January 22, 2008
Potential Sponsored Program
Proposed by Dr Paul S Prueitt
Index
From
the theory, there is a remediation strategy!
Overcoming the point of failure
An
example of the value of teaching about shifts
I am writing various proposals to
develop and refine a bridge program between high school and college. The program should be applicable to any
community college, four-year college, or university.
I believe that the bridge program has
two elements that will make it feasible as a national project, either as a
federal program or as a commercial enterprise.
The delivery system is one part, using web based learning systems such
as Blackboard. The second element is a
specific learning theory, and the grounding of this learning theory in natural
science. [1]
The intent is to demonstrate that a
specific program, having specific pedagogy and curriculums, will improve
freshman mathematics and science retention and can be used to assist in
recruitment of new freshmen. We
conjecture that the improvement will not be incremental, but substantive. The substantive nature is due to an assisted
change in student motivation and expectation.
The program has had initial
developmental phases in HBCU settings; Hampton University (1988-1991), Saint
Paul’s College (1993-1994) and Talladega College (2007-2008); but is not
specific to these settings. The next phase
could be developed at any small four-year college in the United States or at a
community college. Or the next phase
could be developed from one of the major universities.
Of course each person is unique. However, the situation in mathematics
education is surprisingly uniform. It
seems that a transition program is often needed due to the extreme anxiety
feeling experienced by most freshman students.
A transition program may open the mind of a student to the possibility
that mathematics can be learned. After
such a transition, I believe that a proper education can be done within the
traditional six-hour requirements for the liberal arts degree.
To be successful at any college or
university, the mathematics and science faculty at that college or university
will require evidence regarding the central conjecture; that of a ubiquitous
“acquired learning disability”. The
faculty should be aware of conjectured causes and remediation. Many efforts have been attempted in the past
and mostly meet failure, so it is vital that the conjecture is clear and that
the remediation be grounded in methodology.
So this conjecture has to be addressed up front. The conjecture is what is new and what might
make this program different in outcomes.
The focus on a bridge program is on
mathematics but has certain interdisciplinary features. Interdisciplinary reading in areas
supportive of a new curriculum in mathematics and science is suggested, as
discussed below. The interdisciplinary
nature of the program is essential if we are to engage the whole person. The nature of the program is introductory to
what college experience requires from students, and thus a focus might be on
how human thought arises, and in how study habits might be honed. From our experience this year, the students
do see that serious study leads each student to a new understanding about why
he or she is in college. Thus the
benefits from the program, as should be expected, are to the entire freshman
program. We look for a reinforcement of
interest and capability in some students and a complete overcoming of blocks in
other students.
I have conjectured that the
self-efficacy of individual children is shaped in a way that negatively impacts
performance in all areas of academic work, but most particularly in
mathematics. This conjecture is
consistent with what is seen by college and university faculty. In most cases, the faculty of science and
mathematics are puzzled as to the declining performance of each generation of
freshman students. There is; however,
little real scholarship on why this decline is occurring. We all seem to agree that a failed system is
being preserved, but beyond this we seem unable to make responsive changes or
even to agree on what would be responsive.
What might be useful is something that shifts the perspective we all
have about why mathematics is part of the liberal arts curriculum at all. In fact, shifting perspectives becomes the
philosophical principle that students begin to realize. We take on the core philosophical issues so
that these are discussed, but no long become means to slow down the instruction
about the mathematical principles.
The bridge program creates a broad front
that engages students (all freshman) in curriculum and experiences that are new
and relevant. The outcomes should accrue
in all ways that are typically expected of a good freshman mathematics
program.
The first module I developed is about
shifting the representation for numbers between number bases and then doing
first arithmetic and then some matrix algebra; e.g., Cramer’s rule in arbitrary
bases. The arithmetic in arbitrary number bases provides one portal into a view
of mathematics that is relevant to liberal arts students. This includes the concept of solution set, a
plane or cross product, graphs and functions.
A second module is being developed in set theory and probability.
A web-based platform would provide the
novelty suggested by my remediation strategy in a fashion that the students are
used to, and would provide 7-24 access to a process assisting each person in
mapping their own experiences and understandings about foundational concepts.
A virtual bridge program could, if
developed properly, address the problem of student efficacy using a specific
strategy in the mathematics component. The strategy requires clear guidance to
students who must understand what they know clearly and what is not understood
clearly. This strategy is contrasted
with teaching that moves steadily through a surface understanding of arithmetic
and algebra and with students who often claim to know nothing what so
ever. The student behavior shifts if
the rules are changed in specific ways.
The change is not simple, and the processes involved in shifting
behavior cannot be generalized from my experience without some support structure
and training program. So even though I
have created a foundation in experience and theory. There is a great deal to do before a complete theory is
available.
