Part 2: Chapter 6

The Nodal Forest Learning Strategy, the
healing/re-accommodation process

Last edited: Saturday, June 17, 2006

The
conjecture that is developed in the first five chapters is that slow and generally
un-insightful instruction during the pre-college experience leads to an
acquired learning disability. Whereas
earlier in life, the learner had the potential for understanding the
fundamental procedures of arithmetic, this potential is now inhibited, at the
time of the college freshman year, by an adaptation.

Behavioral
adaptation and consequent inhibition is the subject of research [1].
Some preliminary cognitive neuroscience work leading to research projects in
cognitive science is presented [2]
[3]
[4]. However, most of the author’s work on
acquired learning disability was never published. This work was developed from 1985 to 1994 and then picked back up
in 2006. During the years 1995 – 1998
the author developed the manuscript called “Foundations of Knowledge Science,
for the 21^{st} Century” [5]. The manuscript is very technical and quite
difficult, and addresses issues that some regard as being very hard to
follow. It employs detailed knowledge
from more than ten academic disciplines, including neuroscience, biochemistry,
quantum physics, general systems theory, adaptive systems, cognitive
psychology, foundations of logic and mathematics, and philosophy of
science. Part of this work addresses
the issue of coherence and the role that coherence plays in our classical
notions about human and scientific knowledge.
Logical coherence is traced to field coherence and to many types of
compartment processing by biological systems.

In
2002, the author made the decision to rewrite a version of an unpublished
manuscript “A Question of Access” that the author wrote in 1987. This decision came with the full awareness
that a complete cycle of research in educational theory and in cognitive
science would be necessary to produce a marketable book on learning theory. Working notes were posted on the web. But access to a library and the dedication
needed to read the current literature was still not found as of September 2002. It was 2006 before time could be found to
re-express the main themes of the earlier work but in a way that was seen as
entirely helpful even by the mainstream mathematics education community.

Given
the plausibility that cognitive neuroscience, and immunological theory, is
supportive of the acquired learning disability conjecture, one might also look
into the social science. Perhaps there
is an accommodation of a failure whose responsibility is carried both by the
individual and the social system. The
general properties of the accommodation of failure are then made the core
target of investigation. This
investigation is targeted at developing the evidence for the conjecture and in
finding a remediation. Remediation was
found entirely possible, and this is where the unexpected turn of events
started to happen.

We will
frame the issue in this way. The
required remediation is needed at both a social level and at the level of the
individual. Individual remediation can
at times be fulfilling and we found many examples of this early on. The individual can awaken to the fact that
mathematics is something to be appreciated and that the fear of mathematics is
largely due to facts external to the individual.

The
individual can come to appreciate that arithmetic and even the foundational
concepts of number theory and topology are accessable and even
interesting. But the social system can
oppose this newfound appreciation. In a
very real sense the opposition is to change.
The way things are has its own weight, even when the situation is widely
recognized as being not right. So there
has to be a deeper education about the nature of social life and about
individual self-awareness. The student
must be allowed to see him or her self as someone learning higher knowledge
within a supportive community.
Mathematics and Science for the Whole Person is directed at allowing a
remedial program that opens sociology, psychology, physics, chemistry, history,
literature, art up to the learning, and to do this by reducing the uncertainly
and feelings of inadequacy that students often have.

The
result of this new type of education brings the individual into a position to
challenge and to ultimately change the social perception. This is a very difficult thing for one
person to do in isolation. In 1993-94
academic year, the author taught at a small Historically Black college in the
middle of southern Virginia.
Arrangements were made to teach three sections of remediation arithmetic
and one advanced mathematics course.
The results were very successful, with high retention and good student
evaluations. However, the issue of what
is next could not be answered. Where
were these students to go with a new found confidence and with an understanding
that they could achieve much more than they had ever imagined? I did not have an answer in 1994.

What is
needed is a system wide reform of mathematics, computer science and science
curriculums and pedagogy, something like what is envisioned by the SCALE
project [6]. Moreover, the reform has to be reflected in
all college / university courses, including art, history, literature, political
science etc.

Part of
the 2006 solution[7] to the
problem of individual isolation was to develop the concept of groups forming
within the classroom. A complete cycle
of social differentiation was encouraged to develop, including rebellious
leadership and peer bonding between classmates. The student was no longer isolated by peer pressure and by social
expectations. An appreciative social
field was developed as a consequent of suspending belief about what was being
taught.

