(Copyright © 2000-2001 --
this item revised 26 Nov 2000)
| Mini-TOC: | The wrong way to study ... So how can you say math is easy? ... The right way to study ... Conclusion |
You're probably aware you're not alone in this. We see a lot of plaints like this on the newsgroup.
Several things might make you feel this way, but actually being stupid is probably not one of them. One of the biggest problems is studying math the wrong way. This article will focus on that problem. You may find this hard to believe, but math is actually one of the easier subjects -- if you go about it the right way.
Bear with us, and read this section attentively before you move on to the solution below. You need to understand the problem well to understand the solution.
The main mistakes that cause students to fail at math are
In most subjects, you can give up on one particular topic and not lose that much for the next one. Take history: if you don't understand the French Revolution you can still understand the Congress of Vienna reasonably well. And you won't suffer at all when you move on to the independence movements in Spanish America. Come exam time you'll miss the French Revolution questions but you can get the rest right.
But in math it's very different: pretty much everything builds on earlier work. That means if you once let yourself fall behind, it will be very difficult to catch up. And if you try to slide by, understanding something "well enough", you don't understand it well enough to build on it. Picture building a house without first baking the bricks all the way through: the walls will sag, and the house will collapse. In just that way, if you don't thoroughly master completing the square you are going to have a tough time finding the vertex of a parabola, or solving the general quadratic equation.
The problem is, usually you can't work at your own pace, but have to keep pace with the class. There's only a certain amount of time for each topic, so you do what you can in that time and try to convince yourself that it's enough, even though deep down you know better.
And then for fear of appearing "dumb", you don't ask for help, even though half the class might be having the same problem you are. Remember, your teacher can't know he's going too fast unless students speak up. And you can't know whether you are the only one having trouble unless you speak up.
And to make things worse, a lot of students try to approach math like a collection of magic spells. You memorize a bunch of formulas and hope to be able to recall the right one. That actually makes things worse, because you spend time on memorizing that ought to be spent on understanding.
Finally, most students hate homework, so they avoid working problems. This is a big mistake too, because working problems is the only way to learn math. The only way. Period.
It's easy, if you keep up, because each step is just a small one, and there's very little memory work. Yes, really. In history or biology you have to memorize a huge mass of facts. In math you have to memorize only a very few facts, because what you learn in math is a method of working.
Once you know how to complete the square, you simply do that to the equation of a parabola and you can read off the vertex. There's really nothing new there. That very fact, which is a killer if you don't keep up, makes math the easiest subject if you do keep up.
Look back at the common mistakes above. The right ways to study are naturally the opposite of those:
It's usually easier to learn things with someone than alone. Try to form a study group, two to four people. It's probably best if all have about the same level of ability, so no one is tempted to lean back and watch someone else work. Decide how often to meet, perhaps twice a week to start, and stick to the schedule.
Use your study group well. That means do all the work yourself before the group meets. There's no point on wasting group time on things you could have done on your own. You will not learn well by looking at someone else's work; you will learn well by doing your own work and then having someone else cast a critical eye on it while you do the same for her.
Stay focused in the group. There should be very little of "what did you get for number 3?" but quite a lot of "On number 3, I got 11 miles and the book says 4 miles. I've checked but I can't find my mistake. Can you see what I did wrong?" You can get help from the other group members when you just can't see how to work a problem, or if you got a wrong answer and can't see where you made your mistake. Be prepared to give the same kind of help, which will reinforce your own learning.
Group members may want to compare techniques for doing some of the problems. You may be able to learn a useful trick, or share one.
Master each topic thoroughly before moving on.
Everything builds on what came before, so you must make a solid foundation. That means thoroughly understanding each topic in turn, not trying to kid yourself that you understand it "well enough." As Yoda might have said if Spielberg had made Math Wars:
Understand, or understand not. There is no "well enough."
How do you know when you understand a topic? Easy: if you can work problems confidently, and get the right answers, you understand the topic. If you can't, you don't. How do you know whether you can work problems confidently and get the right answers? By doing it. There is no other way to learn math than by working problems.
If you look at the first one or two problems that follow a section, and you don't immediately see how you would start solving them, go back and re-read the section. Be sure not to let your attention wander!
Don't try to memorize formulas. Rather, you'll find that as you work problems, your mind will automatically pick up the formulas that you use often. To facilitate this, each time you need a formula, look it up in your textbook, then write it down on your paper as you need it in working that problem.
You'll find that you'll automatically learn the formulas you have to memorize, without extra effort. For instance, when working quadratic equations using the formula for the first time, don't try to memorize it. Simply work the problems, looking up the formula and copying it to your paper each time. By and by you'll find you know it, without having made any special effort. And if later you don't remember it, you can always derive it on the spot by completing the square.
When studying each section in your textbook, work a representative selection of the problems. That doesn't mean one or two from the group of exercises, it means a substantial number, at least half and maybe even all. Check each answer before you go on. If you got the wrong answer, figure out why, so that you don't reinforce bad habits by making that same mistake on later problems.
How can you know whether your answers are right? Look in the back of the book, if the answers are printed there. This is fast and easy, but it's probably not the best way for learning. That's why many books have answers for only some of the exercises.
You really should form the habit of checking your work. This may seem laborious, but it's actually the best way to learn. Only in this way do you get immediate feedback about whether your solution was correct. And if you did make a mistake, you correct it immediately, rather than continue making the same mistake elsewhere and reinforcing it.
Here are some techniques of self checking.
Can you work backward? For instance, after factoring, multiply out the factors and make sure you get the original expression. After solving an equation or a system, substitute each solution in the equation(s) to make sure it works. After solving a word problem, check your solution in the original problem (not the equation you derived from the problem). After finding the equation of a line, make sure that the two original points do satisfy the equation. And so on.
Can you work the problem in a completely different way? Often (usually!) there are several ways to attach a problem. If you solve the problem using two different techniques, and get the same answer, that gives some confidence that it's correct.
This is a definite last choice, but if you can't figure out a way to check your own work, you can always ask for help. Asking for help is not wrong, of course, but this is a last choice not because you learn best from doing, not from asking.
What if you just don't see how to work a problem? Don't put it aside to bring up in study group. Review that section of your textbook. Try the suggestions elsewhere in this FAQ. Your problem-solving "muscles" are like real muscles: they'll grow stronger with use, and you need to be honest with yourself about when you really can't work a problem without help, and when you just need to come back to it later.
Ask for help, but not too quickly.Okay, you've really done your best, not just taken a quick look at something,. or stared at it for minutes without really thinking productively. Where can you get help?
Ask your peers. If you have a study group, that's great; but you can always ask a classmate.
Ask the newsgroup. You will get the best quality help, and you will learn the most, if you post your complete solution; then we can show you what you did wrong. (And sometimes, in the act of formatting your solution for posting, you suddenly see it in a new way and your own error jumps out at you.) Please see the posting guidelines.
Ask your teacher. Again, do the work yourself first. You learn better by working the problem (active) than by sitting watching someone else work it (passive).
Especially if you are feeling overwhelmed, you may want to work with a tutor for awhile.
You are not a moron. You can do this. Just make sure to target your energy effectively and not spin your wheels. In a nutshell, learn each section, by working plenty of problems, before you try to tackle the next.