Nov 8 Arbitrary and Necessary Article Response

The concepts of "arbitrary" and "necessary" hasn't explicitly occured to me until now. I like how it begins with a scenario about a child's question regarding the name of a square, which leads to a discussion about the arbitrary nature of names and labels in math. The author points out that many aspects of mathematics, such as the names of shapes or the number of degrees in a circle, are arbitrary and are social conventions that have been agreed upon and must be memorized. The "necessary" parts however are those that students can work out for themselves, which are not dependent on memorization but on understanding/awareness. For example, a half plus a half gives a whole (two halves make a whole). A lot of formulae use notation that is arbitrary. I think mathematicians make up a lot of notations for certain notions. These notations are arbitrary. In fact a symbol (whether in Latin/Greek/any other language) is used more than once to denote concepts that are very different (eg. P/p can denote perimeter, pressure, polynomial, primes, and p-value). So knowing the context is important. Richard Feynman came up with his own notation in his teens for sine, cosine, and tangent. He says "While I was doing all this trigonometry, I didn't like the symbols for sine, cosine, tangent, and so on. To me, 'sin f' looked like s times i times n times f! So I invented another symbol, like a square root sign, that was a sigma with a long arm sticking out of it, and I put the f underneath. For the tangent it was a tau with the top of the tau extended, and for the cosine I made a kind of gamma. . .so my symbols were better." 

The fact that certain elements of mathematics are arbitrary (such as the naming of shapes, the order of operations, or the way we write numbers) means acknowledging that these will require different teaching strategies focused on memorization and recall. This is more instrumental approach. The relational approach kind of delves deeper into the root of the necessary truths of a statement. It is emphasized that the "necessary" is not about memorizing facts, but rather about what can be logically deduced or inherently understood from given mathematical properties. So, outlining the necessary elements is key, and it'll be helpful if students could explore some aspects on their own by reasoning from prior knowledge. This not only makes mathematics more meaningful and engaging for students but also helps develop their critical thinking and essential problem solving skills.