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.

Unit Plan

Unit Plan

*updated December 1st

Thinking About Math Textbooks

When I read through the article about first and second person pronouns, I was surprised to read that first person pronouns were not in either of the textbooks that were used for the purposes of this article. When I think back to my personal experiences with mathematics textbooks, most if not all examples of person pronouns involved the word 'we', which would fall under first person, NOT second pronoun. Perhaps it is my recency bias playing a role, and undergraduate math textbooks use these first person pronouns, whereas secondary textbooks still exclusively use second person pronouns. If so, then I must ask, why? The article states that second person pronouns helps to connect the speaker to the reader directly; does this only matter for adolescents, and not for young adults?

I understand that today there is a push to not use textbooks, and some teachers are slowly becoming less dependent on them. I've also talked to some teachers about this topic, and it's apparent to me that it is quite the controversial one. However, I still think that textbooks serve a purpose and shouldn't be rid of entirely. Of course, textbooks are expensive and a new version with minimal changes seems to come out every few years. But they also contain a wealth of information in an organized structure, with many additional resources embedded in them. I will say, I think teachers should opt to make their own examples and not 'teach to the textbook', as it allows for them to adapt to their unique class and their needs, compared to the general examples and rigid format that a textbook supplies.

The Scales Problem

We have four different weights to choose from. These will be labelled as a, b, c, d, with each letter be written in ascending order of the weight. Since the vendor sells herbs ranging in weight from 1 g to 40 g, the smallest weight must be 1 g (i.e. a = 1 g). In addition, the sum of the weights must add up to 40 g, such that

a + b + c + d = 40 g

Now, in order to measure 2 g, we could say that the second lightest weight is 2 g. However, another way of doing this is by letting a mass of 2 g be achieved via a weight on both pans. Therefore, a subtraction is needed, and so we have b - a = 2 g. Since a = 1 g, it follows that b = 3 g. With a and b solved for now, I can measure weights of 1 g, 2 g, 3 g, and 4 g. But how can I measure 5 g? Well, the third weight, c, will allow me to. If 5 g is the smallest weight I have yet to obtain, then I must subtract both a and b from c in order to achieve it. So c - a - b = 5, or c = 5 + a + b = 5 + 1 + 3 = 9. This just leaves d now, which can be solved for from the above equation: d = 40 - 1 - 3 - 9 = 27. Thus, the value of the four weights are 1 g, 3 g, 9 g, and 27 g. Looking at these values, these numbers didn't come from nowhere, as it is quite apparent that they follow a distinct pattern: 3^(n-1), where n = weight number. A quick check shows that these four weights account for all the different weights from 1 g to 40 g.

I don't believe there are several correct solutions, as no other way will allow you to account for each weight distinctly, but I could be wrong as I did not delve too deep into proving it.

This puzzle could be extended by looking at different maximum weights (1 g, 4 g, 13 g, 121 g, etc.) and noticing the pattern for which weights you need, and then trying to deduce the formula that tells you the specific weights you need. This could tie in quite nicely with a unit on sequences and series.