The vocabulary of mathematics is not the everyday vocabulary that you will find in a history book. Some mathematics words look like everyday words, but they have specialized technical meaning in a mathematics context. A rational number is not a number that has complete possession of its mental faculties! A rational number is a number that can be expressed as the ratio of two integers. But in order to understand that, you need to know the technical terms ratio and integer! And on it goes, until you have reduced things to terms you are sure you understand.

When you come across the term rational number as you read, you need to reflect for a moment and ask yourself  "Do I really know what a rational number is?". You usually cannot guess the meaning of a math word from context as you can when reading novels.

When you read the word "cow" in a novel, the distilled essence of "cow" (hooved quadruped half the size of a pickup with a dumb look on its face) immediately pops into your mind.  When you read the words "rational number" in a math book, the distilled essence of rational number has to similarly pop into your mind.  You should be visualizing an image of a fraction whose numerator and denominator are both integers.  As a consequence of that, what you are reading sudenly becomes meaningful.

Mathematics has a very concise and precise notation for expressing mathematical ideas. For example, functional notation (e.g. "f(x)") is ubiquitous, yet widely misunderstood amongst beginning math students. If you haven't internalized the meaning conveyed by functional notation, you may as well be reading Greek! The unfortunate part is this: we know for sure that we can't read Greek, so we don't even try, but because a math text is written in English, we are lulled into the assurance that we must be understanding what we are reading, even when we're not!

Each time a new piece of notation, e.g.

If I don't train my students in the vocabulary and notation of mathematics, then, when I (or some future teacher) refers to the coefficient of the linear term of a polynomial, or the multiplicative inverse of a real number, I may just as well be speaking Aramaic. I believe that we should give more exercises that specifically address vocabulary and notation issues, and that we should do intentional testing on those important facets of a mathematical education.