what is arbitrary and what necessary?

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

 After reading the article, as per my observation, arbitrary appears to be written mathematical learning content that need to be taught to students by teachers or any other external means. The learning of arbitrary mathematics content mainly relies on the memorization of mathematical conventions, symbols, labels and names etc., which have been set up by mathematicians in the past after their inventions. However, the necessary aspect of mathematics curriculum is the mathematical learning content that does not require to be notified to students, but it will be left for them to figuring it out from their previous knowledge.

For example, if I choose to teach students about a percent, I would describe the arbitrary and necessary in the following way:

As per definition,

A percent is a special ratio that compares to 100. The root word is cent. There are 100 cents in a dollar.

In the above definition, remembering the 100 and special ratio is the arbitrary, whereas finding the percentages of any given number or other percentage related problems would be necessary. While teaching arbitrary, I might face the following questions:

Why do we choose only the number 100 for percentage?

Is there any other percentage sign that can be used than %?

Consequently, my job as a teacher would be to teaching students about the meaning of percent by making it less arbitrary.

However, teaching students what is necessary would depend upon students’ learning abilities or their previous awareness about integer or fractional arithmetic. Firstly, I would provide students an appropriate task that would help them to learn how to calculate percentage from their own previous learning.

At the subsequent stage, if I find students struggling to learn the topic, I would try to explain the necessary portion of the topic with easy examples, which would be categorized as received wisdom for students as mentioned in the article. From there, on the one side, if students end up verifying their received wisdom true with their previous awareness, then my mode of teaching would still be educating awareness. On the other side, if students fail to understand the logical reasoning behind their learning, then my role of teaching would become assisting student to learn by memorization, which is no different than teaching arbitrary.

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

This idea would definitely help me in planning my lesson plans as this is what I found myself lacking during short practicum. From my tutoring experience, I always first demonstrate the arbitrary portion of my topic and then push my students to think and solve the problem on their own. I would help them indirectly if they needed it. However, in contrast to tutoring, asking a class of 35 students to solve math problems from the awareness they already have could be difficult goal to achieve. In my opinion, the idea of arbitrary and necessary encourages students’ participation in the lesson. In other words, it suggests teacher to teach the arbitrary in a simple way and then let the students take lead in learning what is necessary for them. While making lesson plan, I would use the idea of arbitrary and necessary in the following steps:

Step 1: Teaching the arbitrary in the ways that students could perceive it easily, but logically.

Step 2: Planning an activity that would engage students in learning what is necessary. During this activity, students would be offered with appropriate task, whereas I would be facilitating them in their work. I would also ensure that students are getting enough time to work on their task.

Step 3: Making assessment on students’ learning. I would first teach struggling learners with very easy examples and then giving them a problem so that they could validate their learning to be true.