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## Improve Your Abstract Thinking Skills

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Yesterday, I wrote about a few different approaches to GMAT Data Suffiency questions. Central to that tip was the notion that, rather than blindly picking numbers, you need to think generally about the specific cases you choose. Doing so is one of the biggest conceptual leaps you can make on the GMAT: without it, you probably can't top a 640 or so; with it, you can drastically improve your Quantitatve score beyond that.

The main point here is that you need to think abstractly. Abstract thinking, as it applies to the GMAT, is the skill of taking what you know about a single case (say, "x is greater than y when x is equal to 7″) and generalizing it to a variety of cases ("x is greater than y when x is positive"). When you look at a concept from a more abstract point of view, you often lose track of some of the specifics (if you know what happens when x is positive, does it matter what happens, specifically, when x = 7?), but you gain a broader perspective.

In other words, you do exactly what the GMAT is asking you to do on Data Sufficiency problems. You think about many possible cases at once.

Abstract Better

Abstract thinking is a skill that correlates strongly to IQ: it's the sort of thing that separates extremely talented mathematicians, physicists, and computer programmers from the rest of society. I can't show you (in this space, anyway) how to ramp up your abstract thinking that far, but to get you a few more points on the GMAT, it doesn't take that big of a jump.

First, as is the case with just about every other math tip on the planet, it's about practice. Every time you try a set of values on a Data Sufficiency question (if it isn't clear what I mean by this, check yesterday's tip) think about what the set is telling you. Let's look at a simple example:

Is x greater than x squared?

(1) x < 1

(2) x is positive

If you're an old hand at this type of problem, odds are that you know that you need to think about both fractions and negatives for the first statement. As an example of fractions, use 1/2: if x = 1/2, x^2 = 1/4. Yes, in that case, x is greater than x squared. What about negatives? if x = -2, x^2 = 4. In that case, x isn't greater than x^2.

Taking the Next Step

In a problem like this, you may be thinking abstractly without knowing it. After checking those two sets of numbers, are you sure that 1/2 represents all fractions? That -2 represents all negatives? (It doesn't matter, of course, since you've already proven that Statement (1) is not sufficient.)

This is where you need to prod your brain to generalize. Instead of thinking about x = 1/2, think about the set of numbers that consists only of fractions. If you have a fraction (a positive number less than one), what happens when you square it? In fact, the square of a number between 0 and 1 is always smaller than the number itself. If you can take that step, you'll never doubt the usefulness of your single example and you can proceed more confidently.

Let's look, now, at Statement (2). We've already considered fractions (positives less than one), so that leaves us with...positives greater than one. Remember that we're looking for a general rule, but it's still useful to try to example to see what happens. x = 2? x^2 = 4. x = 10? x^2 = 100. x^2 is always greater if x is greater than 1.

Recognizing Boundaries

As you may have recognized, a key component of abstract thinking on GMAT Data Sufficiency is recognizing just how much you can generalize. The example problem above is a useful one because its boundaries are so common: the rules change depending on whether x is positive or negative, and whether it's a fraction. Sometimes you may only be able to generalize rules for integers, for evens or odds, or for multiples of a specific number.

Developing This Skill

The number one thing you can do, on any question that asks you to think generally, is to force yourself to do just that. If it doesn't come naturally, well...you'll have to push yourself. Even if you don't do it on your first pass over a question, think about how you could generalize as you review the explanation. The thought process described in the previous paragraphs is a good outline of where you want your thinking to be.

Another approach you can take is to learn to recognize the sorts of statements you will see frequently. The example question above is a great place to start: I've seen a dozen variations on that question, sometimes with x^3 instead of x^2, or -x instead of x, but the same underlying concept being tested. Knowing commonly tested patterns, like x^2 > x when x > 1, is a sort of shortcut to the sort of abstract thinking the test expects you to display.

However, no shortcut will help you on the most difficult GMAT Quantitative questions. For those, you need to be able to recognize patterns and apply them within the framework of the problem–all in just a couple of minutes. It is easy? Heck no. But it is a skill that can be learned, and you can only learn it if you force yourself to practice it. Even small improvements in abstract thinking can net you additional points on the Quant section.

So, what are you waiting for? Practice problems await!

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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