Need for a Mixed Mathematics Curricula
School and college calculus and pre-calculus courses may develop as
easily and as much possible, the algebraic and logical reasoning skills
that the rigourous axiomatic derivation or codification that pure
mathematics requires, without insisting on nor striving for a full
logical codification. Ease of exposition should be the guide in course
design. Calculus and pre-calculus courses should not only prepare
students for possible studies in pure mathematics, but more importantly,
these courses should provide an empirical axiomatic framework and
sanction for a thought-based, operational command of calculus, trig,
Euclidean geometry and the calculations with units and coordinates that
appear in economics and technical trades or disciplines.
Mathematics education is a service industry which aims, we hope, to
develop confidence and comprehension of rules and patterns, steps and
methods, practices, with repeatable, reproducible and hence verifiable
results and conclusions.
Besides taking or developing the properties of real numbers as
axiomatic basis for algebra in senior high school mathematics, we
may assume rules and patterns (more axioms) to support and sanction the
role of the role of pure numbers and quantities in the applied
mathematics that arises from accounting, physics, chemistry and the use
of coordinate systems in single and multiple dimensions. The result
should be an education oriented codification and thought based derivation
not of pure mathematics, but of the applied mathematics needed for the
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the manipulation of units of measurement in applied calculations;
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the geometric role and use of real numbers and units as
coordinates.
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the development of coordinate-free Euclidean Geometry
The rigorous, context free, diagram-free development of pure mathematics
is not for students learning trigonometry, complex numbers and calculus,
and meeting there-in application involving items 1, 2 and 3. Those
applications require some empirical, geometric or physical assumptions
about working with units, coordinates and geometry for the sake of
consistency and completeness not in pure mathematics, but providing a
framework for confidence and skill in basic applied mathematics.
Any full Euclidean style axiomatic codification and derivation of
pure mathematics from assumptions about real numbers or sets, and
optionally, some applied mathematics extension with units and coordinates
might be left to after a mixed mathematics mastery of mathematics
in and before calculus.
Further Reading: See the discussion in Volume 1B, Mathematics Curriculum Notes,
of barriers to comprehension and the failure in pure mathematics
currricula to sanction the use of decimals and coordinates needed
to geometrically introduce and develop analytic geometry, trigonometry
and calculus. It is not possible to have a consistent modern
mathematics curriculum which includes and introduces trig, analytic
geometry and calculus geometrically with the aid of coordinates and
diagrams. That inconsistency implies the need for a mixed mathematics
curriculum dedicated to providing an operational command of skills and
concepts, axioms included, as prequel to any modern axiomatic
development.
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