Reflection on the Presentation.

Overall, I think our presentation went really well. Everyone in our group did a great job with the parts we had planned. I introduced the original artwork and explained the mathematics behind it. Medha talked about our design process and how we created our own piece. Josh explained the area calculations in detail. I also really liked how our interactive activities worked out. Our classmates were very supportive and actively participated in almost everything we asked them to do. The last activity, where they matched the different regions with the correct areas, worked especially well. At first, some people seemed a little confused about what they were supposed to do, which made me nervous for a moment. But after we gave them some guidance, they quickly understood the idea and started thinking in the direction we had hoped. It was really nice to see something we had spent so much time planning actually work in the classroom.

Before starting this project, though, I was very resistant to it. I have never thought of myself as an artistic person, and I am really not good at drawing or making art. So when I first saw that we had to recreate and extend a piece of mathematical art, I felt a lot of pressure. Even two days before our presentation, I was still trying to fully understand how the original artwork was constructed how one curved kite could be rotated and enlarged again and again to create the final spiral. Luckily, my group members were very supportive. We gave each other space to work on the parts we were more comfortable with, while still helping each other throughout the whole process. As the project developed step by step, I started to realize that mathematical art was not as scary as I had imagined. More importantly, I realized you don't have to be amazing at either mathematics or art to create something interesting. Even fairly simple mathematical ideas can produce something beautiful and surprising. Seeing the other groups' presentations made this even clearer to me. So many of their pieces amazed me, especially how they took mathematical ideas and pushed the original artwork in completely new directions. It made me see more clearly that mathematics really is all around us. In my future classroom, I would definitely like to try an activity like this. Students could enjoy creating and experimenting, while still learning and applying mathematics at the same time. I think that combination can make mathematics feel much more alive.

Fibonacci math art project group reflection

From Fibonacci Curved Kites to Our Own Mathematical Art Piece

Group Member: Medha Sharma, Joshua Kanaganayagam, Helen Chen 

For our project, we chose Gauthier Cerf’s Curved Kite Fibonacci Spiral. We chose this piece because it showed us a different way of visualizing the Fibonacci sequence. We were familiar with the usual Fibonacci squares and the resulting spiral, but Cerf’s work showed us how the Fibonacci sequence could be used to make various other geometric constructions while still creating a beautiful art piece. This combination of math and art made us curious about how it was constructed and how we might use the same underlying ideas to make our own artistic design.

Reverse Engineering the Original Artwork

Before creating our own piece, we first tried to understand the math behind Cerf’s curved kite. Each curved kite was constructed from circular arcs, and an interesting property we discovered was the relationship between its dimensions and area. Understanding how these individual kites could be scaled according to the Fibonacci sequence, and rotationally arranged to form the larger spiral, helped us see that the final artwork was not simply a picture of a Fibonacci spiral.  Additional visual patterns emerged from recursively repeating a relatively simple geometric rule.

Helen computed the area of Cerf’s curved kite using geometric formulas available to a grade 9 student. However, we suspect a derivation of this complexity could only be tackled by stronger grade 9 students who are familiar with contest-level math.

After we reconstructed the original design, we obtained the mathematical intuition necessary for our own extension. In particular, our thoughts about the area of the curved kite became important later when we considered what mathematical quantities could be explored in our own design.

Developing Our Own Design

Our attempts at extending the art piece involved many more iterations than originally expected. Firstly, we considered the familiar geometry of Fibonacci squares and transcribed in them various triangular shapes. We experimented with equilateral triangles, circles, semicircles, and Reuleaux triangles.

Although many of these ideas were aesthetically pleasing, none of them matched the aesthetic we were looking for. The original artwork had a strong relationship between its visual form and its geometry, which was challenging to replicate. As the complexity of our designs increased, it became increasingly difficult to find mathematical properties that were both interesting and attainable. We did not want the resulting mathematics to feel contrived, like something added on to the artwork as an after thought. We desired the mathematical structure to be critical to the design.

