(Next Page.. A sample teaching session)
One-Hour Demonstration of
Prueitt's Teaching/Learning Methodology
September 6, 2002
Teaching Objective (in one hour): Engage the audience in the
question: In base-6 what is 1/3 + 1/4?
Anticipated Audience: Professors of mathematics, mathematics
education and university administrators.
There are five “agendas”.
First: We ask the students to engage in the most elementary tasks found
in basic arithmetic. Each student will appreciate the sophistication
that is fundamental to arithmetic when arithmetic is fully
comprehended. The switch to bases other than base-10 is used to
expose the students to an extreme form of novelty while at the same
time a series of problems that at first appear to be unlearnable.
The first example is the question
In base-6 what is 1/3 + 1/4?
Second: We expose the student to what appears to be a series of
very challenging problems. But the challenge of these problems is
due simply to the students not be familiar with arithmetic in various
bases. So a model of learning is taught to the students.
This model is then used over and again to gain confidence in the new
learning theory. It is in fact necessary to teach not only the
mathematics slightly differently than in traditional courses, but to
also teach the learning theory at least just enough to give the
individual student the ability to see that the problem, of poor
arithmetic skills, does not lie entirely with the student.
D/S Model of learning: Things-to-learn are difficult or simple,
depending only on one’s experience. In case of counting and doing
arithmetic in bases other than 10, experience can be developed by
ANYONE over a period of between two or three weeks. The path to
acquiring this experience is at first a private experience with a few
students. If circumstances are proper, this experience becomes a
re-enforced social activity. The author has observed that the
novelty of the learning experience provides positive social attention
to those who learn first. This positive re-enforcement can spread
to those who pick up on the experience. The ease of the first
learning tasks, counting, then addition and then multiplication
represented significant personal achievements that can be shared
between the class members. But these early tasks are followed by
other tasks to lead directly into higher mathematics such as computer
science, topology, number theory, and category theory.
Note on expectations: Positive social attention and increased
interest in the new curriculum was observed (in previous teaching
experiences at Hampton University, St Paul’s University and several
community colleges (1989- 1994)). Given a period of two to
three years, it is perhaps possible to make measurable differences in
the University wide outcomes from mathematics training. We also
expect to make contributions to statewide mathematics education
programs and to the scholarly literature in cognitive science and
Third: We ask that the student come to understand one's own
history regarding experience in mathematics classes. We have used
writing across the disciplines (in vogue in the early 1990s) to ask
that students write about personal feelings towards mathematics (and
science) during the first week of class. At mid-term and then
again at the end of the semester the student is asked to develop a
private log of how they are feeling towards mathematics.
Fourth: We ask that the students realize that not every one will see
the light at the same time and in the same way. So the
Teaching/Learning Methodology puts some rules down in regards to grades.
Rule 1: Any test may be retaken given the student write an essay
as to why the test was failed or done poorly.
Rule 2: Students may reject the notion of learning and then come
to a private understanding of the importance of breaking out of the
inhibition of his/her interest in arithmetic, science, economics,
etc. When that moment occurs, everything should be negotiable.
Rule 3: Any activity that aids students in awaking interest in
arithmetic, science, economics, etc should be supported.
Fifth: The last agenda is related to eternalizing the process of
learning. Most students “learn” that math is to be memorized and
that theory is unknowable. Ask the freshman class and they will
tell you this. Use a polling instrument and one will find that,
on average, on college campus that 60% - 90% of incoming freshman will
say that mathematics theory should not be talked about outside of class
and that good grades in math is merely a question of rote memorization.
But, we hold that the only way to make arithmetic difficult is to deny
the student a clear understand of how arithmetic works.
The author’s proposed developmental mathematics textbook creates
learning tasks that:
1) Can be visualized as a student
is walking to a friend’s house or to the store.
2) Can be rehearsed serendipitously.
3) Have a specific way to check the
answer to problems that are made up randomly by the student.
This last agenda has a neuro-cognitive basis to suggest that if this
agenda become effective within a student body that the students will
positively change behaviors in mathematics, science, economics, etc
classes. We are looking for this type of result.