More info on your write-ups for your math art project

 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.

 

Reflections on Battleground Schools

The first point that made me stop was the Progressive reform associated with Dewey. Before entering this BEd program, I often thought about what kind of teaching method could help students accept mathematical ideas more easily. Dewey’s idea of letting students do mathematics through experimentation, inquiry, and testing hypotheses is very good. However, education is a complex problem. Teachers have different teaching styles, and students have very different levels of interest and attention. In my past two years of teaching, I have found that many students cannot focus for a very long time. Therefore, the actual time that a teacher has to explain important ideas is often limited. My own solution has been to make each part short and repeat important ideas many times. In this way, whenever students focus again, they still have a chance to catch up. I think a good educational idea may be complex, but the way it is carried out needs to be clear, simple, and repeatable. Otherwise, even a very good idea may be difficult to implement successfully.

The second point that made me stop was the failure of New Math. There is a Chinese saying that “if the foundation is not strong, the whole structure will shake.” I think this describes the situation very well. The idea of preparing future scientists by introducing university mathematics into high school education is not necessarily wrong. However, before students learn more abstract ideas, we need to ask whether they have a strong enough foundation in high school mathematics. Does the high school curriculum provide enough mathematical tools for them to understand university-level content? Simply moving faster has no value. If adding university courses to high school could automatically improve a country’s scientific and technological level, then we could simply start teaching complex analysis in kindergarten. Students need time to build their mathematical understanding step by step.

The third point that made me stop was the article’s discussion of math-phobic attitudes. I agree that negative experiences and social assumptions can affect students’ attitudes toward mathematics. However, I also think mathematics is a special subject because of its strong logical structure. Mathematical ideas are connected like links in a chain. If a student misses one important link, they may not be able to continue the reasoning, even if they understand many other parts. I do not think this is only a problem with assessment. It is related to the structure of mathematics itself. Teachers should be honest about this, but we should not tell students that they are not suitable for mathematics. Instead, we need to find the missing link as early as possible and help them rebuild it through clear explanations, short steps, and repeated practice.

Response to “The Three Curricula That All Schools Teach”

The first point that made me stop was Eisner’s discussion of the explicit curriculum. I think curriculum includes two important parts: content and competency. Content is necessary, but it is not the only goal of teaching. In BC, the Core Competencies are as important as content, if not more important. This reminded me of a conversation with my former principal, Eric. He told me that content was not the most important thing. If the actual situation did not allow me to finish every part of the content, this should not necessarily be a problem. However, I should pay more attention to the development of students’ competencies. I totally understand this idea. Knowledge that is not used in the future can be forgotten easily. Competencies may also become weaker if they are not used, but once students have developed them, they can usually be activated again when they are needed. For this reason, I think teachers should pay more attention to helping students develop the ability to think, communicate in standard mathematical language, and learn or even create independently, rather than focusing only on the transmission of knowledge.

The second point that made me stop was Eisner’s discussion of the implicit curriculum, especially the way schools use rewards, grades, and classroom structures to shape students’ behaviour. This affected me deeply because, in my own teaching, I have found that most students care a great deal about marks. Students naturally pay attention to the things that I pay attention to. Therefore, the feedback, rewards, and penalties that I give are very important. In mathematics teaching, if we only give most of the marks for the final answer, students may learn that the answer is more important than the thinking process. They may try to memorize procedures instead of figuring out how everything works. I think we should give more weight to mathematical thinking and the process of solving a problem. Besides, showing students how I assign marks for the steps and the reasoning behind them, and asking students to take notes about that, could also be very helpful. In this way, students can understand that explanations and showing steps are not just something for them to recite. The steps represent a way of thinking in mathematics.

The third point that made me stop was Eisner’s idea of the null curriculum. I think this is especially important in mathematics teaching. In my own teaching, I pay a lot of attention to explaining why students are learning a particular topic. For example, when teaching functions, I emphasize that functions help us describe and solve problems involving related variables in real life. I then give students many examples, and let them give me more examples and explain how to define the independent and dependent variables. The purpose is not only to help students understand the knowledge itself. More importantly, I want them to understand a basic idea: mathematical tools can serve practical life and production. Although some mathematics may be considered “pure mathematics,” history shows that many mathematical tools only seemed to have no application because people had not yet found a field where they could be used. I want students to develop the habit of seeing the things around them in a quantitative way. If this part is missing, the null curriculum may cause students to see mathematics as an isolated system of knowledge. This does not match the original meaning of mathematics as a tool for understanding the world. It can make mathematics more difficult to learn, and even if students work very hard and get good marks, they may still learn it only to get a score, and that would be pointless.

