Buy the right math prep book

Pay attention and read the book’s preface, not only the title.

One of my students was going to take a standardized test specific to some job he wanted to apply for. The test includes a math section, and this is the only part of the test where this student felt he needed tutoring. My student has a friend who already took the same test, successfully. This friend told my student that, if he prepared well for the GRE, he would sure pass this other test, because it is an easier test than the GRE.
So my student went to the bookstore and bought a book he saw there titled “Cracking the GRE* Math Subject Test.” He did this a few weeks before we started our tutoring sessions. He brought this book to our second session for us to work on its problems.
When we started going through the book I was amazed to see double integrals, differential equations, matrices, power series, three-dimensional surfaces, polar coordinates, and almost any calculus topic covered in college, not to mention Abelian groups and topological spaces.
We found only three problems that could actually be in the general GRE test.
I suggested for my student to go back to the bookstore and get the normal Kaplan GRE prep book.
When my student realized he had no use for the book he had bought, he gave it to me.
I was puzzled by this “GRE” prep book because it obviously has nothing to do with the general GRE test. So when I got home I started reading the preface. I found a paragraph with the heading “What is the GRE Math Subject test?” explaining that this test is taken by students who are applying for admission to study math at the graduate level. Now it all made sense!
So, the moral of the story is: be careful when buying a math test prep book. Before paying for it you want to make sure it’s going to help you, and that you are not buying something that has nothing to do with the actual test you are preparing for.

Reasoning, Creativity and Mental Flexibility

The mental equivalent of a boot camp obstacle course

Sometimes after solving a relatively hard problem, GMAT students ask: "Can you give me a few problems similar to this one? Can you make them up? Or, is there a book with several problems just like this one? I want more practice with this kind of problem."
Seemingly, there is nothing wrong with wanting more practice but when GMAT or GRE students make this request they are missing an important point. That exact kind of problem is not going to be on the test. They may encounter a somewhat similar type of problem, but it is going to have its own twist, in some way it is going to be different, tweaked. The companies that produce tests like the GMAT and the GRE work very effectively to make sure that, when you take the real test, you will need to invent a solution on the spot by yourself, for every single problem. Each problem is going to be unique, completely new, something you would have never seen before, no matter how many preparation books and courses you have been through.
Traditional instruction during normal school courses has students drilling through stacks of problems of the same kind, applying the same formulas over and over. And, generally speaking, students can expect to find in their finals and midterms the same kind of problems they practiced with while doing homework.
The GMAT and the GRE do not cater to this drill-based learning model. Granted, you still need to know how to perform the operations, you need to know which properties to use and how to apply the formulas. But that alone is not enough. You need an extra skill that traditional, drill-based homework cannot give you.
Allow me to make this metaphor. Mathematics is a certain kind of mental activity, a mental exercise. Tests like the GMAT and the GRE are designed to measure a specific type of mental performance but, what type exactly? Well, let's compare drill homework (solving a ton of problems of the same type using the same procedure and the same formulas over and over) to working out by lifting weights at the machines in the gym. If you train only at the machines, that can give you muscle mass and strength. But that alone will not prepare you for successfully negotiating a Navy Seal obstacle course. The GMAT and the GRE are the mental equivalent of a boot camp obstacle course, while drill homework is the equivalent of the machines at the gym.
Real preparation for the GMAT and the GRE has to address your mental flexibility, that is, your creativity and your ability to discover patterns you have not seen before. You want to learn how to solve problems by reasoning, not by memorizing routine procedures. Once you understand this, you will be asking for new problems, for problems of a different kind, not for the same kind of problems.

Signs of Numbers vs. Signs of Variables

The inner value vs. the outer look.

A variable does not have a sign of its own. Numbers have signs of their own. Variables represent numbers. Variables may come with different signs in front of them in any particular expression. Still the number represented by the variable has a sign of its own. That is why sometimes I tell to some of my students: "Variables have an inner sign," which we cannot determine from outside just by looking at the sign in front of that variable in a particular expression or formula. I am not talking about astrological signs here but about the tendency some students show to assume a number represented by x in an expression has to be negative just because in that expression there is a negative sign in front of x. Don't go only by the sign in front of the variable to determine whether the number represented by that variable is positive or negative. In most cases we start working on the problem without knowing the value of the variables. At the beginning we don't know whether x is going to end up being positive or negative. Until you find the exact numerical value of that variable, we don't know if it's positive or negative. It can turn out either way, regardless of the sign that is showing "on the outside," in the expressions the variable appears in. For example:

If 7 + x = 4, then x = -3 (a negative number), even though the sign in front of x in the original equation is "+"

If 7 - x = 2, then x = 5 (a positive number), even though the sign in front of x in the original equation is "-"

These are very simple examples but you would be surprised to see how often people make assumptions or jump to conclusions about the mathematical properties of the numbers represented by variables, based only on lexicographic features of the algebraic expressions the variables appear in. In other words, very often students think they know things about the numerical value of a variable based only on how the variable itself looks like when surrounded by all the other symbols in an expression. People do this even in the case of very complex expressions. In general the more complex an expression is, the more reasoning time and effort it will take to infer the exact values of the variables involved, or just to determine some of their properties.

Imagine your friend is telling you a story where at some point they mention their doctor's mother. You wouldn't assume their doctor is a female person just because the noun "mother" is female, and the only time at which your friend mentioned their doctor was in connection with that doctor's mother. Would you?