Skemp Response

I ran a little long on this one, but I was just really vibing with the reading!

When Skemp talks about students being satisfied with instrumental understanding, I am reminded of tutees of mine asking me to “just tell me how to get the right answer.” We have a system which teaches children that, while relational understanding might be superior, we just don’t have time for it in the midst of all the things that must be instrumentally understood. I have largely found myself in the first of Skemp’s two hypothetical situations, with a teacher (myself) who wants relational understanding and pupils who seek the instrumental variety. What a joy it is, though, to have students eager to find the deeper reasonings!

What he is getting at is that our desire to teach in a time-efficient manner has led us to lean too heavily on instrumental understanding, which can never account for the broad array of real-life problems. While it seems easier to teach instrumentally, we would actually take far more time explaining every possible exception and minute variation on a rule than if we explained the base reasoning, at least in my experience.

I think that there is (especially in mathematics for early education) an assumption that instrumental learning is the most that can be achieved within the various constraints a teacher has (time, student interest, teacher competence). Years and years of asking relational questions and being given instrumental answers leads students to stop asking those questions. I’m reminded of a book called “The Game of School” by Robert Fried which talks about how obvious it is that students will take to behavior that is not optimal for learning when learning is so rarely the goal of school as seen by students. Instead, they have goals like “getting a good grade” or “not getting in trouble” or even “being popular.” While some of these certainly could be directly related to learning, they by no means have to be. Anyone who has spent time in a school must know how possible it is to get a good grade in most classes without truly learning much at all…

When Skemp talks about how difficult it is to assess for specifically relational understanding (without some larger, in-depth conversation) I am reminded of my own frustration as a student with being asked over and over to “show my work” on questions which I deemed too simple for anything beyond instrumental understanding. For example, what work does one show for 5x4? I suppose I could have written out a long addition statement, but that felt cumbersome and disingenuous. I hadn’t actually added four fives. I had memorized it!

When Skemp quotes Bondi, I find myself almost leaping out of my seat in excitement. My undergraduate thesis was on how popular media and parental influences affect math attitudes! So much of our paradoxical reverence and fear of mathematics comes from a place outside of the classroom. Students (and indeed adults) are bombarded with constant messages that math is only for the few, a subject known from birth or not at all. It would be a great task for a teacher to correct them in a mere handful of hours a week.