Mixed Maths Curricula

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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

the manipulation of units of measurement in applied calculations;

the geometric role and use of real numbers and units as coordinates.

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.