It is worth repeating what was said in
the previous paragraphs. Entering
freshman students have been so profoundly confused by the instruction in
arithmetic that they, each student, do not know what they know. Some are prepared for college, but most are
not. There is no foundation on which
these students can stand, and little motivation other than to get the college
credit anyway they can. In extreme
cases, students are agnostic about learning the material in the curriculum and
have a deep seated resistance to engaging in any experience that takes
responsibility for any level of knowledge of arithmetic, set theory or
algebra.
By re-teaching arithmetic in bases other
than ten, I have found a means to present very challenging material that does
not have any pre-requisites. The theory
was developed in the late 1980s, and tested at Hampton University and Saint
Paul’s College. However, it was only
this year that the full methodology was used.
The key feature to the particulars of this new curriculum is the
existence of a set of methods for creating an inventory of topics known, topics
that are not know or for which there is an uncomfortable feeling. This feature, e.g., topic inventories,
creates the necessary foundation for a deep study of a few essential concepts.
The novelty of arithmetic in arbitrary
bases creates an affordance leading into an awareness that arithmetic is both
interesting and learnable. Once this
awareness exists in a strong fashion, there is a possibility of shifting the
self-image from one that knows that he or she cannot learn the freshman curriculum. The essential concepts in the traditional
college mathematics courses include sets, the real numbers, an equation, the
concept of a function and the solution set to equations and functions, as well
as the lines and parabolas. In addition
to these concepts; the student becomes open to learning the foundations to
discrete mathematics, computing and probability.
A
strategy for constructing a cognitive map of what is known
A strategy emerges. This strategy is to study what one knows,
and NOT study what one does not know.
The reason this strategy works is due to the acquired learning
disability itself. Because of the
nature of this disability, our students react to material that has become
confused. The reaction is frustration and even anger. By not studying things that are not understood, new orientations
can be produced whereby the self-efficacy shifts. Then, and only then, is the student in a position to take
advantage of the teaching methods. The
strategy is also constructivist in nature. [2]
It is not only the individual that has
adopted specific efficacy. The
situation can be clearly seen in this way.
Students are so uncomfortable studying the traditional mathematics
curriculum in the traditional way that their ability to perform is impacted. Only by relieving all of the students in a
class, or even on a campus, of a fear of failure can the learning process be
legitimized. The social structure is
strong enough to hold all students to old expected behaviors, unless there is a
consistent re-programming of expectations.
This is not to say that expectations are to be lowered, only that the
shift in pedagogy and curriculum comes to be seen by the student population, as
a whole, as deserving of the considerable effort that students must make. Once the shift has occurred, a new dynamic
takes over where individual students begin to really excel.
There are two sides to the efficacy
problem. One side is the collective
body of institutions, with the current generation of entrance exams and other
procedures. These procedures end up
accommodating those aspects of a system that has failed. For example, entrance exams are given in
poor testing environments to students who have not been in a mathematics class
during their junior and senior years.
This procedure is understood by almost everyone as being poorly designed
and poorly executed; but nevertheless the procurement is considered
sacrosanct. Then students are gathered
into large classrooms where expectations are low.
On the other side are students;
individuals linked as a community by common experiences and place in
history. This community sees
mathematics classes as some type of punishment.
Clear leadership is required. Put in simple terms, the decision to adopt
the suggested innovations in pedagogy and curriculum has to be strongly held by
the college or university leadership.
A single faculty member will not be successful by him or her self. It is also possible that individual faculty
will feel threatened by the possibility of deep and profound changes.
Educational leadership will respond to
the promise of a well thought out recruitment program. This response addresses a bottom line
related to incomes and also success in fulfilling the mission of the college or
university. Because of the nature of
the ALD conjecture, recruitment may be is combined with actual enhancement of
the motivation and capability of each enrolled student. Thus dollars spent on
addressing the failure in freshman mathematics are dollars well spent.
Rising waters lifts all boats. The proposal being developed is that a
national system may be established that assists in individualized college
recruitment via a web based high school to college bridge program. Each bridge program will present
opportunities for high school graduates to enrich their working knowledge in
the humanities while opening their eyes to those opportunities provided by a
number of community colleges, four-year colleges and universities.
Those institutions of higher learning
that use the bridge program should expect to see higher enrollments and better
retention rates. Faculty will have
better prepared entering freshman.
Students will not only step into college at a higher level of work, but
will be able to see recruitment presentations by various member colleges.
The bridge program is designed to be a
virtual program and to “lift” pedagogy from the classroom experiences into a
process that involves the development of distance learning content, procedures
for enrollment, and procedure for the conduct of classes.
The program will have modules. Each module could be one-third the length of
a college course. There could be three
modules, one for literature; one for mathematics, and one for writing. The literature module might focus on the
history of mathematics and the writing module might assist the student in
expressing his or her feelings about the personal experience of mathematics
training.
