Hewitt refers to arbitrary as something that is taught to be memorized, something that must be done exactly the way it is taught. For example, using the + sign for addition and the - sign for subtraction is purely convention, and students are required to memorize these signs to progress further in mathematics. Necessary is referred to as content where the students do not necessarily need to be informed. For example, students can learn how to add fractions with or without mixed numbers without the convention of writing it down. Necessary knowledge can come from within or be discovered through inquiry. Arbitrary must be taught to all students, necessary can become aware to certain students through other concepts.
The concept of arbitrary vs necessary is new to me in theory, whoever, I feel that inherently I was always known about it in theory. Using a student led approach or inquiry based approach in the classroom can help students learn the necessary and can help all students understand the arbitrary.
In my own classroom, I could create questions that teach all the students the arbitrary but leave room for certain students to learn the necessary. We want to move beyond teaching and having students regurgitate that information on tests without having any understanding of why the concept works.
A personal example of this for me was in grade 12 when we learned the unit circle. I remember my teacher emphasizing that we must memorize the unit circle and even having quizzes where we would have to fill in the unit circle in 5 minutes. I purely did that, I sat down and I repeatedly practiced writing the unit circle until I knew with certainty that I could replicate it on the test. I had no idea how the values of the unit circle were derived or even what they meant, I just had them memorized. If this was my own classroom, I would have focused on the meaning behind the unit circle and I would have had students find connections between how different trig values create the unit circle.
