Arbitrary vs Necessary

Naming. Dave Hewitt brings up a good point that just because we name something, it's not necessarily what it is and more importantly it doesn't have to be that. The existence of language is a great example of this. What we refer to as a "chair" in English is referred to as something else in every other language however what is important is that we all are naming the same object - this thing you sit on (which is also a limited definition).

This concept of naming is especially tricky within mathematics because there are many abstract concepts we deal with that may not be applicable to our daily lives. For example, the concept of a fraction (like a quarter) is very relevant and almost does not need to be named. The concept of a discriminant however is quite particular and does need to be named. Dave Hewitt differentiates these two concepts by calling them arbitrary and necessary - what one is told and what one is aware of and can come to discover on their own. 

This article is actual helpful to my own journey as a teacher. It gives me direction in terms of which concepts and ideas I can have students do on whiteboards in groups and what things they should be told. In math, sometimes you can be told something "arbitrary" and then make connections and extensions through awareness of said "arbitrary" concept. It also helps me on which order I can deliver these two aspects. For some concepts, it may be better to give students some arbitrary definitions first whereas in others it's better to let students become aware of general ideas before I explain to them the ideas in formal concepts (so that they can be used within mathematical academics). 

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Hewitt explains "necessary" as parts of mathematics that do not need to be informed but it relies upon the awareness of the student. He shows this by using two examples: the first one being the concepts of parts of a whole (fractions) and the second one being using sine/cosine laws to find missing sides of triangles. In the first example, almost all will know that if they face forward and then turn to the right, they have turned a quarter of the way. The second example however, if a student doesn't have the awareness of those concepts (unit circle) then they would not be able to work out the solution. He further differentiates between arbitrary and necessary as arbitrary being something that all students need to be informed of whereas necessary is something that some students can become aware of without being informed by it. 

To me it seems like Dave Hewitt is saying teachers should only give students what is arbitrary and allow them to figure out the necessary on their own. As in, give students the concepts, definitions and terminology and allow them to find the relationships and properties on their own. When he speaks about the realm of awareness it reminds me of the zone of proximal development. Some students certainly will be able to come to the necessary knowledge on their own but what are ways we as teachers can scaffold that awareness in the other students without informing them explicitly of what is necessary? Often this will require starting at the arbitrary points well before whatever concepts they are currently studying. For example, to understand the properties of a line students have to first know what x and y coordinates are. I usually find when working with students that they're having difficulty with their current unit because they are lacking some arbitrary knowledge from a prior unit or grade.