Final Reflection

  Reflecting on this course, I am struck by how profoundly it has shaped my perspectives as a learner and future educator. Initially, I viewed mathematics as a collection of formulas and procedures. This course opened my eyes to the rich historical narratives underpinning the discipline, fostering growth beyond content knowledge.

Exploring the cultural contexts of mathematical ideas, from Babylonian tablets to India’s Golden Age and Chinese ingenuity, broadened my appreciation of mathematics as a collaborative human endeavor. These stories inspired me to integrate historical perspectives into teaching, making mathematics more relatable for students.

Delving into ancient texts and mathematical artifacts challenged me to adopt new ways of thinking. Grappling with unfamiliar notations cultivated perseverance and adaptability while deepening my respect for past mathematicians’ ingenuity.

On a personal level, my confidence grew as I connected historical details to modern pedagogy. I now feel equipped to design lessons incorporating historical narratives, offering students a holistic understanding of mathematics.

This course reinforced the importance of lifelong learning. Mathematics’ history continues to evolve, and I am inspired to remain curious and open-minded as I advance in my journey as an educator and I am very grateful for this experience.

Assignment 3 Reflection

 Working on the project about Maria Gaetana Agnesi was a rewarding experience. Nanxi did an excellent job creating the collage, focusing on the Witch of Agnesi curve. It beautifully showcased the mathematical elegance of the curve and its connection to Maria’s work. Seeing her effort in visually representing such a significant concept added depth to our overall project.

For my part, I focused on preparing the PowerPoint presentation, which gave me the chance to explore Maria’s life and contributions more deeply. I highlighted her groundbreaking work on the Witch of Agnesi and her textbook, Instituzioni analitiche, as well as her role as one of the first women to gain widespread recognition in mathematics. It was fascinating to share how she broke barriers during a time when opportunities for women in STEM were rare.

One aspect that stood out to me was the origin of the curve’s name, which came from a mistranslation of the Italian word versiera. It’s a quirky detail, but it reminded me how even small historical quirks can shape how we remember important figures.

Overall, this project taught me a lot about Maria’s achievements and her legacy in both mathematics and education. Collaborating with my group, especially seeing the creativity in the collage and combining it with the presentation, made the experience even more meaningful. I hope our work helped bring her story to life for the class.

Assignment 3 - The Witch of Agnesi Curve

 Here is the link to our (my and Nanxi's) slides for the presentation 

https://docs.google.com/presentation/d/1XM-YV5SSq2B7yWA_Hjv1Rjf1e3dcNrYoBFZsCR8W1P4/edit?usp=sharing

My Plan for Assignment 3 (with Nanxi)

 For this assignment, I had origianly planned to explore the work of Bhaskar II - specifically his book called Lilavati. This book contains a number of interesting, poetic problems, which give a flavour of ancient Indian school problems. Lilavati  is the first volume  of his main work Siddhanta Shiromani (”Crown of treatises”) alongside Bijaganita, Grahaganita and Goladhyaya. It is the most celebrated work of the traditionsl of mathematics in India. 

One hurdle that I encountered with this was that this work is in Sanskrit and though there are translations available, they are extremely hard to find unless I buy a book that contains them. So, instead I decided to partner up with Nanxi and explore the work of the Italian female Mathematician Maria Gaetana Agnesi. She was the first woman to publish a mathematics textbook and is best known for her work on the "Witch of Agnesi," a curve that holds significant value in the history of mathematics.

We have chosen to create a collage on the Witch of Agnesi.

Assignment 2 Reflections

 For the second assignment of this course, I chose to research history of the origins of Trigonometry. In my presentation, I highlighted the Greek astronomer Hipparchus of Rhodes who tabulated the ratio of the chords of a curcle to the radius of 3438 associating it with the central angle. I was not able to provide a reason as to why he worked with the radius of 3438 at the time of my presentation so I will write th reason in my blog - it was because the circumference of this circle (21600) is the same as 360x60 - this means that every degree of this circle would represent one minute of the hour.

Next, I highlighted the Indians who realized the efficiency of working with half chords and double the central angle. Indians started tabulating half chords instead. There was also a little pice about where we get the name of the sine ratio from, which was very interesting for me. Being able to connect it to Sanskrit and Hindi, I'm able to reaffirm my understanding of sine. Lastly, I talked about Al-Battani, who introduced tangent and cotangent in a right triangle. Researching about him was eye opening for me as I realized that my education in India was full of inplicit bias against the middle eastern and Islamic contributions to math. I was very happy to finally learn greater details about Islamic Golden age not just in my presentation, but from my peers as well.

