'Why' Questions in Math

Date: 04/19/2002 at 13:09:14
From: Sharyn
Subject: "Why" Questions in Math

Is math a bad area for asking 'why' questions?

In science, when one asks why, there seems to be a "real" (to me) 
answer. Such as chemistry, where one can identify a chain of concrete 
events that leads to "now."  But if one asks, when calculating the 
standard deviation, why the sum of deviations around the mean equal 
zero, the answer seems to be a long "proof" that I won't understand.  
(It seems the answer is that someone just discovered that, so we 
square the values to add them.)

I always thought algebra was mystical and there was a whole lot I did 
not "get" (and everybody else did "get").  If I had known to just 
memorize things, I bet I would have been a lot better off. (Maybe math 
teachers should say something about that?)

Thanks for your time!
Sharyn

Date: 04/19/2002 at 14:32:02
From: Doctor Ian
Subject: Re: "Why" Questions in Math

Hi Sharyn,

In science, (we assume that) there is a set of rules (the fundamental 
laws of nature) in operation, and the task is to figure out what the 
rules are by observing the results that occur when the rules are 
followed. Basically, it's an attempt to reverse-engineer the machinery 
of the universe.  

In math, it's the other way around - we get to choose the rules, and 
the task is to discover the results of choosing any particular set of 
rules. 

There is a superficial similarity, which leads some people to confuse 
the two pursuits. In science, the way you test a theory is to codify 
it as a set of rules, and then explore the consequences of those rules 
- in effect, to predict what would happen if those rules were true.  
You do the same thing in math - and in fact, the way it's done in math 
serves as a model for the way it's done in science.  

But here is the big difference: In science, as soon as your 
predictions conflict with experimental data, you're done. You know 
that your rules are wrong, and you need to start putting together a 
new set. In math, this kind of conflict can't arise, because there is 
no necessary connection between any mathematical theory and the world.  
The way you 'test' a set of rules in math is to see whether the 
results they produce are interesting enough to induce mathematicians 
to keep playing with them. 

We might summarize the situation this way: Science is the pursuit of 
_the_ correct description of _this_ particular world; whereas math is 
the pursuit of interesting descriptions of possible worlds. Whereas 
scientific theories are right or wrong, mathematical 'theories' are 
merely interesting or uninteresting.

To get back to your question, there is nothing wrong with asking 'why' 
questions in math! But it helps if you know ahead of time that 
ultimately, the answer will always be 'because we decided to use these 
axioms instead of those axioms'.  

However, there are lots of levels of 'why' explanations before you get 
to that point. To take your example of the reason that the deviations 
from the mean add up to zero, that's a consequence of the way that the 
mean is defined.  

Whether the explanation is long and complicated or short and intuitive 
depends on whom you're trying to convince. Here's one way to think 
about it. To find the mean of a bunch of values, we add up the values 
and divide by the number of values:

         v1 + v2 + ... + vn
  mean = ------------------
                 n

Does that make sense so far?  Okay, now suppose we've found the mean, 
and we subtract that from each of the values:

         (v1 - mean) + (v2 - mean) + ... + (vn - mean)
     ? = ---------------------------------------------
                            n

         (v1 + v2 + ... + vn) - (mean + mean + ... + mean)
       = -------------------------------------------------
                            n

         (v1 + v2 + ... + vn)   (mean + mean + ... + mean)
       = -------------------- - --------------------------
                 n                           n

         (v1 + v2 + ... + vn)   n * mean
       = -------------------- - --------
                 n                n

Now, the first term is just the original mean. And the second term is 
also the original mean. So the right side has to equal zero.

What does this tell us? It says that if we subtract the mean from each 
value, and then add the resulting values, we end up with zero. But 
this is just another way of saying that the deviations from the mean 
add up to zero.

Now, if you think of the mean as an arbitrary formula that you 
evaluate on a bunch of numbers, then this is indeed sort of 
mysterious. Why does it work out this way? Is it magic?

Well, just the opposite, in fact. A better way to think of the mean is 
this: If you have a bunch of values, the mean _is_ the value that you 
would have to subtract from all of them to get the sum to add up to 
zero. That's what it _means_ for something to be a 'mean'.

Do you see the difference? It's not that we make up the formula and 
then this weird consequence drops out. It's that we want to get the 
consequence, and we look around for a formula that will give it to us.  
The problem, of course, is that it frequently gets _taught_ in the 
opposite direction. Hence the kind of confusion that you've been 
experiencing.

For what it's worth, memorizing things is just about the _worst_ 
possible way to go about learning math. If you have to memorize 
something, it usually means that you don't understand it. And if you 
don't understand it, your chances of being able to apply it in 
unfamiliar situations are slim indeed.

In the case of the formula for the mean, if you understand what a mean 
is supposed to do - that it's supposed to distribute the values in a 
certain way around a central value - then you can actually figure out 
what the formula would be (by working out some simple examples), if 
you happen to forget it.  This is the case with an awful lot of math.

To give you another example, I can never remember what the formula for 
the surface area of a cylinder is.  But I know that it's made up of 
three parts:  two circles, and a side.

And I know that if I slit the side open and unroll it, it's actually a 
rectangle. Now, I _have_ memorized the formulas for the area of a 
circle, 

  area = pi * radius^2

and for the area of a rectangle, 

  area = width * height

and the area of a triangle, 

  area = (1/2) * base * height

but those are the _only_ formulas for areas that I've memorized. 
Everything else, I work out in the way that I just showed you. 

Similarly, I can never remember what

    b c
  (a )

is supposed to be.  But I know what an exponent is, so I can use 
a simple example, like

    2 3
  (a )  = (a*a) * (a*a) * (a*a)

           6
        = a

           2*3
        = a

This approach doesn't really take any longer, and this way I know that 
I haven't mixed up something in my memory. 

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/