Arbitrary and Necessary - Reflection

 I am very grateful to made aware of this line between arbitrary and necessary. I recall a lesson I taught during my short practicum.  

The objective of this class was for students to understand how to sum geometric series, as well as how to use (the very arbitrary) Sigma notation. To start the class I explained to students that we were trying to calculate sums, gave them a formula to calculate the sum of a geometric series, and finally provided an explanation of Sigma notation. Although I gave them some necessary information (the formula for the sum of a geometric series), I judge that these students did not have the awareness to arrive at this formula independently. Explaining Sigma notation was required by me as a teacher because it is arbitrary.

After working through problems, I gathered the students to review. Of particular interest was a question I wrote asking students to us Sigma notation to represent a geometric series. At this point, students had a decent understanding of both Sigma notation and how to generate a geometric series. Slowly, we were able to leverage their existing knowledge to ‘derive’ the relevant formula. I had not realized the significance of this question when I wrote it, but afterwards I felt that this was a very powerful moment in the class. I know recognize this as students leveraging our arbitrary notation do deduce something necessary. As a teacher, I feel a ‘correctness’ when I am able to guide students to independently discover the necessary.

As I develop future lesson plans, I suspect acknowledging what is necessary and arbitrary will inform what information I provide students at the start of class, as well as how to minimize scaffolding required for students to arrive at the necessary. I will perhaps even include a section on my lesson plan which explicitly states the relevant arbitrary and necessary information. I am excited to develop my next lesson plan.