Prolog: On Optimizing the Design of Introductory Mathematics

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Physics teachers are universally dismayed by the paltry understanding of mathematics that students bring from their mathematics courses. Blame is usually laid on faulty teaching. But I hold that the crux of the problem is deeply embedded in the curriculum. From the perspective of a practicing scientist, the mathematics taught in high school and college is fragmented, out of date and inefficient!

The central problem is found in high school geometry. Many schools are dropping the course as irrelevant. But that would be a terrible mistake, for reasons already clear to Galileo at the dawn of science.

• Geometry is the starting place for physical science, the foundation for mathematical modeling in physics and engineering, and for the science of measurement in the real world.

• Synthetic methods employed in the standard geometry course are centuries out of date; they are computationally and conceptually inferior to modern methods of analytic geometry, so they are only of marginal interest in real world applications.

• A reformulation of Euclidean geometry with modern vector methods centered on kinematics of particle and rigid body motions will simplify theorems and proofs, and vastly increase applicability to physics and engineering.

A basic pedagogical principle:
The depth and extent of student learning is critically dependent on the quality of the available mathematical tools.

Therefore, we can expect a well-designed curriculum based on vector methods to produce significant improvements in the depth, breadth, and usefulness of student learning. Further enhancements can be expected from software that facilitates application of vector methods.

Whether or not the high school geometry course can be reformed in practice, the course content deserves to be reformed to make it more useful in applications.

Objective of this workshop: To demonstrate with specific examples how geometric algebra unifies high school geometry with algebra and trigonometry and thereby simplifies and facilitates applications to physics and engineering.

References for further study:

D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics," Am. J. Phys. 71: 104-121 (2003).

D. Hestenes, New Foundations for Classical Physics (Kluwer, Dordrecht, 1986, 2nd ed. 1999) {Call it NFCM.}

Websites:

Modeling notes and papers (D. Hestenes)

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