Sept 11 Relational Understanding and Instrumental Understanding

    The words faux amis really resonated with me. Mathematics uses a lot of English words for topics that are nowhere close to the main definitions. When I took courses in Ring theory, in my head I always pictured actual golden rings. Whenever fields and ideals are mentioned, I picture a green flat field and a perfect suitable concept. Radicals too aren’t advocating complete political/social change. They’re just surds. I believe, however, there’s a reason for those words to be chosen. Another mention that stood out to me was how rigorous a student should understand what they’re learning. Personally I don’t enjoy just throwing out formulae. I always try to give a proof so that the students may see how the derivation utilizes previous/current/new (but not out of their scope) knowledge. Stating A = L×W [for a rectangle] without explaining the definition of area (how much two dimensional space is taken up by the object) won’t help students apply the concept of area to other non-rectangular shapes like circles, triangles, or area under graphs. Finally, I collect from the latter parts of the article how much rigor should be taught. It's important to have a balance between instrumental and relational learning. Personally I encourage students to ask "why" and have an intuition of what they're learning, without just glimpsing over statements taken for granted.

    I did enjoy reading this article since it addresses a problem I talked about with a smart junior back in high school. How much emphasis is needed on rigor in the curriculum? Where is that balance between proving statements and throwing them out at the students? Skemp gives pros and cons to both instrumental and relational learning. At times students don't necessarily understand the real meaning behind a concept/formula. Other times students will try to focus too much attention on the background derivation than the equation's formal usage in everyday life. I lean more towards explaining topics on a deeper level, proving some things, but not everything. At times a lot of the background knowledge comes from what is already taught in the current unit, so it is just another form of applying/extending. Skemp's goal is for the students to problem solve on their own, even when somewhere in the middle of the process things don't go their way. That was how I was taught and I hope to present myself in a similar way.