Arbitrary vs. Necessary

 

In this reading, the author presented a very interesting way of viewing the mathematics curriculum. I wasn’t quite sure at the beginning what he meant by “If I’m having to remember…, then I’m not working on mathematics”. But after reading the article, this made total sense. The author made a distinction between what he considered as ‘arbitrary’ and ‘necessary’ knowledge. When something is categorized as arbitrary, it is often something like names and conventions which for most of the time not self-explanatory. There aren’t clear reasons why something has to be called by a particular name and the student just needs to accept this name rather than question or argue about it. Hence, a knowledge that is arbitrary would require the teacher to inform the students and later memorized by the students.

On the other hand, if something is considered ‘necessary’, it is likely something dependent upon the awareness from the student. Unlike arbitrary knowledge that has to be taken as it is, necessary knowledge is something that one can work out by themselves. This requires students to make connections and make sense of what they have learned. Instead of memorizing, students need to reason. However, there are chances that during the teacher demonstrating some math content that is necessary, students mistakenly take them as arbitrary. Students mimic and memorized the steps from the teacher’s demonstration and use them as facts. This is what the author called the ‘received wisdom’. This is a risky thing to do as one can memorize things (or the context) wrongly and have no possible awareness of knowing.

As math teachers, we need to have a good balance of teaching arbitrary and necessary knowledge. We want the students to have the ability to generate new knowledge from adopted conventions. Teacher’s role is to inform students with enough amount of arbitrary knowledge and bring students’ awareness in necessary knowledge so that they don’t rely on memorization. For example, if I were to teach trigonometry, instead of asking them to memorize special trig values, I would ask them to know how to get these values using special triangles as follow.