Reading through the article "Relational Understanding and Instrumental Understanding" by Richard R. Skemp, a number of ideas made me pause and think deeper into their significance.
There are two meanings to the word 'understanding': relational understanding and instrumental understanding.
The distinction between the two is an important one, as being aware of why you are doing something gives you introspection into your own learning. From my personal experience, most of my mathematics education dealt with instrumental understanding, where I would master an algorithm for a particular topic in order to complete problems. I did not understand why it worked, but it did, which only gave me a little satisfaction, as I remained ignorant to the underlying reasoning.
Different students may be much more adept at a particular way of understanding.
This ties in nicely with the different learning styles available, and which one works best for a particular student. Instrumental understanding may work for some people, especially those who just want to know how to arrive at the correct answer; however, for the others who need to rationalize what they are doing, this is insufficient. This makes me question whether I could have a deeper comprehension of mathematics (and other subjects) if my education was delivered in a different manner, as I frequently caught myself asking 'why' we learned a topic a certain way.
Relational understanding can be interdisciplinary.
This point is quite important, especially with the emphasis in the curriculum with how the different core subjects all relate to one another. Understanding mathematics at a fundamental level can aid someone in a seemingly unrelated topic (such as proportionality). This is directly contradictory to what I believed in growing up, as I held the belief that core subjects have nothing to do with each other, and a different set of skills were required to master them separately.
Skemp's issue of understanding
Delving into the main theme of the article, it is apparent that while the two different types of understanding exist, Skemp clearly favours relational over instrumental understanding. In my opinion, I would tend to agree with him, as being a teacher candidate you should have a deep knowledge about your subject area. One of the worst things you can do is have a student question why you are doing something a certain way, and just replying with "well, that's just the way it works". The student will be left with no sense of accomplishment, as they just feel like robots programmed to answer problems (which is where the popular dislike for mathematics may stem from).
However, I also agree with Skemp's decision to highlight the positives that instrumental understanding brings. Not everyone learns the same way, and I think it is important to tailor your teaching to a diverse range of learners. Ideally, when introducing a new topic you would do so with a relational understanding to be your foundation. Only after this, you can implement instrumental understanding. Doing so would supplement a student's learning, showing them different ways at arriving at the same answer, in a more timely manner as well.
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