Overall, Eisner’s article expands my understanding of curriculum. The BC Provincial Curriculum provides important content and competencies, but what students actually learn also depends on our classroom assessment, feedback, examples, and the opportunities that we do or do not provide.

My favourite and least favourite math teacher

 The best math teacher I had was my high school math teacher. He explained everything in great detail. Before introducing a new topic, he would review all the prerequisite knowledge that we needed and make sure that we understood it before moving on. During his explanations, he often proved important results instead of simply asking us to accept them, and sometimes he would even show us more than one way to prove the same result. After teaching a new concept, he would also explain how it was connected to other parts of mathematics and where it could be applied. Looking back, I realize that my own teaching style has been strongly influenced by him. I also like to review the necessary background first, explain why something works, and help students see connections between different ideas instead of teaching each topic separately.

The least effective math teaching I have encountered was actually not from one of my own teachers. One of my former students moved back to China because he was having difficulty in English class. Instead of attending a regular high school, he studied A-level courses at a tutoring school. He recently complained to me that his math teacher's explanations were very confusing. Since I encouraged him to respect his teacher, work hard, and figure out a way to improve himself, he sent me a short audio recording of a statistics lesson. To be honest, after listening to the recording, although I still try to encourage my student without saying anything negative about his teacher, I could understand why he was struggling. The teacher was explaining a statistics problem, but he rarely repeated the names of important concepts such as Q1, median and Q3, and it did not sound like he was writing the important expressions clearly on the board (given the short time he left for the students to think). This meant that if a student missed one step, it would be very difficult to follow the rest of the explanation. The teacher also focused mostly on telling students what would happen under certain conditions, without spending much time explaining why it might happen or giving examples to develop the idea further. Since the student had already made a mistake on the problem, I felt that the part he needed most was probably exactly the reasoning that had been skipped.

These two experiences have strongly influenced the way I think about mathematics teaching. For me, a clear explanation is not simply giving students the correct procedure. Students need enough background knowledge, clear mathematical language and notation, and an explanation of why the mathematics works. They also need connections and examples that help them build a complete picture of the idea.

Locker problems

 

Response of reading "Relational Understanding and Instrumental Understanding"

 I stopped the first time when I read Skemp defining relational understanding as “knowing both what to do and why.” I must say that there is something even more important and earlier: “What is the purpose of learning this subject?” When I was a student, I could hardly learn anything without knowing its purpose. Fortunately, my teachers were all very qualified, and they explained not only how math worked but also what problems people were trying to solve with it. For example, when I was learning about area, my teacher connected the concept to the practical need of measuring farmland and explained why area is calculated in this way rather than in some arbitrary way, such as taking the average of all the sides of a specific shape. Once I knew the purpose, the formula made sense to me.

The second stop was when Skemp argued that instrumental mathematics may be useful in limited situations, but relational understanding is “better” in the long term. I must say that “better” is a kind of value judgment, so we must first define the situation we are talking about before we can weigh its value. In this case, I am not perfectly sure that relational understanding should, or must, be the goal for every student in every part and at every level of math. For example, I am personally very weak at judging fashion, beauty, or visual style, while many other people seem to understand these things naturally. This reminds me that people have different interests and abilities. I do not think a high school teacher is qualified to “decide,” or even confidently “distinguish,” whether a student is capable of learning math through relational understanding. However, I think a high school teacher should be aware of these differences and respect them. As a math teacher, I would certainly begin by teaching everyone the reasons, so that all students have an opportunity to fully understand. However, if some students continue to struggle and show little interest after being given that opportunity, I think teaching them some useful rules and methods may be better than forcing them to study in my way, and it is certainly better than learning nothing.

The third place that made me stop was Skemp’s discussion of the advantages of relational understanding, such as being “easier to remember,” “more adaptable,” and “able to grow by itself.” I felt that he was describing the way I learned math 20 years ago. In high school, I often asked teachers to prove results, explain details, consider possible counterexamples, or tell me whether an idea could work in a wider situation. My memory has always been poor and has never really improved, even now. In China, we could not bring formula sheets or calculators into exams, so I often had to derive formulas again during the exam. When I first learned derivatives, I even found that the “derivative of the derivative” could be useful for determining how a function graph bends, although I did not yet know the term “second derivative.” At that time, many of my classmates criticized me for “wasting their time by asking useless questions” and said, “We don’t need to know that to do the questions correctly.” When I argued with them, the reasons I gave were almost exactly the advantages later mentioned by Skemp.

Therefore, I mostly agree with Skemp about the power and importance of relational understanding, since that is the way I learned. However, ironically, because I understand the reasons for learning the reasons, I do not think it should become a standard for everyone. As a teacher, and as someone responsible for providing students with access to knowledge, I think I must also respect their needs.

Hello World

 Hello World, this is my first post. Good to see you all.