Relational Understanding vs. Instrumental Understanding

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.  

( 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.

( 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

Skyscraper Activity: Mathematical Processes and Constructing Your Own Learning

This week in class we participated in an activity called Skyscraper. For this activity students are given a grid with numbers along the outside (see example below) and a handful of unit snap cubes.  The purpose of the game is to fill the grid with stacks of cubes, each of which represents a building.  The buildings must be placed in a way that each number on the outside of the grid represents the number of buildings (or stacks of cubes) someone standing there would be able to see.  Similar to the game Sudoku, each column and row must have one each of each building height.
  

Example Grid

Solution to Example Grid

When we first were given the activity, the rules of the game were not explicitly given to us.  We almost immediately began the activity, with our facilitator Amy only quickly giving us a brief explanation.  These limited instructions and examples resulted in almost the entire class misinterpreting the rules of the game.  Instead of thinking that the outside numbers represented the number of buildings (or stacks) we can see, we were under the impression that the numbers represented the number of floors (or blocks) we could see.  This led to confusion and frustration when trying to solve the problem.  Amy walked around the room, giving hints to certain groups (for example, she told us that it is similar to Sudoku). However this initial miscommunication of the rules of the game caused a lot of problems when trying to solve the problem.

Misinterpretation (how many floors you see):
Front = 3; Back = 3

Correct (how many buildings you see):
Front = 1; Back = 2

Misinterpretation (how many floors you see):
Front = 3; Back = 3

Correct (how many buildings you see):
Front = 3; Back = 1

 

Misinterpretation (how many floors you see): 

Front = 3; Back = 3

Correct (how many buildings you see):
Front = 2; Back = 2

Initially for this activity, I was going to discuss the importance of being clear with the instructions you give to your students so they are fully aware of what is expected of them.  However, after reflecting on the activity in its entirety, I believe that the purpose of this activity was to experience the many different mathematical processes, as well as realizing the importance of letting students discover and construct their own learning.  If we were given the full and correct rules of the game, less mathematical processing would have occurred.

 

(image created by Lindsey Crawford)

Reasoning and Proving

Through misinterpreting the rules of the game, we were able to reason with each other and prove why either the rules given to us were incorrect, the grid numbers were impossible, or there was something else we were missing.  For example, I proved that with using the incorrect rules it would be impossible for numbers opposite to each other on the grid to be different.  This is because no matter which side of the buildings you are standing on (front or back) you will still see the same number of floors (or blocks).  This started to get us thinking that maybe we misunderstood the rules.

Communicating

Through discussing with my peers as well as the teacher, we were able to realize that the rules of the game were misinterpreted.  When Amy communicated to my group the hint about Sudoku, we were able to problem solve and realize the correct rules of the game.  Once we determined that the numbers represented the number of buildings we could see as opposed to the number of floors, we solved the problem together, communicating our ideas and reasoning.  When we determined where a stack of blocks would go, we would explain our reasoning to the other members in order to prove that it is correct, as well as contribute to our whole group's understanding.

Connecting

Connecting occurred in this activity through relating the rules of the game to a game that most of us would likely be very familiar with, i.e. Sudoku.  Making this connection assisted in our understanding that each row and column should have one of each sized building.  Connecting mathematics to previous experiences is very efficient for furthering one's understanding of a new problem and helping with one's approach to problem solving.

Reflecting

After completing a puzzle, we would look at the solution making sure it was correct.  We looked at all the numbers to ensure they accurately described how many buildings we could see.  We also made sure that each row and column had one of each building height.  This reflection of our solution would help us find any mistakes we may have made.  Reflecting is an important step in solving math problems.  When you reach an answer to a problem, it is important to reflect on it and make sure your answer makes sense before dubbing it your final answer.

 
 

Other mathematical processes occurred during this activity, but these four were the most prevalent in my opinion.  I realized many different things during the activity about being a math teacher.  One thing I learned is that as a math teacher, it is important to sometimes simply step back from the problem and let students problem solve on their own or with their peers.  Discovering mathematics is the best way to truly understand and appreciate it.  I also learned that the final result of a math problem is not nearly as important as how you got there.  The mathematical processes that occur during the problem solving and the discovering of mathematics are most important.

Introduction

My name is Lindsey Crawford and I am in my fifth and final year of the Concurrent Education program at Brock University.  I am currently a Teacher Candidate with teachable subjects in mathematics and dramatic arts.  I believe that this combination of subjects is very unique because these subjects are on complete opposite ends of the spectrum.  Math allows me to access the logical and numerical part of my brain, whereas drama brings out my creative side.  As a teacher, I wish to search for ways that I will be able to bring the creativity of dramatic arts into the very structural world of a mathematics classroom.  I want to show students that math can be learned in many different ways.  I want my students to realize that math is nothing to be afraid of and that math can ultimately be fun.

The purpose of this blog will be to reflect on my journey through teacher’s college, specifically in relation to mathematics.  In my course on teaching mathematics at the intermediate/senior level, I hope to gain more knowledge on different strategies to use in a mathematics classroom. I want to learn how to ensure that I reach the widest range of students in order to let my students know that everyone is capable of learning and understanding math.