A variety of Open Educational Resources providing a gentle introduction to mathematics beyond High School - with plenty of exercises. Some very different styles of presentation are here but with considerable repetition of topics, so it will pay to scan all items first to decide what suits you before launching into detailed study.

mathtutor

An impressive multimedia resource designed by a group of UK teachers, mathematicians and new media producers from the Universities of Leeds, Loughborough and Coventry and the EBS Trust.

"To bridge the gap from school to university study, to revise or find the maths topic you missed, you will want to meet mathtutor. Video tutorials, with diagnostics, summary text and exercises, take you through more than eighty topics in the way you choose."

Note that the videos require Windows, Internet Explorer and Windows Media Player. You can also watch the animations on a video iPod.

Topics covered are: Arithmetic, Algebra, Functions, Sequences, Geometry, Vectors, Trigonometry, Differentiation, Integration

Exercises in Maths Readiness

A collection of exercises by the Department of Mathematics and Statistics at the University of Saskatchewan on high school mathematics topics of most importance for higher education in mathematics, science, engineering and commerce. The exercises are graded as "Introductory", "Moderate" and "Advanced".

Topics covered are: Mental Arithmetic, Algebra, Geometry, Trigonometry, Exponential & Logarithmic Functions, Miscellaneous, Techniques for Proofs

Essential Mathematics Web-book

These exercises on Essential Arithmetic and Algebra designed by Franco Vivaldi of Queen Mary, University of London, UK, come as a 74 page PDF.

"The Essential Mathematics programme consists of over 1000 gradual exercises on integers, fractions, radicals, polynomials, rational functions, linear and quadratic equations. Each exercise is provided with answer for immediate feedback."

The author's suggestions include:

• Working without a calculator.
• Attempting 5 to 10 exercises a day, every day.
• Not skipping exercises unless you consistently get the correct answer at the first attempt.

Arithmetic and Algebra courses

Both courses by D A Santos of Community College of Philadelphia - were written for adults and have an appendix with multiple-choice examination problems.

The arithmetic course, " ... presupposes that the student is familiar with the basic mechanics of arithmetic (for example, knowing the multiplication tables), but it tries to explain why these operations are carried out the way they are."

In the algebra course, "The approach to algebra is classical, introducing addition, subtraction, multiplication, and division of algebraic expressions. It finishes with a chapter on equations."

Reasonable Basic Algebra

Just one of A Schremmer's "Reasonable Texts" with a different approach to mathematics teaching:

"The general intention is to get the students to change from being “answer oriented”, the inevitable result of “show and tell, drill and test”, to being “question oriented” and thus, rather than try to “remember” things, be able to “reconstruct” them as needed."

According to the author Reasonable Basic Algebra is (among other things):

• An introduction to mathematics that appeals to the reader's reason rather than to her/his ability to memorize.
• A way for adults to learn some mathematics—more or less in the same spirit as mathematicians do.
• A presentation that pays pedantic attention to the linguistic difficulties the reader is likely to have in mathematics.
• An anti "Show a Template Example, Drill and Test" manifesto.
• A work by a mathematician who, almost fifty years ago, got interested in reconciling "just plain folks" with mathematics.

TheMathPage

In fact many detailed pages by Lawrence Spector of the Borough of Manhattan Community College at the City University of New York. Main topics are:

• Skill in Arithmetic (complete course)
• Plane Geometry (Book I of Euclid's Elements
  culminating in the Pythagorean theorem)
• Skill in Algebra (complete course)
• Topics in Trigonometry
• Topics in Precalculus
• An Approach to Calculus
• The Evolution of Real Numbers

Elementary Mathematics

A comprehensive set of notes compiled by WWL Chen and XT Duong of Macquarie University, Sydney, Australia.

"The overall emphasis of this set of notes is on examples, of which there are many. The only way to study mathematics satisfactorily at this level is to study many examples and to attempt many exercises ....."

Section A covers Basic Algebra (basics, introduction to matrices, trigonometry, indices and logarithms, polynomial equations, inequalities and absolute values, progressions, elementary counting techniques, complex numbers) while Section B introduces Calculus (functions and lines, introduction to differentiation, further techniques of differentiation, applications of differentiation, introduction to integration)

Mathematical Visualization Toolkit

Designed by the Department of Applied Mathematics, University of Colorado at Boulder, this is an impressive set of visual and computational tools designed to help learners better visualize the concepts of Calculus. It includes: Scientific calculator; Plotting tools; Numerical tools; Linear algebra tools; Differential equations tools; Content-specific applications; Other Calculus visualization tools; Tutorial-style help system.

Calculus Made Easy, by Silvanus P. Thompson F.R.S.

"Being a very-simplest introduction to those beautiful ways of reckoning which are generally called by the terrifying names of the DIFFERENTIAL CALCULUS and the INTEGRAL CALCULUS".

Now out of copyright, all 301 pages of Silvanus P. Thompson's remarkable text book from 1914 are freely available.

From the Prologue - "Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.

Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the text-books of advanced mathematics - and they are mostly clever fools - seldom take the trouble to show you how easy the calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most diffficult way"