Response to 'arbitrary' and 'necessary'

One afternoon I wrote 15-8-2=9 on the whiteboard, just waiting for my grade 7s to come in and get started. Of course they were very keen to tell me that I was wrong. We started the conversation about what could be different if this statement were true. All sorts of ideas were shared: add brackets, change the meaning of the equal sign, change the meaning of the numbers, change the order of operations. This got us to discussing the idea that some things in math are decided or chosen, but some things aren't. We did not frame these findings in terms of arbitrary or necessary knowledge, but we did highlight that some things we just need to know the meaning of if we are going to communicate using mathematics.

Hewitt distinguishes between two main kinds of facts in mathematics: arbitrary and necessary. The arbitrary facts are those that need not be true and cannot be deduced, as such. Arbitrary things are mathematical facts that need to be memorised. Hewitt contrasts this with facts in mathematics that have to be true. These aspects of mathematics can be deduced by someone. The former, arbitrary things are those that need to be memorised: word meanings, symbols, conventions. The latter, necessary things are those which have the potential to be deduced by someone: relationships, properties. 

Considering this division in mathematics has the potential to influence how a teacher teaches. Teaching definitions, symbols, or conventions needs to be shared by teachers and memorised by students. Committing these arbitrary facts to memory will facilitate participation in mathematics, according to Hewitt. Where the mathematics lies is in uncovering the relationships and propositions that hold true despite the words or symbols that are used. (It might be worth noting that arbitrary elements of mathematics have the potential to enable understanding of necessary elements; however, that is a distinct conversation.) These necessary aspects of mathematics have the potential to be uncovered once they are in the 'awareness' of a learner. (The 'awareness,' as Hewitt calls it, seems akin to a learner's PZD.) 

This means that good task design -- perhaps an inquiry-based task -- has the potential to encourage a learner to uncover the relationships that really characterise mathematics. There is no amount of inquiry that can enable a learner to arrive at a mathematically chosen convention; however, relationships and propositions are there to be uncovered if a learner is in an appropriate cognitive state. The specific implications for lesson and unit design are that different activities -- e.g., direct instruction, inquiry tasks -- will be appropriate for different kinds of knowledge. It is a teacher's responsibility to share the arbitrary knowledge through appropriate tasks, and it is a teacher's responsibility (pace Hewitt) to create different tasks that enable/allow for learners to learn necessary knowledge. These tasks will be different in nature because they have different aims and modes. 

Hewitt has the criticism that necessary knowledge is treated like arbitrary knowledge or at least stored as arbitrary knowledge (by some teachers and students). This is a disservice to mathematics and to one's mathematical understanding. While I tend to sympathise with much of what Hewitt claims in the article, I have some reservations regarding graded mathematics education. It can be a challenge for teachers to support a variety of students in a variety of ways. And the propagation of received wisdom may feel like one way to support what could look like student success. However, a criticism of this is that students always take away different things from a lesson. The perceived control of direct instruction is an illusion, and more appropriate task design that honours the role of the learner in their own construction of understanding might not be letting go of any type of control that a teacher thinks they have. Further, with appropriate tasks, students can learn how to participate in mathematics in a more desirable way that supports the competencies that, for example, the BC curriculum aims to develop.