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### Why is Math Interesting?

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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?
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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,

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

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,

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

(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/    ```
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