This insight shaped our progress. Medha, who was our main designer, suggested that instead of trying to force a rotated triangular shape into each Fibonacci square tile, we could work directly with arcs to produce the desired spiral. This insight helped us achieve our exact desire to attain a spiral shape. To produce a solid spiral curve, circular sectors were subtracted from the tiles (in a set-difference sense), which left behind a diagonal region that could combine to form the spiral.

It was not straightforward to reach the final tile design. Medha experimented with many placements of these removed sectors. She also added a special semicircular sector to (almost) every tile, which functioned to smooth the resulting spiral. Our initial design did not include these smaller semicircles inside each tile. Upon adding them later in the design process, we created smoother transitions between the segments in each tile, which made the overall piece feel more connected and therefore cohesive.

Just when we were close to finalizing the design, Josh noticed a small inaccuracy in one of the drawings. After review, we adjusted the design to maintain consistency, and we eventually arrived at our final version.

The colour scheme also involved careful experimentation. Given that regions would be filled, we discussed how colour choices would affect the positive and negative spaces created by the sectors. For our final art piece, Medha developed a blue-and-silver colour combination. The inevitable alternation of colours highlighted the spiral’s curvature while giving our finished artwork a very different aesthetic from Cerf’s original red and white piece.

Finding Mathematics in Our New Artwork

After establishing our Fibonacci tile design, our next question was: “What interesting mathematics can be found in our new artwork?”

We chose to focus on area, as we had found the area of Cerf’s kites to be of interest. Josh focussed on performing analytic calculations and algebraically formalizing our design’s geometry. Josh made some initial errors in his calculation that resulted in much more cancellation of terms than what actually occurs. After Josh corrected his calculations, he eventually organized the full mathematical argument clearly using LaTeX. Even though there was some partial cancellation still possible (using undergraduate index tricks) we did not include this summation cancellation in this project because it exceeded the scope of the BC curriculum.

This helped us appreciate something important about the relation between math and art: changing slight recursive rules can create an entirely different art piece with distinct mathematical relations. We started with Cerf’s artistic construction, but upon making changes we needed to investigate entirely new geometric relationships rather than retaining mathematics from the original piece.

Our Interactive Activities

We want our fellow classmates to experience some of the same problem-solving that we experienced during this project. Instead of simply showing our final calculations, we developed various opportunities for them to participate during our presentation.

We planned interactive activities in four main stages:

1. We would ask the class to consider how the area of Cerf’s curved kite might be related to its smaller radius before we would reveal the relationship.
2. Since one of Medha’s earlier designs contained a small geometric inaccuracy, we planned to challenge the class to see whether they could identify what was wrong. This would allow us to share our authentic design process rather than presenting our final artwork as though everything had worked perfectly on the first attempt.
3. During Josh’s derivation of the area, we would give our classmates opportunities to predict the next step as though they were high school students. Our plan for participation during parts of the calculation should enhance engagement rather than getting them to simply watch us provide the solution.
4. Our final interactive activity involves getting classmates to match terms in the equations with the correct geometric elements in the art piece, so that they can intuit our formula for area. 

If we had more classroom time available, we could consider further extensions. We could instruct our classmates to design their own art piece in a rectangle that is formed by Fibonacci squares. Alternatively, the artwork’s construction could involve physical tiles with different materials and colours that are assigned different costs. We could get students to determine production costs given the area calculations, and perhaps involve constraints to optimize the design. We could connect this artwork to increasingly sophisticated mathematics.

Helen has worked as our project coordinator. She organized the entire powerpoint, and ensured it looked clear and nice. She also considered ways to make the presentation interactive, so allow our classmates to participate rather than just listening to us talk. Helen also connected our project math in BC, while specifically considering how this art could be used in a classroom.

Grade 9 Geometry to Pre-calc 12 Series

During this project, we realized that a single mathematical artwork could be used in a variety of ways to teach. The Grade 9 syllabus requires students to investigate the area of simple geometric shapes, like what was discussed here. At higher grade levels, our design already utilizes series from pre-calculus 12.

