Studying Math: Some Suggestions

Last time, we looked at some discussions we’ve had about motivation to study math. We’ve also had a few questions asking for help with study skills, and some of those answers, too, can be found in our archive. Let’s take a look.

Essential study principles

First, from 1999:

Help in Studying for Math

My name is Lisa. I am currently enrolled in the 8th grade. I have almost always been smart in math, but recently within the past 2 years I have been doing terribly. I was wondering if you had any suggestions or comments that might help me out a little bit.

Doctor Jesse took this, and shared a few basic ideas:

I'm very proud of you for taking a personal interest in improving your math grades. There is no doubt about it; math starts to get hard right around 7th and 8th grade, but you can do it. You just have to take it one homework assignment at a time. Here are some tips that might help you:

- Ask a lot of questions. I know it can feel a little embarrassing sometimes, because asking a question means admitting that you don't understand something, but don't be embarrassed. Chances are that most of the class doesn't understand either, and that someone in the class is secretly thanking you for asking, so that they didn't have to. Sometimes when I feel too embarrassed to ask a question, I tell myself this: "It isn't my fault that I don't understand it; it's the teacher's fault for not explaining it well enough." And then I go ahead and ask that question. Keep asking questions until you get it... Later, you'll be glad you did.

- Don't fall behind in class! In math, especially in algebra, the lessons build on top of one another, so that if you sleep through September, you might be in real trouble come October... Try to be disciplined about doing your math homework, and pay attention in class. If you make a habit of these things, the class will seem generally easier.

- Get a "study buddy." Find someone in the class who cares as much about doing well in the class as you do. Make a habit of getting together to do your homework, or talking about the homework on the phone. Math can seem much less scary when you don't have to go it alone. In fact, when you and a friend are working together to solve problems, it can be kind of fun.

- Find an after-school tutor. This can be a paid tutor, or maybe some friend or family member who knows algebra fairly well. Don't wait until you are getting failing grades to find a tutor, but find one at the first sign of trouble. Many schools have a tutoring system in place, so ask your teacher about it. Some teachers will give out their home phone numbers or emails so you can call or write them when you are having trouble with your homework. Don't be afraid to take advantage of any available help.

- Prepare for tests. Make sure that you know what will be on each test, and practice those kinds of problems. Ask your teacher for a sample test. Study for the tests with your "study buddy." Get a good night's sleep before each test, too.

- Stay calm. Many people suffer from "math anxiety," where just thinking about math gets them feeling underconfident and nervous. This makes them put off studying and homework until it is too late. Don't let the math scare you, or make you feel bad. Math can be hard to learn, but you can do it.

Preemptive studying

In 2001, Bambi asked for the same sort of help:

Tips for Studying

I really need help! I'm in grade 8, so this is the first year of exams for me, and I don't know how to study.

Doctor Ian started with studying for  an exam:

Usually when you have a test, it's going to be about what you've been learning recently.  For example, if you just finished chapter 5 in the book, then the test will almost certainly be about the material in chapter 5. 

So the way you study for the test is to read through chapter 5 and make sure that you can answer all of the practice questions at the end of each section. If you can answer those, then you'll probably be able to answer any question on the test.

But that’s not enough: For best results, you should start preparing for the test before you’ve even been taught the material!

Now, here's the thing: If you wait until the night before the test to try to learn the material, it's probably not going to work. A better way to use your book would be this: If your teacher is going to talk about section 6.3 tomorrow, you should read that section tonight and try to answer all the practice questions. If you can answer them all, then you can just treat the next day's class as a review session. But if you can't answer them all, then the next day in class you can ask the teacher to go over the ones you couldn't do - which shouldn't be a problem, since that's the material he or she is supposed to be covering anyway. 

Trust me, knowing ahead of time what the teacher is going to say makes class a _lot_ less frustrating - and quite a bit more interesting.  (It's a little like when you see a movie again - the second time around, you already know what will turn out to be significant, so you can notice all kinds of details that escaped you the first time around.)

And this way, when the time for a test rolls around, you'll know that you already _know_ all the material, so you won't _need_ to study, except perhaps to go back through the chapter and make sure you can still work the practice problems.

He continues with more thoughts about this method of early studying, which I will leave for you to read.

