This thought process was a ride. At first, I was just walking to my sit spot, but I already knew what I wanted to write about. I could picture the long, infinite walk to the sit spot and the unlimited sky above me. It was cloudy on my way to school earlier, but at this time it was a clear blue sky. Thinking about how much grass there was, I thought about counting them and the impossibility of the task. How could we possibly count grass, or other uncountable things? How are scientists coming up with numbers (estimates?) for things like the number of stars in the universe or the number of grains of sand on all of Earth's beaches? Even if they do get fairly close, the scientific notation truncates so much information that we're just left with a rough estimate on the order of the billions. We could be so far off from the actual truth value, and the idea that EVERYTHING has a true value to it is absolutely wild to me, especially since for many truth values of quantity, we will never get the correct value because we weren't there to count when it was only a handful. There is also a perspective to be had here about our mortal lifespans; even counting to 1000 can take a long time. A person going non-stop would take about a dozen days, and a billion goes into the decades. A trillion? If we're counting the number of ants, then it would never work. The reproduction rate would also ensure that the truth value is dynamic and impossible to settle on, but the mathematical order of existence knows all, even ignoring the theory of relativity that puts a speed limit on information. Perhaps this is what they mean by staring into the abyss to gain infinite knowledge at the cost of sanity. As a last note on this uncountable countable thing, think of a blade of grass and its length. Unless it abides by the rules of integers, whole numbers, rationals, irrationals, or some other confined number system, they are all likely some transcendental number that can't be a root of a polynomial equation. And we'd have to resort to saying "the length of this blade of grass" or some other symbolization, much like we say "pi" for the ratio of circumference and diameter. I digress.
To extend a bit further on what I wrote in my note, the idea of infinities between two finite, defined values. Between two angles (say 45 and 46 degrees), there are an infinite number of infinitesimally small angles. The mere turning of the head because of the wind blowing on you would be a journey across infinite angles, but if you think about it, you're not exactly going at an infinite rate. In the span of zero seconds, all mortals bound to the flow of time cannot achieve anything. So the rate of change would not be infinity over zero, but at the same time, we are going at a rate of infinity over something, which you could say is a smaller infinity than the former. But I digress. Considering the constant shifts of the celestial bodies and our own position happening at near light speed throughout the universe, the question I posed about the cosmic shell and straight lines can be extended here. At any given time, the mathematical order of existence knows which angles can shoot straight lines into the end of the universe with no interference. The order knows this for every organism, every single object or point in the universe. This line of thinking is related to my thoughts on a true simulation of the universe; there is no way that it is possible, with the infinite amounts of infinite exact truth values. With all this fun talk about infinities and incommensurability, I think these would be amazing metaphysical questions to raise in classrooms. While I am mostly atheistic, this talk about an omnipotent mathematical order of truth feels so divine that I can't help but believe in the numbers and truth states, much like the Pythagoreans. This is further exacerbated by my thoughts on how even deities are not immune to the providence of mathematics and its truths. (i.e. It is what it is.) A mere mention and Socratic questioning to a group of students could start to get them seeing the world in terms of infinities: infinite counting, infinities between two defined values, and the infinite number of ways we can define infinity. Maybe even get them thinking about how to divide by zero using analogies, like splitting a cake to zero people or counting how many zeros to add to get a number. But of course, they will never reach it.
Before I continue to talk about what comes next, I just wanted to share a thought I had in the shower: even in cooking and recipes, we use convenient numbers. But if you think about it, every recipe's ingredient list has a true value of optimality if we're going by the Intermediate Value Theorem. Sure, that quantity of milk would work. However, the mathematical order knows what the true optimal value would be for every specific individual. That's impressive. We could get the kids thinking about the mathematical order and ask, "What do you know, almighty mathematical order?" (Though we'd have to avoid getting too cultish.)
