I was helping my brother with math yesterday and while looking his book, I found that I had trouble understanding the wording of their explanations even after passing multi-var calc. I clearly remember when I was actually taking the same class and every single one of my other math classes that I had trouble understanding what the book was saying in explanations and I did most of my learning thru their examples. After looking at the examples and spending a huge amount of time reading it did it finally make sense.
The authors of math text book strive for a perfect definition at the cost of verbal simplicity and coherence. While their definitions and rules are definitely correct mathematically the way they are presented with their exceptions and boundaries, makes it extremely hard for the student. The reader who is usually new to the subject is not trying to prove or refute an argument but rather understand the basis of it. He isn’t interested in when the law is no longer valid, but rather wants to understand what the law actually does.
Not including the exceptions and boundaries would be ridiculous, but a better method of explaining needs to be put into place that doesn’t lose a reader due to the level of complexity at which it is stated. There needs to be a major shift in math books that attempt to be user-friendly while still maintaining a high level of rigor.