Dancing Euclidean Proofs

I will be perfectly candid here - dancing is not my thing nor interest. I was not especially captivated by the Dancing Euclidean Proofs Susan showed us. The article however, was interesting to read. I think my indifference towards the dance fueled the interest with which I read the article; I did not expect there to have been so much thought put into the routine. This is not one of the things that I will write about in regards to what stood out to me but I think it worth mentioning and that is when the authors describe using their second arm in the first proposition. I was a member of the audience watching the performance; I did not consider the work, attention to detail or thought that went into it. Jason Ellis taught our EDST 401 class about the apprenticeship of observation and it seems I did not fully ingest that lesson.

The two things that made me stop and think both came at the end of the article. The first was when the authors mention how there is as much math within our bodies as there is in nature and this made me stop for selfish reasons. Recently I have been paying close attention to the UBC fountain which was such a spectacular allocation of funds on my walks to and from different classes. There is something extremely captivating about the wave patterns; it is like they are moving yet still. I was thinking that if the water comes out of the spout in a consistent manner, will the waves always be the same? Consistent as in the amount of water, the angle at which it comes out, the height of the water, the way it drops back down to the basin, the weather (wind, etc.) and other things - basically if the fountain was always the exact same. I don't know, I think it would. It's like the little waves (they're almost like pyramids of water sitting atop the surface of the water) are just constantly replacing each other. One of these days I'm going to bump into somebody or something while looking at the fountain. That being said, the author's are right when they say there is as much math in/on our bodies as there is in nature. The patterns and symmetry in which my arm hair grows is a testament to that.

The second thing is when the authors were discussing the land and their dance within it. When describing the circumstances they faced with the environment, interestingly enough, they said "...our particular geographical setting became an active limiting agent in our representation." I think, their geographical setting liberated them. They used the term limiting agent to describe the consistency of the sand which "ruined" their initial plans however I argue that whatever the land dictates you do is truly liberating and what is supposed to occur on that land. If we want to pay homage to the land and embrace it, you do so by taking in stride what the land gives you - which the authors/dancers did. 

It could be helpful but I think you are more than likely to encounter students like me that don't want to take part in activities too outside their comfort zone. Although there may be potential benefits I believe in keeping students comfortable within reason. You can't cater to all their wants or else you might get to a point where students do no math at all but you also don't want to push them to a place where they aren't comfortable at all. Experiential learning itself is always going to be enlightening and each type will offer its own benefits. Experiential learning does not always have to be around body movement. Kids can learn about measurements through a baking activity and this might be way more impactful than some exercises on a worksheet. Food for thought, literally.

Individual Presentation Reflection

Pure mathematics has always been the branch of math that I have found most intriguing. I don't think I discovered the name for it until my proofs class in my first year of undergraduate studies. It's like, math from scratch. You have nothing, now create. You have this concept, now prove it. Although I'm awful at it, I find that it is the only thing that I have no problems being stuck at. Which is interesting now that I think about it. 

I feel like I rediscovered this while researching on the history of geometric constructions. It may be the purest of the purest forms of mathematics. When I realized that these mathematicians were creating theorems, proofs and conjectures based solely on concepts and not facts, I was amazed. They were creating factual experiments solely through proofs and conjectures. Euclid does not define a right angle as an angle that is ninety-degrees. He defines it as the two angles created by one line intersecting another line being equal - or "right". I had written a question for potential future students of mine in how I would incorporate this into my classroom where I asked "draw me a square without using a ruler or compass and prove that it is a square". I sat on this question for a long time thinking about how I would answer it and I still have not come up with an answer. But there's beauty in that, right? Why do I need to have the answer to this question - it's meant as an exploratory introduction to geometric construction and it would be informative regarding student thinking. 

Lui Hui and Zu Chongzhi

I think depending on the topic or unit, it can greatly benefit our students to acknowledge and discuss sources of mathematics that aren't Eurocentric. I say this for two reasons. The first, it offers mathematical diversity. We are often plagued by tunnel vision in mathematics and once we find one way to do something or one way on how something is done, it is difficult to step out of that thinking and allow for more possibilities. Exploratory or investigative mathematics does just the opposite; it allows individuals to ask and discover mathematical reasoning. When one comes to these realizations themselves without explicit aid, they have an ability to comprehend other possible solutions. Other routes to solutions are what I mean by mathematical diversity. Teaching concepts like the relationship between the sides of a right triangle, pi and base-sixty number systems gives students the ability to see things differently. Another reason it is good is to combat the obvious bias of Eurocentric views. It is interesting, in the article "Was Pythagoras Chinese- Revisiting an Old Debate" by Ross Gustafson my "spidey-senses" tell me that if history was reversed and the Greeks wrote the Jiu Zhang Suahshu and the Chinese wrote The Elements, Gustafson would have praised the time (earlier discovery) instead of the rigorous proofs. We live in an Eurocentric world after all, and those are just my two cents. I think we can find beauty in both discoveries. 

