Reflection: “Arbitrary and Necessary Part 1: a Way of Viewing the Mathematics Curriculum” by Dave Hewitt

The irony of this reading was that in order to understand the terminology, “arbitrary” and “necessary”, I had to memorize Hewitt's definitions because it wasn’t a natural deduction for myself. I thought of “necessary” in terms of having to lay the groundwork with definitions, and “arbitrary” as the formulas (because quantities are unspecified). That initial confusion helped drive the argument home though, because sometimes the point is to memorize properties and function to succeed in appropriate contexts. 

I really enjoyed the points that Hewitt brought up, and it resonated with both my style of learning and style of teaching. In the recent practicum experience, I had the opportunity to deliver a lesson that was structured around the theme of this reading, and it went successfully for that class. The topic of the lesson was dividing polynomials, and I began with a lay of the land. We went over key vocabulary, reviewed exponent rules, and properties of fractions. Once students had their base covered, we were able to move *together* into the lesson of how to divide polynomials by monomials. 

The students had to know to trust that what we went over in the review was factual, and use those concepts to explore the new topic. Had I waited for each student to explore and naturally derive the process, it may have taken a lot longer and they may not have been able to communicate between themselves and myself as effectively. As an educator, I had to decide what they had the time to derive and what tools they would need to do so. Given the varying abilities in the mathematics 9 classroom, I thought this approach worked really well to ensure everyone was able to progress from the same base and assist their ability to develop the necessary. This lesson format is something I’d like to improve on while moving through the extended practicum, because the organization of information is really beneficial for students that suffer with math anxiety. 

Additionally, in my practicum classroom, there were a number of refugee students who never had a background of mathematics in their home country, and had a harder time trying to pick up the English terminology for any ideas they might be familiar with. By highlighting what exactly is arbitrary, these students are able to gain more independence over their learning by being able to improve their questioning ability and trusting the content to come to the necessary conclusions. 

Final point to mention, another parallel I thought of was the issue of notation for differentiation in calculus and physics! Senior students must simply memorize each of Euler, Leibniz, and Newton’s notations, and understand how to apply the techniques hidden in each symbol for the independent contexts in Calculus and Physics classes. The notations all mean the same thing, but are subject to the preference of the user, which can be really confusing for students that misunderstand accepting the arbitrary and deriving the necessary. When developing lesson plans, especially for concepts that have a similar issue, it would be helpful for the students if we decide what arbitraries they need to accept to put the necessary puzzle pieces together.