in the best cases because the textbooks leave so much going "without saying" that reason has become invisible,
in the worst cases because the textbooks have been gutted down to disconnected "topics" so that no reason is left.
Reducing learning mathematics to memorizing "mathematical facts" does not work. Witness "developmental mathemtics" which, with memorization at its core, has now been in existence for forty some years without having ever shown any sign of success.
Contrary to what most people believe, students and teachers alike, there is nothing inherently difficult in mathematics up to and including the "mathematics of change", also known as Differential Calculus, which everybody should be able to learn.
The truth or falsehood of mathematical statements does not originate in textbooks, or in teachers saying so, but because the real world makes them necessarily so.
So, in contrast with currently available commercial textbooks, the texts to be found on FreeMathTexts.org
emphasize a very few but very powerful ideas on the basis of which everything finds its natural place and can then be questioned as thoroughly as the student feels without hitting a dead end,
appeal only to common sense and make no assumption of any kind about the students,
are focussed on helping the students convince themselves that statements need to be justified and their truth supported with arguments,
pay pedantic attention to matters of language wich often impede understanding.
A further discussion of these facts, and of how the texts on FreeMathTexts.org attempt to take them into consideration, can be found in Chapter Thirteen.
Two more facts govern the development of the texts on FreeMathTexts.org, namely that:
In order to be usable in mathematics classes, the texts to be found here were designed as parts of "bundles" suitable for "turnkey" classroom use in that they include the ancillaries, that is all that is needed, Homeworks, Quizzes, Reviews, Exams, in a form that makes it easy for the instructor to follow and assess their students' progress as well as adjust the materials to their own situation.
Mathematical exposition is no one's "intellectual property". All the materials on FreeMathTexts.org can thus be freely downloaded, used, copied, distributed and modified under the terms of the GNU Free Documentation License.
As of this date, two titles have been completed and one is in progress. All are freely available:
Reasonable Basic Algebra. It is complete but a significant modification will be made in the text "soon". Also, the ancillaries HAVE (FINALLY) BEEN recast into the newly developed uniform format and are therefore usable with ADM, the Graphic User Interface.
Printed copies of the text can be purchased from Lulu for $8.40 plus shipping*.
Reasonable Algebraic Functions. ("Verbose edition".) The following ancillaries, in the newly developed uniform format and therefore usable with ADM, the Graphic User Interface, are available: Homeworks 1 to 14 as well as Reviews I and II. The rest should be available soon.
A "terse edition" of RAF. (337 pages instead of 577) is now available which tries not to answer questions the readers don't have ... yet. (The "verbose" edition will remain available for those who do have these questions.) The ancillaries remain mostly those of the verbose edition but, where needed, an appropriate version of the ancillaries has been supplied.
Printed copies of the text can be purchased from Lulu for $9.57 plus shipping*.
Reasonable Decimal Arithmetic Contrary to what I did in the past, I am making the text available for download as I rework it from previous, unsatisfactory versions. The hope is that this will entice people to get involved in its development. There are no ancillaries available at this time.
The system for managing the ancillaries itself is freely available to anyone wanting to develop their own ancillaries for these or any other texts. The system is described in an article which appeared in the PracTeX Journal on 2011-02-07. It includes the Assessment Document Maker (ADM), the graphical user interface designed by M. Sharpe which allows the user to generate the ancillaries in a variety of formats.
* I would like to acknowledge the fact that Lulu waived its "fees" inasmuch as I do not collect any royalties.
And, for those interested, I have resumed my Notes From The Mathematical Underground (September 19, 2011.)
There are of course other possible approaches to mathematical exposition. For instance,
Santos' texts follow the Edmund Landau style of presentation. More generally,
there are several sites in which free mathematical texts can be found such as Online Mathematics Textbooks, Online Science and Math Textbooks, etc. And then there is iBerry, with Open CourseWare and Higher Education Resources, which describes itself as "the first academic porthole", that is as "a small but cheerful window in the side of the Higher Education ship for purposes of illumination and enlightenment." And there is
Karl's Calculus Tutor, a friendly view of calculus even if, inasmuch as it uses the "standard" approach, it is not as friendly as Lagrange's approach.