Math Art Reflection

 

Our group recreated Melissa Schumacher's "Counting with Knots" by extending the original artwork with the counting from 0 to 15 using base 4. Because we changed from base 3 to base 4, we needed to create a new symbol to represent it. We each created a unique symbol representing the number 3 in this case. Allyssa made 3 correspond to an X symbol, meaning that the crossing has no restriction, but the line does not pass in this area. Asiya made 3 represent a crossing that has either 2 strands or 4 strands. I made 3 to correspond to both horizontal and vertical barriers. Of the three, my idea caused the final artwork to represent something other than a Celtic Knot as the horizontal and vertical barrier cut into my knot. Whoops

While working on this project was hard since each of us made our variation, we also knew the pain and frustrations the students would have when creating the artwork. The easiest part of the presentation was creating the PowerPoint and the actual presentation. Since most of the topics we were going to discuss were somewhat familiar to us or easily accessible online. The artwork, on the other hand, was very tedious. Initially, it took a lot of work to figure out how to start this art project. As we worked on it, I came across a grid with dots that I modified and used Photoshop to create the 3 grids to represent each part of the artwork. Even with the grid, I had to make multiple photocopies of one as I feared my inexperience in art would ruin my one good copy, resulting in a total restart. Out of the three parts, creating the knots with the over-under pattern was tedious. The other 2 parts were quite therapeutic as I worked in a set rhythm. However, since there is a precise over-under pattern with the knot, I had to constantly pay attention to what I was doing, which took a lot of energy. In the end, the Celtic knots were a lot of fun, and it took me about 5 hours once I had the grid to finish it. With an added 3 hours of thinking about how I would start this project, the artwork took about 8 hours from start to finish.

As a teacher, we must try to attempt the assignment ourselves to know what was difficult and how long it took us to finish. Given how different this math art is to many math classes, students would need help completing part 3 of the artwork as it was the most time-consuming and energy-intensive. Knowing the difficulties of that part, I have learned to modify the art project a bit so that students would create a smaller knot that is less vertical and slightly more horizontal, like a 5 by 7. Here, students will explore 7 consecutive numbers, but the numbers are larger in base 3 or 4. As we create Celtic Knots, this artwork will tie nicely with a Social Studies class if they ever cover the Vikings. This project would be a tremendous cross-disciplinary assignment with Math 9 and Socials 9 as we look at various aspects of the knot. For math, we would look at the underlying pattern used to create the knots, while in Socials, they would look at the underlying history of the knots to Celtic cultures. Overall, this was an enjoyable project, and I will keep this art project in mind as I head into my practicum and future teaching career.

The Dishes Problem

 Due to all the influences on me, it was hard not to consider the problem in a non-algebraic method. Thus, I had to solve the question algebraically to find the number of guests that generates 65 dishes... is 60. After doing it algebraically, I modelled and graphed the number of dishes created, given the number of guests attending. Here, I had to make some assumptions to proceed.

My algebraic work as well as working out how an x number of guests affect the number of dishes.

Suppose 5 guests showed up, and following the logic the cook provided, I assumed the following. 3 dish of rice was used as 2 guests used a dish of rice. Then, with 5, 3 dishes are needed, so the 1 remaining guest has a rice dish. Then, 2 dishes of broth had to be used. Once again, this is so the remaining 2 guests who still need to get a dish get one. Similarly, 2 meat dishes are used, so the 1 remaining guest gets something. Thus, we took all the dishes' mathematical ceilings to ensure all the guests got adequate food. With this assumption, one will realize that there are 2 correct answers to this question. 59 and 60 guests are valid, except one will have leftovers. Since the question did not mention leftovers, it is safe to assume both are correct. Instead of using pure algebra, graphing the question has yielded some interesting results, as I realized after seeing 59 guests with 65 dishes is also a valid solution.

Number of dishes needed given the number of guests.

My assumptions came with the idea of the image as I thought what would happen when this many guests showed up. As a host, I would rather have leftovers than a hungry guest because I failed to make enough food to satisfy everyone. Thus, providing imagery to the question can also get students to think outside the box, assuming it is allowed. At the same time, if we restrict students from thinking creatively by adding a handful of restrictions, what is the point of giving the students an image if the stated rules shackle them?

It is suitable to bring up the history of math as it demonstrates to students how/ancient some of the mathematical techniques taught in class are. With a question posed back in the 4th century CE, the student can realize how some of the topic being learned was once used to solve a problem. Similarly, math techniques did not only originate in 1 culture, and I like to believe that people with diverse experiences with different cultures can be a better citizens filled with love and not hate.

