In terms of my inquiry project process, I was unable to make much headway in terms of finding my final source. (I am currently at 7/8, with 5 of them being scholarly.) The reason for this is mainly because I want to diversify the scope of my sources; many of my sources already go into group work, and the Beginning phase of teacher interventions and assistance. On one hand, this is great! I have a lot of information pertaining to that aspect of my inquiry project, but it's been a lot harder to find resources on the End phase. The End phase has to do with the large classroom setting that induces time pressure and competing for teacher attention, but there's an overwhelming amount of educational research done in "large classroom management" related to behavioural challenges. In another project those would've been of great benefit, but not so much for this. There's also a weird difficulty in finding articles pertaining to one-on-one help, because so many intervention-related articles pertain to helping small groups. While I've done extensive field research (and plan to ask more people on their thoughts on the matter) and feel that it provides bountiful feedback that the papers don't even cover, I do want to find something that backs up what I hear and observe from teachers around my school. However, I should note that I have found useful parts that are closely related to the End phase as I've described (e.g. leaving a student and going to the next) in the articles I had found, but most of them look into the entire intervention/assistance process. Still value though. The more I go through the sources, the more it feels like the Beginning and End phases are actually similar, with the only distinguishing factor of the End phase being the awareness of when a student can be left on their own.
I also want to note that while I won't be shifting the focus of my topic, I've found a lot of resources that would be helpful when looking at the Middle phase of assistance, which I've defined as strategies that teachers incorporate to make the process as effective as possible. These would be things like conversational strategies and body positioning. My plan is to continue looking for an article that can balance out the coverage of my other 7 articles; once I can do that I will upload my annotated bibliography. There's also a little bit of perspective broadening that happened; I didn't consider that you could split the Beginning phase further into "monitoring" and "intervening", but it makes so much sense to me because I was already implicitly splitting that phase! To end on a positive note, I am very happy with the sources that I DO have.
On a less serious note, let's talk about division by zero! It's been a topic that I've ruminated over many many times in my life, considering how downright philosophical it can get. To reason with division by zero, my favourite thing to do is to make simple analogies that are used to teach children how division works. Sure, limits are a thing, but when we analyze y = 1/x we just say "the limit does not exist because we approach different values as we head towards zero". There's some insight that could be garnered here; because the limit doesn't exist, 1/0 can't be defined... but this argument falls apart in the face of y = 1/x^2, because the limit could theoretically exist then. So yes, I am inclined to alternative analysis methods.
For example, "6/2" in English could be modelled with "How many times can you subtract 2 from 6 until you reach zero?" or "How many 2's do you need to add together to reach 6?" Obviously, the answer is 3 for both of these answers. But let's try this: "5/0". Using the language framework defined, we can say that as "How many 0's do you need to add together to reach 5?" This is where infinity as an answer breaks down for me. No matter how often you add zeros, you are only left with zero. This is how it's different from the infinitesimally small numbers used to approach zero; those small numbers will eventually reach a non-zero number if you add them by themselves enough time. Zero is special because you can't. Thus, my stance on the matter is that because of the definition of the division operator and its interpretations, there exists a way to model division that does not allow division by zero to happen (as I've illustrated), so I'm on the camp that believes division by zero should be left undefined.
But what about 0/0? "How many zeros do you need to add together to get zero?" Well, let's look at a brief table here:
- add zero 0 times, you get 0; 0 x 0 = 0
- add zero 1 time, you get 0; 0 x 1 = 0
- add zero n times, you still get 0. we get the point here.
As we can see, there is no expression where "0 times anything = not-zero". The idea of inversibility completely breaks down here, which is why we can't just divide by 0. On a number line, if we wrote "x = 0/0" to model "0 * x = 0" (loosely and not-rigorously speaking), we end up getting "x could be anything from (-inf, inf)". If I could make a graph to describe the function "y = 0/x", Desmos would usually give a horizontal line at y = 0 with a hole at (0,0). Okay. But my possibly controversial opinion here is that there should also be a vertical line at x = 0. Think about it:
- (0,0) --> 0 = 0/0: makes sense
- (0,1) --> 1 = 0/0: makes sense
- (0,n) --> n = 0/0: yup, that also makes sense.
- But what about negative numbers? What about (0,-1)? -1 = 0/0? I would argue this also makes sense, because even if you have "negative one zeros", that's still... zero. It works out with negative numbers, at least for me.
The reason why this makes sense to me is because when you enter a more bizarre equation in a graphing calculator that deviates from the y = [expression with x] form (maybe x^2 = y^2), it will give every single point where the equation would hold. So based on my definition of 0/0 (which is "any number... perhaps every complex number, every quaternion, and every other extension of the number systems; infinity included from all directions, which is important to state because complex numbers have an infinite number of infinities depending on the direction you approach the edge of the complex universe), the calculator should try to include every point that satisfies "y=0/x", which should be the x and y-axes. EDIT: I want to note that maybe it's less so "every point", but 0/0 is actually ambiguous. So perhaps 0/0 should be defined as "an ambiguous number from any number system". Perhaps that definition has its uses in some field of mathematics? The idea of an "ambiguous" number. Quantum superpositions anyone?
I've clearly gone on for a long time about the topic of division by zero, but you could probably tell it's a very fun topic to think about. I could even make comments on things like rates of change; what does it mean to go 1 metre in 0 seconds? It completely violates the flow of time within a frame of reference, because nothing can happen if time doesn't flow... at least by my understanding and within this scope. This is probably why stopping time can't really happen, because that's the only situation I can think of where that rate of change occurs. But maybe the impossible math is the thing that defines the cold hard laws of the universe; time stopping is completely impossible. There's also connections that could be made with Lorentz-thingy equations in physics that pertain to the speed of light and time dilation, but that's way off the deep end here. I'll cut off the blog post for now and say that I'll keep looking for that last inquiry resource :)
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