We took on different roles through this project. Helen researched and explained Cerf’s original art piece and explored the Fibonacci foundation. Medha focussed on the artistic design and the physical productions, while Helen and Josh contributed feedback during this iterative process. Josh performed much of the mathematical analysis, which predominantly included the area calculations. We collaboratively developed the various interactive activities.

This process was much more iterative than we could have imagined. Since many of our first ideas did not work, we had to change our design repeatedly. There were so many iterations that we needed to correct a nearly finished version. Despite this, these challenges were a valuable part of this project, which made the project fulfilling.

Despite our distinct roles, we worked as a team throughout. This involved sharing ideas, giving feedback and solving problems together, since many things did not work out as expected. We all contributed in different ways, and by helping each other throughout the process, we created the final result that we wanted.

Very Hard Working Students:

Late-night meeting working on the presentation. 

Happy faces after all the things are mostly done. 

Rethinking How We Learn and Teach Mathematics

Reading Dr. Gerofsky’s article on the history of mathematics education made me realize that many questions I am asking as a mathematics teacher have been debated for over a century. Understanding, memorization, inquiry, and technology are all part of this complicated history. What surprised me was how often similar conflicts returned, even as society changed. Three moments made me pause and reconsider both my own learning experiences and the classroom I want to create.

My first stop was Table M.1, which compares conservative and progressive stances in mathematics education. I recognized many connections between the progressive column and what I have been learning about BC’s curriculum: inquiry, flexible problem-solving, communication, and understanding. These are qualities I want my students to develop. However, the table also made me wonder whether our assessments always support these goals. If we encourage students to explore different approaches but mainly reward speed and correct answers, what are we really telling them matters?

This tension feels familiar from my teaching experience. Although exams can assess understanding, preparing for them can easily become practising recognizable question types and remembering the expected steps. I initially found myself thinking that BC’s approach seemed more supportive of learning than the emphasis I associate with exams such as the SAT and AP. However, I cannot judge an entire education system through particular exams. What attracts me most is the possibility of using ongoing formative assessment to understand students’ thinking and help them improve. When exam preparation does not dominate every lesson, teachers have more space to follow students’ questions and invite different solutions.

The more I teach mathematics, and the more I learn in this program, the less satisfied I am with describing it as simply a right-or-wrong subject. Answers and reasoning still need to be mathematically valid, but there can be several meaningful ways to solve a problem. The quickest method is not always the one that helps a particular student understand best. The discussion of Dewey’s emphasis on inquiry helped me recognize that this concern is far from new. Why, after so many years, is it still difficult to give exploration enough space in mathematics classrooms?

My second stop was the discussion of mathematics anxiety among elementary teachers and their reliance on memorized procedures. Although this passage describes teachers in North America, I immediately thought about my own education in China. From middle school onward, I often learned mathematics by remembering the steps. Sometimes I did not understand why a method worked, but my memory was good enough to help me answer questions successfully. This gave me the impression that I was good at mathematics and that learning it was fairly easy.

During my final two years of high school, mathematics became much harder. Looking back, I wonder how many earlier ideas I had never fully understood. This pattern continued into university, where I could sometimes use formulas without understanding the concepts behind them. It was only when I began teaching that certain ideas became much clearer. I found myself thinking, “So this is what a derivative describes,” or “Now I understand what integration is doing.” Preparing to explain mathematics to someone else made me revisit gaps that good grades had allowed me to overlook.

This is one reason my classroom feels different from the classrooms I experienced as a student. Sometimes I feel as though I ask a hundred questions in one lesson. I want to know what students think a concept means, why a method works, and where the idea might be useful. They may not immediately solve every problem independently, but I want them to have some sense of what they are studying. This reading also reminds me that asking questions is not enough by itself: I need to listen carefully and give students time to think.

It also brought back an online discussion I had encountered about whether anyone could learn calculus. After these first few weeks in the BEd program, I am more willing to question assumptions about who is capable of learning advanced mathematics. However, believing in students’ potential does not mean expecting everyone to learn at the same pace. Prior knowledge, time, teaching approaches, and support all matter. My own understanding developed much later than my ability to reproduce procedures. That makes me hesitant to treat a student’s current difficulty as evidence of a fixed limit.

