All the major test prep books for the SAT, ACT, and GRE -- published by companies like Kaplan, Princeton Review, Barron's, and Manhattan Test Prep -- are poorly written, conceptually deficient, and, worst of all, riddled with serious errors. Students can't be expected to learn from books that aren't even right! And I don't mean the books are riddled simply with typos, which unfortunately is also true, because they are so poorly edited; I mean they really are riddled with serious conceptual errors. Here's a simple example from the very beginning -- the diagnostic test, of all things! -- of Princeton Review's "1,014 GRE Practice Questions." The problem is on page 24, and the answer key and explanation is on page 38. Not only is their answer wrong; what's worse, their *explanation* is wrong, too! I'll set off the problem by dashes (----) and then add more commentary after. NOTE: The question is a classic GRE "quantitative comparison," so it's hard to... read more

All the major test prep books for the SAT, ACT, and GRE -- published by companies like Kaplan, Princeton Review, Barron's, and Manhattan Test Prep -- are poorly written, conceptually deficient, and, worst of all, riddled with serious errors. Students can't be expected to learn from books that aren't even right! And I don't mean the books are riddled simply with typos, which unfortunately is also true, because they are so poorly edited; I mean they really are riddled with serious conceptual errors. Here's a simple example from the Introduction (page 23) to Manhattan's Strategy Guides for the Revised GRE. This passage appears in all eight of Manhattan's strategy guides, so it somehow went unnoticed after at least eight rounds of editing by allegedly "expert" readers and test-takers. See if you can spot the error! ---- "If ab=|a|x|b| which of the following must be true? I. a=b II. a>0 and b>0 III. ab>0 A. II only B. III only C... read more

The most lasting way to improve your vocabulary is to learn new words (1) in context (by looking up unknown words when you read and keeping a journal of their definitions) and (2) in thematic groups -- NOT by memorizing huge lists of unrelated words. These are some of the resources I use with my students; feel free to comment to add your own favorite vocabulary book! BOOKS -- English Words from Latin and Greek Elements An excellent etymological resource that helps students learn how to recognize Latin and Greek roots in modern English words and use them to predict the meaning of a word. Useful for students of all levels, from high school to college. -- Roget's Thesaurus of Words for Intellectuals This is the best book of advanced thematic word lists I know; I assign it to all my GRE students. The vocabulary is a bit too advanced for most high school students prepping for the SAT, but it's still an excellent resource for anyone -- in high school or not --... read more

Most students taking the SAT, GRE, or GMAT know their algebra fairly well, but many find they can't complete all the problems in the allowed time. Why? It's NOT because those students are just naturally slow: it's because they're doing more work than they need to! It's not their speed but their very approach --- the very way they conceive of the process of problem-solving --- that's flawed. To ace the math sections of standardized tests, you have to learn how to attack problems in new ways so that you get the right answers by doing as little work as possible! (Part of the reason so many students don’t already know how to do this is that it’s not taught well throughout middle and high school math classes. Learning how to think quickly and deeply often requires UNLEARNING habits your math teachers instilled in you in school!) To see if you’re up to par, try the following problems, which test your ability to make deep algebraic connections that will save you time. If your... read more

Many first-year calculus students fall into a common trap: they tend to make bad assumptions about how functions behave. In particular, they tend to think all functions are "nice," in the sense of easy to draw and understand -- because most of the pictures their teachers draw in school to illustrate examples tend to be of nice, familiar functions they are comfortable working with, like polynomials. But functions, in general, are extremely unwieldy, and to truly master differential calculus, you have to learn to be on guard against making simplifying assumptions: what we often imagine to be the case turns out to be false on closer inspection. Here's a simple example in the context of critical points and local extrema. A classic application of derivatives involves finding the local minima and maxima of a function. You may recall that the first step to finding these values is to find the function's critical points. Here's the common trap: most students mistakenly... read more

Of the vast amount of math taught in high school, combinatorics is usually the most baffling for students. In my ten years of teaching, I've never had a student who felt totally confident about counting problems -- I myself didn't feel I really understood them until I went to college! -- and the most typical reaction to them is immediate fear or frustration: students often give up as soon as they see one, before they even attempt a solution. Why? Probably because many high school math teachers don't do a good job of explaining the basic concepts with concrete examples; instead, they often present a bunch of formulas to be memorized, without conveying any intuition about when to use the formulas or where they come from. But you won't earn a stellar score on the SAT, GRE, or GMAT if you can't master the basics of combinatorics. To see if you're up to speed, take a look at the following challenging problems, all based on the following scenario: PROBLEM Luigi's Pizza... read more

Many of my students preparing for the SAT, GRE, and GMAT have decent algebraic intuition when it comes to EQUATIONS, but most are much weaker when it comes to INEQUALITIES. On the one hand, this is entirely natural: inequalities capture less information than equations -- they establish merely a relation between two quantities, rather than their equivalence -- so they are inherently trickier to think about. But on the other hand, it's crucial to have a very solid grasp of how inequalities work to do well on the SAT, GRE, and especially the GMAT (which tends to love data sufficiency questions that deal with tricky inequalities). To test yourself to see how up-to-speed you are, try to decide whether the following statements are true or false. (I have intentionally made the problems very abstract and seemingly confusing to see if you really know what's going on, so DON'T WORRY IF YOU'RE TOTALLY LOST OR INTIMIDATED!) 1. If a+b=c+d and e+f=g+h, then a+b+e+f=c+d+g+h... read more

Many of my students preparing for the GRE or GMAT have decent algebraic skills, but most have trouble with statistical reasoning --- for a variety of reasons. Some have never had statistics; others have been away from it for years. In either case, it's crucial to get up to speed on the basics! To get a sense of how prepared you are for some of the more challenging statistics questions on the GRE and GMAT, check out the following worksheet I've developed. When you work with me, you'll gain exactly the skills you need to ace these and similar problems --- you'll learn to complete this entire sheet in fewer than five minutes, in fact! --- and have access to a wide range of specialized materials I've developed over the years, materials full of strategies and problem sets you won't find in any published prep book. I guarantee you won't find a more helpful or expert tutor, so send me an email today! PROBLEM SET: NORMAL DISTRIBUTIONS Exercises: Percentiles & The Empirical... read more