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.