Beyond Getting the Right Answer

Three things made me stop while reading Skemp’s article. First, his distinction between relational understanding and instrumental understanding really stood out to me. I understand relational understanding as not only knowing how to do something, but also understanding why it works and what it means, while instrumental understanding is more about knowing the procedure without necessarily understanding the reasoning behind it. This made me think about how easily I may have assumed that a student “understood” something just because they could get the correct answer. Second, I was particularly interested in Skemp’s discussion of the mismatch between the teacher’s and students’ goals: “Pupils whose goal is to understand instrumentally, taught by a teacher who wants them to understand relationally,” and the other way around. I have experienced this tension in my own teaching. For example, when teaching area, a teacher may tell students that the units must match before calculating the area. Students may follow this rule, but if they don't understand why the units need to be consistent, they may still have only instrumental understanding. Third, the statement “Well is the enemy of better” really made me stop. As an AP teacher, I have often seen that once students know they have mastered everything required by the exam, they are very satisfied with where they are. Without the pressure of an assessment, many students are reluctant to spend extra time exploring ideas more deeply. This made me think about how exams and the desire to make every lesson “efficient” can sometimes push both teachers and students toward instrumental learning. More content or more efficient learning does not necessarily mean better learning.

I personally stand more strongly on the side of relational understanding, although I do not think instrumental understanding is useless. Instrumental understanding can give students a sense of achievement because they can quickly get correct answers, and sometimes this is necessary. However, if students become too dependent on this immediate satisfaction, it may not support long-term learning. I believe relational understanding requires more patience and sometimes delayed gratification, but the deeper connections students make between new knowledge and what they already know can lead to a stronger, longer-lasting sense of achievement. I have also tried to create opportunities for this in my AP classes through projects, especially after the AP exam, when students had little motivation to learn new content. Projects allowed them to use calculus they had already learned and apply it to real-life situations. Although not every student was deeply engaged, those who invested time in the project often gained something that simply getting another correct answer could not provide. For me, this is why relational understanding is more valuable in the long run: the goal is not just for students to know how to get an answer, but to understand the connections behind it and eventually become able to use their knowledge in new situations.