Over time, an accreditation of some sort might be
considered necessary, in the usual sense; and the desired results might be
measured using performance outcomes. A virtual bridge program might be governed
by a board composed of educators and might have a mission that is oriented
towards service to the high schools, the colleges and to society.
My PhD was granted by University of Texas at
Arlington Mathematics Department in 1988, and was followed by a number of
journal articles and book chapters. [3]
I had been Assistant Professor of Mathematics at the elite HBCU, Hampton
University (1989- 90) and Associate Professor of Mathematics at a small HBCU in
Virginia (1993-94). Between these two
appointments I was a Research Professor at Georgetown University, where I
worked on issues related to artificial intelligence, collective intelligence,
and biological models of intelligence.
Between 1994 and 2005 I worked as an independent consultant on various
projects involving the design and development of intelligence systems. This
work allowed me to stay engaged in issues related to cognitive science and the
sense of self that often defines individual performance.
I am asserting that the normal
functioning of a human action perception cycle has been, in most instances,
interrupted by poor instruction in mathematic in high schools and by media
programming. One may consider the
concept that there is a utility function governing the evolution of social
attitudes towards higher learning, and towards the mathematics and sciences in
particular. The resulting biological
response is modeled from theoretical immunological models combined with neural
cognitive modeling. This response is
conjectured to be an acquired inability to think about the materials in
standard mathematics curriculums.
The first principles of this strategy
involve shifting the responsibility for knowing what the individual student
knows, from the textbooks and professors to the individual. This shift involves specific strategies and
has a distinct social component to it.
The blank paper test and the modified R.L. Moore classroom pedagogy are
two elements of the learning strategy.
Reassignment, and student advancement provides additional motivation. Reassignment provides clear goals and
enhances the student’s expectations.
The blank paper test requires that
students treat knowledge of the curriculum in a way similar to knowledge of the
content and themes of a novel read for English class. Students are shown how to rehearse an exposition of the topics
covered in class. Each test is
comprehensive, going all the way back to the beginning of the semester, and
presenting all concepts in a minimal fashion.
The student takes responsibility for knowing what is known and also even
to describe on blank paper what the student does not understand properly. Each test is graded based on a subjective
evaluation about the clarity of the student’s perception of materials that the
individual has taken responsibility for.
The learning community concept may be
evolving to now include shared experiences from one college campus to another,
thus creating a type of virtual campus environment. The bridge program fits within this virtual environment and
creates an interdisciplinary framework for student advancement in his or her
understanding about formal systems.
Student recruitment from high school as well as a high quality first
semester college experience for each individual student is possible.
Part of the ideal support process is
a procedure that allows students performing at roughly equal levels to be in
classes by themselves. As has been
shown from the experience in fall 2007, an attempt to implement this ideal can
be a point of failure. The institution
may not be able to support the movement of students from one course of study to
another in the middle of the semester.
However, this movement may be essential to a continuing alignment
between a class and a curriculum. As
students assume responsibility for knowing what they know, the teaching effort
can be directed at extending from what is known to what is not yet known. If there is no flexibility, students
sometimes get bored and quit attending class.
It is sometimes the case that the
institution just does not get it.
Students of mixed capability and having various kinds of learning
disabilities will be herded into over crowded classes where the primary issue
is class attendance. Students who miss
several classes show up and demand to be taught under the assumption that
absolutely nothing has been learned and absolutely nothing will be
learned. The individual crisis controls
the conduct of the class.
The point of failure is systemic. The traditional two semester freshman
courses in mathematics do not recognize acquired learning disability, nor the
possibility that individual student motivation may change radically under
certain positive circumstances.
The bridge program is designed so that
student engagement and level of curriculum are handled separately. The result of the bridge program will be a
proper placement of the student into mathematics, humanities and science
courses, depending on inner student motivation and demonstrated
capabilities.
Distance learning environments provide
new means to support learning experiences.
The bridge program will exploit these means. As specific acquired learning disabilities are indicated by class
performance, students may be moved into mini-courses designed to meet specific
classes of disabilities, or to reward success.
Remember that the pedagogy moves responsibility for learning from the
teacher and textbook to the individual student. With the assumption of
individual responsibility, the student engagement means that students can
express inner interests in various topics in mathematics, humanities and
science.
The fact that a diversity of shifts of
viewpoint lead to a correct answer, creates strong psychological barriers to
students even when learning, or not, how to multiple two simple polynomial
expressions such as:
(x^2 + x – 1) ( 2x – 1)
A confusion over how to solve this
particular problem blocks understanding because there is a non-agnostic
underlying cultural belief. The
cultural belief is religious in nature, but even individuals having strong
rejections of religion will have the cultural belief that “truth” can be found
in only one way. This belief is
enigmatic in nature, and is sorted out only with some deep understanding about
the nature of particulars and universals.
