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. 

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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.

Debunking the "Math Person"

In my Introductory post I brought up a famous quote by John Dewey, "If we teach today's students as we taught yesterday's, we rob them of tomorrow".  One of the most common misconceptions about mathematics, which teachers seem to believe, is the notion of a "math" person.

Retrieved from: https://www.youtube.com/watch?v=Ukt4A5GCfQU

As Professor Jo Boaler from Stanford University explains in the video above, no one is born with a "math" brain but we all have the potential to grow and change our brain all the time. This notion is very important because from my experiences tutoring high school students in math, one of the most common statements I get from pupils is the fact that they are not a "math person" and therefore need help in math. This statement speaks to me because as we have just learnt, there is no such thing as a math person. When the students say they are not a math person, they are setting themselves up for failure. They are already creating an excuse for failure right from the beginning because according to them it is not in their nature to be a successful math person. In the education field, we call this type of thinking a fixed mindset. 

Retrieved from: http://carriekepple.com/2015/04/24/growth-
mindset-vs-fixed-mindset-which-do-you-have/

In order to fix this misconstrued idea of a "math person", teachers need to teach students the difference between a growth and a fixed mindset. If the teacher can get the students to change they way they think from a fixed mindset to a growth mindset, not only will the student be successful in math, but they will also be able apply this problem-solving in real world situations. In addition, by getting students to start thinking with a growth mindset, the teachers are getting students to use mathematical processes that are part of the Ontario curriculum such as problem-solving and reasoning. 

Several techniques teachers can use in the classroom to change the mindsets of their students from a fixed mindset to a growth mindset can be found on mindsetonline.com. Furthermore, a study conducted by Mercer and Ryan (2010) explains how teachers play a big role in fostering a growth mindset in students. The way teachers provide feedback is crucial due to the fact that it may promote a fixed mindset if the teacher focusses on grades, or it could promote a growth mindset if the teacher pays special attention to the learning process, belief about developing the student's ability through hard work, and commenting about the learner's efforts during the feedback (p. 442). 

In the end by moving away from the idea of a "math person", or in general one particular subject persons, teachers are enabling their students to broaden their abilities, while at the same time increasing confidence in the classroom. The teacher is providing a positive learning environment and more importantly, creating an atmosphere in which all students can thrive and enjoy math, rather than being disengaged and feeling discouraged.

References: 

Mercer, S., & Ryan, S. (2010). A minsdet for EFL: learners' beliefs about the role of natural talent. ELT Journal, (64)4, 436-444.

Welcome To My Blog!

Hello everyone and welcome to my blog!

Before I begin my educational posts and reflections about math, I thought it would be better suited if you all got to know a little bit about me. My name is Bevan Fernandes and I am currently a teacher candidate at Brock University. I hope to become a secondary school teacher with teachables in Math and History, which most people find odd considering the fact that they are not complimentary in the conventional school system. However, I feel like there are many ways that you can connect the strands between the two subjects, for instance, the process you take to tackle new problems in both subjects. It is my goal to get students to find an appreciation for both math and history, two subjects that most students traditionally find disengaging.

Retrieved From: https://editorial.rottentomatoes.com/article/get-
down-with-the-get-down-15-things-to-know-about-netflixs-newest-drama/

Outside of my aspirations to become a teacher, I enjoy playing various sports such as soccer and volleyball, and I have a growing interest in music and musical instruments. Lately, I have been more and more interested in different types of music such as jazz, 90s hiphop and music throughout the 60's to the present day, which has allowed me to connect with multiple students on different occasions during my volunteer experiences. This idea influenced the title of my blog, since it not only brings the idea of the basic necessities in mathematics, but also the popular culture reference to music and Djing, a growing industry in the twenty-first century and among highschool students.

The purpose of this blog is so I can discuss and improve key insights in the mathematics field, hopefully collaborating with my peers on future posts in order to find better methods and techniques in teaching math in the future. A notable quote comes from John Dewey who says:

"If we teach today's students as we taught yesterday's, we rob them of tomorrow".

Retrieved From:http://www.nigeriancuriosity.com/2009/06/chaos-
status-quo-and-nations-future.html

My goal is to challenge the traditional and conventional ways of teaching math, and more importantly reflect on how I can be an advocate for change and reform in the classroom. I feel that the quote by Dewey encompasses this idea as it hints towards the notion that we as educators need to take heed to the constant developments in education. The best way to do this is to reflect on the current practices in math and work together as a community of educators, in order to change the traditional system so that we can create a more inclusive learning environment that helps all students achieve their full potential.