Watercolour Painting Techniques - Micro Teaching Lesson Plan

 

The Locker Problem

 

Image 1. Thinking

Image 2 Excel Simulation

In the introduction of the book “Thinking Mathematically” by Mason, Burton and Stacey, you can read: “Probably the single most important lesson to be learned is that being stuck is an honourable state and an essential part of improving thinking”. This phrase gave me the courage to share my approach to solving this locker problem.

              As I have learned to code in the past, the first impulse was to try to find an algorithm that would help me find the answer. I couldn’t find any, and I decided not to look for AI solutions or suggestions. I left the problem for the next day. The next day I scribbled my reasoning on a paper and didn’t have luck. I started to listen to some classmates talking about the problem, and I avoided them. I wanted to think about the solution by myself.

              Finally, I decided to solve the problem manually. I went into Excel and started simulating row by row the action of each student. I coloured the cell yellow if the cell represented a closed locker, and I left the cell without any fill if the locker was open. After the 4^th student, I could deduce a pattern, but I kept the simulation until student #49.

              My conclusion was that the lockers that will be closed are:

1 ², 2², 3², and so on until 31², which corresponds to locker 961. The rest of the lockers will be open.

During the exercise, I found myself playing to find patterns of distance between numbers, and the beauty of the figure formed by the yellow colour. (See image 2)

I still don’t know how to express the solution in a mathematical expression, but I had a lot of fun playing with my model and sharing it with my family.

Art and Math Project , Personal Response.

I created this logo 10 years ago, when I started a small atelier where I taught art for kids. The name was "Art for Success," and the logo represented the letter "e" with a paintbrush. "e" because it is transcendental and irrational. I thought those adjectives were perfect for my art space and the art experiences of the kids.

 Here is my reflection after working with the Art and Math Project for our 342 class:

The artwork: 

12 Golden E-Chains Spiral

by Philippe Leblanc

40.0 x 24.0 x 24.0 cm

Plexiglass, colored glass beads, nylon thread, metallic squeeze beads

2025

As a Student, I learned about the concept of catenary curves, which I don’t remember having learnt before. I reviewed the concept of the Euler number, its origins and applications. I reviewed the golden square. The Euler number is an answer to the need to connect real and imaginary numbers.

I confirmed the importance of detailed observation and collaboration. Some details about the artwork were called to my attention by my peers. Like the double spiral where the beads were hanging from, and the fact that each chain was circumscribed inside a golden rectangle.

As a teacher, I confirmed that art can help to visualize concepts differently. Math can be beautiful. Sometimes we can do repetitive activities like beading and knitting to send our mind to a meditative stage. Maybe beading, as a repetitive and mechanical activity, doesn’t contain a math concept itself, but we can plant the seed of curiosity during these moments.

The most fulfilling part was being able to re-create a piece of art with accessible sources. Based on a concept that I could never have imagined could be an inspiration for an art project. This opened my mind to use math as a source of inspiration for my personal art projects.

At the beginning, I doubted how to connect this piece of art with the BC curriculum. This was my main difficulty, especially because I am not familiar with the curriculum: fortunately, Shannon has been teaching in Australia for so many years that even with a different curriculum, she was acquainted with the subjects in secondary school.

The practical difficulty was the process of calculating the length of each chain, due to the lack of information. We didn’t have the size of the beads the artist used. So, we decided to count them from the original piece of art. This way, we could determine the length of the longest 6 chains and extrapolate the rest.

Personally, I internalized that a piece of art could be a strong source of curiosity. After having replicated the piece, I wanted to know more about the Euler number. I spent a couple of hours reading about the history of this constant and the practical applications of this constant in Finance, Statistics and Calculus.

This kind of work can help me to design creative lesson plans around complex math concepts. Focusing on creating curiosity for learning what is in the curriculum. I can start with art exercises that approach a complex abstract concept, such as Pi, or compound interest. It will also allow different perspectives among the students to participate and understand.

 The materials were affordable, and I think there are no limitations with the budget. I would keep this as a group activity, in case some students don’t enjoy the meticulous job of beading, measuring and counting.

