Extrinsic Origins
Mathematics has an extrinsic or external origin. Over time,
we have learnt to describe physical quantities in terms of coefficients -
counts and numbers - of a unit. Modern mathematics with its assume
patterns (axioms) as starting points for a deductive arrangement and codification
of mathematical operations with numbers - whole to real or complex, etc -
gives an instrinsic (non-extrinsic) development
independent of the extrinsic origins. In the past, Euclidean geometry with
its definition, theorem and proofs provide an axiomatic model for the rigorous,
logical development of mathematics and further subjects. But the
development of Euclidean geometry depends on generic drawings, drawings
inspired say extrinsic, approximate and precise use of maps, plans and
designs and construction and navigation, and generic drawings may be
faulty. So there has been a movement in mathematics for the sake of
greater rigor and certainty to a more secure intrinsic and abstract
development and organization of concepts apart from drawings and the
geometric and physical assumptions there-in. That movement provides the
content of graduate and undergraduate studies in pure mathematics. That
movement provide motivation for modern (pure) mathematics curricula in the
period 1955-80 or so which in aiming to represent the axiomatic foundation
clearly and properly introduced some inconsistencies or incompleteness in
its secondary school development of mathematics from arithmetic to
calculus. Modern mathematics curricula did not sanction and so was
inconsistent with the use of decimals in arithmetic, the use of drawings
in Euclidean Geometry to arrive at results, the further use of drawings in
trigonometry and calculus to define and an analyze calculations.
Further more, following earlier traditions, it expect mastery of the
algebraic way of writing and reasoning by exposure instead of explicit
development. Furthermore, applications of mathematics and even
instruction in it applied subjects (geometry, trig and calculus) require
an extrinsic viewpoint to facilitate skills and concept development. Thus
an extrinsic view is unavoidable.
LLAMP aims for a consistent, accessible extrinsic
development of geometric and quantitative skills and concepts in an
empirical and thought-based manner. Due to the possibility of faulty
drawings, instruction offers a drill and practice based development for
solving a wide variety of culturally relevant problems in routine and then
perhaps more adventuresome, non-routine ways, repeatable and
reproducible, if not well-described, recorded and observable, for
the sake of verification or correction. Verification and testing of
solutions remains an empirical part of applied mathematics despite and
besides all deductions or logic in it that suggests the methods and
results in question. For students, their fellow-students, their
teachers and tutor are (optional) part of the peer review process
present in skill and concept development during instruction. Part of the
empirical development of science and technology is based on methods which
in practice produce repeatable and reproducible results alongside
theories, dare we call them stories, to describe and connect the pieces of
the practice and to provide a framework for comprehension absolute
not, and further repetition of the practices. While the empirical
development of science and technology requires labs and equipment too
expensive for in school use, LLAMP provides an operational command of
mathematics that may generated and verified in the classroom, and
also accompanied by a nearly full-thought based, extrinsic, development of
its skills and comprehension. The development will be nearly full
except for tables of values for key functions and/or the use of electronic
calculators to also provide and combine function values. In schools, the development of
mathematics and its applications may be self-contained and peer review an
immediate possibility
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LAMP
(first
draft, June 2008) a program for adult
and teen mathematics education
Mathematics education standards implied by calculus should
be a factor, not the only one, yet not a forgotten nor hidden one in course design
Area Intro
Introduction
Arithmetic
Geometry
Algebra
Logic
Calculus
Musings - More Ideas
More About LAMP
Evaluation
Maths Cultural Origins
First Nation Education
Modern Mathematics
Before LAMP
Problem Solving Skills Routine to Non
Instructional Concepts
Student Cooperation
Maths Extrinsic Origins
Science Education
For further musings or thoughts see site books.
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