Nov 13 - Dancing Proofs

 Reflecting on "Dancing Euclidean Proofs," two aspects stood out and made me pause. First, I was struck by how the authors highlighted a shift in learning perspective—from passive observation of a proof on paper to active, physical participation in creating it through dance. This dynamic approach to proofs made me rethink how much learning geometry could benefit from physical engagement, helping students visualize and internalize mathematical concepts. It was eye-opening to see movement transforming abstract geometry into something deeply tangible.

Second, the authors' integration of natural elements, like sand and shells on a beach, showed how environments can become active participants in learning. This reminded me of the many conversations I've had in my classes about integrating land and nature into our learning. I can see how this kind of activity can make geometry—and even math history—more accessible to high school students. By embodying ancient methods, students might better appreciate the historical and cultural contexts of mathematical discoveries.

In a high school setting, this approach could engage students who struggle with traditional methods, allowing them to learn by “doing” instead of just memorizing steps. However, there could be obstacles. Space limitations or students' self-consciousness about performing might restrict the activity’s effectiveness. During my practicum, i have noticed the unwillingness of students to raise their hand and participate. Still, overcoming these constraints, perhaps by using small groups or allowing students to express ideas through minimal movement, could offer a powerful way to connect with math’s logic, creativity, and historical evolution.

Nov 6 - Was Pythagoras Chinese?

 I have always felt really strongly that acknowledging non-European sources of anything, not just mathematics, is highly important. In today's time, we are aware of Europe's past of attempting to erase cultural and historical identities of a lot of countries, I feel we should make an active effort to acknowledge where certain ideas and concepts originate from. This is also making me think of the ELL students in our classrooms. Surely, they feel more accepted, welcome and important if the history of their native country was brought up in their new classroom in Canada. Moreover, we would be encouraging our students to think critically when we take a moment to mention non-European sources of mathematics. When we pose questions like "why is Pythagorean theorem called the Pythagorean Theorem?" we encourage them to truly think about the history and discuss with their peers what if the names of theorems they learn are appropriate. 

In regards to the naming of the Pythagorean Theorem, I do believe that knowing all that we do now, thanks to researchers and historians, we willfully participate in ignoring the contributions of the people that deserve credit for it. I understand that we, as a society, have agreed on this name when we refer to the right triangle theorem and changing it now would be a hassle (much like changing the whole curriculum). It would be meaningful though, when we refer to this theorem, to sandwich it like so: "The right triangle Theorem - the Pythagorean Theorem - The right triangle theorem." This will get the students to make connections between the two names without losing the meaning of it. This is a strategy I learned in LLED 360 for introducing new words to ELL students. I determined to try this out in my classroom!

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.

October 2 - Market Scales Puzzle

 

I came up with the above solution to the market scales puzzle introduced in class. I thought that the weight I use would have to add up to the max weight being measured, and the first 2-3 weights have to be close together so as to account for the consecutive numbers. For the first, I was surprised to find the answer to be powers of 3 as I was working with powers of 2 for the longest time but accidentally ended up solving part b.

Sept 25 - My further thoughts on Word Problems

 The topic of how to incorporate word problems is a hard one for me. I have personally always enjoyed word problems and I was able to see the beauty and the imagination in them. Perhaps I've also enjoyed them because I was good at them and I was able to show off my skills. But I realize that not everyone shares my sentiments and I do recognize that word problems are often not related to the real world in any way. Acknowledging this, I would love for my students to see the imagination and perhaps partake in creating their own crazy word problem in my future classroom. Depending on the demographic of my classroom, and students opinions, word problems might even be optional in my classroom so that the students who enjoy them can do them and the students who don't, can skip them and focus on something else in the unit that resonated with their learning. 

Additionally, I was impressed by the anecdote that Dr. Gerofsky toldin class about her colleague(?) who brought the students outside their classroom to collect data points and then had them work on a relevant word problem. Doing this would change students' impression of word problems and they would probably be more inclined to give it a chance.

Sept 23 - Surveying in Egypt

 The article 'Surveying in Egypt' talks about the two ways in which surveys took place in Ancient Egypt. We talked during our class about some of the ways Ancient Egyptians measured their fields to report losses and taxes, which was confirmed in the article. It was surprising to me that the methods ancient Egyptian employed to measure distances were so sophisticated that these methods are still used in today's time. I had not comprehended before taking this course how mathematically advanced some civilizations were.

While reading this article, I wondered how did ancient Egyptians keep track of the cubit system to measure in a wide population? Surely, different people must have had different lengths of their forearms and fingers. Secondly, the article mentions that less is understood about how ancient Egyptians understood angles and which tools they used to measure angles and yet, one of the ways they created a right angle was with the help of two equilateral triangles. So, I wonder how they managed to get this far without the tools needed. Perhaps they did have the tools and we haven't been able to uncover them yet.

