Joe + Kieran, Hiu Fung TAI

 We chose to work on a piece by Hiu Fung TAI, linked here and shown below.

Hiu Fung TAI - Bridges 2026 Exhibition of Mathematical Art, Craft, and Design

An initial glance may not grasp the depth to this rather simple submission. The common factor between these shapes is that they share the same integer value of area and perimeter. Further interestingly, each side/arc length is also an integer value, which seems unintuitive when we look at the pie piece. Tai ends his article with a somewhat obvious statement that two unique 2D shapes can share an area and perimeter value. To the more analytic reader, this raises a few questions: Are there infinite unique area/perimeter pairs? Can circular objects be paired with linear objects? Does this statement generalize to 3D? To higher dimensions?

We aimed to answer the penultimate question and examine the relationship between volume and surface area in 3-dimensional objects through a similar lens. It was quickly decided between us that we should create a few shapes and working out a lesson through them. However, glancing at an object in your hand and determining the volume is significantly less obvious than looking at a shape on graph paper. How can we make the demonstration visually intuitive then? 

Joe initially proposed that we 3D print these objects to make them watertight and submerge them within a clear container of water, clearly showing volume via displacement and careful measurements. Surface area would then have to be calculated by hand, which we thought wasn't too much of a hassle, both for us and the audience. 

A cube and sphere, sliced and ready to print

However, due to logistical complexity and issues with the printer, we moved to a demonstration revolving around Gabriel's horn and paper mache.

We chose Gabriel's horn to highlight the complex relationship between surface area and volume, since, of course, this famous example has an infinite surface area paired with finite volume. We chose a scale for the horn and set bounds of y=20/x from x=1 onward, rotated about the x-axis to find a workable volume of 200π. Kieran had an idea to create a hanging display that we could leave in our classroom, and so decided on a final design involving three objects with a volume summing to 200π suspended above the horn. The dimensions of the shapes, a cube, sphere and octahedron were carefully calculated before we got hands-on.

Profile of horn sketched onto cardboard

The finished product

After the outstanding work to create such a beautiful display, we moved onto the interactive component. There was a realization at this point that our demonstration would likely involve a few calculations from the audience. To avoid a boring talk about geometry, we aimed to engage the audience in open-ended, discussion-prompting questions (some unanswered!) with a goal to hold some real estate each students' curiosity after we end our lesson.

Art Project!

 

Reflections

This project allowed us a great deal of creative freedom. For me, above all, this left me inspired. Getting to have some choice in our adaptation of the art piece inspired me to jump head-on into this project. Mathematically, it prompted me to review many things and use a wide range of skills. First, I had to recall some calculus for the volume and area calculations of Gabriels horn. I used hands-on skills measuring the exact dimensions of the horn, sphere, cube, and octahedron. I also used some euclidean geometry with my compass to make equilateral triangles for the octahedron. In this regard, the project prompted me to use and explore a range of mathematics to calculate and construct the final art piece. 

In my own classroom, I think about setting students up with art projects which cater to their natural curiosities but also the curriculum I'm trying to teach. I would hope that students find a similar amount of inspiration as I did with this project. It makes me realize the power of being deeply engaged in something. Of course, this comes at some cost. The drawback of doing a project like this might be that you sacrifice some in-class time where you can be teaching new material in a slightly faster paced manner. Balancing the use of engaging projects like these and more traditional lecture style, or even discussion based teaching is critical.

Sep 23 Reading

I first stopped when the influence of the Bourbaki group was discussed in the New Math Movement. The idea of introducing abstract algebra, calculus, proof, set theory, etc. and the high school level is an interesting one, but the exclusion of the use of diagrams and geometry feels like it would miss what I’ve experienced to be useful intruments in the development of my own understanding of mathematics. I laughed when they discussed the implementation. Of course, math teachers don’t know how to teach real analysis to high school students. First, they probably know very little real and analysis and secondly, they’re expected to teach it to students who have hardly developed whte algebraic skills necessary to study such a topic from a classical perspective. This implies the teachers were demanded in developing an entirely new method of teaching an advanced subject!

 

I next stopped during the discussion of the involvement of the right-wing evangelistic Christian religious lobby groups. This steered any conservatism into associating with such groups, developing massive amounts of polarization on the topic. I think this is a great example of how media can exacerbate polarization, preventing productive debate. Anyone who may be (correctly) in support of a more conservative approach on a specific topic will never be heard and immediately associated with having extremist political views. 

 

Finally, I was excited about the idea that studies showed that a “deeper conceptual understanding of mathematics was key to success in international rankings.” This makes me feel optimistic about my teaching goals around developing relational understanding over instrumental understanding, and to take that responsibility on in both my teaching and assessment styles (Of course it’s easy for any math teacher to teach and assess instrumental understanding, while the outsider does not realize it’s how they’re teaching. Thus, the onus is on the teacher to take the responsibility of teaching relational understanding). 

Sketching measurements at work - Applying the how and why of math!

