As fall approaches, and the beginning of a new school year for many, let’s take a look at some of our past discussions of how to study math. We’ll start with some perspectives on being motivated to study, since you are not likely to do well if you hate math (as so many students tell us when they write to us). How can you get interested in the subject?
Find out how interesting math really is
Doctor Ian has often answered these questions. The following is a very long answer, in which he put together many of his thoughts on it; I will only excerpt a few parts to get the flavor of what he has to say. You will want to read the whole thing!
The question, in 2001, was a simple one:
Why is Math Interesting? Hi Dr. Math, Why does math have to be so boring and hard sometimes?
I suppose that could be answered as it stands, but it wouldn’t be as helpful as turning it on its head, as indicated by the title:
Hi Amanda, We get this question all the time, and here are a few of the answers that I've given in the past. I'm sorry if they seem long, but it's an important question, and I've put a lot of thought into the answers. I hope you can get something out of at least one of them. 1) Here is one of my favorite stories (from _The Little, Brown Book of Anecdotes_):
At Columbia University, the young professor Raymond Weaver gave his first class in English literature their first quiz. The young men, who had been trying to make things hard for the new instructor, whistled with joy when Weaver wrote: "Which of the books read so far has interested you least?" They were silent, however, when he wrote the second, and last, question: "To what defect in yourself do you attribute this lack of interest?"
Math _is_ interesting, and once you've figured out that it's interesting, it's the easiest thing in the world, and more fun than baseball or video games or going to the movies. So the real questions are: Why don't you find math interesting? And is there anything you can do about it? Why don't you write back and tell me about some of the things you _are_ interested in, and we'll see if we can't find a way to get you interested in math, too.
Some students, given this request, have written back in response, which led to some good interactions. We still welcome such discussions.
2) Think of something that you _have_ been able to learn easily -- baseball statistics, playing the guitar, the names and powers and evolutions of 49,000 different Pokemon cards, or whatever. What do you suppose made it easy to learn? The answer is almost certainly i-n-t-e-r-e-s-t. If you're interested in something, it's easy to learn. If you want to make math easy to learn, you have to find some way to make it interesting to you. There are lots of ways to go about this. One is to find some relation between math and something that you're already interested in. That might be target shooting, or building electronic devices, or betting on horses, or playing with model trains. It's a good bet that no matter what you like to do, learning about math can let you do it more easily, and can even increase the amount of enjoyment that you get out of it.
Take charge of your own education
On the one hand, it's unfortunate that your teacher doesn't seem to have a gift for making math come alive. On the other hand, this is an opportunity for you to learn a very valuable lesson, which is that you don't need to wait around for teachers to teach you math - or anything else - in order for you to learn it. My advice to you is to take charge of your own education in math. So, how do you do that? You'll never go wrong by starting at the beginning! I would suggest starting with your present textbook and working through it. Read each section, and work the problems for that section until they seem too easy. Then go to the next section. When you're done with that book, find another book, and do the same thing.
He gives advice about finding books that work for you — not everyone benefits from the same kind of teaching.
One final thing you need to learn is the importance of _practicing_ what you've learned. The more you practice the material at each stage, the more quickly you'll be able to learn the material at the next stage. Think of practice as a pain management game. A little pain up front is often the key to avoiding a lot of pain later on. (If you think about it, the main task of becoming an adult is learning, and applying, that lesson.) I know this advice sounds almost too simple to be useful. But I can assure you that almost everyone who has become any good at math has learned it in this way. And whenever you come across a particular problem that you can't solve, write to Dr. Math, and we'll see what we can do to help you get past whatever's blocking you.
Play the game!
3) This is a real problem! My own feeling is that a lot of kids are frustrated by math because it's taught incorrectly. Imagine if we taught baseball the way we teach math. Kids would never get to see an actual game, let alone play in one... in fact, they wouldn't even be told that _is_ a game. They'd be asked to learn to compute statistics like batting averages and fielding percentages, "because you'll need them someday." They'd be asked to learn formulas for computing the ballistic arcs of balls flying through the air, even though those would _never_ be useful. They might be asked to learn to catch and throw, and hit balls tossed from a machine, but they'd never be told how any of those skills related to one another (for example, that one person tries to hit what another throws, and others try to catch what he hits). And so on, and so on. It would be a foolproof way to make sure that every kid would grow up hating this stupid "baseball" thing, wouldn't it? But isn't that exactly what we do with math? Here's the big secret that everyone is keeping from you: Math is F-U-N. It's more fun than video games, more fun than sports, more fun than just about anything you can think of... once you see it for what it is, which is a huge, international, cross-generational, fantastically whimsical, collaborative game of "What if?" played by people who get paid good salaries to do something they'd do anyway, for free.