We needed
to stay within policy standards imposed by the state government and by the
federal government. So standard curriculum was taught in a standard way. However, groups were allowed to form outside
of class and ask the instructor for a one to three weeks exploratory
workshop. These workshops often started
with the two curriculum outlines developed by Prueitt[8]. One outline is an introduction to computer
science, data structure and discrete mathematics. This introduction is also preliminary to curriculums on process
management and general systems theory, as well as linguistics and cognitive
neuroscience. The other exploratory
course is a three-week re-learning of arithmetic. Re-teaching counting, addition, multiplication and division of
positive integers using arbitrary number bases, other then the base 10, was
shone to accomplish a re-learning of arithmetic task.

In
addition to re-teaching arithmetic, the exploratory course leads into the
history of mathematics, advanced topics in number theory, topology, real
analysis and foundational issues related to the Internet use of web-based
ontological modeling. Students were
given support for exploration about fundamental insights, into chemistry,
physics, biology, and social science.
Other workshops were developed by faculty to open access to students
into new areas. These workshop outlines
are given in the appendix.

Students
were encouraged to develop groups to attack various problems that seemed at
first to be difficult, like the addition of two fractions in a base other than
base 10. Other groups formed to develop
a philosophical justification for their mastery of some theorem proving skills,
or of some part of the history of mathematics.
These were not simple make work projects that allowed students to pass
the course without developing the skills required to take trig or the
calculus. The course was a time
consuming and challenging course where some failed to make it because they did
not make the commitment.

The
initial part of the program was not mathematics in nature at all. The early part of a workshop was very much
about cognitive science. Each student
had to come to understand that after they accepted an adaptation/accommodation
process, the student’s brain system simply would not selectively attend to the
standard curriculum. This understanding
was strongly resisted by some students.
The image of a non-mathematics learner had become part of their image of
self. It was “who” they were. Other students got right to work. So we had all of the elements of a
partitioning of the classes into small self-organizing groups. These groups were then given the opportunity
to engage a faculty member, for a number of disciplines, in one of the
workshops. No college credit was given
for the workshops.

Let is
return to the core of the conjecture about acquired learning disability. The conjectured behavioral response is
similar to the behavioral response of the immune system when given a series of
under-critical-dose vaccines.[9] The conjecture was developed in 1986 while
the author reviewed the literature in bio-mathematics and theoretical
immunology. The key element to
immunological theory is the replicator mechanism, at the gene, cell and
anatomical region level. How does the
immune system recognize something that is foreign to the host? The answers to this question are still open
to science. However, it is safe to
suggest that the recognition process involves the development of natural
categories via replicator mechanisms at all level of biological organization,
not merely at the level of gene expression.

A
simplified and metaphorical form of the underlying theory about “conjectured”
acquired learning disability was presented in class, during workshops, and on
an individual basis.

At core
the thesis is presented, to the student, that there is an inhibition of an
otherwise natural mental/behavioral response.
Why would anyone not be interested in the nature of science and the
natural world? In some cases, students
were asked to write essays on why not everyone is interested in science and
mathematics. These essays put all of
the issues on the table, and because they were expressed the reasons for
individual resistance to learning could be addressed individually.

In
summary, the conjecture asserts that the student’s mental system was simply
making an intelligent response to an entrenched set of expectations from social
experience, as manifest by the experience with former teachers and the way in
which the curriculum is presented in textbooks. The students’ viewpoints often expressed a perspective about that
personal experience.

This
type of thesis can lead to huge difference in commitment because there is
finally an answer to a deeply personal question, “why am I not better than
average?”.

One
must say that there are many very good mathematics teachers, and one may even
excuse the textbooks since the textbook industry is the consequence of a much
larger framework. But we all know from
personal experience that the system reinforces the notion that the system
works, without allowing a acknowledgement of the processes that produce the
failure. Without acknowledgement, there
can be no effective re-structuring. It
is even controversial to make the statement that the system fails. Since the system is bigger than the
individual, it is the individual that fails.

In the
workshops, the individual student is given an opportunity to challenge this
failure, and with the help of peers is able to create profound philosophical
statements regarding a type of liberation from this failure. Learning arithmetic is then the easy part.

