In response to Skemp's musings, I first paused at his mention of the young, intelligent boy who stumbled in his studies when he was forced to learn mathematics instrumentally. This stood out to me, likely contrary to the author's intent, and made me consider if some students would actually benefit more from an instrumental understanding of mathematics. Perhaps those who have a great difficulty with math can better understand this method, as I may not grasp the deeper meaning of a piece of art, but could benefit from studying the methodology.
Next, I paused at his listing of how the methodologies affect the teachers (notably how instrumental understanding is easier to teach, grade, etc.). This made me curious about which ages and subjects would benefit from each method and how the progression would come about. Is it sometimes better to teach them the method and subsequently help them discover the reasoning? Or is it always best to build their knowledge from scratch? His analogy of the two methods in relation to a man mapping out the city was quite eye-opening. This is where I stopped reading, reprocessed my ideas of understanding, and became a stronger proponent of relational understanding.
To elaborate, I always felt that to learn a subject such as math, you need to "understand" what you are doing. When I used that word, I had been unwittingly using Skemp's meaning of "relational understanding". Without it, you can simply solve the task at hand with an "instrumental understanding" but you will not have the ability to modify or create new methods when additional problems arise. I am of the opinion that, even though it is easier to teach and grade instrumental understanding, you will not be tapping into the potential of your students without giving them the tools to have a relational understanding.
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