I read an article this week about the difference in instrumental understanding and relational understanding and what each of these means (Skemp 1978). I have learned about these concepts in multiple courses throughout my undergraduate and graduate career, but did not know they had a name until now. In short, instrumental learning is more ‘surface level’ or the "what," whereas relational learning is deeper, understanding the "what" as well as the "why."
This article made me think about the ways that I have experienced instrumental and relational learning in different courses and I made a few connections back to some “ah-ha!” moments I experienced in my own education as well as teaching experiences (did you know "completing the square", is not just a formula, but comes from physically filling in the missing part of a square!?). In another article I read this week, a student named Benny had trouble with understanding fractions (as many students do) (Erlwanger, 1971). He had a bit of relational combined with some instrumental understanding of fractions. His understanding of fractions was a result of how he was taught. Benny had the proper foundations and he showed signs of relational understanding (he knew ½ was equivalent to 50 cents), but was unable to apply this understanding to all fractions. In one of my undergraduate courses we learned different ways to teach fractions for relational understanding (using lots of images and diagrams as representation of fractions). Although I was unaware of the title, this was building my Mathematical Knowledge for Teaching (MKT), as described by Hill & Ball (2009). The instructor was actively trying to portray that mathematics is "more than being able to solve the problems" (Hill & Ball, 2009). While this was all wonderful and I understood the representation of fractions in this way and how to teach fractions to students using these learning tools, I still found myself reverting back to the instructional techniques for computing answers that I have relied on for so many years. At the moment, I didn’t think much of the ways that I was learning and processing information. Much like Benny, I just wanted to get the ‘"right" answer and pass the class with a good grade. I learned what I needed to know, and let my brain go on autopilot to help me with what I already knew. I think this is an issue that many older and adult learners face. If something is initially learned in an instrumental understanding, practiced this way for many years, and suddenly you are expected to understand the content in a relational way, it can be very hard to switch the method of learning. This is why MKT is so important. As stated in Hill & Ball (2009), "most adults remember a 'rule'" and proceed with mathematics teaching in this way (Hill & Ball, 2009). Skimming through the Common Core State Standards, I came across a handful of standards that I wasn't even sure I understood, let alone knew how I would teach them to a student- for example, standard 4.NF.A.1, which involves explaining fraction equivalence using visual models instead of mathematical processes. For students to develop a relational understanding, the teacher must teach with this goal in mind. This requires that the teacher has a thorough understanding of the content and combines this knowledge with effective pedagogy. References: Common Core State Standards Initiative (CCSSI). (n.d.). Mathematics standards. Mathematics Standards | Common Core State Standards Initiative. http://www.corestandards.org/Math/ Erlwanger, S. H. (1971). Benny's conception of rules and answers in IPI mathematics. Journal of Children's Mathematical Behavior, 1(2), 7-26. Hill, H., & Ball, D. L. (2009). The curious - and crucial - case of Mathematical Knowledge for Teaching. Phi Delta Kappan, 91(2), 68–71. Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15. Whytok, K. (2014). Educational Quotation: "A change in understanding is neened to change instructional practice" [Photograph]. Flickr. https://www.flickr.com/photos/kenwhytock/15106104349
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