Reflecting Upon My Reflections

In the first couple weeks of the course, we talked about learning mathematics and what we can do as teachers to enhance our students' understanding of mathematical concepts. The articles read helped me to realize the best approaches to use and to look at my own teaching philosophy critically. We were also periodically given math puzzles to work on and post our own solutions to. This was one of my favourite aspects of the course, because I feel like this is a great way to gather interest in mathematics, which has traditionally been seen as a 'boring' subject, and I will definitely consider implementing this in my own classroom. In addition, seeing all the different types of approaches different people use helps to highlight that there are many ways to solve a problem, not just one. 

The first major project was the math art project. After having completed it, I came out of it with a greater appreciation for math and its intrinsic artistic beauty. It was also a great way to work collaboratively with other teacher candidates while working towards one common goal. It reminded my of myself in Grade 9 Math Honours, where we were tasked with creating a picture on our graphing calculators using common functions. If there was a project aspect of a math course I am teaching in the future, I would like to do something similar.

The next theme was looking into the curriculum and had us do microteaching lessons. Although I found the amount of time devoted to looking at the curriculum to be lacking, I did like the microteaching lessons. The one thing that stood out to me was the timing associated with teaching, as I believe it is one of the most important things to consider when planning a lesson.

We as a class also participated in two math fairs: one we visited, and the other visited us. I thought that these were interesting as it gave me a lens I could look through and see how different students were able to mesh their 'creative' side with their 'mathematical' side. Also, the passion each student had while presenting was great to see!

In the end, I am happy with the course and the activities I participated in. However, since I have two teachables, I was only able to take one math curriculum course, and I wish I could take more! 

Off the Grid

Overall, I agreed with the theme of the paper. Why are so many places separated by straight lines? Although this may seem to be not as important at first, if you look deeper into it the nature of this is problematic. Consider Africa for example, where straight lines or other unnatural boundaries were created by Europeans during colonization. This has ultimately lead to violent ethnic conflict in certain regions, as feuds and relationships between different peoples was not taken into consideration. Or on a lighter note, the Canada-U.S. border is infamously a straight line for the most part along the 49th parallel. This has lead to weird citizenship situations, notably Point Roberts, which is connected to Vancouver Metro, but is part of the United States since it technically is below the 49th parallel. Since the citizens of this city are American, they have to attend school in Washington state, which takes at least 40 minutes!

A point in the article that struck my interest was when the author was talking about the Principle of Relativity and related it to grids in the style of Riemannian geometry. Arguing that there is no right geometry to use, and as such we should broaden our horizons when it comes to implementing different geometries in grids was an interesting statement. They say that "variety is the spice of life", and I feel like giving grids the ability to be designed in mathematical patterns would be fascinating.

When talking about incorporating Indigenous content into classes, I always wonder "how", especially in a mathematics course. However, this article sheds light Indigenous traditions and their relationship with geometry. Being able to use this in a class could help both Indigenous and non-Indigenous students. Indigenous students would be able to learn in a way that they are accustomed to, whereas non-Indigenous students would be able to explore new ideas of how to appreciate and solve problems in math, and being able to solve problems in a multitude of ways helps the student understand the content better.

Reflection: Math Fair

I thought that this math fair was quite interesting. I thought that representing mathematics in a way different than that taught traditionally was a breath of fresh air, especially since many people claim that math is boring. The students also seemed to be enthusiastic about presenting their hard work, as it's apparent they worked very hard on their project, and were eager to display their work they invested so much time and effort into. One thing that caught my eye was the project about changing algebraic questions into work problems. When I think of common student difficulties, what comes to my mind is word problems. But these students found it useful to first translate the question into a word problem, and then solve the question, which was an interesting insight into how different students may approach the same problem.

Drinking Party Puzzle

Starting with 2 bottles, a rat can choose to drink from either the first or the second bottle. If it drinks from the first bottle and dies, then that one is poisoned. If it drinks from the first bottle and lives, then the other one is poisoned.

Anything above 2 bottles will require additional rats. The greatest amount of bottles tested with 2 rats is 4 bottles. Let

• rat #1 tests bottles #1,2
• rat #2 tests bottles #1,3

then

• bottle #1 is poisoned if rats #1,2 die
• bottle #2 is poisoned if rat #1 dies
• bottle #3 is poisoned if rat #2 dies
• bottle #4 is poisoned if NO rats die

For three rats, I found the maximum number of bottles tested to be 8. Let

• rat #1 tests bottles #1,2,3,4
• rat #2 tests bottles #1,3,5,7 
• rat #3 tests bottles #1,2,5,6

so that all rats share one bottle (#1), all rats don't drink from one bottle (#8), all rats drink from one bottle which no other rat has drank from (#4,6,7), and two rats share the same bottle (#2,3,5). This covers all the bottles. Then

• bottle #1 is poisoned if rats #1,2,3 die
• bottle #2 is poisoned if only rats #1,3 die
• bottle #3 is poisoned if only rats #1,2 die
• bottle #4 is poisoned if only rat #1 dies
• bottle #5 is poisoned if only rats #2,3 die
• bottle #6 is poisoned if only rat #3 dies
• bottle #7 is poisoned if only rat #2 dies
• bottle #8 is poisoned if NO rats die

A pattern is starting to emerge. I notice that rat #1 just needs to consume half of the bottles, #1-n/2. Rat #2 needs to drink from every other bottle. Rat #3 needs to drink from every 2 bottles. With 1 rat, it can test 2^1 = 2 bottles. With 2 rats, they can test 2^2 = 4 bottles. With 3 rats, they can test 2^3 = 8 bottles. Therefore, I claim that with 10 rats, they can test 2^10 = 1024 bottles, sufficient for the 1000 bottles of the problem. Continuing from above, rat #4 would drink every 3 bottles, rat #5 from every 4 bottles, etc.

*This works fine for 10 rats, but breaks down for 9 rats.

After thinking more about this, I can show that the number of bottles follows 2^n, where n is the number of rats. Let R = {the set of rats}. Then the power set of R, P(R), lists each bottle and which rats drank out of them. For example, for n = 3, we have R = {1,2,3}, and so

P(R) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

From this, we see that each subset matches up exactly with the hand-written solution above. We can then use a theorem that states: "a set containing n distinct objects has 2^n subsets". Therefore, the power set of a set containing n distinct objects has 2^n elements. And so for 10 rats, n = 10, and the set of bottles has 2^n = 2^10 = 1024 elements, which means that 1024 bottles can be tested.