Almost two years later, another student asked for an explanation of the last sentence. Here is a small part of Doctor Ian’s answer:
So let's take the main sentence of the paragraph a little at a time: [Math is] a huge, A lot of people play it. international, They come from every country. cross-generational, People alive now are still trying to solve puzzles posed by people who have passed on, using tools developed by people who have passed on. fantastically whimsical, Mathematical objects aren't constrained at all by the rules of the 'real world', and some of them are pretty bizarre. collaborative Each mathematician makes use of what the others have learned, and returns the favor by sharing what he learns. game of "What if?" Mathematics is about answering the question "If I make up such-and-such rules, what possible things can happen?" played by people Mathematicians. who get paid good salaries Not rich, but quite comfortable. to do something they'd do anyway, for free. If every professional mathematician lost his job tomorrow, and had to drive a cab, or sell insurance, or whatever, he'd _still_ play the game in his free time! And mathematics would continue to develop, although more slowly. So that's what I meant.
Obviously no single approach can work for everyone, but I can't help wondering how many students who hate math as it's taught now would embrace it if it were presented in the way mathematicians see it, i.e., as a game where you make up and answer questions that are interesting to _you_, rather than a collection of techniques to be memorized so you can solve problems that are interesting to someone else.
Here are some other answers to related questions of motivation:
Making Math Interesting Do We Really Need to Learn Math? Why Is Math Important? Math is Power?
After writing this post, I ran across an article that may be important for you if you think your interests are what they are, and you can’t change them: Instead of ‘finding your passion,’ try developing it. This is another aspect of the “growth mindset” concept, which we have discussed in my post on avoiding careless mistakes.
Why should I care about math?
A common question along the same lines is, Why do I need to learn this? A good answer can help motivate you. Here is a question from 2007:
Why Do I Need to Study Math? I am learning how to solve quadratic equations, graph quadratic equations and am also learning about complex numbers. Why are all of these concepts important to learn? How do they relate to real life? I am a high school student who is college bound, but I am not going into a major involving math, so why should I care?
We have given specific answers relating to particular topics in math, but I answered the bigger question:
Perhaps you shouldn't care! But I can suggest a couple reasons why it might still be worth having studied, even if you never touch a variable the rest of your life. First, let's turn the tables and ask the same question about another field. I DID go into a major involving math. My major did NOT involve literature. Yet I had to take courses in English, studying novels or short stories. Why should I have been made to do that?
You never know
One answer is that one never knows where one might end up. Although literature was never one of my great loves, I have (in part because of interesting courses in which I had to write essays) developed a love of writing, which is one reason I'm a Math Doctor. I also enjoy reading with my kids, including some of the great books to which I was introduced in those courses. And maybe someone else in my position would have ended up writing a novel! Similarly, you might end up either getting hooked on some kind of math, or simply getting into a job where, like it or not, math is part of the territory. Even if you don't end up DOING math, you may need to be able to communicate effectively with those who do--or to decide whom to trust to do your math for you!
It’s a great achievement of humanity
Another answer is that it is good for all of us to be introduced to the great cultural achievements of our civilization. It gives us a fuller understanding of the ideas on which civilization is built. This is called "liberal education"--getting a generous helping (the literal meaning of "liberal") of general knowledge, rather than focusing on one small part of what is known. And math, including all the topics you mention, is one of the great achievements of culture. To take simple concepts like numbers and shapes, and turn them into a complex structure of provable facts including many complete surprises like the Pythagorean theorem or the existence of complex numbers, shows the power of creativity. Just knowing it's there broadens your horizons.
