Dave Hewitt: Arbitrary and Necessary

1) What does Hewitt mean by "arbitrary' and "necessary"? How do you decide, for a particular lesson, what is arbitrary and what necessary?

2) How might this idea influence how you plan your lessons, and particularly, how you decide "Who does the math" in your math class?

By Hewitt’s definitions, names and labels are normally arbitrary to students as the reasons things are named and labeled are because of conventions we inherit from ancestors. Names are things teachers need to inform students so that they can share the same information in communication. On the contrary, necessary things are usually workable and understandable by students as students do not have to be informed and can be figured out by some reasons. 

What Hewitt indicated was necessary to me as I agree as well that there are things in mathematics that are “given” and provided in advance for use and discussion. The list on page 4 indicates that there are things that are arbitrary in mathematics. Names, and the meanings of these names, need to be memorised for future usage. To me, what are needed to memorize are the bounded relationships that a name is a “short cut” for things with some given properties.

I always believe mathematicians and mathematics users are lazy in the sense that they tend to use a name to indicate things with similar attributes. Although a name can be changed to some other names, the meaning of the name is not changeable. Thus for arbitrary it means the relationships: both the name and its meaning. 

In my class, I strongly believe it is important to show students what are arbitrary and what are necessary. However, I need to be very careful as some necessary knowledge is arbitrary for some students. In the article Hewitt considered this as this is not the right time to teach these students the knowledge. To me this means that necessity and arbitrary can be internally transformed based on students’ backgrounds and mathematical levels. If a class is full of arbitrary, I may not be surprised why students do not like this class. To me, even though introducing arbitraries is essential, minimizing the amount of arbitrary in class is crucial to maximize students’ learning outcomes.

 

Reflection on Dave Hewitt's teaching video

The counting numbers strategy is wonderful! Dave first knocked the blackboard to make noise to attract students' attention. He then start to count "1" with a knock, "2" with a knock right to the previous knock, and so on. After counting "7", he started to knock the next one without counting the number. The next moment was amazing: students started to count "8" with him. He then knocked the next one and students responsed by "9". After several knocks and responses, he orally called "dot, dot, dot" and knocked the wall with calling "20". Students responsed the next knock with "21". Dave repeated the game until he moved to the back of the classroom and said "1,000,000". He started to knock backward and students quickly figured out the trick. When he knocked a place left to the zero, students started to called "-1". Then Dave started to knock multiple places and waited for response. Students were able to response correctly as well. Then this is the moment he started to introduce plus and minus integers.

Dave provided students long pauses after questioning, and actively performed to engage students. Even if some students had their hands raised, Dave still waited a few more seconds for the whole class to digest the information and make sure everyone has been on the right track. This is a strategy I need to use in my class.