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LAMP and Modern Mathematics
The modern mathematics curricula of the period mid-1950s to 1980s
determined what would be taught and how, at least to the advent of the
constructivist approach and dogma for instruction in mathematics and
science.
LAMP could be have been named LEMP, an acronym for Lean Extrinsic
Mathematics Program.
LAMP offers an extrinsic rather than intrinsic
development in which quantitative skills, concepts and principles
(patterns or axioms) are derived or inferred from figuring and drawing practices
in daily life that appear to give repeatable and reproducible results.
LAMP offers a replacement for the modern mathematics curricula in secondary
instruction from arithmetic to advanced calculus. Yet in doing so, it provide an
extrinsic context and the algebraic-deductive maturity (we hope) to enable
interested students to obtain a clearer and better understanding of modern
mathematics.
Mathematics originated from extrinsic considerations before
its more rigorous, intrinsic, pure mathematics codification in terms of
axiomatic set theory - the basis of modern mathematics and the motivation for
modern mathematics curricula. The intrinsic or context-free
view of pure mathematics starts with axioms for sets or real numbers
to provide a
logic-based codification and development, context-free. So extrinsic
considerations - the drawing of diagram as in Euclid's elements or its
successors - are excluded from advanced mathematics for the sake of rigour.
The modern mathematics curricula duplicated some of that rigour its course
designs and materials, but did so inconsistently without stating nor
acknowledging the tacit assumptions needed to illustrate and apply
mathematics.
The LLAMP extrinsic
development envelopes the tacit assumptions needed to apply mathematics
and in doing so arrives extrinsically at the field properties of real
numbers, in other words, axioms for real numbers assumed in the modern mathematics curricula.
LAMP does more. It arrives extrinsically or geometrically at the field
properties of complex numbers. The extrinsic development in LLAMP is no
less and no more rigourous than the extrinsic development of geometry and
trigonometry present in the modern mathematics curricula. Further LLAMP adopts
and sanctions the decimal representation of real numbers from first use to the
discussion of limits and convergence in calculus where as the modern mathematics
curricula avoided all mention of decimals in its theoretical development while
employing a mastery of decimals in the representation of irrational
numbers and in the numerical evaluation of function and limits from trig to
calculus. Whence the adherence of the modern mathematics curricula to an intrinsic
(context-free) development was inconsistent.
This LLAMP approach furthers implies the algebraic and deductive
maturity needed for a later mastery of intrinsic axiomatic
development of mathematics from assumptions about real numbers or
sets. The LLAMP extrinsic approach also provides a context and
motivation for the intrinsic axiomatic approach. Yet LLAMP alone may be
sufficient for people without the time and inclination to meet and master
pure mathematics.
Summary: The modern mathematics curricula intrinsic approach to
mathematics was mixed with extrinsic approaches to real numbers, to geometry,
trig and calculus inconsistent with the intrinsic or context free approach to
modern mathematics and represented in the algebra portion of modern mathematics
curricula. That realization in fall 2007 led to the notion of providing a
replacement LLAMP based on inductive principles for skill and concept mastery
from arithmetic to calculus in ways that will be supported by and be motivated
by applications and the availability, but not imposition, of a thought-based
development of all or almost skills and concepts.
Remark: For students for whom calculus preparation is
not a possibility or not wanted, the thought-based development is optional.
Yet the operational mastery of rules and patterns, one at a time, one after
another, alone or in combination, may eventually make the thought-based
development accessible to those for whom it initial imposition may be a
burden.
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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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