Sept 18 - Babylonian Word Problems

 I really enjoyed this week's reading as it explores the topic of word problems, something that, in my experience, I have never struggled with and most of my students have. I noticed this pretty early on in my tutoring career that students find word problems difficult because they have trouble representing the scenario using math. I like to think about it as translating from one language (often English) to another (Math). To help my students overcome this barrier, I explored storytelling - I would not show them the written word problem, but rather make up characters and stories to present the scenario. Sometimes it helped, but often students got frustrated because the story was not realistic.

This reading, in a way, comforted me in the sense that I was not doing wrong as the tutor. I explored creative avenues to make word problems slightly easier for my kids and it is just that word problems historically have not been realistic. It is a funny thing to comprehend but it makes sense because word problems are imaginative. You make up scenarios that help you visualize, for example, a boy buying a 100 cantaloupes. For the Babylonians, they had no means to measure a grain pile 18-24 (referring to word problem mentioned on page 6) but they were willing to imagine. Perhaps word problems need not be realistic today, but we can hope for them to be so in the future.

Babaylonian Multiplication Table for 45

 

This is my attempt at making a Babylonian style times table for 45. I really enjoyed the process - helped me get a deeper understanding of the "decimals" in sexagesimal system.

Sept 16 - How we measure time

Before our discussions in class about Babylonian math and the sexagesimal system, I had not really put much thought into how I perceive time. Of course, when I was younger and just beginning to learn to read the clock, I thought it would be convenient if a quarter of the hour was 25 minutes instead of 15 minutes. When I asked my teacher as why it was different, they told me this was how time was "set up" long long ago and I accepted it as a little blip along the way. I am only now realizing the significance of 60. 

I found it really interesting that every eight years a minute has 61 seconds in order for the atomic time to stay consistent with the astronomical sign. There is also an inconsistency between the two articles as the MacTutor one mentions that no civilization came up with base 12 for their counting system but the other article mentions that Egyptians used the duodecimal system as it was easier to count up to 12 on the fingers joints of each hand.

September 10 - The Crest of the Peacock

 I really enjoyed reading this chapter, especially because of the focus on Indian and Arab contributions to history of mathematics. Having completed my high school education in India, I already knew a lot about the content covered in this chapter, so I was surprised that I wasn't surprised by the European "cover up" of the rest of the world's rich history and limitless knowledge. But I am happy to find out through this reading that the non-European history of mathematics is now gaining the recognition that it deserves. 

I was, however, surprised to find out that I didn't know anything at all about the contributions of Spain towards the advancement of mathematics. There wasn't much about Spain in this chapter but I hope to learn more about it in this course. Another surprising fact that I encountered in this reading was the Mayans were able to make such advancements in astronomy with no glass or optical devices at all. To get a perfect estimate of synodic period Venus without such equipment is beyond impressive. 

Sept 10 - Why base 60?

 This is in response to the in class activity on Babylonian math done on Monday, September 8. My classmates brought up some really interesting reasons as to why the Babylonians chose the sexagesimal system to count. One of my reasons that I can think of related to what was already brought up in class is that they counted on one hand - a fist representing "zero" and the five fingers representing the numbers 1-5. Another reason I can think of is that it has something to do with the position of the sun throughout the day. I'm not sure exactly what - but I suppose I will find out after researching. Today, we see 60 being used in a lot of concepts - an hour has 60 minutes, a minute has 60 seconds, 60, 120, 180 and 360 degrees are some of the special angles in trigonometry. 

Upon researching why Babylonians used base 60, I realize that I had been overthinking about an obvious thing. 60 is a composite numbers that has factors that are themselves composite. It is divisible by 2, 3 and 5, thus making it easy to work with fractions. This is such a sophisticated way to do math!

September 6 - Why teach math history?

 

Before reading this article, I believed that teaching history of mathematics in a classroom could be highly beneficial to the students. I have experienced first hand higher engagement and enthusiasm to learn the material. When I was doing my B.Sc. in Applied Mathematics, my teachers and the head of the math department worked alongside to organize Math Movie Fridays for students and they were some of best days I've had at my university. We would watch a movie based on a mathematician's life and discuss their work and make connections of their work with other work that advanced the field.

This is not to say that as an educator I did not have questions on how I can incorporate history of math into my own teaching. I realized that a high school classroom is far different for a college lecture room and teachers do not usually have as much resources. The objections posed in the article on why history of mathematics should be incorporated seemed very valid until I read further along and realized that teaching history can be accomplished as easily as making posters, watching a youtube video or simply having a discussion in class. Section 7.4 of the article especially struck me because some of the examples shown in this article are in my teaching practice but I never stopped and realized that I was teaching history. I believe these practices are central to gaining a deeper (relational) understanding of mathematics.

This article has solidified my belief that teaching history of mathematics is very essential. It helps students understand the "why" of math, which satisfies their curiosity and encourages them to keep going.

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