Comment on Skemp’s article

According to Skemp’s article, “Relational Understanding and Instrumental Understanding”, there are many points that I agree with, but I also have different opinions in some aspects.
There are three points that I agree with the author. First, in the article, the author says, “There are two kinds of mathematical mis-matches which can occur: 1. Pupils whose goal is to understand instrumentally, taught by a teacher who wants them to understand relationally. 2. The other way about”. I totally agree with that. Based on my working experiences, I have met the two kinds of students. I worked with an adult student, Linda, as a math tutor in a college last year. While I was explaining the reason of a2 + b2 = c2, she told me that it was too much and difficult for her to remember the reason, and she just wanted to memorize the formula and use it. Actually, in her practice, she did very well for using the formula to solve the questions. On the other hand, when I taught a grade 8 student about the formula, she asked me “why” as soon as I gave her the formula.
Second, in order to avoid the mis-matches mentioned above, I agree with that point in the article—“All of these imply, as does the phrase ‘make a reasoned choice’, that he is able to consider the alternative goals of instrumental and relational understanding on their merits and in relation their merits and in relation to a particular situation.” A math teacher should use different ways—relational way or instrumental way to teach based on the students’ demands.
Third, I totally agree with “If what is wanted is a page of right answers, instrumental mathematics can provide this more quickly and easily” and “Just because less knowledge is involved, one can often get the right answer more quickly and reliably by instrumental thinking than relational”. “That relational understanding of a particular topic is too difficult, but the pupils still need it for examination reasons.” Linda, the student mentioned above, is a good example of this point. At that time, she is working on her GED math test. Her purpose is to pass the test. Since her basic knowledge of math is poor, the relational understanding is more difficult for her than the instrumental understanding. In fact, the instrumental understanding that we used made her pass her test smoothly and successfully.
However, when talking about the “blame” of the negative attitude to mathematics, the author thought it was the widespread failure to teach relational mathematics—“If for ‘blame’ we may substitute ‘cause’, there can be small doubt that the widespread failure to teach relational mathematics…”. I think the failure is one of the causes, but it is not the main causes. I think that if math teachers can apply the relational and instrumental understanding to students based on the students needs, the negative attitude to mathematics would change to positive attitude.