How to retain information

In 2001, we got this more specific question from a student who understands independent learning, but struggles with an imperfect memory:

Math for a Future in the Sciences

I've had a hard time with math for the last three years. The theorems, equations, patterns etc. all seem to pile up - it's impossible to memorize them all. Or sometimes I won't understand a concept.

Well, I've become obsessed with a field called Nanotechnology (or Molecular Engineering/material sciences). This field is intense in math and physics (I love physics; I can't get enough of it). My problem is, I want to be able to understand as much math as possible (geometry, Trig, Algebra, Calc, etc.) and keep this information with me for the rest of my life. But once I learn something, I forget it quickly - I can barely remember ANY algebra from last year!

Is there anything you can recommend for someone who wants to learn math on his own, and be able to retain information for the years to come?

Doctor Ian took this one, too:

I can sympathize with you, because I have a similar problem. I've _never_ been able to remember formulas, and the way that I've had to compensate is that whenever I learn a new formula, I have to learn how to derive it from first principles - by which I mean, the things that are so basic that I _can't_ forget them.

(One of the luckiest breaks I ever caught in my life was when I took second-semester calculus in college.  That's the semester when you have to memorize about a zillion formulas that all look pretty much the same - one plus or minus the sine or cosine of plus or minus something, under or over the square root of something similar - so I fully expected to fail. But on the first day, the professor said: "I don't expect you to remember anything that I can't remember, and the only thing that I can remember is that cos^2 + sin^2 equals 1." So you're not alone!)

One trick you can use when trying to remember patterns or principles is to encode them as examples. For example, for the life of me I can never remember whether (a^b)^c is a^(b+c) or a^(bc).  So whenever I need to know, I drag out this example:

  (a * a) * (a * a) * (a * a) = a^6, so (a^2)^3 = a^6

which means that it must be (bc), and not (b+c).  Does this get tiresome? You bet. Is it preferable to guessing wrong? You bet!

I do things like this all the time, particularly when I’m helping a student with a topic I haven’t recently used or taught, so I don’t trust my memory. I will have an idea what the rule is, and I check it out by trying a quick, easy example. (Or I’ll do a quick online search, when it’s not face-to-face tutoring!)

Two other tricks I can recommend are:

1) Try teaching what you've learned to someone else. This is probably the single most effective way of learning anything, especially if the other person is having difficulty learning it. It forces you to think of new ways to understand the material, in order to avoid presenting it in the same old way. 

2) Try imagining yourself _using_ what you're learning at any given moment. The more vivid your imagination, the stronger the effect it will have on your memory. With math, the thing to do might be to try to find examples in physics where you would use any new concept or technique.

In the end, you may find that your memory just isn't going to work as well as you want it to, in which case... there's always paper and ink! That's what they're for, really, so don't be ashamed to make use of them. If you have to choose between remembering things you don't understand, and understanding things that you can't remember, I strongly recommend the latter.  :^D

An idea I use, similar to his first, is to be sure to write out my work on a problem in such a way that someone else could follow it; that way I’m communicating clearly to myself, too — which helps me avoid mistakes,  makes it easier to check, and also teaches me what I need to be taught.

As to his second point, we could add to it that we learn best by actually using what we are learning. I would write out the formulas or facts I need to learn, and refer to it as I do the problems. Then, if things work out as I hope, I will gradually need to look at the paper less and less, and I will have learned them. Memorization does not have to be a separate activity.

The value of repetition

Here’s a question from 2011, about the same problem of forgetting:

Forgetting and Remembering, Re-inventing and Explaining

I can quickly learn math things I don't understand, but I always forget a few days later. Do you have any way that can help me remember what I've learned?

For example, one day I learned about ratios and now I forgot how to do them until I Googled it. Can you help?

Doctor Ian starts by pointing out a semantic issue:

So one might say that you haven't really 'learned' them, right? This is normal for things you don't keep doing on a regular basis.

A couple of months ago, I was making bread every day, so I knew the recipe for that -- how much of each ingredient to use, how hot to make the oven, and so on.

But now that I haven't done it in a while, I'd have to look it all up again! So did I really 'learn' it? 'Yes' in the sense that I was able to do it. But 'no' in the sense of permanently committing it to memory.