My third stop was the description of reforms that emphasized flexible problem-solving, multiple representations, technology, and mathematical communication. I was struck by how familiar these goals sounded. Ideas that feel current in my teacher education were already central to earlier reforms.

With today’s calculation and visualization tools, I can see why students might spend less time performing lengthy calculations and more time interpreting relationships. However, I also feel uncertain. A strong foundation in arithmetic and algebra can make later learning much more accessible. If students rely on a calculator whenever it is available, they may miss opportunities to develop fluency or recognize an unreasonable answer. Students may not yet know which skills they need to practise independently and which tasks can usefully be supported by technology.

I found the reminder that these reforms still valued computational fluency especially helpful in thinking through this tension. Technology and basic skills do not have to compete. As a teacher, I need to be clear about the purpose of a task: are students practising a procedure, exploring a relationship, or interpreting a result? Sometimes working without a calculator will support that purpose; sometimes a graphing tool will reveal something that calculations alone would obscure. What I am still learning is how to make those choices thoughtfully, so that tools extend students’ mathematical thinking while they continue developing the foundations they need.

What Are We Really Teaching?

One thing that really made me stop while reading Eisner was the idea of the implicit curriculum. Eisner writes that one of the first things students learn in school is to give teachers what they want or expect. This made me reflect on something that seems very small in everyday teaching. As teachers, we may genuinely want students to explore, take risks, and think independently, but our unconscious responses can sometimes communicate a completely different message. For example, when I ask a math question, I might naturally give positive feedback to the first student who answers correctly. Even if I encourage the students who answer incorrectly, they may still interpret the experience as “I am not fast enough” or “I got it wrong again.” This made me realize that what students learn from us is not always what we intend to teach. As a math teacher, I especially wonder whether some of our classroom practices unintentionally teach students that being good at mathematics means being fast and correct, rather than being willing to explore, struggle, make mistakes, and think deeply.

Another point that caught my attention was Eisner’s discussion of grades and competition. He describes how grades can become “commodities” with different values for different students. This feels contradictory to something I often experienced as a teacher. We constantly tell students that grades are not everything and encourage them to focus on their own growth and discover who they want to become. Yet when university applications begin, grades suddenly become extremely important again. In the international schools where I taught, I saw students taking many AP exams and evaluating themselves by the number of AP courses and the 4s or 5s they earned. Some students began planning their entire testing schedule as soon as they entered high school, sometimes before they had really explored what subjects or future pathways interested them. It makes me wonder: if our words tell students that learning and personal growth matter most, but the structure of schooling repeatedly rewards scores, competition, and credentials, which message will students actually believe?

The idea that stood out to me most, however, was the null curriculum. Eisner argues that schools have consequences not only because of what they teach, but also because of what they do not teach. This made me realize that curriculum design is not only about deciding what students should learn; every decision about inclusion is also a decision about exclusion. As a mathematics teacher, I might think I am simply teaching algebra, functions, or calculus, but if I continually emphasize procedures, correct answers, and exams while leaving out mathematical history, creativity, multiple approaches, or different cultural ways of knowing mathematics, I may also be shaping students’ understanding of what mathematics is. This connects strongly with my recent thinking about bringing more history and culture into mathematics teaching. It leaves me with a question I want to carry into my future classroom: when students say they do not like mathematics because it is only about formulas and right answers, is that really mathematics they dislike, or is it the version of mathematics that their previous explicit, implicit, and null curricula have taught them?

