Response to "Euclid of Alexandria"

The first thing I want to say is this: it's a good thing the whole modern textbook publishing industry was not around to get in the way of Euclid's "The Elements". Could you imagine of this vital piece of knowledge and wisdom were paywalled? That aside, I should probably address the questions for this reading. 

After seeing what Euclid's hit book series, bestseller of the historic world, was all about, I feel like it only makes sense that his ideas are still studied today. A lot of the ideas that he compiled are essential to the foundations of geometry, and I feel like a lot of it would coincide with mathematicians from different eras and civilizations, much like how a bat and a bird both develop wings but have never influenced each other in their evolutionary paths. After all, Euclid's book series included simple things like the properties of a circle, number theory, basic properties of various simple everyday shapes, and irrational numbers. The ancient Babylonians had something similar to the irrational numbers with their quadratic solving ideas. The ancient Egyptians also did a lot of work with geometry as we've seen from past readings on measurement, although it could be that Euclid's ideas were responsible for a lot of ancient Egyptian mathematics given that he was spending a lot of time there. With modern mathematics education, we see these topics arise all the time. We've seen the practical aspects of these geometries in fields like architecture, astronomy, computer science (particularly cryptography, where number theory is vital), and more, so it stands to reason that we would continue studying the foundations that have uplifted civilizations for centuries. We should continue to take these ideas and proof methods and keep discovering new things about them.

As for why The Elements might've been so important and popular in the past, it's probably because there weren't late-stage capitalistic opportunists trying to monopolize the textbook industry and sapping the finances of poor, starving scholars. (I jest.) I theorize that back then, mathematics appeared as a subject of fun and wonder, and the ideas in the books provided a hot new perspective on how to approach the subject. Would you look at that -- proofs! Rigorous proofs, if I may add! Now, there's an interesting thing that I noticed was missing from the reading, and that's the mention of "inductive proofs". Doing some quick Google searching, I found that inductive proofs were done as far back as the ancient Greeks, but it wasn't as widespread and certainly wasn't formalized as a method until the latter half of the 2nd millennium. But I digress. There was also just a social dynamic at play here: The Elements was simple and easy to parse! Consider this: you are interested in learning about mathematics. Your only option for research is to sift through piles of archaic texts and sort it all out yourself. Utterly cumbersome. Then comes along Euclid, the man, myth, or legends, and he gives you the equivalent of a Khan Academy or Paul's Math Notes -- nice and easy to decipher, easily communicated, and sorted in a comprehensive manner that saves you weeks of mind-numbing scholarly work. No wonder this book series was so popular -- it was a huge convenience! The other reason I can hypothesize is that the subject matter is just interesting, agreeable (logical), and reputable. Mathematicians likely continued seeing successes with the books, and the ideas are practical, so it makes sense why it was so popular.

In terms of the Euclidean postulates, notions, and principles for proofs, there is certainly a beauty to them. The beauty is basically along the lines of taking a small particular and creating the ideas of a whole discipline through the recursive process of creating new ideas and using them to prove more ideas (which in a way is like induction). In this case, the simplicity of Euclid's postulates is beautiful on their own. Oh, we can draw unique lines between a unique pair of points. Cool. These lines can be straight. Also, we can draw circles. These postulates are so simple, and we have such simple axioms like "a = c and b = c, so a = b". Then somehow, through a process of logical parkour, we can safely create a whole series of books of proofs, which were beautiful and inspiring on their own. Not only was this series providing neat ideas about geometry, but it was teaching people how to create their own ideas. Why was I doing direct proofs the way I was during my classes? It could very well be because of Euclid's work. But I digress. Based on how I'm describing the beauty of Euclid's ideas, I would summarize beauty in this way: beauty is about working with the micro to create the macro. It is about going from simplicity to complexity in an easy, flowing manner. It is unforced and natural, and that's what Euclid's work exactly is.

