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.
.
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.
