Pape & Tchoshanov (2001)

Pape and Tchoshanov define mathematical representation as both the internal and external manifestations of mathematical concepts in the "The Role of Representation(s) in Developing Mathematical Understanding" (2001). The Skemp article had me thinking about the different ways students understand and this article has me thinking about the different ways students get to their understanding. 

I mostly agree with the opinions of the authors. I also believe that representation is a vehicle for discussion and that it is an inherently social activity. However, I think there are often cases where students have a wildly different internal manifestation of the concept in question. Differences in internalization may be influenced by the student's cultural and personal experiences. I think it is important for students to have the opportunity to explain how they understand concepts in order to further their understanding and add discussion to the classroom. Communication is an important step of the learning process where representations are co-constructed by teachers and their students.

The clearest example of representation that comes to mind is from balancing forces in the vectors unit of calculus in grade 12. We had problems with static objects experiencing or exerting three forces in total. It helped me to go from the word problem to two diagrams: one with the object and the forces and the other with the three vectors to form a triangle. In order to balance the forces, the triangle needed to be complete. There were trig equations to determine the answer but seeing the triangles and the angles boosted me through the problems.

I especially like the example of how the number 1123581321345589 is only able to be 'memorized' once the student understands that the Fibonacci sequence is utilized. This made me realize why I often preferred math classes to science classes on test days: When I understood how something works, through a representation or otherwise, there were seemingly less things to remember because on a test I could quickly rebuild the pieces to the puzzle I knew I could solve. I would like to find a book that outlines the best practices as instructors for which representations to utilize by concept in K-12 math education. 

My History as a Math Learner

As I simultaneously reflect on the past and ponder the future, I can't help but draw parallels to the most recent movie I've seen: "Tenet" by Christopher Nolan. I highly recommend the quest to a silver screen to see the film. I write this blog post as I listen to the soundtrack at a reasonable volume. 

 

Past Teachers

I honestly can't remember much from my elementary school learning experiences, so I will reflect on my learning experiences in high school math. I'm fortunate that almost all of my math teachers were good so I will instead share my memories of one not-so-inspiring and one inspiring.

The not-so-inspiring: my grade 9 math teacher. It was a boring class. The lesson plans were clear, but the majority of the time was spent with dimmed lights and a squeaky marker shrieking y = mx + b on the sheet of an overhead projector. They taught well and I improved over the course of the semester, but I was far from interested in the material. To utilize vocabulary from the reading from Skemp (1976), this class felt entirely instrumental, which engineering students especially like to call "plug and chug". During class I was doing math, but I wasn't thinking about it. Perhaps these negative experiences are due to my 14-year-old self still getting used to high school, but I think there are better ways to get grade 9 students thinking about math.

The inspiring: my grade 12 functions teacher. On the contrary, there was less instrumental understanding in this class. We talked about more real world examples of the math we were learning and consequently these types of problems came up in word problems on tests. Class started with examples on the smart board, followed by independent work time on the homework problems. We could ask for help if we needed it, and sometimes these questions would lead to discussion involving the whole class to further clarify a concept. However, we were usually able to work at our own pace, which I liked because I could get into my own groove - meaning it was okay to rip through a handful of problems and then get pleasantly stuck on the next few. In addition to the class structure, they also utilized math puzzles. Each test, the bonus question was a different math puzzle which we could practice throughout the unit. My favorite was the KenKen.

(A close second place is my grade 10 math teacher. They wore a button on their dress shirt with a math pun Monday to Thursday, and on Fridays they dressed casually, usually with a t-shirt with a math pun. Some of the puns that come to mind include "math is the path" and "I can't get no satisfraction".)

Future Students

It's exciting and intimidating to think that we will be teaching as many as 6,000 students during our career. Here are two hypothetical emails from future students.

Hi Mr. Hamilton,

I'm in my final year at the University of Waterloo in mechanical engineering. The program has been a lot of work, but I know I made the right choice. The math classes we've taken have been tough, especially ODE's and PDE's. However, I felt prepared for these challenges thanks to your class in high school. You taught me to be patient with problems I can't solve and to ask for help when I need it. 