The Dishes Puzzle

Pictured: My work for this problem

To address the first question, I believe that I used very minimal algebra already (if any). Instead of creating a formula or some kind of algebraic notation monsterpiece which I feel like would be harder, I decided to go for a more direct, brutish approach. After getting through the initial interpretation phase -- for some reason I had trouble deciphering "every 2 used a dish..." -- it was just a matter of writing down which guests would be receiving which things, filtering out guests that wouldn't receive anything, and tallying up the number of dishes as I went up the guest numbers. I think that this problem could be solved without algebra if we just did the brute force process the entire way if we really despised any semblance of algebra. However, as you can see, I noticed that the pattern of 1 + 1 + 2 + 2 + 2 + 1 + 1 + 3 = 13 repeated every 12 numbers. From there, I extrapolated the number of repetitions needed, and multiplied it with 12 in order to get the answer: 60. My definition of algebra is still a bit hazy, but since there was basically no algebraic notation or "let letter be meaning", and all the math could be converted to words (e.g. "65 / 13 = 5" could be stated as "there will be 5 groups of guests, and each group will identically take 13 dishes").

As for the second point, I would say yes! The first thing that came to mind when I read "examples, puzzles, and histories of mathematics from diverse cultures (or from 'their' cultures)" is that it sounds an awful lot like the advice being given in my LLED 360 textbook, which is all about how to make ELL students feel welcome in the classroom. This exact idea was suggested as a strategy in that textbook as a means to incorporate and celebrate the cultures of the students in the classroom. This doesn't necessarily need to be an ELL thing either, as a classroom will inherently have students of diverse backgrounds. In addition to this benefit, I would also like to admit as a partially Chinese person that when I saw the word "China" in the problem description, my subconscious made me say out loud, "WOO China mentioned!!" This also happened in my number theory course when the mere mention of "Chinese Remainder Theorem" had my eyebrows raised. If even a basic name drop could evoke such reactions, imagine what incorporating various cultures could do to make students feel welcome, connected, and celebrated.

For the final question: I think in this case, the word problem had a frustrating component (at least for me) because of the vague and confusing wording. "every 2 used..." made me confused. "Every 2 what? Every 2 guests? I guess?" Not only that, but "every 4 used a dish of meat between them" felt like a red herring was at play here. "Between them? Them, as in the rice and broth? But that wouldn't make sense, why would we put meat between rice and broth? The rice and meat would just sink together!!! Ok, this is silly. I'm going to just assume that the meat between them is just another item, this time associated with every 4th guest." Other than that, the word problem itself is fine once the premises are understood. It's not particularly flashy outside of the historical background, as it has a simple scenario with a mundane setting. The numbers are also relatively grounded, so it's not as funny as the Ahmes' loaf problem, or the other example I provided in a previous blog post about a Wal-Mart customer pushing a shopping cart of 100 Twinkies boxes, which I imagine would have fantastic imagery to go along with it. As a word problem, it does carry the benefit of being a humanizing medium to pose mathematical scenarios. To that I'll give it credit.

On the note of imagery, I think it absolutely helps and am surprised that it doesn't happen more with problems. Remember my mural from my Assignment 1 for this course? Think back to every word problem that you've done in math class where a picture is provided. Yes, all 5 of them. In my experience at least, the images are a complementary form of art that enhances the flavour profile of the problem. It's also great for accessibility purposes; going back to the ELL students, they benefit greatly from having visual aids to help them learn our language, which is particularly important for an intermediate language-involved subject like mathematics. Depending on the execution, humour, and a bit of artistic talent, these story problems can be enjoyable. It can feel like we're trying to solve a problem that someone had ages ago, someone whose struggles aren't so different from our own. 

Assignment 1 Reflection

Click here to access the presentation slides.

Our assignment group was responsible for presenting Ahmes' loaf sharing problem, which was a problem created for entertainment purposes in ancient Egypt. It had two conditions: 1) 100 loaves for 5 men, with the allocations being an arithmetic progression, and 2) 1/7 of the 3 largest portions combined equals the 2 smallest portions combined. For the presentation, we discussed the context leading up to the problem, organized a toy example activity, discussed our modern and ancient solutions, and proposed an alternative strategy as our extension.