The
systematic opposition to this liberation has been extensively documented. In Part 2 the author will bring forward a
number of citations from the mathematics education literature that builds the
case that no system could be successful in producing a non-phobic reaction to
arithmetic for most students. There is
an extensive literature that is very clear about this point. It is argued in plain English that the
average student simply does not have an aptitude for arithmetic, and that
society should be pleased with what is achieved. The literature review will demonstrate without any question that
an irrational viewpoint was adopted by a great majority of the mathematics education
community, and that this irrational viewpoint was based on false notions about
human nature. The deep framing, G.
Lakoff’s term, supporting this irrational viewpoint is embedded within other
social viewpoints.

The
most direct evidence of a systematic inhibition of interest and ability is that
freshman students do not daydream about the mathematical procedures that are
being taught in the classroom. This is
an abnormal behavior whose causes are not reflected in the student’s innate
capabilities to learn or to be aware of knowledge. By conjecture, those capabilities but turned off due to the
repeated low and poor dosage of mathematical knowledge. The cause of this behavior has been traced
in case studies (Prueitt, unpublished 1987 – 1994) where his students wrote
about their feeling about mathematics.

But
beyond all of this documentation of what appears to be something truly odd, we
have simple common sense. Other daily
activities are rehearsed, but not the mathematics.

A
general strategy for measuring the phenomenon of daydreaming and the
distribution of day dreaming occurrence was developed while the author was
teaching at Hampton University (1989-1990).
Our students developed personal diaries where they expressed types of
experience rehearsal. It was observed
that the experience of the mathematics class was not rehearsed. These results were then discussed in
class. He advised the students to be
aware of when their subjective experience was about one of the problems that we
were discussing in class. What was
reported was that when the learner begin to reflect on mathematics that an
automous inhibition was also experienced.

What is
the nature of this inhibition? Can the
student fight back and become aware of the subtleties of this inhibition? Can the student begin to regain control over
that part of him or her self that would normally reflect now and then about
problems that were actually interesting?
Wow!

The
students and the author talked about this phenomenon in class and we decided
that the division of topics into “known, and comfortable with” and “unknown”
was helpful [10]. The notion was that those things that the
individual was comfortable with could be thought about without feeling
frustrated. Things not understood could
be just listed as a topic. Maybe all of
a sudden the topic’s meaning would be come clear. But the focus should be on what was understood. The students have to be given the confidence
that they will not be failed from this course, again, if they let go of how
they have oriented towards arithmetic and algebra in the past. The student found a comfortable ground on
which to stand.

We
agreed that each individual, on blank paper in class, would develop questions
and then provide the answers. This was
to be a major test grade, but if the grade was not a good one; the student
could write an essay on what happened and redo the test.

The
student was able to design the test and then take the test strictly based on
those topics that he or she was comfortable with. This was similar in nature to allowing a homeless person to get
comfortable with some location where no threats were imposing. Once this comfort level is found, then things
begin to change. Notice that nothing
about a teacher specifying a curriculum has been emphasized. The student has to find a means to select
topics perhaps quite different from the person next to him or her what is of
interest and what is comfortable. This
is where the self-organizing groups can be facilitated. The expression of “me”, what I am interested
in, was what the student was doing; not “taking a test”.

The
test day brought with it a remarkable experience. Students came in and we passed out sufficient paper. They each developed material and solved
problems. Some of them had memorized
most of the material that they where to provide. Some were comfortable with just writing the test and solutions
like one would writes an essay in an English exam. Some, very “poor” students, wrote furiously for the whole class
period and gave up the paper only with great reluctance at the end of 50
minutes. These were remarkable sets of
papers. Several very good students took
on one problem, stated it well and demonstrated a mastery over that one
problem. The process was very creative and the students did remarkably
well. More importantly was the sense of
empowerment at the beginning of the class.
This was not a normal freshman mathematics class test environment!

It
should be said that the author has talked about this experience to academic
groups, who expressed great reservations about almost everything
described. “How did you find time to
grade those papers and all of those essays?”
The truth is that each of these papers was a wealth of information about
individuals deeply engaged in “self”.
They were not about the adding of fractions, or the finding of the slope
of a straight line. They were about an
individual, the “whole self”. I
remember looking at a test where the student had filled up 12 pages. The express was art. If you turned the papers up side down and
framed them they would be expressions that could be sold in an art
gallery. Why would I give anything but
an “A”, no matter how many mistakes in arithmetic were made? The correctness of the answers is a separate
issue, and that is where a deep and reflective examination of the details could
be made. But “whom” was I grading at
that point?