Art and Math Project Group Response

 Authors: Juma, Shannon, Jimena

Artwork, Artist: 12 Golden E-Chains Spiral, Philippe Leblanc

We met to discuss whether this would be a viable task, what materials would be required, and to outline the production process for the piece itself. We also looked into how to extend this for the classroom. We tried to understand the way the artist calculated the lengths and positions of each chain. There were also practical considerations about the available box sizes and how the bead sizes might relate to this.

The original artwork was recreated with purchases from Michael’s and Amazon. Some aspects of our piece were chosen for practical reasons (for example, Leblanc’s work utilized a square-based rectangular prism with lateral faces in the proportions of the golden ratio). We began with spreadsheets to calculate bead colour amounts. As we were determining the lengths of each chain and the position of each chain, we found that we were trying to honour the golden ratio that appeared in Leblanc’s work as well as the golden spiral that determined each chain’s place. In reality, we were able to mimic the bead count in the five longest chains; however, the remaining chain lengths were more arbitrary. We worked within the constraint that we wanted 700 beads in total and projected possible lengths that seemed plausible for the shorter chain lengths. To determine the placement of the chains, a printout from Leblanc’s information on the Bridges Math site was used. Another alteration of the piece is that we chose our own colour scheme of 10 colours.

We decided to follow Leblanc’s inspiration of irrational numbers. The first thought was to build a project around pi. However, this didn’t play out in a way that we anticipated. So, we pivoted and focused on square roots. Using either a straight edge and a compass or paper folding, we could create irrational lengths. We were also guided by the Grade 11 curriculum, which requested that we order irrational numbers. From this, we decided to ask the students to fold a Theodorus spiral from a single, long strip of paper. The plan was that the spiral served as the base for our class-focused extension to the art project: a square root mobile. The hope was that this would mimic the beads of Leblanc’s chains. (In a real class, we think that we would have also asked students to create them using different colours for each digit of the decimal expansion of each root, provided it didn’t get too confusing visually.) We have them hung from a ring so that the magnitude of each chain is reflected in relation to the bead chain that represented root one. (We felt this provided a visual to the ordering aspect that is required by the Grade 11 Pre-Calculus syllabus.)

Images from work in process:

Art Project lesson ideas

12 Golden E-Chains Spiral, Philippe Leblanc

https://gallery.bridgesmathart.org/exhibitions/bridges-2025-exhibition-of-mathematical-art/philippe-leblanc

Mathematical Concepts in the artwork itself


The artist says he is inspired by irrational numbers: e, phi, root 2, and root 3 are specifically mentioned

e → decimal expansion

links to irrationality


Catenary → equation of shape:



Phi is also present

The box is in the golden ratio


Each catenary (each strand of the 12 chains) is made to the appropriate length so that they are framed in a golden ratio → so this models the self-similarity of e as well as the recursive nature of the golden ratio (involving phi)

BC Curriculum Links

Square roots of perfect squares (Grades 7, 8)

Pythagorean theorem (Grade 8)


Exposure to irrational numbers pi, trig ratios (Grades 7, 10)
** Pre-calculus 11: This seems to be where we can make the most of it

Lesson arc (this might be one or multiple lessons in the “real” classroom)

Real numbers

Think- See-Wonder with the artwork that we made

(possible alternative to help visualization)

• 

https://www.weartxl.be/fr/2025/philippe-leblanc-2/#pll_switcher
This one (right) is just called Golden E-Chains

Activity for the classroom

We only have 5 -7 minutes for the interactive element.

Students Make Theodorus Spiral

https://mathworld.wolfram.com/TheodorusSpiral.html 

With the help of this video

https://www.youtube.com/watch?v=LxVzhxJMqVE

In our “real” classroom, we would pair the activity with this worksheet (modified).

https://ferrington.weebly.com/uploads/3/5/8/5/3585877/chapter_8_quilt_project.pdf 

Later in the lesson or a later lesson

This is where I see the bead construction. In the classroom, this could be the expansion activity – provided it’s not too much busy work – but for this artwork assignment, I would like to give them one that is pre-made to an appropriate length, and they can put the pieces together to make an analogous artwork of square roots.

