Reflection: “Relational Understanding and Instrumental Understanding” by Richard R. Skemp

Over the years I have had many opportunities to assist students struggling with mathematical concepts. Before and after the shift in curriculum for math assessments (2016-2018), I have found that teachers that communicate and assess mathematics according to the instrumental understanding are generally considered “better,” and teachers that opt for structuring their classroom around the relational understanding are deemed as more difficult. I agree with Skemp that this is a partial consequence of “over-burdened syllabi”, “backwash effect of examinations”, and “difficulty of assessment” (page 11).

As a math major, I myself have opted for the instrumental approach in a number of math courses due to the structure of assessments, but enjoy the relational approach when it is taught with enough enthusiasm and purpose. The most notable courses where I loved having a relational understanding were: Euclidean Geometry, Coding Theory, Game Theory, and Number Theory. On the contrary, courses that I only cared for enough instrumental understanding to succeed included: Calculus III, Linear Algebra, and Differential Equations. 

What separated both categories of courses for me (and some of my students) were time to comprehend topics, relevance to my perception of the world, structure of assignments / examinations, personality of the educators, and connection between students in class. The better those five qualities were, the more I enjoyed relational understanding. The less prevalent it was, the more I desired instrumental understanding to compute and move on. As a result, I empathize with the perspective presented by Skemp, having experienced the back-and-forth as both a student and teacher. However, given that more students nowadays are learning how to learn and have access to math-solving technologies, I think transparency in instruction and assessment when the teacher is shifting between relational understanding and instrumental understanding is crucial for the student to be more curious and mindful. 

The three lines that I stopped at while reading, not due to excitement, but rather sadness, were:

• "The encounter is compulsory, on five days a week, for about 36 weeks a year, over 10 years or more of a child’s life." (page 4)

• "The reply was: “300 square centimetres”. He asked: “Why not 300 square yards?” Answer: “Because area is always in square centimetres.”" (page 4)

• "That he is a junior teacher in a school where all the other mathematics teaching is instrumental.” (page 11)

This was because the first quote depicts the duration of time in their life that some students spend fighting mathematics rather than enjoying it; the second is because this blind trust- this lack of critical thinking and reflection is more common than it should be; and the third because it seems as though there is little hope for junior teachers to inspire new methods. In a good way, the third line refreshed me; it reminded me why I started the journey to become a math teacher in the first place and I feel hopeful that it doesn’t have to be the case for my future students.