Pictured: Ahmes' loaf sharing problem in the style of an ancient Egyptian mural

For the background, Raymond was responsible for looking into the ancient Egyptian mathematical developments that were necessary for the solving of this problem and presenting the problem description to the class. As for Zain, he was responsible for researching and presenting the solution that would've been possible with ancient Egyptian mathematics, as well as coming up with the extension which was a way to solve the problem by combining our understanding of linear systems with the ancient Egyptian strategy of false position. Finally, Carson was responsible for generating the toy example numbers, and writing and presenting the modern solution as well as linking it to the modern BC mathematics curriculum. He was also in charge of the artistic direction, which includes the artistic interpretation of the problem and the incredibly funny joke at the start of the presentation. All three members were involved in facilitating the toy example activity.

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Preparing for the presentation of this assignment, I found myself pursuing the modern solution of the problem. Working through the problem, I found that my instrumental framework of word problem to algebraic mathematical representation to solving algebraically worked perfectly. First, model the equalities given by the conditions. Then, figure out the terms we need to find by reducing the number of variables. Finally, apply what I know about systems of linear equations and bam! The problem is solved, and Ahmes is happy. The nice thing about this is that I reaffirmed to myself that I can indeed still do this kind of math. There was also an aspect of visual and textual communication that is important with this, especially since we had to present it. How do we make "showing your work" fleshed out and interesting to an audience while being mindful of the time? I will address this when discussing the actual presentation, but I feel confident that I did a good job.

In addition to that, I was able to get practice with switching around numbers in order to make a toy example with considerations in mind. Doing the original problem with its original numbers would take too long, and the numbers weren't exactly the nicest to work with because of things like /6 fractions and double digit numbers. We also had a 5 minute allocation for the activity, but we needed the premise to be close to the original problem. Thus I had a goal: create a version of the problem where the numbers have a fraction with a denominator of at most 4.

Pictured: the thought process behind the toy example

I started by changing the 1/7 into 1/4, and reducing the number of men to 3. On the left side of the image above, I made a formula with the number of loaves and the number of loaves for the first man, which had 'n' being divided by 5. This is how I decided on n = 10, since 10/5 = 2 which was a clean number. From there, we had nice fractions that were easier for the audience to work with. This whole process of meticulously deciding on numbers for a problem opened my eyes at how much consideration there is in designing questions; there are so many variables, such as appropriate difficulty, ease of numbers (contriving numbers, if I may say), and even time constraints. It has implications on how I design problems in the future when I teach my own classes, and it increases my appreciation for the problem banks that exist as resources for everyone.

On presentation day, I recited the joke and got a good audible laugh from the audience. Perfect! They also liked my artistic interpretation of Ahmes, so that was a good start. I think that a good sense of humour can get your audience in a good starting mood, which is so important especially when heading into the more contextual, reading-at-the-audience, minimal interaction sections that follow. I've always been a fan of bad puns, and I can see myself making them in a teaching role as a way to introduce a different kind of art -- the art of comedy -- into my teaching.

Next for my part, we had the class activity! This was also an exercise in formatively assessing students by seeing how they are all doing, and I feel like I could've done this a bit better. I was standing around a lot, just hovering with not much input (though I would try to nudge people in the right direction once in a while, and remind them to check their answers with the problem conditions). Thinking about it, I generally stuck to one side of the classroom to address one group, but if I were alone (as teachers tend to be), I understand that I would have to give attention as fairly as possible to all my students. On the note of the activity, I want to note that I was inspired by the Thinking Classroom paper that I read a while ago, as well as my own classroom volunteering; the advantage of whiteboards is that students are more willing to engage since erasing is easy to do. It's a mistake-friendly environment, and it lends well to on-the-fly presentation. This was further exemplified when Jacob's group started on a tablet, but they realized that it was not working well with a group that wanted to work together, so they switched to the whiteboard. Seeing this, I became more convinced of the power of whiteboards, and I now see it as an absolute must in my own future classrooms as a fundamental element of collaboration and engagement.

