Reflection on Relational Understanding and Instrumental Understanding

After reading the Skemp’s article, I realize how I have always been taught instrumentally. However, I never felt the instrumental understanding was forced on me, as I automatically tried to understand the material relationally. Thus, prior to hearing about the two different meanings to the word ‘understanding’, I have always regarded ‘understanding’ as identifying why a method works and realizing how it works in practice. I do not see how instrumental understanding is a form of ‘understanding’. To me, instrumental understanding is just pure memorization and an imitation of relational understanding.

Upon reading Skemp’s article about relational understanding and instrumental understanding, I realize the difficulties in teaching relational understanding to high school students. Most high school students are more likely to tend to instrumental understanding as it is easier and faster to achieve good grades on exam. I recall my friend telling me, “Who cares? I got it correct and that is what matters.” This gives rise to the question, “how do we make students learn relationally?” Maybe we can try to lay the basic foundation with instrumental understanding and build on it with relational understanding. Maybe we can try relational understanding right off the bat. Maybe we should just teach instrumentally if that is what the students need. Maybe how we teach the material is dependent on the topic we are teaching. I guess the answer will come as we gain more experiences as teachers.

Lastly, I like how Skemp use the route-finding example to describe the two types of understanding. This example clearly demonstrates how instrumental understanding is useful for quick information, though unhelpful if one get lost in the middle of a step.

Interesting quotes from the article:

“[Instrumental understanding] is what I have in the past described as ‘rules without reasons’” (pg. 2)

This quote describes directly about what instrumental understanding is.

“If we were now to say to him “You may think you understand, but you don’t really,” he would not agree. “Of course I do. Look; I’ve got all these answers right.”” (pg. 2)

This quote shows how instrumental understanding gives a false delusion to students in making them to think they understand. In reality, they are just repeating the memorized instructions. If we give the same student the same question with alteration in the wording, he may not have been able to get the answers correctly. The rigorous step-by-step methods will fall apart in this type of situation.

“All they want is some kind of rule for getting the answer. As soon as this is reached, they latch on to it and ignore the rest.” (pg. 4)

Some students fail to notice the fact that the process of getting the answer is just as important as the answer itself. Those students will be thinking, “If something is not broken, why fix it?” or “If I can get it right this way, why learn anything else?”

““[Relational mathematics]” is easier to remember.” (pg. 9)

Indeed. I used to have trouble remembering what formula is used for calculating the surface area of a sphere. However, after learning calculus and realizing calculating integral gives us “the area under the curve”, I can now use this same idea to surface area of a sphere. By thinking about how surface area is like a “curve” and volume is like the “area”, I can derive the surface area directly from volume without having to memorizing another set of formulas.

“Difficulty of assessment of whether a person understands relationally or instrumentally.” (pg. 12)

I agree with this quote. When a student answers a question, we can only infer to what the student is thinking by what was written down. Whether the students understand relationally or instrumentally, we won’t know unless we ask each of the students directly about how they come up with a solution. Students with instrumental understanding could have answer correctly because they were simply following the steps. Students with relational understanding could have answer correctly because they truly understand how to work out a problem. At the end, we won’t know unless we talk to the students individually which can be time-consuming.