The Why, The What, The How

and

The Switcheroo

Math isn’t about remembering everything. 
It’s about knowing a few things and figuring out everything else.

Allow me to introduce you to the three main players in the math scene.   They are The How, The What and the Why.  The How is the low level thug on the street and The What is his manager.  But The Why is the king pin, the one calling the shots!

Most of teaching and learning math revolves around The How.  That is, how do you find the vertex of a parabola, or how do you solve for x, steps one, two three, weee!   I get it.  Math is the least favorite, and often most terrifying, subject for many students.  They have to take math, it’s required.   The How is easy.  It’s easy to “learn,” looks great for the teacher if they are being observed by an administrator, and it’s very comfortable for student and teacher.  Sometimes The How is all the teacher knows.  I know I have been in that situation before.

An example of The How follows.  I will show you how to calculate products of numbers close to 100, in your head.  Not only that, with a little practice, you’ll be able to do it more quickly than you could using a calculator.  Ready?

Let’s look at the product of two numbers close to, but greater than 100.  For example, the product of 102 and 107, which is 10,914.  Without doing column multiplication, here’s how you can calculate that product quickly, in your head.

The answer will be a one followed by  four digits which we will refer to by WXYZ.  It will look like 1WXYZ.  The WX will be the two digit sum of the 2 and 7, which is 09.  The YZ will be the two digit product of 2 and 7, which is 14. 

So all you do is add and then multiply the two numbers at the end and place them properly. 

Let’s try 104 and 108. 

104 X 108 = 1WXYZ

WX =  12 (because 4 + 8 = 12).

YZ = 32 (because 4 X 8 = 32).

So 104 X 108 = 11,232

Why don’t you try a few and then check using a calculator:

1.        101 X 103 = ________________

2.       109 X 102 = _________________

3.       105 X 107 = _________________

There are similar methods for finding precise square roots of prime numbers and products of 4 digit numbers and so on.  But I say, big whoop.  First off, if all a student knows is The How, they’ll forget.  Without The What and The Why, they’ll never be able to recover their lost “learning.”  That takes this trick out of the realm of learning, in my book (haha), and into the realm of training. 

In math classes there is some, but not enough of, The What.  That is, what does it mean to find the vertex, or what does it mean to solve for x?  What does it mean to factor a polynomial? 

The What, in this example you’ve just seen is so obvious that there’s little to explain.  The What here, is finding the product.  You already know how to do that, although in a different way.  Usually you need to know the what so that if The How you know doesn’t work, you can perhaps find another.  Beyond that, if you know The What, you can use the good ole internet to help you! 

I often ask my students, What does it mean to solve for x?  Most do not know the first time they’ve been asked.  These same students are those following lists of steps, the same that will fail to recall how to solve for x after the briefest of vacations from the topic.   To be good at math a student must know what it is they’re being asked to do, then they must know how to do it. 

The Why refers to, Why does this work?  The Why is the conceptual foundation.  Without it you cannot connect ideas or apply previous knowledge in a new way.  All three parts How, What and Why, are needed, and sometimes just knowing The How is acceptable.  That said, The Why is the granddaddy of them all.  What it is you’re doing in math, and how it is you’re doing it, both come from The Why.  The Why explains the inner workings, it justifies that what you are doing is valid and true.

The Why in this trick you’ve just learned involves the order of operations, distributing and some other things that I will not draw out with great detail as the point is to explain how it is The Why drives The How. 

·         In the product of 104 and 108, the first digit will always be a 1 because 100 x 100 is 10,000. 

·         The reason the next two numbers are the sum of 4 and 8 is because we are adding 4 to 100 (to get 104) and multiplying that by 8 added to 100 (to get 108), as shown below here:

104 X 108 = (100 + 4)(100 + 8)

·         By distributing, which you may not yet know (it’s okay if you don’t), we get:

(100 + 4)(100 + 8) = 100(100 + 8)  + 4(100 + 8)

·         Which gives us the following sum:

10000 + 800 + 400 + 32

·         If we write this in a column it’s easy to see that we add the 8 and 4 to create the second two digits, and of course the last two digits are the product of 8 and 4. 

10000 800 400     +         32

11232

Now, if you really understand The Why (it works), then perhaps you can come up with a similar short cut for multiplying two numbers slightly less than 100, like 97 and 98.  If you can, then you’re so much better equipped than had you just learned the simple trick I showed at first.

On a side note, I once started the school year off with this, and a few other tricks (Just The How, nothing else).  Word got out that this amazing teaching was taking place in my room and that students were doing great math on the first day of school.  The principals heard and soon all of the mucky-mucks in the administration office stopped by my room.  They witnessed and participated and were wowed…and I was ashamed because it is simply a parlor trick.  The point of it was to get students to engaged in class, to do a little math and have fun and to know that every day, even on the first day of school, we will work.  I was awarded the school teacher of the year award that year.  I hope it had nothing to do with this trick!

The How (to do it), without understanding The What (you’re doing) and The Why (it works), is just a trick.  A computer could be programmed to do it and computers understand nothing. 

In this course, while still remaining practical, I will try to address The Why far more than I address The How.  That’s the switcheroo, if you were wondering.  It is my sincere desire that you know why it is that you do certain things, why it works.  If you also understand why things work and what it is you’re being asked to do then you can always figure out how to do it.  That would make you a powerful math student and I suggest it will change how you think about problems outside of math.

I hope you are convinced of the value, and appropriate places, of The How, The What and The Why.  If so, you will no longer settle for just getting an answer.  Someone telling you how to do it, without explaining why, is just as useful as someone telling you the answer.  If you don’t first understand what it is you’re doing and second why it works, then you simply do not understand.