To the office staff guarding the school supplies... I'm coming for you :)

The second last thing I want to touch upon is the insertion of how our modern solution ties into the very modern BC mathematics curriculum! It was a fun idea intended to contextualize not just the work shown, but also the entire loaves sharing problem. Going through my modern solution, I noticed common themes of linear equation systems and arithmetic sequences, and it made me think of a problem that another math teacher that I volunteered for had on his test for the same topics. Basically, he combined skill requirements from both systems of linear equations and arithmetic sequences, and it proved to be quite challenging. By going through the BC curriculum pages to find the grade level that this problem matches, and finding the related content and big ideas, I got more experience in that. Not only that, but it made me realize that Ahmes bestowed a wonderful resource for whenever I teach FoM and PC 10: a historical word problem that challenges young minds. Only half the groups from our exercise today were able to solve the toy example within 5 minutes, which shows that it's not exactly the easiest problem. However, going by the BC curriculum outlines, we know that it is in fact an appropriate question for them, even though it's challenging! I will definitely reuse this problem as a way to introduce mathematics history and give a chance for students to demonstrate their learning.

Finally, I want to note the importance of visuals and art that I feel even more strongly about in terms of mathematics education. When I was "unearthing" the murals for the presentation, it was not a forced process, but more so one where I felt like I was having whimsical fun. During the presentation, when we revealed the murals, the immediate reactions were gratifying. I also asked Leon if he felt that the art helped guide his thinking for the problem, and he said that the scale part of the mural helped him understand the "1/7 of the ..." condition better. There's also a humourous aspect to just doodling out the content, so the benefits are also similar to the comedy from above. While I don't see myself making pictures for EVERY word problem, the importance of visual presentation cannot be overstated. Whether it's a slide show animation or a graphic adaptation of a wall of text, it goes a long way in getting students interested and understanding.

I would like to thank my group for having me, and thank the assignment for giving me so many great ideas for my own teaching while shedding a humanizing light on the mathematicians of the past. Until next time, perhaps when my own mathematics could be considered in a mathematics history course... 

Response to "Word Problems as Genre in Mathematics Education"

Reading this excerpt about word problems in the past, the main thing that comes to mind is that while everything changes over time, from beliefs to values, to religion and science, and power structures, our needs stay the same. Our need to eat, drink, seek shelter, breathe, and most importantly: make mathematical word problems. Perhaps the universal constant of inconsistency throughout the times is what allows the format to flourish as it has, as the real-world conditions and inspirations were constantly refreshed. The ancient Babylonians scribes were worried about agriculture, commerce, and other things that benefitted greatly from applied number crunching. The ancient Egyptians had to worry about land surveying and monument construction. As for the ancient Greeks, their philosophers seemed to have a lot of free time to wonder, so word problems became a matter of play to them. Then there's our modern day civilization, mathematics is applied in every corner of our achievements, from construction to computing to business. 

Even for myself, I have found the need to word-problem-ify many situations in my life. What's the tip that I should pay? If I'm making a scale model of my high school, what should the length of this wall be in the model if I measured this length on Google Maps? (This was a real thing I did in Grade 12. If you are curious and own a copy of Minecraft, I can post a download to the map files.) One time, I was in Hanoi with my dad and he asked me how high the bridge we were on was above the water, to which I responded, "Can you spit into the river for me?" (The idea was to get my stopwatch out, count the air time, and use the d = vt + 0.5at^2 kinematics formula.) Word problems are also a way to articulate play in mathematics; once again taking a page from Lockhart's Lament, maybe we want to pose a proof-like question to ourselves and others. "If we take a circle, draw the diameter line, and from the ends of the line we make two more lines that meet at any point on the circle, what will the angle be between the two lines formed?" The versatility of word problems has allowed it to flourish as not only a mathematical thing we've come to accept, but also as a subset of language. Much like language, there is a formal and serious way to use it, but also a casual and unserious way to use it.

