A Question of Access


Part 2: Chapter 6

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

Last edited: Saturday, June 17, 2006



Purpose of the chapter


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 21st 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In 2002, the author made the decision to rewrite a version of an unpublished manuscript “A Question of Access” that the author wrote in 1987.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 learned 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.

[5] URL: http://www.bcngroup.org/area3/pprueitt/book.htm

[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


[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:


[14] URL:


[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”.