In
Chapter 2 we looked at models from cognitive neuroscience and immunology as a
means to ground the conjecture. In
Chapter 3, we continued to look for a neurochemical and neurofunctional
justification for the conjecture. But
this work on grounding the conjecture is largely the material expressed by the
author in his other book, “Foundations of Knowledge Science”. A research program is suggested, and some
new program proposals have been submitted to the National Science Foundation
and to other funding institutions. But
the work is largely confined to the author’s teaching experience and to the
author’s review of literatures. The
work that began in 2006 would change all of that.

(Author’s note: I am projecting
myself into the future with the hope that I will be able to use the methodology
I have developed at a small college during the Fall of 2006. )

Student
essays gathered in the early 1990s strengthen the plausibility of the
conjecture, but could the success, as measured by test scores and student
evaluations, be duplicated by average instructors using a textbook? Clearly this was the challenge undertaken
when we started the “Mathematics and Science for the Whole Person” text.

It is
not easy to criticize one’s own social systems’ deeply held beliefs and
practices. In Chapter 4 we developed a
theory of social reinforcement that can be claimed to be a partial cause of the
social acceptance of mathematics dislike.
Often one is wrong in these criticisms, simply because the issues are so
complex and so little is really understood about social autopoiesis. So it is, perhaps, only after three decades
of individual reflection that the author’s own analysis might be positively
expressed. The challenge was to convey
this analysis to the student indirectly as a result of the learning
experience. Because the challenge is
one that must involve the whole person, there are many research topics for PhD
students in the fields of mathematics and science education. This is a long discussion, which we hope
will be revealed in the near future.

So, for
now we should return to a discussion of the conjecture. Natural origins of behavioral problems and
or learning disabilities must also be understood, but understood in the context
of a more powerful set of causes, and that is the social field in which
education in mathematics and science occurs, or not. The conjecture is that learning disability is generally imposed
on individuals. In many cases, natural
learning disabilities and behavioral problems are altered by the social field
where it is asserted that only a very few have ability in mathematics.

The
author’s training in mathematical models of intelligence seemed to get him
started on a good path. This training
was in place by the time he was awarded his PhD in Pure and Applied Mathematics
in December 1988[11]. His experience, during the years 1991 – 1993
at Georgetown University as director of the Neural Network Research Facility,
was a blessing that will forever offset the many challenges that the authors
was given and failed at. Here the
understanding of the chemistry and functional aspects of neuron and immune
systems was matured.

Between
1995 and 1998 he was allowed to work with and understand the Soviet school of
applied semiotics developed by Victor Finn and Dimtri Pospelov[12]. This work on applied semiotics lead to the
concept that the computer can develop sign systems and that the human is
absolutely needed to create the semantics for these signs. This concept is then expressed as part of
his work on international standards for service oriented architecture (SOA).
The key to his contribution in SOA standards is in the development and
refinement of the notion that choice points are necessary to align computer
based ontological modeling with the stated goals and objectives of communities
of interest [13].

Knowledge
management issues and attempts to translate computer science to knowledge
science have been helpful. Standard
knowledge management (KM) methods were used to help individual students
discover and reveal who, where, what, how, why and when facts relevant to a
personal path towards higher knowledge about the world and him or her
self.

In
Chapter 5 we developed the first of a series of models that may help understand
what in fact goes on in the freshman mathematics class. The first model is a process model with
three states of student orientation

{ motivated, bored, fearful }

We
presented case studies acquired in the early 1990s. Additional case studies are presented in the appendix.

Chapter
6 has been a review of the issues. In
this chapter we looked at a strategy for learning and we built both the current
context and deep framing for a transformation of the situation found in
freshman mathematics classes. This
transformation is dependant on and linked to a transformation of how society
views itself and what we wish our American dream to become. The entire book, “Mathematics and Science
for the Whole Person” seeks to “walk forward”, to express an understanding of
the failure of American education and to build a solid foundation for renewing
our American culture.