Do you have any new favorite math puzzles?

Sincerely,

A student who absolutely loved your class 

Hi.

Lots of my friends took your class and for some reason they really liked it, but I feel it's necessary to send this email to tell you that I thoroughly despised it. Even though got a decent grade, it was the lowest of my classes in grade 12. I don't think the applications to engineering and medicine were interesting because I was just in class to get through the math so I could go to business school.

Just wanted you to know.

And math puns aren't cool.

Cordially,

A student who didn't like your class

 

One of my motivations for becoming a math teacher is to provide an enriching learning experience like my grade 12 functions teacher did for me. I hope my future students will feel how I feel about this teacher. If that were the case, they would say things like my positive letter. However, my main concern is that after several years of undergraduate and graduate studies I am quite rusty with the material in secondary math and I am anxious about my abilities to teach as well as I have been taught.

This has been an interesting thought experiment and I will continue to think about what skills I need to develop during our time together over the next year.

September 21 Exit Slip

Today in class we reflected on our key takeaways from the reading of the week by Skemp, "Relational Understanding and Instrumental Understanding". I enjoyed hearing the opinions of my peers on the reading. We realized that our own experiences influence our teaching, but this can change throughout our career as we embrace new ways to look at math and new was to teach those new ways to look at math. As much as I may have grown up with a strong emphasis in instrumental understanding, especially in elementary school, I am now cognizant of this distinction in looking back and looking ahead as I put my experiences into my teaching philosophy and put that into practice. Another takeaway from class I have is how introducing puzzles in our classes (much like we have in this very course) is a way to naturally spark curiosity and foster relational understanding.

The Locker Problem

I was pleased that I had not yet met "The Locker Problem". My first step was to rewrite the puzzle in my own words so that I could work on it in my notebook. From there I broke the problem into manageable chunks of N = 1, 2, 3, 4, 5 students to see if any patterns emerged. I noticed that once the Nth locker has changed states from the Nth student, it no longer changes states because no other students will be eligible to touch it. I decided to investigate the idea of N with odd factors because an uneven number of state changes results in a final state of closed.

I checked to see if there are any square numbers with even number of factors. By nature of square numbers, they have an odd number of unique factors.

 

 

Thus, I concluded that perfect squares 1, 4, 9, 16, ... 961 are the doors that are closed after the 1000 students are done their shenanigans. 

 

It has been too long since I've been captivated by a math puzzle. I loved having them in my classrooms throughout my high school education. However, in university I often had to go out of my way to get lost in thought with puzzles. I will be accumulating my favorite puzzles so that I can provide my future students with endless scribbles in the margins of their notebooks.

Skemp (1976)

I'm impressed that this piece was published 44 years ago. After three weeks of readings throughout my classes in my first month in teacher's college at UBC, it immediately stands out as one of the articles I will continue to think about. "Relational understanding and instrumental understanding" is still remarkably relevant in 2020, in math education and throughout STEM education. 

Skemp compares and contrasts the two meanings of understanding: relational, knowing both what to do and why, and instrumental, using rules without grasping the reasons. He goes as far as stating that math taught in each of these ways ought to be thought of as its own class. I agree with the supporting points for each type of understanding; such as the immediate rewards the students feel through instrumental understanding and the adaptability to new tasks in relational understanding. However, I am optimistic that the best of each approaches can be applied to our classrooms.

I think there is material that must be taught in an instrumental manner, such as the memorization of multiplication rules and times tables. However, over the course of a student's education, I think it is our responsibility as teachers to build the schemas of what we teach so that students can engage in relational understanding. My teaching philosophy is centered on an interdisciplinary approach to problems in STEM. Though subjects are neatly packed into their own discretely labeled entities, I think it is important to show our students how concepts from one lesson apply to other units and to other subjects. The fire may need to be sparked by some instrumental kindling but it needs relational logs to flourish.

Hello, World!

I like to think I am one of the few, and therefore best, photographers with an iPhone 4S, my current phone. Here is the view atop La Campana in central Chile from 2018.