In regards to the naming, this requires a long discussion. Unfortunately, we live in a post-colonial world. This affects many factors of our day to day lives that should not go overlooked. For several years, world maps used exaggerated the size of North America and inaccurately depicted the sizes of South America and Africa. Cultures that do not align with Western traditions are spoke of with negative connotations. Something as simple as eating with your hands is considered barbaric. Along these lines we can also find naming conventions. Naming is often a reframing of history so that it starts somewhere in Europe. So on this point, I don't like it at all. We should know the actual earliest discovery of math concepts and celebrate all who stumbled upon something related (so long as it was not stolen). 

Euclid

Euclid's Elements is widely regarded as one of the most important works in mathematics. It has been in use for over 2000 years and remains important until this very day. Although it is not unknown that the content found in the book are not uniquely Euclid's, Elements was groundbreaking for its organization, clarity and exposition. How Euclid proved his theorem's and proofs have become a standard. Aside from his remarkable communication in mathematics, I think the topic itself is why Elements has stood the test of time. The concepts touch on both basic and advanced, and geometry is ever present in our lives and in education/academia. His proofs can be learned as foundational understanding for kids in the public school system. Elements can also be a core part of those pursuing higher education in mathematics - in fact, one might argue that it is imperative to learn about Euclid's work for anyone pursuing such a field of study. 

If there is beauty in Euclid's postulates is subjective - it depends on who you ask. I think I can find beauty in the simplicity of it and certainly, I can appreciate Euclid's thought process in using these simple (but proved rigorously) facts to prove much more complicated theorems. I think where I have difficulty finding the beauty in it is its specificity to mathematics. A line segment isn't such a common aspect. 

What do they say, "beauty is in the eye of the beholder"? That's my best answer to the last question, "how can we define beauty if these are considered beautiful?" It's always about perspective, and one's opinion on what's beautiful cannot be held with higher importance or validity than someone else's opinion. We may bring in the aspect of popularity or the opinion of the masses. For example: if one hundred people think something is beautiful and one person does not, then maybe we have an argument but what is important to note is that it would still be something to be argued - it is not a fact. 

Something I find beautiful is a small aspect of relativity. Stephen Hawking in his book "The Universe in a Nutshell" speaks on Einstein's theory of relativity. He brings up an example of two planes starting at opposite ends of the earth. Both airplanes are set with extremely accurate clocks and they set off at the same time - one flies east and one flies west. Both planes arrive at their initial point having recorded slightly different times! The rotation and speed of the earth contributes to the plane flying east. I find that the beauty in this lies in the fact that it goes against basic human understanding. If you describe this scenario to someone and ask them what the clocks will show, they will most likely say the same time. I don't know, my sister gave me this book many years ago (I think I was in middle school) and I never got past that page with the airplane because I couldn't comprehend it. At my big age, I still don't think I do. 

Maybe that's where the beauty lies; beauty in the sometimes amazing yet incomprehensible nature of science and mathematics. 

The Dishes Puzzle

 

There were 60 guests. It is important to note that technically, there could have been 61 guests as well as this would also yield 65 dishes (59 guests would give 62 dishes and 62 guests would give 66 dishes). 

The only way I could think of to solve this question without using algebra was to draw out a table of values. I actually did this on Excel and I was thinking, well it's easy to figure out how many dishes based on guests but it's difficult to go the other way without using algebra. After 12 guests, the number of dishes always exceeds the number of guests so I was thinking you could also do trial and error by starting with (n-1) if n is the number of dishes 

It definitely is beneficial to offer these kinds of math puzzles highlighting a diversity of cultures however I can't help but think of the concept that if someone is excluded, then someone is included. It's impossible to grasp all cultures in one problem. As a teacher, if I included a problem like this then there will certainly be students from other backgrounds that may feel excluded. Obviously as I introduce more puzzles as the school year continues, I would ensure all cultures represented in my classroom are presented. It's about patience and building that trust. I do remember as students we would get giddy when some of our names appeared on test questions and the teacher made sure to rotate and use all of us. Even if we weren't on one test, we knew eventually we would be represented, and that was important. 

