Tuesday, June 15, 2004
The Open
Access Program in Mathematics and Computer Science
Tuesday, June
15, 2004
Some strategies for Opening
Access
The formation of supporting communities and individual
mentorships
A community grounding for Open
Access philosophy
The Safe Net for extended scholarly discussions
Obtaining credit for freshman
mathematics and computer science
The compositions
of university liberal arts mathematics curriculum have evolved under a set of
assumptions related to the high value that engineering and accounting has to
our society. These values will always remain
important. Yet exciting aspects from
the history, philosophy and foundations of mathematics are left outside of the
freshman mathematics courses.
The consequences
are profound. For example, many college
graduates are not aware of the difference between arithmetic and the core
concepts in higher mathematics. In this
sense, the freshman mathematics course fails to be liberal in nature. This liberal component might help in a
democratic society where the issues related to abstraction have a important
role in decision making, particularly in regards to information technology.
A new liberal arts
curriculum is justified. The current
curriculum is less that effective in open access to a liberal understanding of
higher concepts. This same curriculum
has not been accepted as something interesting to most freshmen. More can be expected from our educational processes;
both in terms of a satisfying freshman curriculum and as a preparation for core
concepts that are needed if computer based Human-centric Information Production
tools are to be widely adopted.
Our recommended curriculum
fully addresses the nature of computation and the nature of human
knowledge. The nature of computation is
simple. The nature of human knowledge
is complex.
Some things that
we can do to make this clear are:
1) The relevance of formal constructions to
understanding computer science and information technology can be established by
pointing to differences between an engineered system and a living system.
2) The nature of computation is critically
reviewed from a historical perspective that includes illustrations about the
successes and failures of information technology.
3) Formal models of some of the basic properties
of complex living systems introduce the new sciences of biology, psychology and
social systems.
We believe that a new
curriculum can be advanced within a social view that is found accepted by most
people.
Given proper
grounding in a socially acceptable philosophy, we may re-define the purpose of
freshman mathematics in the following terms:
1) Higher knowledge about mathematics can be
de-mystified.
2) The liberal art mathematics class will more
clearly lead into the life sciences as well as into the natural sciences and
technical disciplines.
3) The liberal art mathematics class will
provide a new and exciting path to higher knowledge about social philosophy,
the life sciences, and one own sense of self.
The technical
justification for a re-newal of the liberal arts curriculum can be found in the
scholarship of leading thinkers in theology, political science, social science,
and the life sciences.
New mathematics is
being created that has a focus on the foundations to computational theory, and
the applicability of computational theory to psychological, social and
ecological sciences. Liberal arts
survey courses can open access to the average student by properly developing
the foundational concepts for this new mathematics.
For students
interested in computer science, the new mathematics is more highly relevant and
makes a more direct path to topics that are of interest and a re likely to be
relevant in emerging marketplaces.
This curriculum
has as its primary objective the development of a mature and comfortable
understanding about elemental concepts of computer science and pure
mathematics. The objectives of the curriculum
are met when a student has a profound appreciation about the nature of formal
science and about its historical contexts.
In this sense, the objectives are not skills in a specific set of
problem solving exercises, but an appreciation of how knowledge fits into one’s
private life.
Nodal Forest Learning/Teaching Strategy
The Nodal Forest
Learning/Teaching Strategy works by allowing students to make lists of topics
that they see appearing in any textbook.
The pedagogy
involves allowing a student to make decisions regarding what is in a chapter,
regardless of whether the student understands the topic completely, partially
or not at all. The process of listing
the topics by name from a reading of the words in the chapter is mechanical and
does not trigger negative responses that the learner might have from past
experiences with similar curricular materials.
However, this listing process does start a familiarization process that
leads to learning opportunities.
Ideally the listing
is on index cards, but consistent with the philosophy of the Nodal Forest
Strategy (NFS), the student is given maximum freedom to make lists and
manipulate lists of topics.
We discuss in
class, the Nodal Forest Strategy and its purpose. The purpose is for each student to create a framework for
studying the topics of the chapter. How
this purpose is achieved is left to the creativity of the student. In some cases, the student will want to make
index cards so that the cards can be organized in various ways. However, other students will feel a need to
write and re-write lists of topics into a daily diary.
The framework has
three categories
Topics that are
known and understood
Topics that are
not known or not understood
Topics that are
not known that it is not known
The NFS framework
is dynamic and it is from this dynamic nature that a student learns, often for
the very first time, what mathematics is.
Students also learn how to study the concepts in mathematics. Mathematics is seen as a way of thinking
rather than a set of static concepts that have to be memorized. A sense of self-accomplishment is created.
Distance learning
modules should be available for students to see into more advanced notions from
pure mathematics; not as a requirement for the course but as part of an
exploration process. The student
becomes a member of a community that is being supported in a discovery
process.
Using web sources,
the freshman curriculum can be extended to on-line learning resources about
pure mathematics like topology http://at.yorku.ca/i/a/a/b/23.htm. It should not be assumed that
a non-mathematics major could, in no circumstances, immediately begin to
inquiry about higher pure mathematics while taking the freshman course. Access to some interesting curricular
modules is needed when this personal inquiry is made.
