Tutoring For All Ages

Every year I work with science and math students who come to me thinking they just need help learning the material, without realizing that their main problem is that they don’t know how to study. They’re generally very good students who make the common mistake of assuming that the methods that work for them in history and literature classes will work for them in chemistry or geometry. In fact, these are such different classes that the same study habits rarely work for both.

So if you’re taking your first hard math class, or your first physical science class, here are some tips to help you study more efficiently:

• Know when to stop memorizing. Every subject in school involves some amount of memorizing. Sometimes you have to spend hours a night drilling dates and names into your head, and sometimes you just have to memorize the Central Limit Theorem. Memorization is a skill that will serve you all your life in some capacity or another, so don’t neglect it.

  However, another important skill you need to learn is when to put down the flash cards and pick up a pencil and some scratch paper. In math and the hard sciences especially, there is just no substitute for practice problems. (more…)

What does it mean to get the right answer on a math question?

Some may give the quick answer and say that a right answer is one that earns you full credit. But that answer, while necessary, is not sufficient. For any given problem, there’s usually more than one way to get to the right answer, even if you’re not taking a multiple-choice test, where you know the right answer is staring back at you before you even start the question. There are textbook right answers, there are clever right answers, there are inefficient right answers, and there are lucky right answers. Of course you always want to get the answer right, but you want to do it in the best way possible.

It’s worth examining some of the different paths to the right answer:

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Consider the equation:

x² + y² = 1

Math geeks can appreciate its elegance and simplicity: the unit circle, a circle with a radius of 1.

For anyone who has taken pre-calculus or above, you also (hopefully) know it as it relates to trigonometry. You probably griped about having to memorize it. But if you got to the point where pure memorization gave way to real understanding and internalization, you probably now also appreciate just how useful it is.

For anyone who is about to take pre-calculus, examining this circle in a bit more depth will give you a preview of what’s to come:

• The Unit Circle from Humboldt State University
• SparkNotes on the Unit Circle
• An Interactive Unit Circle

These links are just a few of many sites that extol the virtues of the unit circle. For my part, I will just say that your life (or at least calculus) will be considerably easier if you take the time to memorize (and then really understand) this “simple” little function.

Here’s today’s medium-level SAT Question of the Day (note that if you saw it on the SSAT/ISEE, it would be a Hard question):

In the figure above, the large rectangle is divided into six identical small squares. If the perimeter of the large rectangle is 30, what is the perimeter of one of the small squares?

A) 5
B) 8
C) 9
D) 10
E) 12

How do you solve this problem in the quickest way possible?

Here’s how the College Board thinks you should solve it. That’s the “right” way to do it: use the ratio of length to width to set up an equation using the overall perimeter. And assuming you don’t make a careless algebra error, that will get you to the right answer. For some students, it may even be the fastest way to get to the right answer.

But not every student is going to see this way immediately. If you’re one of them, consider another way: working backwards, sometimes called guess-and-check. Start with the middle answer choice and assume it’s the perimeter of a square. Figure out what each side of the rectangle then has to be, and see if that matches the given perimeter of the whole rectangle. If it does, you have your answer. If not, you should be able to tell whether you need the perimeter to be bigger or smaller, so you can figure out which answer choice to try next.

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We hear a lot of reasons for why students don’t like to write down their intermediate steps on standardized tests. Sure, it may not be necessary to show your work in the same detail as when you’re graded on that work, but you should think twice before leaving your test booklet blank. Our most commonly-heard excuse is:

Habit. It’s just what I’ve always done.

Now, sometimes this is just another way of saying, I’ve never been allowed to do it. Students taking statewide standardized tests aren’t always allowed to write in their test booklets. That’s because some schools re-use their test booklets from year to year, and erasing is time-consuming and can be damaging to the paper. (Whether or not we think that’s a good policy is a subject for another post.) So when students go from using separate scratch paper to having no separate scratch paper at all, they often just think they’re not supposed to write anything down.

If this sounds familiar to you, know that the ACT, AP, ISEE, SAT, and SSAT exams all expect that you’ll be using your test booklet as scratch paper. Part of your test fee is paying for that test booklet, which will be thrown away after you take the test. So, write away!

But for those students who are just in the habit of doing all their work in their head, consider the adage: “If you always do what you’ve always done, you’ll always get what you’ve always gotten.”

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