  This paper is one of a series in which elementary-education practice is
analyzed by comparison with the history of mathematics, mathematical structure,
modern practice, and (occasionally) cognitive neuroscience. The primary
concerns are: Why do so many children find elementary mathematics difficult?
And, why are the ones who succeed still so poorly prepared for college material
needed for technical careers? The answer provided by conventional wisdom is
essentially that mathematics is difficult. Third-graders are not
developmentally ready for the subtlety of fractions, for instance, and even
high-performing students cannot be expected to develop the skills of
experienced users. However we will see that this is far from the whole story
and is probably wrong: elementary-education fractions are genuinely harder and
less effective than the version employed by experienced users. Experts discard
at least 90% of what is taught in schools. Our educational system is actually
counterproductive for skill development, and the reasons for this are an
important secondary concern.
