minimum effort.  Instrumental mathematical understanding caters to this by deadening the topic - only skimming the surface and omitting the overall schema of knowledge in favour of individual (often detached) components.  This practice can result is errors on an exam by students who otherwise are quite competent (the area units when l is given in cm and b is given in yards).  Moreover, a base in instrumental understanding could yield errors resulting in much larger consequences later in the workforce (e.g. confusing 1% with 1bp when dealing with large sums of money).  I’ve seen many circumstances where students churn out an answer (involving many correctly executed steps to get there) and assume they are correct overall only to find out later that they made a small error somewhere along the way.  Everyone is prone to mistakes, (we are human) but if the learner had a more complete ‘mental map’ they would realize that although the steps seem correct, their final answer is nonsensical (e.g. a negative length).  It’s not worth the risk to only teach instrumental understanding/‘plug and play’, because so many math problems can only be solved by straying off the beaten path from A to B; without the proper map you could end up lost.  Relational understanding, on the other hand, enables you to traverse most landscapes since it connects (seemingly) disjoint components together to create a web of knowledge.