Saying that you understand
something does not have just one meaning.
There are many different ways someone can understand. In mathematics, there are two main types of
understanding: relational and instrumental. Relational understanding is “knowing both
what to do and why”; whereas instrumental understanding is “described as ‘rules
without reason’” (Skemp, 2006, p. 89).
In other words, instrumental understanding is simply memorizing the
steps that get you to a final answer; whereas relational understanding includes
knowing why you do each step.
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| ( http://www.clipartkid.com/ i-don-t-understand-cliparts/ ) |
Through reflecting on myself as a mathematics learner, I
realize that in my education career I have been more focused on the
instrumental understanding rather than the relational understanding. I realize that I was not concerned about
knowing why we do something or what it means; I was simply good at memorizing
the steps to get to the correct answer.
Through learning this way I was able to more effectively and reliably
reach the correct answer; however, this type of learning would not contribute
to my overall understanding of the material.
This would often lead to me memorizing the steps, regurgitating the
information onto a test or assignment, and then almost immediately emptying my
brain of all the information.
For example, in second year university calculus we were
taught how to find line integrals. To
this day, I could not tell you what a line integral is, and I would definitely
not be able to solve one without looking over my notes from that calculus
course. I simply memorized the steps,
regurgitated the information, and wiped it from my memory once it was no longer
needed. I strongly believe that if my
course was tailored to a more relational understanding rather than
instrumental, I would have a better understanding of what a line integral is. In addition, I believe I would still have a
general understanding of how to find a line integral, even without knowing the
specific steps. Skemp (2006) mentions
that “instrumental mathematics is usually easier to understand”, as well as
quicker to learn; whereas relational mathematics is more difficult to learn but
“easier to remember” (p. 92). A goal in
mathematics is for students to truly understand and remember the material to
use in their future, rather than forgetting the material the second they put
their pencil down after writing their final test.
Through reflecting on myself as a mathematics learner and my
past experiences in a mathematics classroom, I am able to better reflect on
myself as a future mathematics educator.
Although I was able to achieve high marks through an instrumental
understanding, I think it is important that I incorporate relational
understanding in my future classroom. I
want students in my classroom to understand
why we do certain things in mathematics rather than just following the
rules. Skemp (2006) mentions that there
are arguments for and against both types of mathematical understanding
(relational and instrumental). I
personally believe that it is important to incorporate relational understanding in a mathematics classroom in order to maximize learning and
understanding. Students should use steps
to assist them in their mathematical process but make sure they are not simply
memorizing the steps, again emphasizing the why. The following video further describes Skemp's article (2006) and describes the different combination of teachers and learners.
This idea of relational understanding can also support the
concept of a spiral curriculum. Contrary
to a staircase model of education where students never return to a topic
learned on a ‘lower step’, in education we want to emphasize a spiral
curriculum in which you are constantly returning to the same concepts but you
are building on your previous learning (Debra McLauchlan, personal
communication, 23 September 2016). Relational understanding will allow students
to more effectively build on their prior knowledge and participate in this
spiral curriculum.
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| ( https://www.quora.com/What-is-the-meaning-of-spiraling-curriculum ) |
Prior to reading Skemp’s article (2006), I would not have thought
about these two different types of understanding, nor would I have recognized
the importance of relational understanding.
I believe that this knowledge of the different types of mathematics will
assist me in better teaching the students of my future classroom, and thus has
helped me grow as a professional educator.
Reference
Skemp, R. R. (2006). Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 2, 88-95
Hi Lindsey!
ReplyDeleteI really enjoyed reading your post on relational vs. instrumental understanding. I thought you raised some excellent points that I am sure many students in our class can relate to. After reading your post, it made me think about my own educational experiences and how I also focused on instrumental understanding. In high school, especially in grade twelve, I was always focused on achieving the highest mark possible without actually caring why I was learning something. I can admit this was not only the case for math, but for other subjects as well such as biology and chemistry. In University, my mindset slightly shifted and I began to invest more time in figuring out why we were in fact learning particular concepts. Especially in math, a subject where many students come in “disliking,” it is important to teach students the relevance behind the numbers. This topic also goes along with making material meaningful for students, an area I hope to further develop my knowledge on. Considering all of this, I am also very appreciative that we had the opportunity to explore this article. Lastly, I want to let you know I really enjoy the set-up of your blog! It is extremely visually appealing and is set up in a way that is very “reader friendly.” Great post!
Rachelle
Hi Lindsey!
ReplyDeleteI really enjoyed your post this week. I related to your experience in second year calculus, I felt the same way during that course and in other moments throughout my undergrad as well. I often would memorize a concept without truly understanding what the concept. Similarly to your experience, I would memorize a concept to use it on a test or exam, and then clear it from my brain as soon as I no longer needed it. I believe that if more meaning was attached to these concepts, I may have been able to hold on to these concepts a lot better. This is something that I know I must kept this in mind when I enter the classroom. It is so important to make my students learning meaningful. I look forward to discussing is more.
Jordan Black
Hi Lindsey!
ReplyDeleteGreat post this week. I definitely relate to your experiences with instrumental understanding in high school and university. Looking back at my own experiences, I find that I too forget everything I have learned after the exam is over and do not understand why I was using certain techniques. Although this needs to be changed, from tutoring high school math four years later, I found that once I look at the problem and the solution I can easily figure out the problem even though it has been so long and I was learning through instrumental understanding. I believe this is because instrumental understanding is part of relational understanding. I do agree with you that student's learning should be meaningful, and I hope you will be able to instill relational understanding in your future students! I look forward to your future blog posts!
Bevan Fernandes