Nov 4 - Euclid and Beauty

 Euclid and Euclidean geometry are still studied to this day because his contributions laid the foundational framework of mathematics. His postulates are set the ground work for more logical deductions to come later on. Euclid starts his work with basic definitions and every concept to come next is built on the previous one, making it rich in simplicity. In my opinion, any work rich in logical reasoning and simplicity will always endure through the centuries.

Euclidean geometry is not only simple, but also rich in beauty. I remember introducing my student to Euclidean geometry and her enthusiasm to come to class everyday was tenfold. My student appreciated the universality of Euclidean postulates and found it very inspiring that these concepts were intuitive rather than complex. When I myself was introduced to Euclid in grade 8, I thought his proofs were logically harmonious, almost like a poem. The use of logicals reasoning to prove geometric concepts is the reason why his work is considered beautiful. One cannot forget that it was this beauty that inspired great minds like Newton and Descartes among many others. Euclidean geometry also embodies visual symmetry and structure using circles, triangles, and polygons.

Euclid's Elements and the appreciation of its beauty come from its ability to combine logical rigor with simple, universal truths that resonate both intellectually and aesthetically. It has provided a basis for how to think, argue, and deduce that has shaped mathematical thought for centuries.

October 16 - The Dishes Puzzle

 My solution (and the process) to the dishes puzzle without using modern algebra is this - when I first read this problem, I want to define x as the number of guests and build an equation to solve and I am so used to thinking in algebra that I had to pause and think of what mathematical concepts and ideas I am employing when I use algebra and I realized it's all about the LCM. So with smart guessing and checking, I began thinking of a number that would be divisible by 2, 3, and 4 since we're given that every 2, 3, and 4 guests share dishes of rice, broth, and meat respectively. The solution beautifully turns out to be 60. In the following image, I use algebra to verify my answer.

By relying on my past experience as a math educator, I can confidently say that offering rich histories and background to a problem does in fact make a difference to our students. Story telling and adding context is one of those things that lets people connect real life with mathematics and we should absolutely continue to incorporate history from all the cultures into math. Moreover, doing so will make the ELLs in the classroom feel included if teachers incorporate history of mathematics from their culture into the classroom. For the same reason, puzzle story and imagery matter as well. They create excitement in a topic that would otherwise be boring and frustrating. As well, keeping the First Peoples' Principles of Learning in mind, embedding histories, story-telling, imagery and other context into learning is important.

Assignment #1 Write up + Reflection

 

For this assignment, my group (JJ, Nanxi and myself) presented the problem 1.2.4 that deals with a 3x4 rectangle and its diagonal. We presented the modern solution, the ancient Egyptian solution and extended it to the Binomial Theorem.

For the modern solution, we decided to solve the problem using Pythagorean theorem as today it is arguably the easiest way to solve a problem like this.  I showed ancient Egyptian solution on the slide as a translation of the original text and was explained using modern algebra. I offered a few limitations of the Ancient Egyptian 'formula' as well. Nanxi also offered a beautiful geometric solution what combined 4 of the congruent rectangles to the original one and using a bit of algebra and the area of a rectangle, we were able to arrive at the correct length of the diagonal. Then JJ extended this to apply to the Binomial theorem and explained some benefits of visualizing binomial expansions geometrically.

Here are the slides that we used.

My Reflection:

I thought I did well overall. A couple of areas to improve would be use my media smartly. I found myself turning my back to the audience in order to look at the slides and point things out. I should perhaps find a way to set up my computer in a way that its easier for me to look at the slides while giving attention to the audience as well. Moreover, due to momentary nervousness, I wrote a wrong math statement on the whiteboard but I think I was able to correct it and managed to not confuse the audience. This goes with my EDCP 342A reflection as well that I believe for my initial lessons that I deliver in the classroom(just until I build enough confidence) I would like to explore writing a script for myself so that I have something to fall back on in moments of nervousness.