Instrumental and Relational Understanding

This week in my teaching journey, my University math class uncovered two different perspectives of math that I never thought about before. We looked at an article by Richard Skemp that assessed math in the twentieth century, however, what he concluded in his paper is still extremely relevant today. In his article, Skemp addresses the idea of two different types of understanding, instrumental and relational.

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Now if you are like me, you may be familiar with instrumental understanding, but not that accustomed to relational. Instrumental understanding means a child knows a specific rule or procedure and has the ability to use it. In other words, the student knows how to follow instructions and specific steps to get to an answer. Skemp insists instrumental understanding are "...rules without reasons" (90). Take dividing fractions for exmaple, in order to get the right answers, students blindly follow steps not knowing why they are doing it. 

https://buildingmathematicians.wordpress.com/tag/nixthetricks/

https://alearningplace.com.au/2-ways-to-teach-and-learn-maths-2/

 Relational understanding is when the student knows exactly why they are doing those steps. In essence, instrumental understanding is part of relational understanding, however, the key fact here is that the student knows why they are doing it. This is important because relational understanding allows students to tackle questions that are out of the ordinary or that look unfamiliar. Relational understanding helps facilitate mathematical reasoning. With instrumental understanding, students are just trying to get to the answer, and if the teacher asks a question that does not quite fit the rules they have learnt, the student will get it wrong. (Skemp 90).

A notable analogy that Skemp uses to understand the difference between instrumental and relational understanding is taking a walk in a park. Imagine you are in a park, and you want to get from point A to point B. You learn from someone else a certain path to take and you get to the destination fairly quickly. Eventually, you add more points to your locations you want to visit and you know how to get there. Step off any of the known paths, and you are quickly lost and can even develop a fear of losing your way (in this case math anxiety). You never really develop an overall understanding of the park, and you may not know about other connections between points that might be quicker. This is instrumental understanding. 

Now instead of walking around the park through specific paths, you get to wander all over. For some parts of the park you are guided, for others, you walk around aimlessly. In time you get an overall picture of the park and this allows you to figure out a shorter path than you are accustomed to, or how one point in the park is related to the other. If someone showed you a short-cut, you would understand why it worked and why it was faster than the path you took before. Now you wouldn't be afraid of walking off the path, because even if you did, you can find your way back easily. This is relational understanding. 

http://www.gocomics.com/calvinandhobbes/2011/03/09

The above comic highlights the importance of relational understanding. Teachers need to allow students to 'wander around the park aimlessly' in order to tackle math anxiety and better prepare their students to tackle diverse problems. The real question is how do we as teachers facilitate relational understanding in our students, and how do we assess if they understand a situation relationally or instrumentally? Furthermore, for students who do not plan on having math as a primary factor in their future career, would instrumental understanding be better suited for them? Maybe for the future educators and policy makers should analyze the high school math streams and possibly advocate relational understanding and instrumental understanding based on their future careers. 

Skemp, Richard. "Relational Understanding and Instrumental Understanding." Mathematics Teaching in Middle School 12, no. 2, 88-95.