Why is Math Interesting?

Date: 01/16/2001 at 18:16:45
From: Amanda Blair Channell
Subject: Why is math boring?

Hi Dr. Math,

Why does math have to be so boring and hard sometimes?

Date: 01/18/2001 at 10:30:55
From: Doctor Ian
Subject: Re: Why is math boring?

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.


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.

Another way is to start doing math-related puzzles. Some people find 
them incredibly boring, but others find that they make math come 
alive. Another way is to learn to appreciate math the way many people 
appreciate art, or music - that is, because it's beautiful. I would 
_strongly_ recommend that you read Doug Hoftstadter's book _Godel, 
Escher, Bach: An Eternal Golden Braid_. Forget about the main chapters 
at first. Just read the dialogs _between_ the chapters.

A second possible answer is that you've been able to learn other 
things easily because you've been able to instantly form lots of 
connections to things that you already know. If you've worked with 
wood a lot, then you already have a feel for much of what you'd learn 
in a class on trigonometry. If you've worked with pottery, or 
sculpture, then you already have a feel for much of what you'd learn 
in a class on integral calculus. That is, you'd be learning new 
syntax, but you'd already have a handle on the semantics. The best way 
to understand _anything_ that you're told in a math class is to come 
at it from the direction of something that you understand intuitively. 
That way, each new pattern or formula isn't a new, isolated fact to be 
memorized, but just a new way of looking at something you already 
know.

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.

When you're looking for a book to work through, you want to find one 
where the problems in the first chapter seem too easy, and the 
problems in the last chapter are too hard. That means that the stuff 
in the middle is stuff that you need to learn.

Once you've found a few books that seem to be at the right level, open 
each one to somewhere in the middle and see how well you can follow 
the author's explanation of something you don't know yet. Each author 
has a different style, and you'll learn faster if you find authors 
whose style of teaching matches your style of learning. If you have 
trouble following what's written in one book, don't assume that it's 
entirely your fault. Look for another book that better suits your 
personal learning style. One nice thing about math is that, no matter 
what you're trying to learn, there are a zillion books (to say nothing 
of Internet sites) that are waiting to try to help you learn it. (This 
in itself is one of the keys to 'making math easy'.)

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.


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.

So the question is; why do we do this? It's possible that we're just 
inept at figuring out how to teach things. (As someone once pointed 
out, if we had to teach children to walk, they'd never learn!) On the 
other hand, there might be a method to the madness. If _everyone_ 
learned to be good at math, then there would be more competition for 
all those high-paying jobs. As it is, by making math seem much more 
difficult than it is, we ensure that we'll always have a steady supply 
of people to work at fast food restaurants, pick up the trash, turn 
wrenches in factories, and so on.

As for why kids so often "freeze on tests," my own feeling is that 
it's because they aren't fooled at all by the pretense that the tests 
are being given to help them figure out what they need to learn. 
Adults deny it, but kids know that every test is another opportunity 
for society to decide who is going to end up on top, and who isn't. 
Tests are competitions, kids know that, and so they freeze - for 
exactly the same reason that they'd freeze if they were forced to do 
something trivially easy (like toss a piece of paper into a trash can 
five feet away), with the promise that they'd be flogged in public for 
screwing up.

Here's another way to think of it. If I put a 2-by-4 on the ground, 
could you walk on it from one end to the other without falling off? Of 
course, it would be no problem at all.  

But what if I placed the same board between two ledges, 500 feet above 
a busy street? Now could you walk from one end to the other? It would 
be somewhat more difficult, wouldn't it? But it's the same task, isn't 
it?

There is a philosophy whose purpose is to help people get past this 
"freezing" business. It's called Zen Buddhism. (I know, that _sounds_ 
like a religion, but it isn't.) You can read a really nice, short, 
simple introduction to the ideas of Zen Buddhism in a book called _Zen 
in the Martial Arts_, by Joe Hyams. (I know, that _sounds_ like it's 
going to be a book about karate, or judo, but it isn't. At least, not 
mostly.) It's a pretty popular book, so you should be able to find it 
in most libraries or bookstores. It would be well worth your time to 
read it.

