Reflections on Skemp

Richard Skemp's "Relational Understanding and Instrumental Understanding" prompted a stop or two and a few smiles and nods. 

Where I did stop in Skemp's article was when he presented his analogy of learning (and teaching) math as a person knowing selected routes between places and that same person wandering around with the aim of developing his mental map. This analogy seemed to make a lot of sense for me. To help explain why, I will share the following anecdote.

While reading this article, I could not escape my own epistemic relationship to math and the curriculum that I was responsible for teaching in my first few years as a teacher. My very first math class was a grade seven group of girls (twinned in a co-ed school). I was super keen to take them, but I really was taking it unit by unit. (And, if I'm honest, other issues -- classroom management -- were really demanding more of my attention.) The next year, I had another grade seven class and a grade eight class (among others). At this stage in my teaching, I was fumbling around in survival mode. It wasn't until my fourth year in a classroom that I had the capacity to productively reflect on the broader scope of my own teaching. By that stage, I had been exposed to enough grade levels and courses to start wandering around/exploring the campus, as Skemp put it. I was absolutely that junior teacher mentioned by Skemp -- despite her best intentions -- who walked the routes. I walked routes (I feel) in my own math learning and definitely in my own math teaching. But, somewhere in year four, I had started to explore (following the metaphor) in my teaching and in my understanding of mathematical relationships. I gained a modest level of confidence, and I concluded that the best thing that I was able to do for my grade nine teaching was teach grade eight, the best thing for my grade eight teaching was to teach seven, and so on. This long-view meant that I could see features of the landscape -- the relationships that the curriculum sought to share -- and find ways to draw them together. Perhaps that is what draws me toward concept-based teaching now.

Generally speaking, I agree with what Skemp discusses. While there is a feeling to want to prioritize relational mathematics, I am not willing to set aside the role that instrumental mathematics plays and has. Sometimes mathematics is done for utilitarian purposes. And I can sympathize with learners who see something that they're not sure about and just want to know what to do. So I feel that until we adjust what it means to be successful in a math classroom, then it will be even harder to prioritize relational mathematics learning.