In addition to the
topology module pointed to in the hyper link, we
suggest that the following short
on-line curriculums be
developed and made available to students on a demand basis (e.g. within a right
size, just in time learning theory).
1.
Learning to
Learn Formalism is a curriculum
that is about the nature of abstract thought; philosophy, social theory and
cognitive theories
2.
Analytic
skills directed to problems
that are relevant to cultural understanding.
3.
Discrete
structure as a foundation to
computer coding.
4.
Introduction
to Elementary Number Theory as
a foundation to referential information bases
These short on-line curriculums are designed
to bring the social scientist, political scientist, religion major, or
philosophy major to an appreciation of modern techniques in computational
theory. The new mathematics curriculum
is designed to give a more direct path to pure mathematical concepts that have
elegance and applicability to notions needed by scholars on the non-engineering
sciences.
The essential nature of acquired learning disability
has both community and individual aspects.
Both aspects are addressed in the Open Access program for freshman
mathematics and computer science. The
individual will find knowledge about the standard freshman mathematics
curriculum and various opportunities to demonstrate skill over this
curriculum. Each individual will also
find intellectual challenges, including:
1)
The opportunity
to be part of a new community of individuals who commonly share a social
philosophy related to Open Access.
2)
The opportunity
to interact socially and intellectually with a group of scholars within a
structured on-line resource (called the Safe Net)
3)
The opportunity
to obtain (accredited) university credit for freshman mathematics and freshman
computer science courses.
We can talk each of these three challenges
and discuss them separately.
Human social philosophy
shares structural natures with the formation and use of natural language, in
exactly this sense: “individuals use
words in order to create specific understandings in the minds of others.”
Language owes its existence
to the use and acceptance of word meanings, but only in what the Scholars refer
to as the “Later Wittgensteinian Sense”.
Ludwig Wittgenstein is
known to have illustrated the notion that human language is used “as a game” in
pointing to things in the world. In
exactly this sense, Wittgenstein departed from Logical Positivism and the
Whitehead/Russell programme. The
programme for formalization of arithmetic had to be abandoned. The grounds for extending the strong
formalisms discussed by Hilbert and Godel was
established.
So the descriptive
foundation for playing language games, using on-line recourses, is grounded in
a foundational discussion about mathematics and logic. The playing of these games creates social
community, and this social community then creates added value to those who
would participate. The new mathematics
curriculum identified the issues to freshman liberal arts majors while also
developing skill at learning mathematics.
The doors to higher mathematics, relevant to modern challenged in
information theory, are pushed open.
The Safe net
concept was introduced in February 2004.
The Manor software, a
Multiple User Domain (MUD), is being used to develop the notion of a scholars Glass Bead Game. The Manor code base is available to the
BCNGroup, a not for profit foundation, to evolve into a dedicated distance
learning environment.
The development of
knowledge representation using Orb technology is being staged at one of the
Manor “sites”, with Manor address:
manor://bcngroup.excaliburhosting.com:12025/?1205. This is an experimental deployment of advanced collaborative
software.
The practical
consequence of most freshman experiences is that students dislike mathematics,
or what they think of as “math”, more after completing the freshman
requirements than they did before enrolling for the first time. A number of reasons exist for this outcome.
This is not
everyone’s experience. Some find the
current instructional pedagogy fine and the curriculum interesting and
relevant. However, in surveys conducted
by the BCNGroup, we find that adults remember the mathematics class experience
with frustration and often with deep resentment. The resentment comes from two sources. The curriculum is the same as what is taught in high school, and
this curriculum is one that is rejected by the peer community of students as
being boring and not relevant to private interests.
The flip side of
high school peer evaluation, by students, is that private interests in higher
mathematics and science is present but is repressed by a feeling of
powerlessness to learn. The
immunological and neural grounding of our theory of learned disability suggests
that specific re-enforcement mechanisms are involved in producing the sense of
learned helplessness. The sense
is specific to the high school curriculum, and thus not clinically what is
referred to in the literature as learned helplessness. Also see work on attribution theory.
Prueitt’s work on
instructional design has lead to some new results. His early work appears to demonstrate that the introduction of
some specific novel elements into a learning module will remediate the acquired
learning disability and renew the individual student’s sense of discovery. The theory has considerable work to be done
before it is published, but seems promising.
What the theory promises is that a major part of the poor performance in
freshman college mathematics classes is artificial and can be completed
remediated.
It is at this
point that acquired learning disability is seen to be quite different from
attribution models. The attribution
models talks about a cognitive re-enforcement via self image attribution. Acquired learning disability is more
complex, in that the individual is likely to not realize that there is
something that could be done. Acquired
learning disability is not global, and is re-enforced also with a social
philosophy that allows individuals to be good in some areas and poor in
others. Thus one has to address both
the social philosophy and the remediation process. One also is able to rule out a natural cause of the poor
performance in simple mathematics tasks.
Specifically the
strategy can involve the re-teaching of addition, multiplication, and division
in bases other than base 10. This is
not an easy task. Novelty engages
selective orienting mechanisms involved in learning. So the model of biological reactions in immunological circuits is
used to understand learned learning disability.
If real learning about
the nature of mathematics can be achieved while these mechanisms are engaged,
then a new beginning can be developed of the student. This is the hope we have.