How to Prepare for Standardized Tests in Mathematics

Summary: Alas, to get that perfect score, you have to reconsider everything you have been taught at school.
My system of preparation for the standardized tests in mathematics (ACT, SAT, SAT2, GRE, etc.) is somewhat unique and unconventional. In fact, it goes against the grain of most of what you have been taught in school... and likely even in a test-prep class, if you have taken one. Sound a bit unnerving? Perhaps. But consider this: those same math teachers who tell you what to do, had most certainly not scored well themselves when they were your age. What is more, chances are good that they cannot score that well now, either. That's because their ways are... well, unduly complicated.

What is a better approach? First of all, I will teach you how to solve 95% of all questions mentally, without writing a thing. Why bother, you may ask. Several reasons.

One, it will teach you—anew—what you once knew but have since forgotten: the mathematical imagination. Its importance is hard to overemphasize. The better you can imagine what is going to happen when you do this or that to the math expression at hand, the farther ahead you can see, the faster you can find a solution.

Two, a calculator or paper can take virtually unlimited amount of arithmetic, no matter how complicated.  But nobody is thrilled about doing too much work in one's head.  Thus, doing math mentally teaches you to look for a shorter, simpler way to solve the problem—and eventually to see such a short solution. A typical multiple-choice problem can be solved mentally in under 10 seconds—once you've slowly read it, of course.

There are other [inter-related] benefits in mental math: improved short-term memory, reduction in the number of 'silly mistakes,' increased accuracy and speed, etc.  (I write about these in detail in a separate article on this site.  Please read that article, too!)

Next, since you are not writing, you cannot 'show your work.' But to take it further, I will not let you 'show your work' even in articulating your solution orally, either to me or to yourself. Why not? Because 'showing your work' prevents you from short-circuiting the steps that at some point in the past should have become trivial. For example, years ago, when you were learning how to read, you had to first learn how to recognize the shapes of individual letters, and the sounds that each of them made. But since then, these—formally formidable—tasks have long become superfluous. Thus, when reading today, it would be preposterous if you were required to 'show your work' by calling out the sound that each and every letter made. You would never accept such an 'educational' approach to reading. Why should one accept it in mathematics?

As in reading, or sports, or any other human endeavor, the process of learning consists of combining formally separate steps into a single new step. 'Showing your work' prevents you from doing exactly this, and so from becoming better in mathematics. Hence, it must be rejected.
(Of course, the ability to write down the non-trivial steps of one's solution in a coherent narrative is a useful skill.  But 'non-trivial' is the key adjective here.  In the solutions presented to a teacher (such as in the free-response sections of the AP exams, for example) milestone steps must be included.  Under those circumstances, my rule of thumb however is that if a mathematically literate person can follow my train of thought, I have written enough.  Why keep on proving to the teacher that you know how to solve say a linear equation, in separate steps, when you have mastered linear equations 5 years ago?)

Next, I will ask that you 'throw out the clock,' i.e. have no concept of time when you are preparing. "But I have a very limited amount of time on the test!" you may object. True. However, the more time you spend 'dwelling' on a problem during preparation, to painstakingly slowly build a quality mental model of it, the faster your thoughts in the future will be able to access and reuse the model for other problems of that type.  In other words, speed will come naturally if you take your time.

There are other tidbits to my approach, dovetailing to what is described above. For example, for the vast majority of questions, even of the form "which of the following...", I will ask you not to be looking at the answer choices. The reason for that is simple: by covering the answer choices, you are forced to think about the problem deeper, sometimes even putting yourself in the shoes of a problem maker: a higher level of mastery. Of course, this contradicts what you may have heard elsewhere about the process of elimination. Alas, I'd much rather teach you how to dominate problems, than to be defending yourself against their onslaught.  Besides, having solved the problem—and only then—you may continue analyzing it, along the lines: "If I got this problem on the actual test, would it be of benefit to see the answer choices?  Would it be faster to go through the process of elimination?" 

I should add that the proof is in the pudding. I had taught a number of my young students to achieve the perfect score on the math portion of the SAT long before they would reach high school. I had also tutored a number of high-school students to score either perfectly or near-perfectly on the math portions of the ACT, SAT, and SAT 2. 
If you buy into my method, I can teach you how to do it, too.


Andrei K.

Yale Math PhD: SAT prep, gifted-child math and physics

3000+ hours
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