More info on your write-ups for your math art project (for September 22)

Our group members are: Eric, David, and Henry. For this project, we choose Caroline Bowen's 2026 artwork <y, x>. This artwork is a 30 x 45.5 cm drawing made with gesso, coloured pencil, and graphite pencil. Bowen describes it has a revisiting of a series of vector field drawing that she made in 2019. What interested us most about the origin of the artwork was Bowen's decision to create the vector field manually. She calculated the direction of each vector using a handheld calculator, and then drew each vector individually on a grid of circles made from a compass. She describes this process as creating a sense of "mathematical intimacy" with mathematics that we would normally let computers calculate for us.

To remake the artwork, we approached the process in the opposite direction. Instead of calculating and drawing every vector by hand due to a lack of time, we used MATLAB to generate the vector field using Main.m and Regenerate_graph.m. Main.m allows the user to choose either differential or implicit equations and generate their corresponding fields. Main.m also allows initial conditions to be entered so that solution curves can be displayed. The program generates vectors over a grid and uses numerical integration to determine how the field moves from each starting point. We then created Regenerate_graph.m which was adjusted specifically to reproduce the behaviour of Bowen’s <y, x> artwork by making sure that the arrows pointed in the correct direction in different regions of the plane rather than simply producing lines with the correct slope. We also had to deal with the behaviour near the origin, where the usual slope-based method becomes problematic, so the program uses nearby starting points and checks how they behave within the grid.

Regenerate_graph.m

One difficulty occurred around the origin, where the vector behaviour could not be handled in exactly the same way as the rest of the grid. Working through these problems helped us see the difference between simply graphing a differential equation and representing an actual vector field. It also gave us a new appreciation for Bowen’s original process. MATLAB can calculate hundreds of vectors almost instantly, while Bowen deliberately performed this repetitive calculation and drawing by hand. Her artwork therefore makes the process of doing the mathematics itself part of the artwork, whereas our recreation explores what happens when that same process is automated.

We also slightly changed the piece to make it our own. Rather than trying to create an exact digital copy of Bowen’s physical drawing, we developed a MATLAB “Vector Field Studio” application that can generate other vector fields using the same basic visual idea. The program uses a dark background, a coloured gradient across the vectors, and allows the user to enter different equations and initial conditions. This change allowed the project to take in different user inputs and see how the graph is altered. 

Our interactive activity with the class is still being designed. Our current idea is to connect differential equation vector fields to functions that students have already learned, then allow them to enter their own functions and observe the resulting vector field patterns. However, we think this activity may be a little too difficult, especially for students who are unfamiliar with differential equations, so we may simplify or change the activity later.

Homework reading and blog response: Battleground Schools (for Wednesday Sept 23)

One pause I had was when the comparison between conservative and progressive approaches to math education. I have never thought about how teaching methods reflect different beliefs on what math is? The conservative approach focuses on more fluency, facts, procedures, and applying algorithms correctly (I think this is how I was mostly taught?) On the other hand, the progressive approach focuses more on understanding, inquiry, and problem solving. I think both approaches have their places in math education, and good teaching is not necessarily choosing one approach only. 

Another pause was during the discussion of math anxiety and how negative experiences with math can actually keep going for generations. Students who learn math mainly through conservative approaches, and struggles, may grow up believing that they are simply bad at math, and these negative attitudes can then be passed down into later generations. This reminds me: being a math teacher, explaining how math works is essential, not just something to add on to a lesson.

Lastly, I was very surprised about how politics and national events play their roles in math education. After the Soviets launched Sputnik 1, concerns about the Soviets led to the New Math movement, which introduced more advanced topics like Calculus to high school classrooms, hoping to help future scientists. Later debates about this "Math Wars" also sparked wider debates about school standards, teaching methods, and education as a whole. I had always though math was a politically neutral subject (since when 1+1 involved politics), but turns out I was wrong!

(edit) To add on to last paragraph, I think China had something similar back in the 1960s-1970s as well. During the cultural revolution, normal schooling was heavily disrupted and curriculum was replaced in part by political study and manual work. 

research: https://www.sciencedirect.com/science/article/abs/pii/S030438780700003X

What is meant by 'curriculum'? (Homework reading and blog response for Mon Sept 21)

My pause #1: Eisner argues that when schools rely on things like grades, prizes / rewards, students may start treating school like a RPG game: do quests (homework, tests, etc...) and get rewards for it. I now think that simple things like grades might have unintended affects on students. Us teachers might be just trying to encourage learning, but students will only care about bumping up their grades. (in my opinion, this is definitely true, even for me) This is a definitive example of implicit curriculum because nobody ever said school is about getting A's, it kind of just naturally develops based on how schools are structured. 

