Response to "Skemp on two approaches to teaching and learning mathematics"

(This first section is not completely related to the response demands, but more so a funny story that I remembered while reading the article.)

Before getting into the deeper issues of the article (a fantastic one exploring issues I found myself struggling with before), I wanted to share the repressed memory that surfaced when I saw the first definition on "faux amis". Back in my elementary school years I loved drawing (and even now, if you look at my first blog post!), but I struggled with the lack of consistent crayons in my house. They were either chipped, on their last legs, or the handling was poor. I asked my parents one day, "I want crayons!!!" 

After a few days of waiting, my dad came home with a bag from Staples. I was so excited at the prospect of being able to colour my drawings with whatever hue I wanted... only for it to be a bunch of pencils. We were both confused, since the box clearly said "crayons"! I'm not sure how we found out, but eventually I learned that "crayon" was the French translation for "pencil". Either way, I was disappointed, and I don't think I ever ended up getting the crayons I wanted. Little did I know, those pencils would've come in handy during high school when we started shifting away from the arts and crafts side of things... though by then, I had misplaced the box, never to be seen again. C'est dommage, I suppose.

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Going into the article, I already had my own beliefs of the issue raised in the paper. Something along the lines of, "it sure would be nice if I could teach for understanding (relational), but students seem to not have the time, and it's so much easier to just show them the steps and have them practice it like some kind of basketball drill." As such, even simply seeing the issue of relational and instrumental understanding and mathematics felt like my struggle was being acknowledged by Skemp, and it piqued my interest in the article. After all, this is not a unique struggle, and as he continued giving analogies and his own perspectives on the issue, I found myself entering the same internal debate as him as a relational-believer on the teacher side of things, but as a student, preferred to just save time and learn the rules. Even if relational understanding/mathematics feels so right, why are we still teaching along an instrumental paradigm?

A lot of my pre-reading thoughts were guided by being in a transitionary period where the conventional class syllabi were slowly being changed. One case of this is the integration of why rules worked - for example, exponent laws can be derived by simply expanding out each exponential term involved. Knowing this was great in cases where I needed to derive a rule on the fly, and it carried the advantage of being able to understand my own steps. When the article raised concerns of effort, and preparation, and the paradox of time-efficiency, it made me think of the exponent example. If we could selectively choose which concepts to teach more relationally, we could save a lot of time for the students without sacrificing much short-term time. Even something like the times tables is learned through short-term derivations (ex. all the 10s end with 0) until a person eventually skips them and simply recalls from memory. This was further reinforced by the section on organic growth, which I have experienced in my later university math courses where I would "lab out" a concept to explore its possibilities. Many hours were spent on graphing calculators, just playing. This section in particular evoked ideas of getting students to not only understand, but also get thinking about their own ideas; to synthesize, and maybe even to love mathematics.

After reading the article, I still stand by the opinion where relational understanding and mathematics are the optimal form, but come with a tradeoff of heavy short-term work and time consumption. Until the tradeoff disappears completely, there will still be room for instrumental methods to ensure that everything in our packed curricula can be covered in the short duration of a school course. That being said, we should still try to inject relational understanding as much as possible, wherever most appropriate and applicable while keeping timelines and student attention in mind, since it is key to developing a richer grasp of the material versus training children to become robots that process simple algorithms.