The pedagogy I use, teaches that the nature of mathematics is
“artificial”, and is part of the creation of the human mind. Thus the nature of mathematics, and the
nature of other cognitive constructions can be seen to be not of the same kind
of knowledge as religious knowledge.
One simply must understand the nature of
theory to see why mathematics works.
Without the ability to shift viewpoint, to see more than one way to
obtain a solution, one is kept away from this experience. The first part of our curriculum is designed
to give each student this experience.
The shifting from one number base to another number base in solving
elementary problems from college algebra creates an experience with “theory”.
Creating shifts in viewpoint is a
central focus of my freshman program. Only after the student is able to command
a small but clearly coherent theory about number bases, is the multiplication
of polynomials looked at. The full intention
of the pedagogy is then honored as students are given a series of problems that
they have not seen before and which go at the nature of the variable as well as
the notion of equality.
Critical to the application of this
pedagogy are (1) a sense of novelty and (2) a feeling that a well-delineated
boundary separates what is understood clearly and what is not understood. In other words, one needs to know what one
does not know. The student also must
have a foundation, no matter how small, in which he or she is absolutely
comfortable.
The facts are clear. First, students present challenges to our
teachers in our classrooms. Second,
these challenges are themselves not well understood by the teacher profession. The universities and colleges have not
understood the crisis in mathematics and science education, and have in the
past developed resistance to approaches that changes the current system.
It is tacitly assumed that students, who
do not learn, cannot learn due to an absence of proper motivation and/or due to
an absence of ability. Nature verses
nurture controversy arises and this controversy is not settled. The limitations themselves serve real cultural
purposes, related to the restriction of social groups. Well-established cultural norms put
individuals into a proper social order depending on cultural histories and
expectations. These cultural
limitations are the subject of studies in sociology, politics and history. The linkage should be made, making it
explicit that the current system is, in general, not performing well.
There are only a few in the mathematics
education community whom are prepared to examine closely why remediation during
the freshman year of college is required, or why remediation efforts most often
fail. So what
can be done? My
experience suggests that the development of a web based high school to college
bridge program will be seen as a means to outsource the necessary tasks related
to preparing all incoming students into college systems. Colleges will welcome this.
The bridge program presents
opportunities for high school graduates to enrich their working knowledge in
the humanities while opening their eyes to those opportunities provided by the
best community colleges, four year colleges and universities. The psychological and social dimensions
expressed in acquired learning disabilities should be openly revealed to the
students and methods made available so that each individual may learn how to
overcome these acquired traits. Such a bridge
program cannot, of course, replace the college experience. What the bridge program is designed to do is
to is to overcome the institutional limitations imposed by high school
training, and accommodated by college systems.
[1] Prueitt, Paul S (2008) “A Research Project on Mechanisms known to be involved in learning”. Unpublished paper developed as part of application for emplyment.
[2] Constructive theory is linked to the neural and immune theory of mechanism in my unpublished paper “A Research Program on Mechanism”.
[3] Selected publications:
Prueitt,
P.S. (1994). System Needs, Chaos and Choice in Machine Intelligence. Chaos
Theory in Psychology (A. Gilgen and F. Abrams, Eds.) Contributions in
Psychology Series. Westport, Conn.
Prueitt,
P.S. (1995) A Theory of Process Compartments in Biological and Ecological
Systems. In the Proceedings of IEEE Workshop on Architectures for Semiotic
Modeling and Situation Analysis in Large Complex Systems; August 27-29,
Monterey, Ca, USA; Organizers: J. Albus, A. Meystel, D. Pospelov, T. Reader
Prueitt,
Paul S. (1996a). Is Computation Something New?, published in the Proceedings of
NIST Conference on Intelligent Systems: A Semiotic Perspective. Session:
Memory, Complexity and Control in Biological and Artificial Systems. October
20-23.
Prueitt,
Paul S. (1996b). Semiotic Design for Document Understanding, in the proceedings
of the Workshop on Control Mechanisms for Complex Systems: Issues of
Measurement and Semiotic Analysis: 8-12 Dec. 1996.
Prueitt,
Paul S. (1996c). Structural Activity Relationship analysis with application to
Artificial Life Systems, presented at the QAT Teleconference, New Mexico State
University and the Army Research Office, December 13, 1996.
Prueitt,
Paul S. (1997a). Optimality and Options in the Context of Behavioral Choice, in
Daniel Levine and W. Elsberry (Eds). Optimality in Biological and Artificial
Networks?, Erlbaum
Prueitt,
P. (1997b). Grounding Applied Semiotics in Neuropsychology and Open Logic, in
IEEE Systems Man and Cybernetics Oct. 1997.
Prueitt, P. (1998). An Interpretation of the Logic of J. S. Mill, in IEEE Joint Conference on the Science and Technology of Intelligent Systems, Sept. 1998.