To continue on paragraph one, sadly, I personally do not really know how to solve this issue at the moment. Most school systems (that I am aware of) use a grading system of some kind, because how else can we evaluate students' performance in classes? If we remove grading system entirely, then isn't school just a glorified playground with no goals in mind? Would students just work on their own projects? How would be have enough teachers to facilitate that kind of learning? This is something I will think about during my time at BEd. 

My pause #2: Eisner discusses subjects & experiences that schools leave out. For instance, schools never teaches us financial literacy or how to do taxes. A valid argument could be that schools have limited time for teachers, so some things have to excluded. However, things like financial literacy are very important topics too, and students will be better prepared for their lives after high school if we teach these things. The importance of null curriculum is that they can shape students just as much as what schools teach students. 

(Edit: I just thought of this while eating food) I guess math does teach compound interest? But that is barely anything in the world of financial literacy.

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Eisner expands on the idea of curriculum beyond just the subjects and outcomes that we are required to teach. He argues that students can also learn values and behaviours through school. The BC Provincial Curriculum connects pretty directly to Eisner's explicit curriculum because it formally outlines that kids should "know, do, understand". (refer to picture) At the end, both implicit and null curriculum also shape students' understandings, do's, and knows.

Locker problem

# lockers = [1] * 1001

# for student in range(1, 1001, 1):

   

#     # on second students

#     if student % 2 == 0:

#         for i in range(2, 1001, 2):

#             lockers[i] = 1

#     # on third student

#     elif student % 3 == 0:

#         for i in range(3, 1001, 3):

#             lockers[i] = not lockers[i]

#     else:

#         for i in range(1, 1001, 1):

#             lockers[i] = 0

# print(lockers.count(1))

# theres a bug in this code: person 4 will actually be counted as second person instead of first

# and i misunderstaod the problem, oops..

# ---------------------------------------------------------------------------------------

lockers = [1] * 1001

for student in range(1, 1001):

    for locker in range(student, 1001, student):

        # Student 1 -> change locker 1, 2, 3, 4, 5, ...

        # Student 2 -> change locker 2, 4, 6, 8, ...

        # Student 3 -> change locker 3, 6, 9, 12, ...

        # Student 4 -> change locker 4, 8, 12, 16, ...

        # Student 5 -> change locker 5, 10, 15, 20, ...

        lockers[locker] = not lockers[locker]   # each student's action is just changing the state of the locker

                                                # so doing not lockers[locker] is fine

print(lockers.count(1) - 1)

# this version works!

Introspective writing: Favourite and least favourite math teachers (for Wed Sept 16)

One of my favourite math teacher is definitely my mother! She was a middle school math teacher back in China before coming to Canada. Ever since I was a kid, my mother has been teaching me various things in Mathematics, like quick mental math practices, or just general tricks that I can use to speed up calculations. Whenever I had problems, I would always ask my mother for help. Beyond math, my mother has also taught me how to be a good person in general. My behaviours, actions, the way I speak, a lot of what defines me today came from my mother.

In terms of my least favourite math teacher, it has to be my discrete math 1 teacher from my first year undergrad class. His moved through materials very quickly (going to slides and not allowing students to ask questions), and most of the time he did not go in much depth. His exams also placed a lot of emphasis on memorizing from slides, which made studying more like making predictions on what will be on the exams. From this experience, I learned that as a future educator, I need to create a classroom where students are given opportunities to ask questions, and make assessments that reflect mathematical thinking. In addition, I must also regularly check for student understanding, making sure no one falls behind. 

Sept 14 Entrance Ticket

First, Skemp’s pointed out that a student can get the right answer without understanding the mathematics is totally reasonable. His idea of instrumental understanding is knowing the rules / algorithms without knowing the reasons behind them, this idea made me think about how often teachers do this without realizing it. I am definitely a victim of this, and I need to be more mindful of this. Second, I thought it was interesting that Skemp does recognize the benefits of instrumental learning, despite it being somewhat against what we have discussed in class so far. We as teachers should be explaining why methods and algorithms work the way they do, rather than just memorizing them, However, getting a right answer quickly can certainly help build student confidence. Therefore, instrumental understanding still has its uses. Lastly, Skemp’s example of comparing learning mathematics to navigating a town is a very interesting example, definitely was not expecting that while reading the article.

 

For me, I think an approach I could try is: first do instrumental methods to build algorithmic skills, and just to make sure students get the right answers first, then connect to simple proofs or real life examples to reinforce learning.

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