Reflection: Instrumental vs. Relational debate (Richard Skemp)

      Asking everyone to take a stance on a topic and then defending that stance was something that I hadn’t experienced in a classroom setting for many years.  Initially, I was surprised how many people were on the ‘instrumental’ side of the coin (approximately one third of my colleagues).  It was entirely possible, however, that those that (in class) were on the instrumental side of the debate didn’t actually have that belief (and vice-versa).   

      During our discussion, which I thoroughly enjoyed, many different viewpoints and counterpoints were presented from the groups’ brainstorming lists. Many items were subsequently argued accurately from both sides, but (for brevity) I’ll focus on just one.  The ‘achieves that results’ argument (pro-instrumental) involved teaching the Pythagorean Theorem via simple exercises involving right triangles yielding immediate results and thus fluency.  The 'branch out to other problems' pro-relational rebuttal hypothesized: what if a student turned the paper over only to encounter non-right triangles and then proceeded to apply the PT incorrectly (where the Cosine Law is necessary)?  Motivating PT by introducing the more general setting and explaining how PT is a degenerate case of CL¹ gives the learner a better understanding of the meaning behind the raw calculations.        

      This brings us back to the original question that Skemp posed: should instrumental and relational mathematics be considered two different subjects?  The jury is still out…

           

¹ For a triangle with sides a,b,c>0 and corresponding angles A,B,C we have:

            c² = a² + b² -2abcos(C)        [CL]

            c² = a² + b²                         [PT]

   Hence, [CL] = [PT] <=> C = (pi)/2.