Update, 9 December 2012: Don't miss Alan Macdonald's Linear and Geometric Algebra, which is recommended enthusiastically by Hestenes.

Although New Foundations for Classical Mechanics (NFCM) is primarily a physics book, it's also intended to demonstrate the usefulness of geometric algebra (GA) in solving any sort of problem whose data and unknowns can be formulated as vectors.

Several previous reviewers were more qualified than I to discuss the advanced aspects of this book. I review it from the viewpoint of someone who was considering Hestenes' advice, expressed elsewhere, to employ geometric algebra in high-school classes. Of course I didn't expect that New Foundations would be suitable for high schoolers. Instead, I wanted to decide whether GA might save students enough time in college to be worth introducing in high school. To that end, I worked many of the problems in the first 3-1/2 chapters, then skipped to chapter 5, where I have worked on only the first section. I also attempted, with mixed results,* to solve classic geometry problems via GA, especially those involving construction of circles tangent to other objects.

That amount of experience is probably necessary to decide about trying GA in high schools. My own decision is a cautious "yes", with some caveats regarding both GA itself, and this book.

First, NFCM is definitely not a stand-alone textbook. Although Hestenes' explanations of many topics are not only lucid, but genuinely thought-provoking, few people who tackle NFCM on their own will find it easy. But then, Hestenes never said it would be. As he noted on p. 39 of his Oersted Medal paper (see first comment, below, for all references in this review),

"... I had to design [New Foundations] as a multipurpose book, including a general introduction to GA and material of interest to researchers, as well as problem sets for students. It is not what I would have written to be a mechanics textbook alone. Most students need judicious guidance by the instructor to get through it."

By the way, anyone who's considering teaching GA anywhere should read that paper to learn from Hestenes' own travails.

Since I had no instructor to give me judicious guidance, I read several papers on GA by Hestenes and others. The lectures and problem sets from Cambridge University were helpful up to the point where they became too advanced for me. Another good reference was Ramon Gonález Calvet's "Treatise of Plane Geometry through Geometric Algebra". The chapters from the previous edition of NFCM that Hestenes maintains online offered many valuable perspectives.

However, all of those resources couldn't make up for the lack of a good solutions manual, with plenty of additional worked-out examples. If I could make just one suggestion to Hestenes for facilitating adoption of GA, this would be it. Ideally, the manual would also show how to explore GA using computer software such as GAViewer, or even CaRMetal (which I plunked along with). I suspect Hestenes would agree with all of these recommendations.

IN SUMMARY

This is a good book for learning to use GA, if used as Hestenes intended. I'm convinced that GA is worth trying to teach at the high school level. I don't expect that it would be any easier to teach than the geometry and trig that it would replace, but it should pay off better down the road.

Please note that Hestenes and his colleagues have also done extensive research on teaching physics. The "Modeling Instruction in Physics" method they developed has given good results. (See links.)

* EDIT 1 August 2014: Geometric Algebra actually works quite well for solving "construction" problems. When I wrote this review in 2010, I hadn't yet learned to frame the problems as needed to make use of GA's strengths. The contrasts between how to frame problems for solution via GA and via "ruler-and-compass" are thought-provoking. See the link "Classical vs GA Solutions" in the first comment.