To future me

Hey Mr.Tse,

I appreciate your attempts to make math more fun and accessible for all students in class, but sometimes, we don't take math to enjoy it. Instead, some of us take it as a graduation requirement. I did not feel engaged in any of your classes, and in the end, I felt more lost than anything else. I preferred if you taught like all the other teachers, how we can solve the questions and that's it. All the extra fluff caused me not to understand the concepts next year, which caused me to fall behind in math. Thanks for ruining my chances of getting into university.

Student X

One of my biggest fears is that because I wish to teach math more relationally than instrumental, I would end up with students not caring about my lesson as they treat math as a mandatory class to enter university. Also, as I want to vary up my class and not heavily rely on pure classroom lessons and on some outdoor teaching or vertical class time to teach lessons, many students might not be ready for a shift, especially if all they have experienced in math is the idea of copy notes, doing homework, and doing exams.

______

Hey Mr.Tse,

I want to let you know that your class has really changed my views on math. While most teachers still teach in the traditional ways, I really appreciate your varied teaching methods to encourage more learning from us. This made me always want to show up to class as I am no longer always sitting in a stuffy classroom listening to the teacher go and go about things. You made math to be more than just memorizing numbers. While I did not score the highest marks in your class, I appreciated the use of math/logic puzzles in your class. I never got many of them on the first try but I did try my hardest and it helped me develop a more logical thinking method which I thought only smart people did.

I hope you keep up the good work best wishes,

Student X

As I change how we approach math in classrooms, one of the biggest things I want to accomplish is encouraging more students to explore different ways to solve problems. I also want to help students foster a growth mindset in my class and never feel too down when they struggle with something. While I understand that most people take higher levels of math in high school for university, I also want students to be willing to try and fail yet finish the course like they have learned something regardless of the mark they finish the class with.

______

As I look at my strengths and weaknesses, my worries stem from a sense of imposter syndrome about whether I should be qualified to teach in a classroom. After I start my teaching career and build confidence, my response to this activity would be very different as I would better understand how students react to certain things. As of now, a lot of what I imagine is what I have experienced as a student in high school and post-secondary. Yet, my idea of changing the way students are taught math will be a tough challenge to reach, so I hope that I am prepared to start slow with the changes. As I have been told, year 1 is for survival, year 2 is for change, and year 3 should be smooth sailing.

Lockheart's Lament

Lockheart proposes some exciting ideas regarding math education in schools, which I wholeheartedly support. Looking back at my high school math education, it has always been a monotonous cycle of lessons, reviews and exams. This rigid way of teaching has sucked away and made math boring to many students. As Lockheart points out, most students take math to improve their college application or get college credits to get math over with. Coupled with the need for marks or standardized testing in the US, I can see how the current education system emphasizes instrumental mathematics in the classroom, as detailed by Skemp.

While Skemp's argument of relational mathematics may seem similar to Lockheart's way of changing the mathematical education system, Lockheart's methods are more aggressive. Skemps's idea of relational mathematics still relies on the fact that there is a set curriculum that a teacher should follow, but rather than teaching the "hows of math," we as teachers should teach the "whys of math." Meanwhile, Lockheart argues that the rigid curriculum restricts the creative juices of math teachers. To truly embrace a new way of teaching math, Lockheart argues that we should thoroughly teach math without a proper curriculum so students can embrace mathematics and allow teachers to explore various topics.

While I appreciate Lockheart's view of dismantling the curriculum, he eventually still tied the dismantling of the curriculum to lead the students to take calculus in a formal setting. Yet, calculus is not all of math. Many different math disciplines still need to be more represented in our current math curriculum, such as discrete mathematics, elementary number theory, rings and fields, and many more. These math disciplines can be taught at a secondary level, albeit more conceptual, to introduce students to the abstract topics of these different math disciplines. Regardless of everything tying back to calculus, as a new teacher candidate, the best way to make math somewhat not dull is to break out of the current shell of classroom lessons and add more interactive activities for students to engage in and learn. Overtime, as we push for change in how math is taught in school, only then can we consider a proper reform, as mentioned by Lockheart.