 

My Math Teachers

 

My Dad and I, a long time ago.

My Math Teachers

I will tell you about my favourite moments learning math because I had different special moments with several people around me, and I identified that what made those moments special were additional aspects beyond the person who taught me.

My friend Jaime, during my first year of engineering, taught me matrix transformations. We have been in the same class, but I failed the first exam. For the final exam, I sat with Jaime to study and try to review the problems in preparation for the exam. After reviewing the concept of transforming the matrix, the rest of the subjects made sense, and I felt like a revelation. The essence of the revelation was to draw the matrices as bidimensional arrays instead of linear notation separated by commas, which the teacher used to use. It was a simple difference, but it became a game-changer for my matrix algebra from that moment.

My Friend Jaime and I

I didn’t have happy moments with my math teachers during high school. I think my teachers had an ok performance, and my parents overshaded their performance in some way. My parents were teachers and intellectuals. My dad taught Literature at the university, and my mom did a lot of Chemistry research on the sugar cane plantation efficacy. Both my parents loved math and science, and my brother and I were always surrounded by questions about Math concepts, sometimes after dinner, or while we were doing our homework. If we had doubts about our homework, my parents were always there to solve it with interest and curiosity.

My memory of my least favourite teachers lies with those who dedicated more than 20 minutes on the chalkboard to write a long integral calculus procedure. I think I had difficulties following along, and I remember panicking when I was trying to understand, and it was time to erase the board to keep writing.

As a Math Teacher, I want to build bridges with the students, to identify their different ways of learning, and accordingly prepare my lessons. I will take advantage of the technology available to create visual scaffolds that support the learning of abstract concepts.

 

Explicit, Hidden and Null Curricula

 

One surprising idea from the reading is the notion that schools build compliant behaviour into students through the use of a reward system. When you consider the vast amount of time kids spend in school, this reveals the significant power of prolonged, manipulative behaviour. While this power can be used to benefit students by encouraging healthy habits, it becomes dangerous when it's used to instill stigmas or prejudices.

The Importance of Teacher Education 

Another crucial concept is being aware of what's truly being taught in school. We often only think about the explicit curriculum—the subjects and units formally written in the program. However, students also absorb values and traditions during their schooling hours.

Licensed by Google

This realization highlights the immense importance of educating teachers. Society has a responsibility to invest in teacher training to ensure a healthy, supportive environment where students can thrive.

The Power of the Null Curriculum 

Just as powerful as the explicit curriculum is the null curriculum. This refers to the concepts that are intentionally or unintentionally ignored or left out of the formal curriculum. The decision not to teach a particular subject or topic can be as impactful as teaching it. In some cases, deliberately ignoring a concept is akin to teaching apathy or bias about that subject. It's a phenomenon of "opposite poles"—what is left out can be just as significant as what is included.

One of my Students of VFX at Delta Film Academy 2024

Dogma vs Freedom, Instrumental Undersstanding vs Relational Understanding

 Dogma vs. Freedom

Relational Understanding and Instrumental Understanding

My daughter doing math in the kitchen

This could sound very strong regarding the learning process, but after reading the article, I correlated Instrumental understanding with the prison of living surrounded by dogmatic principles. The only advantages of this kind of understanding would be the faith that the pupils would have in the system. On the other hand, Relational understanding is related to a state of internal reflection about the way we learn new things and maintaining an open approach to the unknown.

These different approaches have their own benefits and disadvantages, as the reading explains.  Instrumental Understanding of Mathematics can provide the student with faux confidence based on the practical procedures learnt, but once the conditions or the framework rules change, the pupil would feel lost and disoriented, as in the town map example. Relational understanding would demand that the student leave the comfort zone of certainty and explore different ways to understand. This will take more time and cross feelings, but from a life perspective, the skills acquired will be transferable to other subjects and real-life problems.

As a teacher, I want to be able to be part of the autonomy building of the students, and I think that requires getting out of my comfort zone and becoming creative and efficient in my lesson planning in order to maintain a relational understanding through the Math learning journey.