On that note, the excerpt also had a mention of the word "contrived" -- wow! This was particularly funny to me because I remember during our classes, I brought up the term as if I had never read the excerpt (if you are grading this blog post, you may notice that it was completely omitted by accident). It really is a key part of many word problems, however. Due to the side of word problems intended to be for practical skill development, one possible design philosophy is that learners should not be focusing on the annoying arithmetic, but instead focusing on other ideas posed by the problem. "If problems were not contrived, they would be impossibly hard to solve for mere students," the interviewer said. However, I would argue that we don't need word problems to be contrived, especially now that we have the technology to skip the boring arithmetic. Sure, it's nice to have and we can still keep problems within the bounds of realism. Why would a non-contrived problem be unrealistic? The usual contrivances usually come from numerical adjustments, but what's stopping us from simply taking numbers from the real world and putting them in the problems? That's what we do all the time! Pedagogically, there's also the benefit of the person solving these problems being able to use their intuition as a verification tool for their answer (ex. standard problem about the cost of a McDonald's meal, and your answer comes out as $100 -- this is clearly wrong, at least in Canada in the year 2024). While I'm at it, I would go as far to say that every word problem is equivalent to a contrivance of some sort, because they only exist as the product of a game of telephone between a thought or real-world situation and a problem description. Word problems are inherently made up.

Another thought that I had was that the purposes of word problems seemed to line up with the dichotomy of conservative and progressive mathematics, and by consequence the dichotomies of relational versus instrumental, artfulness versus sterility, and so forth. This time, we are introducing the two purposes of word problems: 1) as a teaching/learning tool for practical skills, and 2) no real-world applications, only to tickle the brain. On one hand, like Lockhart lamented, word problems have a sterility to them in our modern schooling contexts. This is akin to the conservative approaches to mathematics education that he scorned so much, whereas his ideas of just making things up (rather, contrive) and playing with them are more progressive and lean towards the latter purpose of word problems. Writing this out, it feels like these are really just two side of the same coin, given that they hold contrivance as a commonality. Sure, Lockhart may have been right about these word problems being complete nonsense and how they often had wacky givens and unknowns (for example, a word problem may give information like "the sister is twice the brother's age" and "the brother was born in 2000" while asking you to figure out the easily ask-able nugget of information that is the sister's age). However, I would disagree with his sentiments after reading this article, as it's not a new idea to make up completely unrealistic truth states in word problems, considering that it's been done all this time! There is a fun to be had, and I argued in a previous blog post that we can keep up this silliness while injecting a sense of humour into our word problems. To tie in more of my own writings, I recently mentioned in an EDUC 450 blog post that dichotomies aren't necessarily true, and I think it applies greatly here. There is no real line between word problems designed for practicality and absurdity, and I think they can go hand in hand, much like how the ancients did it.

As a final short note, I liked that the ancient Babylonians not only had problem texts, but they also had formula sheets (reference tables) and problem banks for teachers to pull problems from (teachers' lists). If you didn't frame the context within the ancient past, I would think that you were talking about a high school mathematics classroom, not a scribe school. 

Multiplication tables to 45, base 60

 

For this table, we have:

• 20 (1/3) --- 2,15 (135)
• 30 (1/2) --- 1,30 (90)
• 40 (2/3) --- 1,7,30 (67.5)
• 45 (3/4) --- 1 (60) (note: it was done incorrectly on the picture above)
• 3 --- 15
• 4 --- 11,15 (11.25)
• 5 --- 9
• 40 (40) --- 1,7,30 (1.125) (this was particularly interesting, because I worked out the division on the right side of the picture above and it ended up giving the same result as 2/3 and 67.5 from row 3.)

I did want to try to tackle a question I had: "What if we had a repeating sexagesimal and rounded it?" Unfortunately, I have no examples at this time, but I suspect we could get one if we tried dividing 45 by 7 and taking the result, for example (due to 7 being a prime that is co-prime to 60 and all its powers).