Monotonous to unique teachers

Throughout my time in secondary and post-secondary schooling, I have realized that most math teachers/professors follow a monotonous routine of teaching/lecturing, assigning homework, and, once we finish a unit, reviewing followed by an exam. If you look back, you would realize that most, if not all, teachers did something similar. Rarely have a teacher done something out of the ordinary, which is why I loved learning discrete mathematics and number theory in my programming classes in secondary school. I solved these problems by coding up a program which helped me understand the relational mathematics behind the way it works rather than the instrumental mathematics or how it works. It was also why I shifted my mathematical views from caring about the "how-to" to wanting to know the "why it works."

Similarly, in post-secondary school, the professor who stood out to me was the one who did not follow the monotonous routine of lectures, homework, and exams. I went to Simon Fraser University, and there was one professor whose classes I took introduced a different way to teach. The main goal of the course was to teach us group theory. However, rather than taking the traditional monotonous approach to a relation mathematics approach, we have to first play with games that emphasize the learning material without explaining it. He had us explore the things and then formally introduced them to us. The final was also not the traditional style of an exam. A poster and report asked us to explore other games (not covered in class) and teach us the math behind them. Most of it was a variation of the Rubik's cube, and for the odd one out (us), we analyzed a computer algorithm for solving a 3 by 3 Rubik's cube. For those interested in the course, http://www.sfu.ca/~jtmulhol/math302/  (He made a free textbook to go with the class.)

Locker Problem

Looking at the description of the locker problem, my gut told me to investigate a smaller problem to see if an interesting pattern emerged. So, I simulated a smaller problem (10 students, 10 lockers) to see what I could turn up.

After simulating my smaller case, I noticed that only 1, 4, and 9 are the only closed lockers in my simulation. While 2, 3, 5, 6, 7, 8, and 10 are the ones that remained open. This observation led me to investigate the factors of these lockers further and see if perfect squares are the only lockers that remain closed after all the students go by.

• Looking at perfect squares such as 25, the factors are 1, 5 and 25. Or 841, the factors are 1, 29 and 841. Perfect squares have an odd number of students opening or closing the lockers since they all have an odd number of factors.
• Comparing it to non-square composites such as 96, the factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96. For prime numbers such as 101, the factors are 1, 101. Or 887, the factors are 1, 887. These all have an even number of students opening or closing the lockers as they all have an even number of factors.

Perfect squares have an odd number of students opening or closing the lockers because to create a perfect square, a number has to multiply by itself, adding only 1 student(count) to open the locker (factor). While in most cases, all other numbers, prime and non-square composites, every number needs to multiply with another. Hence, the commonly taught idea of the rainbow in classrooms.

Rainbow factor of 96

Technically, 0 is considered even. Thus, our initial case of all open is represented by an even number. Since any even number incremented by 1 becomes odd, we have a pattern of open represented by an even number of students who have gone by and closed represented by an odd number of students who have gone by.

Thus, all perfect squared number lockers will be closed. These lockers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900 and 961—a total of 31 lockers. (1² up to 31²)

The remaining lockers will be the ones remaining open, and there are a total of 1000-31 = 969 lockers are open.

Skemp Disscussion

 

Sept 11, 2023 Entrance Slip ... Relational and Instrumental Understanding

Looking back on my life and how I ended up understanding math compared to many of my friends, I realize that I shifted from an instrumental understanding to a relational one at some point in my education. Throughout the reading, I frequently stopped at all the parts on how he described students understanding math instrumentally. I compared it to how many of my friends only cared about how I got the solution rather than why when they asked for my help. Seeing how many of my friends' attitudes are represented in this paper brings a sense of relevance to the current math education system today.

It was pretty intriguing as Skemp tries to justify why so many teachers teach instrumental mathematics. Looking back on my math teachers, they taught math in a mixed view of relational and instrumental. Yet, in the end, the biggest takeaway with my classmates was usually the instrumental understanding, as everyone mainly cared about the how and not the why. Since this paper is quite old, or I was lucky to have the teachers incorporate some relational mathematics, nowadays, I think there is an effort to show some why rather than only the how in mathematics. While Skemp tries to justify a shift into relational mathematics, presently, many teachers incorporate a mixture of instrumental and relational. However, given the need for exams to assess students, many teachers teach math instrumentally since the student's primary goal is to get a good mark. I learned mathematics with a weird hybrid of relational and instrumental math. I understand how relational math can more easily allow a student to see the beauties of mathematics. However, unless there is a clear shift in how we do assessments in math education, most students' goal is to answer as many questions correctly, which leads them to embrace the "How" over the "Why."