Response to "The Role of Representation(s) in Developing Mathematical Understanding" by Stephen J. Pape

The author argued that "representations must be thought of as tools for cognitive activity rather than products or the end result of a task"(p.124), and I agree with this statement. When we represent mathematical concepts, we are making sense of the internal and external meaning of them, and representation is used as a tool to reduce the cognitive load for this activity. If we treat representation as an end goal, then it is easy to produce it according to sample templates without actual understandings. One example is to find the slope of a graph, where most students are taught to connect two points on the graph. The resulting "line" representation of the slope is rather pointless if the students are ignorant about the physical meaning of the x-axis and y-axis, and students will not be able to use the idea of slope in a wide range of applications. 

 

This article discusses a variety of representations:

- Internal representation: abstractions of mathematical ideas, cognitive schemata

- External representation: numerals, algebraic equations, graphs, tables, diagrams, charts, symbols and etc. 

 

I also want to add phonetic representation to the above list, where music, math and patterns can be taught to enhance the understanding of one another. Reading musical pieces are similar to reading math symbols, and the construction of music sometimes also follows mathematical patterns. Teachers can show students some beats constructed following mathematical patterns and analyze the aesthetics in those sequences of beats. 

 

(A saxophone demonstration of mathematics by Marcus Miller)

Response on Discussion of the Skemp Article

Here is one discussion that I took from our course blog: 

Earlier in life it doesn't make sense to do relational - include relational at a higher level. Relational helps with problem solving. Consider learning a new language and how we teach language. Instrumental can be important in the beginning to set some rules before expanding and introducing the why. 

I didn't want to agree with this opinion about implementing relational learning only at a later stage of math education, but this is how I was taught and I still ended up loving math despite the fact that I spent my entire childhood struggling with math. In my opinion, if math education is unescapable, some degree of instrumental learning is required to make sure there is progress, but relational learning should be added gradually to keep students interested. 

Letters From My Future Students

Dear Ms. Xu,

I was one of your students in Grade 12 Math and I absolutely loved your class. I was inspired by your enthusiasm and I also decided to pursue a career in math-related fields. I remembered it when you showed us a lot of applications involving math, and those were the moments when my perception of math changed. The more I learn, the more exciting mathethematics gets. I also appreciate your help outside of classroom. I am a slow learner, but being able to drop by your office and get some questions clarified really helped me to keep up with the materials. 

Student A

Dear Ms. Xu,

Hope this email finds you well. I guess I am not a math person after all. When you taught me K12 Pre-calculus, I always wondered why I have to learn this. Today as an adult, I still don't find math interesting or even useful in my daily life. I remembered the weekly assignments that were given to me, and I feel the time that I spent on those assignments was rather wasted. I could have used that time to practice guitar or play more sports. I struggled a lot when attending your class; maybe math is only designed for those extremely smart people, but not for me. 

Student B

My Favourite And Least Favourite Math Teachers

When it comes to my favourite math teacher, I always have this professor on my mind. He taught me MATH 320 & MATH 321 at UBC, which are the two most memorable courses I have ever had. Ironically, I did really bad in these two courses, but this won't stop my admiration for him. He is definitely a good teacher. He knows the subjects well and explains everything smoothly despite the content difficulty. Sitting in his classroom is the most exciting as well as relaxing moment throughout the academic term. Compared to teaching techniques, I put more emphasis on a teacher's passion about mathematics. A passionate mathematician can easily affect me in the same way, and I feel a lot more pleasant when learning under this atmosphere. 

I don't really have a least favourite math teacher, but I did felt a tremendous amount of stress when I studied math at kindergarten. I was required to learn abacus at that time, involuntarily. The teacher is not the one to blame because the abacus is part of the curriculum. However, the teacher did get really upset and mad when I was unable to use the abacus. Nothing bad really happened; it's just the feeling of making my teacher disappointed stayed with me for a long time, and I was upset about it too. 

Locker Problem Notes

Response to "Relational Understanding and Instrumental Understanding" by Richard R. Skemp

Before entering the education program, I already have a general idea about how am I going to teach math. When the author said that it may be hard to realize how widespread the instrumental approach is (page 3), I paused and reflected on myself. And the answer is, yes, I didn’t realize that my intended way of teaching is very instrumental. I never question myself why multiplication of two negative number becomes a positive number because it was also taught to me as a rule. And I proceeded to ask myself: do I regret about not being taught in a relational away? The answer is no, and it can be explained by the author’s view on the over-burdened syllabi (page 11) — with the number of topics covered on exams, I simply didn’t have time to learn everything and truly understand it. 

The author also discusses the two types of confidence students get: one is the self-satisfaction from getting problems right in a speedy manner (instrumental understanding), and the other is the confidence from a complete schema (relational understanding). As a student who has studied math for 18 years since kindergarten, I have experienced both. I agree with the author in the sense that relational understanding is important in order to engage students. In fact, I only fell in love with math after I entered university where I gained a more systematic understanding of mathematics. But, my confidence from instrumental understanding in the early years is the reason why I can enjoy mathematics at the university level. In my opinion, a teaching style that uses the combination of instrumental and relational understanding is ideal for the current curriculum, and teachers should not fear to explore the methods for relational understanding.