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...
Great work and an excellent project!
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