I stopped the first time when I read Skemp defining relational understanding as “knowing both what to do and why.” I must say that there is something even more important and earlier: “What is the purpose of learning this subject?” When I was a student, I could hardly learn anything without knowing its purpose. Fortunately, my teachers were all very qualified, and they explained not only how math worked but also what problems people were trying to solve with it. For example, when I was learning about area, my teacher connected the concept to the practical need of measuring farmland and explained why area is calculated in this way rather than in some arbitrary way, such as taking the average of all the sides of a specific shape. Once I knew the purpose, the formula made sense to me.
The second stop was when Skemp argued that instrumental mathematics may be useful in limited situations, but relational understanding is “better” in the long term. I must say that “better” is a kind of value judgment, so we must first define the situation we are talking about before we can weigh its value. In this case, I am not perfectly sure that relational understanding should, or must, be the goal for every student in every part and at every level of math. For example, I am personally very weak at judging fashion, beauty, or visual style, while many other people seem to understand these things naturally. This reminds me that people have different interests and abilities. I do not think a high school teacher is qualified to “decide,” or even confidently “distinguish,” whether a student is capable of learning math through relational understanding. However, I think a high school teacher should be aware of these differences and respect them. As a math teacher, I would certainly begin by teaching everyone the reasons, so that all students have an opportunity to fully understand. However, if some students continue to struggle and show little interest after being given that opportunity, I think teaching them some useful rules and methods may be better than forcing them to study in my way, and it is certainly better than learning nothing.
The third place that made me stop was Skemp’s discussion of the advantages of relational understanding, such as being “easier to remember,” “more adaptable,” and “able to grow by itself.” I felt that he was describing the way I learned math 20 years ago. In high school, I often asked teachers to prove results, explain details, consider possible counterexamples, or tell me whether an idea could work in a wider situation. My memory has always been poor and has never really improved, even now. In China, we could not bring formula sheets or calculators into exams, so I often had to derive formulas again during the exam. When I first learned derivatives, I even found that the “derivative of the derivative” could be useful for determining how a function graph bends, although I did not yet know the term “second derivative.” At that time, many of my classmates criticized me for “wasting their time by asking useless questions” and said, “We don’t need to know that to do the questions correctly.” When I argued with them, the reasons I gave were almost exactly the advantages later mentioned by Skemp.
Therefore, I mostly agree with Skemp about the power and importance of relational understanding, since that is the way I learned. However, ironically, because I understand the reasons for learning the reasons, I do not think it should become a standard for everyone. As a teacher, and as someone responsible for providing students with access to knowledge, I think I must also respect their needs.
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