After this brain drilling, we went into an activity on measurements using our bodies. Using a table of body measurements, we paired up and thought about how many seeds we can sow in a garden bed. The first thing I want to bring attention to is the fact that the chart is based on a 6 foot tall adult male. Me being a 5'7.5" on an average day (I say average because the human body's height shrinks a little throughout the day; I've measured my shrink to be from 5'7.75" in the morning to a humble 5'7.25" in the evening, and this shrink can affect the embodied measuring process). As someone who has had a lot of casual height insecurity as a teenager (and a little bit now, now that I realize that there will likely be eighth graders taller than me... how dare they), I've thought a lot about height. Posture is very important for me as a result, since hunching would just add to my shortness. When I took the bus during my undergrad, I would people watch, but specifically look out for heights. From this, I identified common reference points by looking at the reflection in the glass while standing next to the bars, and used them to compare people's heights to my own. It's a connection that I've made, and I think it could be a neat activity to combine with my "make a bar graph" activity from my curricular teaching topic. "Find a reference point, watch and count how many people are taller and how many people aren't taller." That or we can the concept of marking reference points of height in some other way. As a last note on the topic of height, Manveen or Madison telling us about how some fine dining establishments have height references on the wall somewhere was incredibly funny. I guess there is a need for accountability in the dating scene?
As for the garden bed measurements, I found that because I was such a short person, I had to depend on Josh to measure out the garden bed with his shoes. We picked the thyme bed behind one of the east-side benches and a packet of cabbage seeds. We found out that since we only had a 48" by 60" area (rather, a 4 Josh shoes by 5 Josh shoes) with one of the 60" sides being open, we could only plant up to 4 seeds. That was still pretty neat though. Even though we *could* just pull out a measuring tape, I think there's a good connection with history here, like the ancient Egyptians and their whole cubit system using body parts. I especially found Leon's fun fact about using your elbow and wrist to measure your shoe size particularly devastating to my world view. Was it really that simple? Did we really not need to jump through all the hoops of measurement???
Branching off in another direction from the measuring unit of "Josh shoes", there was a fun discussion about football fields and various other funny American makeshift measurement units. While it was a cause for laughter because of how ridiculous American measurements get, you have to admit that it's a lot easier to picture than something like "360 feet" or "110 metres". The average person can picture a football field. When we're talking about the surface areas of our lungs in a fun fact, oftentimes we would say it is equal to a tennis court. Tennis courts may not be a conventional measurement unit, but they sure are good for getting a sense of scale for surface area, which is much harder to conceptualize because a surface area can take infinitely many shapes! Thus, I think we should be crediting all the makeshift units made up on the spot, as they are not so different from the embodied measurement that has shown to be surprisingly reliable. The same process that many rice cooks use when measuring the appropriate water level with their fingernails. I think the concept of makeshift measurement units could be worthwhile for students. In addition to giving them a table of body parts, we should challenge them to make their own units especially since many students are still growing, so a body measurement may not hold after several months. They could use pencils, specific distances, and more. Distances are also funny because we use time to measure them, and there was a whole discourse about Manhattan distances and the diagonal of a square. Isn't it neat that a square will never satisfy a^2 + b^2 = c^2 where all of a, b, and c are natural numbers? Only when we reduce it to a line or point would it work. But why is a diagonal zigzag approaching total straightness always equal to a length of 2 when an exact diagonal is square root of 2? I have my disagreements with my classmates on this, which basically boils down to this: each inductive step of the zigzagging doesn't make it approach anything new. On each L of the zigzag, there is still a shorter diagonal, so nothing's really changing here. The comparison doesn't hold. Incommensurable, if I may.
Finally, we discussed some ideas for our Inquiry 1 topics. There were cool ideas like Saiya's classroom layouts, Mark's topic on textbooks, and I remember Leon had a cool idea aside from his cheating one but I could not for the life of me remember it. Personally, after verbalizing a bit, I think that I could narrow down my math phobia topic as such:
Topic: Math phobia classification -- are all forms of math phobia the same, or are there differences? Causes and effects of each one
Concept 1: Assigning criteria to math phobia in general, and to different subsets
Concept 2: Causes of math phobia and which causes link to which subsets
Concept 3: The effects of math phobia types; motivation and ability
I've been told that the direct literature is relatively scarce and old, and I would need to lean on adjacent subjects like psychology. I've also been suggested that gathering primary sources of information (students, math teachers, counsellors, community members; underperforming and high-performing students too, as you can be good at something while disliking it) is one of the better approaches to this. Thus, I would have to create appropriate surveys, which I think is where the literature review part comes in as a means to make helpful hypotheses.
I hope that I've been able to narrow it down enough by focusing purely on the diagnostic aspect of math phobia, and if anyone wants to join me, I would be more than happy to take on team members :)
Very interesting musings, especially your thoughts on body height and infinities and uncountability!
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