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C00002 00002	Discussion  inspired by "Computer Science, Federal Programs and Nivrana"
C00005 00003	         "...it  is   time  to  start   thinking  about
C00008 00004	One of the most fertile and elegant fields  of mathematics is that of
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Discussion  inspired by "Computer Science, Federal Programs and Nivrana"
by Kent K. Curtis, Division of Computer Research, NSF.

 can c.s. become foundational in sense that math is?

 tell what's wrong with new mth
  not fundamental
  bad teachers

problem solving ---structurre

problems with papert and coding mentailty

coding --programming
calcualtors-- mathematics

improtance of ds. relation to num thery. (level of detachement)

usefully in h.s. and  other educational instutions.


good quote
"...it is time to start thinking about teaching mathematics in the context
of computer science rather than teaching computer science in the context of
mathematics."

disagree on limitations of machines.

he related to minority education...show approach appliccable to
any...not a question of high hardwaare  costs...simplle  system is
quite satisfactory.

dijkstra's problem....numerical examples

hum.... why study math? ways of thinking or necessary for computational skills

computational skills => calculators
ways of thinking still open and claim better handled in less regimented way

why study propramming? way of thinking or necessary for job
job =>  how many really want to be coders.. for that's what is taught
unless  it's ways of thinking.


note 
in h.s. algebra no idea of computational scheme for functional notation.
in calculus not shown that diff is recursive


historical accident: numbers rather than d.s.

reinforce, not frustrate, intuition
         "...it  is   time  to  start   thinking  about
         teaching   mathematics  in   the   context  of
         computer science rather than teaching computer
         science in the context of mathematics."

Discussion  inspired  by  "Computer  Science,  Federal  Programs  and
Nivrana" by Kent K. Curtis, Division of Computer Research, NSF. 

His paper investigates the plausibility of proposing computer science
as the foundations of sceintific thought. His general feeling is that
there is  sufficient substance  to the  field  to seriously  consider
such. 


I  too  feel that  there  are deep  fundamentals  in  what is  called
computer science,  and  the point  of  this note  is  to lay  out  an
argument  for such  a  program.  We believe  that  we can  show  both
philosophical  and pedagogical  arguments for  such a  campaign.   The
piece of computer  science which  is fundamental to  our argument  is
what is called abstract data  structures and abstract algorithms.  In
the   philosophical   portion,   we  will   argue   that  algorithmic
computations  on  such data  structures  properly  include  those  of
elementary  number theory.    In the  pedagogical  discussion we  will
argue that such computations are more natural and inviting than  what
is  typically required  in  conventional mathematics.  The  intuitive
motivation  and reeforcement  of realizablility  on machines  is also
examined. The approach  is contrasted  to that espoused  by the  "new
math". 

One of the most fertile and elegant fields  of mathematics is that of
number  theory   and  its  computational  counterpart  of  elementary
recursion theory.   The superficial  aspects of  this discipline  are
covered  in  hgh school  algebra.    Namely  the bare  essentials  of
algorithmic  processes and the properties of integers. Unfortunately,
such an endeavor presupposes a grounding in basic mathematical skills
and that  is usually accomplished  by grade school  exercises in rote
memorization.  There are at least two questions here: are ther better
ways to  teach  mathematics, and  what is  the  reason for  requiring
mathematical training of all students who graduate from high school. 

The question  of relevance is frequently  raised by students studying
mathematics.  In is not sufficient to dismiss these questions  as due
to lack of  maturity or understanding on the part  of the student. We
should  be able to respond.  The primary  reason, it seems to me, for
studying mathetics is  to cultivate ways  of thinking: precision  and
logic.   We  are  not in  the business  of  training arithemeticians.
Indeed with  the  advent  of pocket  calculators,  the  questions  of
relevance become much more pressing.  Perhaps a clearly example stems
from Plane  Geometry; we do not teach  geometry, with the intent that
students become geometers.  Geometry courses again  cultivate thought
patterns;  here  the  ideas  are  axiomatics  and  proof  techniques.
Perhaps  one of the  motivations for advocating the  New Math program
was that it  too was a way  of cultivating thought, dimisnishing  the
punishing rote  work of typical mathematics courses  and relpacing it
with an understanding of  the structure of  number systems. What  was
missing in the approach was reinforcement  through intuition. We will
say  much more about  this later.   What is  really needed then  is a
fundamental discipline  which can  give the  intellectual meat  which
mathematics has supplied, which can be shown to be relevant to modern
society and is reasonably free from unnecessary drudgery. 