Personally, the reality of the question helps. I'm not too particular on imagery if the concepts are too farfetched - as we discussed in class, no, Susan will not be buying 83 watermelons. However, a good balance between imagery and reality, as well as representation always makes mathematics more fun.

Ancient Problems in Modern Ways - A Reflection

For our presentation, Caris, Brandon and I tackled the volume of a truncated pyramid. To solve it using modern mathematics, the truncated pyramid was presented as the summation of 3 different shaped, a cuboid, four corner pieces (which formed a pyramid when combined), and four triangular prisms (see picture). I was having immense difficulty condensing the formula after adding the different volumes together and I was not able to come up with the correct formula for the volume of a truncated pyramid when collecting like terms. I couldn't see what I was doing wrong and after discussing with my group mates, Brandon was initially able to solve it by using the ratio of b and a to the ratio of the height of the pyramid H to the height of the truncated pyramid h. The entire method requires solving it through a limit process by using integrals but I was really confused why I wasn't able to just add up the different volumes and combine like terms to get the correct formula. I was sitting there racking my brain over the difficulty and I couldn't help but admire the simplicity of the method of ancient Egyptians. As I discussed in class, all they did was average the areas between the base and the top and multiply it by h. Then, they realized they didn't do it properly so added a "median" area of ab and took the average of all 3 and multiplied that whole result by h. 

I recall a unit sometime either in elementary school or high school where we learned about estimation and I vividly remember sitting there and thinking this has got to be the dumbest unit. Why don't we just do the same amount of work (because educated estimates require some thinking) and actually solve for the correct value. Estimating cannot be overlooked and honestly, it might be the concept we use most in our every day lives out of everything we learned in public school mathematics. It needs to be mentioned that the Egyptians were not regular mathematicians - there is no proof for the logic behind their incredibly accurate mathematics despite the lack of modern mathematics knowledge. But their ability to lean on
"rough", educated estimates is a testament to its applicability considering its success. I went down a YouTube rabbit hole just yesterday watching a stonemason create different shaped stones using a chisel and hammer. His accuracy was remarkable. At one point I thought, "but that isn't a perfect triangular prism" and then it hit me that these stones would still be used to make driveways, stone houses and other things. And if I had seen it in real life as a completed project I would be able to admire its beauty without thinking "wow but this side of the triangle isn't perfect". There's a lot to be said about that. At the end of the day, practically we don't use high-powered lasers to create perfectly level sides when we're shaping stones - we're creating roughly accurate shapes. I don't know, it's something I'll continue to ponder on for sure. 

Market Scale Puzzle

Going through elimination (and with a hint from Saiya), the four weights have to be 1, 3, 9 and 27 grams. We can weigh up to 4 grams with the weights 1 and 3. We can weigh up to 13 grams using the weights 1, 3 and 9. We can weigh up to 40 grams using all 4. The weights would have to be used in a variety of ways such as weights on both sides, omitting some weights (ie. for 30 grams we would only use the 27 and 3 gram weights), etc. I was talking to TsáKtalay’pa a little while after receiving this problem and he said he didn't use the weight 3 (if I remember correctly) so the wording in the question "must" is interesting - I wonder if there is only one combination or if there are multiple ways to weight 40 grams. 

On a one pan scale you would need the weights 1, 2, 4, 8 and 16 to weigh up to 31 grams. This problem was surprisingly much more straight forward since there was only one pan that can be used to weigh. As soon as you maxed out the amount of weight, you knew the next weight needed - for example with the weights 1, 2 and 4 you can weigh up to 7 grams and so you know the next weight needed has to be 8. 

I feel like the two pan scale problem is really counter-intuitive in the beginning. Although I can't think of any explicit connections to the secondary curriculum I do believe it could induce a larger theme around exploratory mathematics. If students took this on in a classroom activity with a real two pan scale and different weights, I strongly believe it could challenge their initial perceptions about how to combine numbers. It's almost like a physical manipulation of left side equals right side, and all the different ways that weights (terms) can be moved around and utilized to have both sides be equal.