This probably isn't what you wanted to hear, but it's better that you 
should find out sooner rather than later.

I hope this helps. Write back if you'd like to find out more about how 
to make math fun for you, or if you have any other questions.

Anyway, Amanda, those are my thoughts on what makes math seem so 
boring and so hard to so many kids. If you'd like to talk about this 
some more, please don't hesitate to write back. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 12/13/2002 at 05:16:03
From: Marc
Subject: Why do they make math seem hard?

Why do some people or teachers make math seem hard to understand?  Why 
do some teachers teach math backward? 

Also I want to know what Dr. Ian means by this paragraph:

"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."

A million thanks!

Date: 12/13/2002 at 14:12:32
From: Doctor Ian
Subject: Re: Why do they make math seem hard?

Hi Marc, 

One of the things it took me a long time to realize was that math is
largely a game, and the game works this way: We make up a set of rules, 
and then we try to see what consequences follow from the rules. 

For example, one game is called 'arithmetic'. We define what it means
to be a number, i.e., the first number is zero, and if we have any
number, we can define another number by adding 1 to it:

   0 + 1 = 1
   1 + 1 = 2
   2 + 1 = 3

Of course, to do this, we also have to define what we mean by
'addition'.  Now, having made up these rules, we start playing with
them to see what happens. One thing that happens is that we notice
certain patterns, and we use these patterns to add new rules. We also
make up new notations to make things easier to write.  

For example, we notice that


  1 + 1 = 2

  1 + 1 + 1 = 3

  1 + 1 + 1 + 1 = 4

and we realize that we can shorten this to 

  1 * 2 = 2

  1 * 3 = 3
  
  1 * 4 = 4

So now we've got the concept of multiplication as repeated addition.
Now we can do this

  3 * 4 = 12

instead of this

  (1 + 1 + 1 + 1) + (1 + 1 + 1 + 1) + (1 + 1 + 1 + 1) = 12

And we can start seeing where that leads. One place is this: Some numbers 
can be obtained by multiplying different pairs of numbers, 

   24 = 2 * 12

      = 3 *  8

      = 4 * 6

while others can only be obtained by a single pair of factors,

  13 = 1 * 13

This is kind of interesting! We give the latter numbers a name -
'prime' - and now we can start to think about what things are true of
these numbers: How many are there? How are they distributed? How
can we find them? If someone gives me a really big number like
2194359484329019, how can I tell whether it's prime or not?  

Now, here's the important point: All of this stuff is presented to
students as if it's always been known. But in fact, it was all
discovered by mathematicians playing around, making up questions and
then trying to answer them, and then sharing their answers with other 
mathematicians who would use the answer to make up new questions, and 
so on. 

Of course, over time, the questions get more complicated, and you have
to understand a lot of what's happened in the game already to even
understand what's being asked. And at a certain point, the questions
start to be about the very nature of games:

   What is mathematics?
   http://mathforum.org/library/drmath/view/52350.html

But it's still the same game. You invent objects (numbers, sets,
groups, fields, points, lines, planes, manifolds, algorithms), and
rules that have to be obeyed by the objects, and then the game begins.  

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. 

While we're on the topic, there's another point that I think is
important to make, which concerns the popular idea that mathematics
should be made 'relevant' by showing how it can be used to solve 'real
world' problems.  

If I take some silkworms and put them in a warm place with plenty of
mulberry leaves to eat and no predators to eat them, they'll make
silk. Now, I can take that silk and use it to make all kinds of 
things - scarves, boxer shorts, surgical thread, parachutes, or 
whatever - but the silkworms don't care about any of that. They make
silk because it's what they know how to do.  

Mathematics is like silk in that sense. If I take some mathematicians
and put them in nice offices and give them money to take care of their
physical needs, they'll create mathematics, i.e., they'll make up
puzzles and solve them. Now, we can take their solutions and use them
to make all kinds of things - bridges, computers, budgets, animated
movies, flight simulators - but the mathematicians don't (necessarily)
care about any of that. They make mathematics because it's what they
like to do.  

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.

Maybe someday we'll find out.

I hope this helps.  Write back if you'd like to talk more about this,
or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/