One of the points of this note is that computer science has presented
us which an even  richer environment that elementary  number theory. 
And  what is  equally important  the structures  are as  intuitive in
their development and their relevance to society, and therefor  their
motivational character  to eduation  is significantly  greater.(gasp,
gasp) In particular we are talking about the study of data structures
in computer  science.  The  analogous discipline  to that  of ent  is
LISP. 

The formal development  of LISP equally impressive.  We  begin with a
primitive domain  of atomic elements, and a binary successor function
named cons.  The domain of elements we consider in LISP is called the
symbolic expressions  and consists of  the atoms and  all the objects
constructed using  the  constructor, cons.    In LISP  we  also  have
recursion, now  over symbolic  expressions; and  indeed the class  of
computations includes those of elementary number theory. 

What  takes this endeavor  form a  mathematical exercise to  a viable
tool is the recognition of our ability to represent many phenomona as
computations  over  symbolic  expressions.   The  key  idea  here  is
representation. Conventional mathematics is constrainted to represent
any  phenomonon as  a  numerical  problem;  the data  is  encoded  as
numbers, and  the dynamics which we  which to model is  forced into a
relationship over numerical  functions.  Now  clearly there are  many
phenomona which  are  not naturally  numerical; the  problem is  best
described  as a  structural one involving  interrelationships between
non-numerical data.  Thus  a candidate for a LISP-like  computational
scheme. 

It is interesting  that the idea of non-numerical  computation had to
wait  until the advent of the digital  computer. There is no a-priori
reason why non-numerical  algorithms and the ideas  of representation
could not have been developed  in Greece as the numerical ones were. 
Perhaps the  complexity of  the computation  is too  demanding to  be
enticing. Be that  as it may, we  are currently in a  very desireable
position.  We  can apply this hindsight to begin a very revolutionary
development in education. 

This  sounds  like  the  propoganda  for  the  ill-fated  "new  math"
experiments.   The current  proposal is  quite different.  First, two
criticisms of  the "new math" which are relevant to our discussion. I
feel that  the "new math"  was not  founded on fundamentals;  granted
that set-theoretic  ideas are fundamental in  the strict mathematical
sense, they are not fundamental in the intuitive sense. There are  at
least two  problems here:  sets and  set relationships  are just  too
denuded  of structure.   The  human mind sees  and likes  patterns or
reasonable complexity.  To flush  out this structure for the sake  of
logical parsimony is unnatural. A  strong criterion for a fundamental
educational  discipline thus is intution.   Whatever the endeavor, it
has a significantly haigher chance for success if it  is based on, or
can relate to,  human intuition.  Indeed, at  the beginning levels of
education we  have little  else  to depend  on  but common  sense  or
intuition.  Recall the  constant complaints of the parents  about not
understanding   what  their  children  were   doing.    Something  is
definitely worng here.   People  are not stupid;  the difficulty  for
both the parents  and the children was lack of  intuitive support for
the set-theoretic ideas.  We must remeber this difficulty when making
our case. 

The second  criticism indeed  is only another  manifestation of  this
anti-intuition; that  is, the difficulty with the  teaching of such a
discipline. 
One of the difficulties with the new math program and a difficulty
inherent in the prososed scheme is the lack of trained and dedicated 
teachers. We feel however that such defficiencies can be minimized
by a well-prepared text  coupled with a viable computer-based system.

Indeed we are preparing such a text( toot, toot) for advanced
students and  are considering the problems of writing one for
more elemenatry purposes.


******rip education*****