On the other hand, there are things I haven't done for a long time that I'd still be able to do right now if I had to. Hitting a golf ball, for example.

What's the difference? I've hit a lot of golf balls, over a long period of time; but making bread, I've only done a few times, all over a short period. In a word: I've practiced golf a lot, but I haven't practiced making bread very much.

The same thing applies to the math you're learning. If you don't keep practicing, it will fade. But that has nothing to do with the fact that it's math. The same is true of just about anything.

It takes time to really learn something new. How do you know you’ve learned it?

One other difference between really _learning_ something and being able to do it with help is that when you really know something, you're able to explain it to someone else who doesn't know it.

In the case of something like ratios, it's one thing to be able to set up and solve a problem, by following some steps that someone else showed you. It's quite another thing to be able to explain to someone else, not just WHAT the steps are, but WHY those steps work.

There are two things here: When you can do this, you are putting it into your own words, and you are going deeper than just following steps. By processing the information more fully, you are making it more a part of yourself.

Another way to think about this is: if you can re-invent something that you've forgotten, then you really knew it. Otherwise, you didn't.

So how do you make use of this information? First, practice the things you learn -- and in small sessions over a long time, not just a lot in a few long sessions. It would be better to use a skill twice a day for a week than to use it 14 times on a single day, even though that works out to the same number of uses.

The second thing is, when you've learned something, try to teach it to someone who doesn't already know it. Younger brothers and sisters are perfect for this kind of thing. Other students in your class are also good.

The important thing about this kind of teaching, though, isn't so much what you say at first: "Do this, then do this, then do this, and you get the answer." The important part is when the other person doesn't understand, and starts to ask questions. It's when you try to answer the questions that you realize that maybe you don't understand what you're teaching as well as you thought you did! And that's when a lot of real learning takes place: when you have to answer your OWN questions in order to answer the OTHER person's.

That, by the way, is one of the reasons volunteers like to answer questions for Ask Dr. Math. It constantly reminds us of how much we don't actually know about things we thought we 'knew.' :^D

Yes, one reason I love doing this is that we get questions I’d never have thought of myself, and they force me to deepen my understanding.

How to teach yourself

That last question was from a 10-year-old. Our last question is from an adult:

How Do I Learn Basic Math as an Adult?

I am a counselor, and I would like to take a course on Statistics in Psychology, but I have not studied math for over 20 years.  Math was also never a strong subject for me.  I need to take an entry test to get into the stats course, so I'm wondering what the best way to brush up on my math skills is?  I'd also like to get over my dislike of math.  What do you recommend?

Doctor Mitteldorf answered:

First, recognize that there are no shortcuts.  If you want to learn 5 years of middle school and high school math, it's unreasonable to expect yourself to do it in one year.  With intense study you might do it in 3, but it is more realistic to think of it as a long-term project for a full 5 years.

How much time is needed depends, of course, on how much has been forgotten. But it is important to be realistic.

Second, get out some books that make the stuff interesting.  Don't try to pore through your old 8th grade math text.  I remember the first math book I read, by Isaac Asimov, called "Realm of Numbers".  It came from a "gee whiz" attitude.  Isaac Asimov was a person of wide-ranging curiosity who constantly asked questions and found relationships, and was interested in whatever it was he was learning, from biochemistry to Gilbert and Sullivan.  That was 40 years ago, but there are other, more recent books that make math fun.  Try "Number Devil" by Hans Magnus et al.  There are other books as well, probably directed at kids, but don't be proud.

I started out with the very same Asimov book, and a couple others like it! (Learning outside of school, because it was interesting, is the best way to begin.)

Third, solve problems.  Find them in the books, think them upyourself, create them from the world around you.  Work on them in your spare time.  Dream about them and wake up writing down numbers and equations.  Go for long walks where you puzzle out loud how to solve a problem that has stumped you.  The time you spend trying out solutions that turn out to be dead ends is not time wasted.  In fact, that's where all your real learning is going to take place.  No one ever learned math from reading a book, or from watching someone else do it. The book is there just for inspiration, for posing problems.  You have to do the work yourself, and put in the time exploring your ideas wherever they lead, to their logical conclusion.

Yes, this is the way to really learn.

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