It teaches you to think
Finally, studying a wide variety of subjects strengthens your mind. Just as athletes may do exercises that have little to do with the particular sport they are planning to participate in, just to help their bodies become strong and well balanced, much of what you learn in school is meant to strengthen all parts of your mind. Studying literature may have helped me in my mathematical studies by giving me different sorts of things to analyze intellectually, providing me with a broader set of experiences I can use in solving problems. Studying math may help you learn about various methods of solving problems, giving you insight into how to solve problems in the future that may not directly involve math, but may require some similar modes of thinking. Or maybe you'll just have more confidence, knowing you've been able to learn something hard, and therefore can do so again when you need to.
Sometimes when a student in one of my college’s “social” majors asks me why they need to learn math, I tell them that their math course may be the one course they take in which there are right and wrong answers; they need the experience of being required to think carefully.
Advice for teachers
Let’s close with another question answered by Doctor Ian, this one from a teacher in 2008:
How to Motivate Students to Learn Math? I am a teacher and need to give students a compelling why for learning about properties of special quadrilaterals. In other words, how would learning about properties of special quadrilaterals help students apply this knowledge to a previous use, a current use, or a future use? I need a reason to motivate students to participate in the geoboard activity I have planned. I do NOT want the reason to be because it will help to know when you get to higher math classes or to do well on tests like the state assessment or SAT. Please help.
As he often does, Doctor Ian turned the question around:
It's just my opinion, but in my opinion, the harder you try to "motivate" students this way, the more it works against you, because their experience will tell them that you're not being straight with them. That is, any one of them could go to a public place, like a mall, in a relatively affluent neighborhood, stop 100 randomly-selected people, and ask them what they know about the properties of special quadrilaterals. They would find that only a handful would even know what a "special quadrilateral" is; even fewer would be able to name a property of one; and still fewer would be able to name some practical use for that information. And yet, the majority of these people would be leading relatively successful lives. Which would say a lot about the ostensible "value" of this information. Your students know this, and pretending that they don't just undermines your credibility.
This is why specific answers to such questions are so hard. We all know that higher-level math is not needed by most people in most of the things they do.
So why study it? Because of the bigger picture:
Except to a very small number of people, the properties of special quadrilaterals aren't of any importance at all. What IS important, though, is learning how you go about solving a problem that you haven't seen before. If the focus of your activity is on that, and not on the particular results, then you have an easier sell. I like to use this analogy: When you were very young, you spent a lot of time doing things like stacking objects up so you could knock them over. That's not a skill you use anymore, but the time wasn't wasted. The point was never to learn a particular skill, but to develop hand-eye coordination, which is useful in lots of different contexts. Math is the same way. Scientists and engineers use it as a tool for understanding and designing things, but for most people the value of math is that it's like a little jungle gym for the mind. It's a medium in which we can set up problems of increasing complexity, so that students can get practice at learning to solve problems using both creative and organized thought. Once the particular problems have been solved, most people can just forget about the solutions, at least once they graduate. The valuable residue isn't the math itself, but the ability to do things like break problems into sub-problems, the ability to follow a logical chain of reasoning (and detect breaks in the chain), the ability to work backwards from a desired state ("If I knew this, then I could solve the problem; so how can I find that out?"), the ability to abstract or simplify a problem to make it simpler to work on ("What if I had only a dozen of these, instead of several thousand?"), and so on.
These techniques are things we try to demonstrate frequently in our answers, modeling for students how to think logically and creatively.
What's unfortunate is that many teachers emphasize the answers rather than how to deal with the questions; so their students, once they get out of school and forget the answers, are left with nothing. To put it briefly: There's no real reason to know this material, or even to learn it; but there's great value in getting practice at facing a problem that you haven't seen before, and working your way through it. Because that IS something you're going to be doing over and over in your life. To put it another way: The point of solving math problems isn't to be able to remember or use the solutions. It's that by solving lots of math problems, you get a chance to develop strategies for solving ANY kind of problem, and to learn how to work with your own personal strengths and weaknesses as a problem-solver. Buddhists have a metaphor for this: You use a boat to get you across a river, but then you leave the boat behind. You don't continue to carry it with you. It's done its job.
I should add that for many students, the particulars of what they learn are very important in their future. We’ve had many people write and say, “I wish I’d paid attention back in algebra class, because I need it now!”
For detailed answers about the uses of math, see our FAQ,
Why Study Math?
On the other hand, much of what is commonly taught is not what should be taught. For some personal thoughts on changing the curriculum to better match up with real needs, see
A Better Mathematics Curriculum, by Tom Davis