I am suggesting that we provide some historical and paedagogical background as part of the actual (non-prefatory) content – including also some discussion of the mathematical background material (eg ideas from geometry and familiarity with some? level of whole number arithmetic) which needs to have been covered previously in order for this approach to be successful.
- Ed Lit on Difficulties
- Historical Background
- Arithmetical Prerequisites
- Geometrical Prerequisites
It is often remarked [refs primarily from k-8 educators] that the definition of multiplication of natural numbers in terms of repeated addition (ie repeated use of the addition operation in an iterative process, not repeatedly computing the same sum!) causes difficulty with the transition to an understanding of what to do when the factors are not positive whole numbers (and especially when they are not even rational).[more examples here of how children get stuck and/or go wrong]
Perhaps it should not be surprising that individual learners have trouble with this transition as it is something that we took several centuries over as a species, and those who led us through it [ref the mediaeval math stuff] had access to a well developed version of the necessary geometrical motivation.[a bit of summary of that book]
In the classical world, as summarized by Euclid [see eg Joyce], it seems that the “arithmetic” of geometric proportions was developed to a large extent independently of that of numbers.
So even while they were learning a lot about the arithmetic properties of whole numbers the classical Greek mathematicians were also building a model within which it is possible to develop an intuition for fractional and even irrational numbers. But (because of the lack of the necessary geometry in the k-8 curriculum?) it seems that modern students don’t really have such a model at hand when it comes to extending their arithmetic concepts to non-integer rational and real numbers.
It is the intent of the present work to remedy that situation by providing an account of real number arithmetic that is based entirely on geometrical operations.
The goal of the project is to create a book that builds arithmetic from a geometric perspective (i.e. no limits or algebra needed) with a strong emphasis being placed on multiplication. The intended audience of the book are undergraduate STEM students and prospective math teachers.[a bit of a summary of what children do learn about geometry and an indication that may be enough on which to build the numerical ideas]
The ideas here are not new. In fact they correspond quite closely to the defining model of real numbers given by David Hilbert in his 1899 classic ‘Foundations of Geometry’. What we offer is both an introduction to those ideas for those not familiar with them and some suggestions as to how they might be introduced into the current elementary math curriculum.