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Part 2: Chapter 7

Re-teaching arithmetic using number bases other than base
10

August 15, 2002

The Nodal Forest Learning Strategy
asks the learner to list the topics that are in a specific curricular
unit. The curricular material is
defined to be that unit of material that is the next learning task.

The learner sorts these topics
into two categories.

New topics may be added to the
categories.

Topics may move from one category
to the other category.

In fact, some students will list a
topic as “known” and then later come to understand that the topic is really
“not known” and perhaps even come to understand

1)
That he/she has a specific
deception that is imposing confusion experienced during the taking of a test.

2)
That the topic was not
understood in a way that is characteristic of a specific confusion that he/she
has about the topic, or even more generally the nature of abstraction,
arithmetic or the non-arithmetic parts of higher mathematics.

Remember that the first topic
category is to be rehearsed, but the second topic category is to be left alone
unless the student feels an interest in one of these topics.

As mentioned in Chapter 6,
cognitive neuroscience and academic theories of self (autopoiesis) suggest that
an inhibition of natural interest has been effectively turned on by the
learner’s experiences and by the reinforcing effect of the ubiquitous social
philosophy supporting mathematics dislike.
How can this natural interest inhibition be turned off?

Various exercises are provided to
turn off this inhibition if the inhibition is not critically sever. However, with as much as 60% of entering
freshman, the inhibition of natural interest is systemic and unrelenting.

So can anything be done to fully
remediate the learning capability and interest in mathematics?

Is this remediation always
possible even with adult learners?

The author’s claim is that yes remediation
has been shown in a high percentage of the student that he has taught. Moreover, the conjecture is that a web
based learning system can be used to provide a low cost relief from the
mathematics anxiety that most adults have learned to live with.

Generalization of the learning
system can be used to teach other subjects and in general knowledge management
processes in business and education.

The process must also allow the
experience of some knowledge about cognitive processes and the image of
self.

In
addition to a process model of three states of student orientation

{ motivated, bored, fearful }

A second process model is
discussed

{ easy to understand, difficult to understand }

Stories from the Author’s
collection of student papers are to be included in this chapter and then the
strategy used at Saint Paul’s College by the author in 1994 will be
developed.

This strategy involves the
learning about the Nodal Forest Learning Strategy, and then the use of this
strategy in the re-introduction to the students of the fundamental notions of
counting numbers, including addition, multiplication and division.

Some history of the counting
number is given in Chapter 10 (Part 1).

In cases where the entire class is
faced with learning arithmetic as college freshman, the behavioral problems of
students can be a factor.

Again, the use of case history
helps to set up the context in which the re-introduction of arithmetic is
made. The 1994 study showed that the
class would periodically move between the motivated, bored and fearful
states. Individuals would also move
between these states.

The instructor can be aware of the
state transitions. Bored and fearful
class-states can be converted into a completely motivated state, as expressed by
the entire class.

Again
some stories help to illustrate.

One young
man attended for the first three days of a college arithmetic class. He was keen on motivating the class as a
whole to mis-behave in every way possible.
Then he did not attend class for two weeks. Meanwhile the class had been challenged to learn to add and
multiple in any base { 2, 3, 4, . . . ,
25 } and to check the answer by converting the beginning of the problem and the
end of the problem to base 10. Some
interesting theorems are actually discovered by these students, and the novelty
was so great that some students began to attempt to engage the upper level
mathematics majors in “the issues”.
{This is another story.}

The young
man came back into class just as the class was struggling with a new bombshell
that the author dropped. The class was
asked to multiple two fractions when the fractions where in an arbitrarily
selected base (other than 10). The
class moved immediately from motivated to fearful. In fact the class was pretty upset.

The young
man was genuinely pleased with this state of the class. But the class accepted the challenge, and
this seemed to be sufficiently interesting to the young man that he returned
the next class period.

The
beginning of the next period was remarkable.
The students were mostly ordered and attentive. The young man was talking with several
others. But it was clear that he did
not understand what was going on in general.
When the author walked in, he kept attempting to organize a student walk
out to protest the unreasonable demands that had been placed in the class.

For a
minute or so there was some tension in the class, and between the young man and
the author. However, several of the
other students turned on him and made it clear that his behavior was not
acceptable. This actually seemed to
startle the young man. He sat back and
observed, and in his attentive state he was deeply changed.

All most
all of the students knew how to multiple factions in arbitrary bases, and the
few who had not figured out the procedure were very quickly in line. The class even accepted a much more,
actually radically more, difficult challenge of adding two fractions in base
17.

After the
students left the class, the young man stayed behind. A discussion followed and some agreements where made; to forgive
the absences, to allow the previous grades to be replaced with the average of
all grades from then on out, etc. This
man made an A in the course. Later that
year, he and the author meet one day and sat down and talked about the
experience.

The full story will have to be
told with some care. The author’s
racial background is English, Irish and Russian. Saint Paul’s College is a small Historical Black College in rural
South-central Virginia. His contract to
teach arithmetic classes for two semesters was controversial for a number of
reasons.

These notes are
formative towards Prueitt’s new book on learning theory.

We expect that the
book will be developed over the next four months.

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