Sep 21 Reading

  The Educational Imagination by Elliot W. Eisner made several interesting points about curriculum design. The first that stood out to me was around the use of extrinsic rewards. It makes sense to me that extrinsic rewards help with student compliance, but making them “reward junkies” could be ultimately unhelpful for them. Most students will end up working jobs which are often mundane, and having an extrinsic reward dependence would be a hinderance. The author does note, however, that part of the “implicit” curriculum is to train students to do mundane and boring tasks. This may serve some of these students which will have a career of this sort. My take on it, is that perhaps they’ll learn that the act of doing something at all has intrinsic value, no matter how mundane or boring. The fact that your activity is outputting something tangible into the world has value on its own.

I like that the author pointed out some of the positive aspects of the implicit curriculum: cognitive flexibility, punctuality, and deferred gratification are extremely useful tools which can help a person succeed in many areas of life. Finally, I was interested in the point made on page 99: That intellectual processes used in school cover only a small portion of the kinds of intellectual processes of which a human being is capable. This is a framework where I’d like to expand my ideas around lesson planning and curriculum for mathematics education.

My Favourite and Least Favourite Math Teachers

Among all the math teachers I've had over the years, one is the clear favourite and one is the least. I'll begin with the bad, and end things on a high note. 

I really enjoyed the first real analysis class I took at university. I thought the development of the real numbers was fascinating, and a rigorous view of calculus felt right. I was excited, then, for the third year real analysis class, where we'd be working in R^n. I thought it would be great to learn some of the rigorous math behind what I'd done in multivariable calculus. Needless to say I was left disappointed. What began with high hopes was met with skepticism and my professor stumbled through his lectures, sans notes. While this might've been a manageable situation, the course notes contained hardly any of the proofs which we were discussing in class. Moreover, the professor brought his dog to class on a couple of occasions. Unfortunately a student was terrified of dogs, so the professor had to hold the whining dog in one arm while they tried to lecture on the board with the other. Needless to say his complete lack of effort left most of the class in the dark, and we had to resort to having post-lecture meetings in the math lounge to work together to unpack the results and teach ourselves the course - a silver lining, perhaps. 

As these things often do, this teachers reputation preceded them, and my experience was not such a surprise. The lesson I'm left with now is the importance of caring. When a teacher cares, they put a little effort in, and the students will notice. Caring about your students and the kind of learning experience they have, will take a teacher so far from the get go.

My favourite math teacher I ever had was also in university. I took a third year linear algebra class with him, as well as commutative algebra in fourth year. He spoke dryly, and with slight monotone, but as the lecture got going he came to life. He cared deeply about the mathematics he was teaching, and would often pause to prompt the class to join in and ask questions. He would also pause, and give brief historical digressions about the origin of a certain branch of math. He wrote perfect lecture notes in chalk on the board, all without the use of his course notes - which were comprehensive and followed the lectures closely. The homework problems were unique and interesting, and whenever I went in for office hours he would take more time than he needed to help me, answer my questions, and discuss mathematics outside of the context of the course. It was through this teacher that I saw the beauty of the deeper side of mathematics which I felt like I had been pursuing throughout my degree. Even so, he opened my eyes to the world of mathematics and I realized I was barely even scratching the surface. This experience left me with a sense of awe and wonder at the brilliance of humanity, and all we could achieve when we put our mids to something.

This feeling of wonder is something I hope to impart on my students one day. I want them to feel like it's a big world out there, but they're capable and ready to take it on one little bit at a time.

The Locker Problem... Spoilers!

 I hope that my writing is somewhat legible. I had worked out some ideas already in class, but use of the diagram really clarified things for me!

EDCP 342 Reading 1

Skemp's article on relational and instrumental understanding elaborates on a basic idea which many of us are familiar with in mathematics: simply being able to 'do' the problems, and properly understanding what they mean. Skemp's analogy with 'faux amis' introduces the idea in a very natural way. 

The first thing that made me pause to reflect was when the initial analogy was drawn between the term 'faux amis' and relational and instrumental understandings. I reflected on the fact that perhaps in assessment, relational and instrumental understandings show up in similar ways, but are very different in nature. The second place I paused was ni the devils advocate section. I had assumed that instrumental understanding may only convenience the student in being an 'easier' way to get through a math class, but I had not considered that there was a case to be made for teachers to teach instrumental understanding. Finally, the idea that practicing different pathways between points in a students schema does not necearilly develop that schema. Not only does a student need to always develop their schema, or relational understanding, but they need to be able to navigate with proficiency and efficiency. Going back to map analogy, a student should know the map of the city relatively well, but also be capable of taking transit or walking or biking to navigate throughout.

I believe that in education we should always be striving for relational understanding over instrumental understanding in mathematics. I believe relational understanding not only lends itself to better learning math in the future, but opens the doors for relational approaches to learning throughout life. 

Hello World!

Happy September!

Here is a film photo of a formation in Yosemite national park in California.