I have just finished chapter 1, so my review is for my impression of the book thus far. I will try to update this review as I complete subsequent chapters.
The book is huge. The pages are as wide as A4 and a little shorter in height. There are 634 pages, including the answers and index, so a lot of material is covered.
This is exactly the type of maths book I have been looking for. I bought Edexcel GCSE Mathematics Higher and found it to be a standard secondary school maths book - adequate, not bad. I also bought Core Maths for Advanced Level because of its high Amazon rating. The authors of Core Maths have made excellent choices in the questions they ask and the order in which they are asked. Core Maths' main failing is that little explanation is given. It often gives one example question, loosely shows how the answer can be derived, and then gives 30 exercises, lots of which cannot be done from the information in the example alone. It is sometimes satisfying to figure these out for yourself, but this takes a lot of time.
Maths: A Student's Survival Guide is a self-help book designed for those teaching themselves without attending classes. It succeeds in this. The author has written down what she would say if she was sitting next to you, explaining the subject matter. Even the answers at the back of the book are explained. In the other two maths books I have, the final, simplest answers are provided, but in this book the answers mostly show the question, show each step in how to work it out, and then show the final answer. Furthermore, the author occassionally writes entire paragraphs in the answers section explaining the problem, and some common pitfalls that students tend to make.
The text itself is amazing. It's a maths book with more words than numbers. EVERYTHING is explained, right down to 2+2=4, and that's not a joke. She literally explains that 2+2=4 is notation for 'positive two in addition to positive two equals positive four' when she introduces negative (minus) numbers. You can preview the first few pages on Amazon, and I recommend you read those pages and do the included sums to see if this book is what you are looking for.
Chapter 1 starts with a self-test. If you correctly answer all of the questions then you can obviously skip the chapter if you wish. This is helpful to prevent time wasting, but I still recommend you read Chapter 1 from start to finish because it deals with more than is covered in the test, such as simple binary numbers. Adding, multiplying, subtracting and dividing are all explained - yes, these simple little things are EXPLAINED, and the explanation makes sense too.
Chapter 1 is about algebra. Algebra is fundamental because you are always trying to solve a problem, like 3+a=7. You learn why a+a=2a and axa=a^2 (a to the power of 2). You then learn how to simplify problems by factorising them, and how to do the opposite, multiply the factors back out again.
It moves on to explain fractions, and does so visually. The author uses visuals where necessary, like when explaining the scary term 'the difference of two squares'. She often shows why things are not arbitrary, why the wrong way is wrong and not just different e.g. by substituting numbers in for the following letters to show why a/b+c does not equal a/b + a/c.
There are exercises for each section, which start easy but get more challenging each time. There are not too many exercises, usually around ten, which seems enough. Core Maths often has 30 exercise questions, if you prefer that.
I feel like what I have been taught in school was not really maths, and that this is what maths really is.
I want to talk about section 1.E - The different kinds of numbers. It gives a history of how we got from having no numbers to giving them names, inventing zero, negative numbers, fractions, surds (roots), and complex numbers. Every child should be taught this, because simply saying 'two plus two equals four' makes it a game with arbitrary rules, instead of a rigorous system with some historical influences still present.
Binary is a short but joyous section, and makes me want to do all my sums in binary because it's the simplest system. Prime numbers and factors is another fantastic section because it shows how prime numbers are the atoms of all numbers, how any whole number can be made from a prime, like 12 is 2x2x3, but no numbers make 11 (except eleven 1's). The chapter ends on surd fractions, which is enjoyable.
One thing I don't like is constantly having to flip to the back of the book to see the answers. When examples are given, the answers come after the example. Why isn't the whole book like this? Instead I need two bookmarks, one for the questions and another for the answers. It's a big book and my desk is small, so flipping to the back after each question is completed, which I find to be the most rewarding, is not easy. Maybe people have problems preventing themselves from peeking at convenient answers, but I don't, and putting the answers at the back doesn't prevent this if you check if your answer is correct after each sum. Make them convenient and on the same page as the questions, not in the back of the book after the last chapter but before the index.
Overall I love this book. Maths should be taught like this from the first day children start learning maths in school.