Favourite and Least Favourite Math Teachers

When I think about my favourite math teacher, I think about my middle school math teacher. I went to school in China, where a very important graduation exam takes place at the end of Grade 9. Because of this, math classes were often very teacher-centred. Teachers usually explained how to solve problems, students took notes and practiced, and the main goal was to get good grades. Students had little time to ask questions or explore ideas. My teacher was different. Sometimes, when he had the chance, he would stop the math lesson and tell us stories from Chinese history or even introduce us to The Book of Songs. Our class had around 60 students, so inquiry-based learning was not always easy, and our homeroom teacher often reminded him that exam results were the most important thing. However, he still brought these things into our classroom. I really enjoyed these moments because he helped me see that mathematics was connected to history, culture, and the world around us. More importantly, he made me curious about learning.

My least favourite math teacher was probably my father. He has a background in science and finance and has always loved mathematics. He enjoyed solving difficult problems and often helped me with my homework. However, he would sometimes say, “How can you not know this? It’s so easy.” Although he probably didn't mean to discourage me, these comments often made me feel I wasn't good at math. I started comparing myself with him and thinking that I was simply not smart enough. It wasn't until I studied mathematics at university that I realized I was capable of learning math; sometimes I just needed more time and a different way of understanding it.

These two experiences have shaped the kind of math teacher I want to become. I do not think every student needs to be “good at math” or get a high grade. I want my students to feel comfortable trying, making mistakes, and learning from them. A wrong answer does not mean a student is not capable. As teachers, we should guide students rather than judge them, and help them develop ways to think and solve problems. My favourite teacher taught me that math can connect us to the wider world, while my father taught me how easily a teacher’s words can affect a student’s confidence. I want to carry both lessons with me as I take up the mantle of a teacher.

The Lock Puzzle

Enumeration was the first method that came to my mind. At first, it was difficult to see a clear relationship, but as I listed out the changes in the lockers’ states, I gradually started to notice a pattern. I noticed that after a certain student changes a locker’s status, its state remains fixed because no more students change that locker. As the number got larger, we can see there is a pattern that only square-numbered lockers change status. After finding this pattern, I started to think about why it happens. It took me a while to connect the pattern to the factors of each locker number.

Beyond Getting the Right Answer

Three things made me stop while reading Skemp’s article. First, his distinction between relational understanding and instrumental understanding really stood out to me. I understand relational understanding as not only knowing how to do something, but also understanding why it works and what it means, while instrumental understanding is more about knowing the procedure without necessarily understanding the reasoning behind it. This made me think about how easily I may have assumed that a student “understood” something just because they could get the correct answer. Second, I was particularly interested in Skemp’s discussion of the mismatch between the teacher’s and students’ goals: “Pupils whose goal is to understand instrumentally, taught by a teacher who wants them to understand relationally,” and the other way around. I have experienced this tension in my own teaching. For example, when teaching area, a teacher may tell students that the units must match before calculating the area. Students may follow this rule, but if they don't understand why the units need to be consistent, they may still have only instrumental understanding. Third, the statement “Well is the enemy of better” really made me stop. As an AP teacher, I have often seen that once students know they have mastered everything required by the exam, they are very satisfied with where they are. Without the pressure of an assessment, many students are reluctant to spend extra time exploring ideas more deeply. This made me think about how exams and the desire to make every lesson “efficient” can sometimes push both teachers and students toward instrumental learning. More content or more efficient learning does not necessarily mean better learning.

I personally stand more strongly on the side of relational understanding, although I do not think instrumental understanding is useless. Instrumental understanding can give students a sense of achievement because they can quickly get correct answers, and sometimes this is necessary. However, if students become too dependent on this immediate satisfaction, it may not support long-term learning. I believe relational understanding requires more patience and sometimes delayed gratification, but the deeper connections students make between new knowledge and what they already know can lead to a stronger, longer-lasting sense of achievement. I have also tried to create opportunities for this in my AP classes through projects, especially after the AP exam, when students had little motivation to learn new content. Projects allowed them to use calculus they had already learned and apply it to real-life situations. Although not every student was deeply engaged, those who invested time in the project often gained something that simply getting another correct answer could not provide. For me, this is why relational understanding is more valuable in the long run: the goal is not just for students to know how to get an answer, but to understand the connections behind it and eventually become able to use their knowledge in new situations.