Responses to First Reading: Relational Understanding and Instrumental Understanding

As I was reading Skemp's article, I felt genuinely inspired. When he contrasted instrumental and relational understanding, it was like he was describing my own experience as a student. In class, professors often just introduced a formula with a typical example, which made it seem perfectly fitted. But when I later worked on assignments and quizzes, I often ran into "surprises" where the formula did not quite apply. At that point I had to practice many similar but slightly different problems, and by comparing them I would slowly discover new patterns. That is why Skemp's " 20 cms by 15 yards" example struck me. He mentioned about instrumental understanding involving "a multiplicity of rules rather than fewer principles of more general application" also helped me realize why math education often relies so much on endless practice, because no single example or formula can really cover it all.

What also stood out to me was Skemp's analogies. The football story is interesting, but the music one made the most sense to me. Thinking about students just memorizing notes without ever hearing the sound reminded me of my own math learning. It was lots of symbols but not much meaning. I can still remember teachers saying, "Even if you don't get it now, just memorize it, you'll understand later." That way of learning shaped a lot of my early math and it is exactly what Skemp is pointing out as a problem.

Finally, I agree with Skemp's call for relational understanding. For me, math makes more sense when I can connect rules with reasons, see how topics fit together, and try methods on new problems. His article made me think about my own experiences and also about my future as a teacher. If I only pass on rules without reasons, I might help students get through exams, but I will not give them the confidence, flexibility, or joy that comes from really understanding math.