A
freshman level textbook is offered to colleges and university across the
nation. The textbook reveals both new
curriculum and new pedagogy. The
teaching learning strategy happens to be ideal for computer mediated learning
where the subject domain can be converted into a set of topics, or a machine
representation of human knowledge called machine ontology. So the means for liberation of individuals
from the repression of acquired learning disability can be placed into
software, and information commons developed that support the emergence of small
communities of learners.

The
material for this remediation software will come from a prototype distance
learning system [14], and from
hand notes and remembrances of the author.
Our freshman text will contain the reformulated computer science curriculum
and mathematics curriculums outlined by the author in 2005.

The
strategy requires that the student list the topics in curricular materials,
after having read or skimmed over the material. For example the student might be asked to start at the end of
Chapter 1 in a text book on college algebra and just write down on index cards
those word phrases that “pop out” as the student turns the pages (turning pages
from the end to the beginning). These
index cards are then the student’s first view of a nodal forest. The concept of a node is discussed, and the
elementary understanding of what is knowledge taxonomy is provided to the
students. A workshop was offered on
linguistic parsing using computers.

The
students are asked to see the nature of an abstract concept. It was suggested that all topics within the
curricular material could be put into three categories.

Students
observed that the placement of the topics in categories became dynamic and
depended very much on the passage of time, the student’s experiences and the
skill level that the student develops.

The
three categories are:

{ known, not known, not known that not known }

In 1998
the author started to refer to this curriculum and pedagogy as the Nodal Forest
Learning Strategy. The Nodal Forest
Strategy is to list the topics that are in the first two categories and sort
these topics into two categories. 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 those topics. The listing and separation into categories
opens up a set of “affordances” [15]
which involves the rehearsal of experiences at random times throughout the
day.

The
author’s citation into the cognitive neuroscience literature suggests that an
inhibition of natural interest has been effectively turned on by the learner
experiences and the social philosophy of mathematics dislike.

Various exercises are provided to
turn off this inhibition.

__ __

[1] Prueitt, Paul
S. Neuro cognitive basis for learn__ed__
disability in the learning of arithmetic, in progress.

[2] Levine, D. & Prueitt, P.S. (1989.) Modeling Some Effects
of Frontal Lobe Damage - Novelty and Preservation, *Neural Networks*, 2,
103-116;

[3] Levine D; Parks, R.; & Prueitt, P. S. (1993.)
Methodological and Theoretical Issues in Neural Network Models of Frontal
Cognitive Functions. *International Journal of Neuroscience* 72 209-233

[4] Prueitt, Paul S. (1996a) Optimality and Options in the
Context of Behavioral Choice, in D. S.
Levine & W. R. Elsberry, Eds. Optimality in Biological and Artificial
Networks?, Erlbaum, 1996.

[6] From:

http://www.wcer.wisc.edu/projects/projects.php?project_num=155
(2006)

SCALE brings together mathematicians, scientists, engineers, social scientists, and education researchers and practitioners to improve the math and science achievement of all students at all grade levels in the four participating school districts by engaging them in deep and authentic science and mathematics instructional experiences.

[7] The approach was suggested by one of the chemistry professors at the small college.

[8]
Prueitt developed two types of proposals for
curriculum refinement:

http://www.bcngroup.org/beadgames/LiberalArtsCore/home.htm and

http://www.ontologystream.com/beads/QuestionOfAccess/Syllabus.htm

[9] The author describes in Chapter 2 this “low zone tolerance” behavioral response by an immune system. Eisenfeld and Prueitt also published the basic research at the Santa Fe Institute in 1987. Additional preliminary work is in the dissertation by Prueitt in 1988.

[10] See the curriculum outline at:

[11] Prueitt, Paul Stephen (1988) Mathematical models of
biological intelligence exhibiting learning.
PhD thesis, University of Texas at Arlington.

[12] Prueitt, Paul Stephen (1997). “Quasi Axiomatic Theory and physical
stratification in nature”. Invited talk
given at a VINITI conference in Moscow.

[13] For an examination of relevant SOA standards see:

[15] Prueitt’s concept of affordance is derived from the usage of this term by J. J. Gibson, the father of the ecological psychology school, now centered at the University of Connecticut. Karl Pribram and Paul Prueitt discussed Gibson’s concept and Prueitt came away form this discussion (1992 – 1994) with two terms, “internal affordance” and “external affordance”.