Skemp on two approaches to teaching and learning mathematics

When Skemp discussed about the two kinds of mathematical mismatches in student learning, it reminded me of my experience teaching and how I used both relational and instrumental teaching on different students. One of the problems I had was that different students had different goals in their learning. Some students wanted to understand the "why's" behind everything I did on the board, while others wanted to pass the class by memorizing formulas and methods, ignoring all proofs and explanations of formulas and theorems, and then forgetting them once the test or exam is over. This raises the question of how we are to teach a classroom if every student has different goals and are looking for different things in their learning?

Skemp's discussion on the advantages of relational and instrumental mathematics made me ponder and reflect of my experience of both being a student and a teacher. I remember that as a student, I had to take exams and standardized testing, all of which are under heavy time constrant. I was a curious student who always looked for a reason behind concepts, formulas, and theorems, but because of the time constraints, I often resorted to instrumental mathematics. I memorized formulas and short cuts to get to the answer, often leading me to better results. I realized that although relational mathematics are better for one's critical thinking and problem solving skills, instrumental mathematics often produced better results. On the other hand, I taught mathematics through relational understanding during my 1.5 years of teaching experience under an LOP. I heavily believed that relational mathematics trains problem solving skills (which it does), and taught students that nothing has to be memorized. After reading Skemp's advantages of both relational and instrumental mathematics, I realized that it is important to find a good balance of both. Although Skemp seemed to side more with relational mathematics, standardized testings and post-secondary institutions are telling us that relational mathematics is insufficient for students to succeed. As math major, I have used a mixture of both. For example, we studied Galois theory to prove that the quintic is insolvable. Obviously it is a long proof that I do not quite remember anymore, but I still do use the fact that it is insolvable in mathematics. However, learning the proof in the class certainly strengthened my problem solving skills and allowed me to be a better thinker.

Overall, I think that as teachers, it is important teach a balance of instrumental and relational learning. Although some advantages were addressed about instrumental mathematics, it seemed as if Kemp was a tad biased toward teaching relational mathematics. The followup question I have is how do we teach a balance of relational and instrumental mathematics, how do we get students to understand the importance to knowing both, and how do we accomodate students with different learning goals in the classroom? I want to explore that area more.