Here we give the details of each attribute in our book reviews, and how we evaluate them. Scores are 1-5, with 5 being the highest.
Quality of Exercises (1-5)
Good exercises are a key to learning mathematics, especially if you are trying to self-study. Exercises should be motivated by the chapter or section text. They should also carry a range of difficulty, from the simple concept-checking to problems that extend the topic covered in perhaps interesting ways. No exercise should require looking forward in the book for material to aid in solving it. Exercises that ask the reader to prove something should be able to be proven from first-principles and prior material rather than some “clever trick.”
We rate a book highly on self-study if the book is fairly self-contained – that is, you don’t need tons of additional reference material to learn the topics the book aims to present.
Proof Quality (1-5)
The proofs in the text should be clear about their motivations. Why did they choose the approach they did? Does the proof build naturally, as if the reader is helping to construct the proof, or does it rely on a weird lateral trick to keep the proof concise? Proofs should be well-explained, with few references to other places in the book, e.g. “Apply Theorem 3.1 to equation (2) and the result is obvious.” In addition, there should be very few proofs in the text that are just left to the reader.
The language shouldn’t be so lofty or dated so as to confuse or distract the reader. It shouldn’t consist almost fully of symbolic language, and descriptions of topics and attempts to relate the topic and motivations via language should be present.
Good for Teaching (1-5)
A text that is good for teaching should have lots of examples. It should also have different topic choices that can be combined in various ways to construct a customized course given the needs of the instructor’s students.
The difficulty is evaluated based on the prerequisites required prior to starting the book, as well as the breadth of knowledge required to understand the topics presented.