Math: A victim of its own success

Why is mathematics so hard?

Because math is hard.  How hard it is varies from person to person, but by and large everybody finds math hard.  For some people, math is hard like climbing a mountain is hard:  there is a lot of work, and a few muscles pushed beyond what you thought they could do, but the end has an amazing view and leaves you feeling great.  For other people, math is hard like swimming in sewage is hard: its a lot of hard work, there’s nothing worth seeing, and you don’t get anywhere you really want to be.  Either way, though, the math really is hard.

So why is it so hard?

The answer is that math is complicated.  Really, really complicated.  Buy why?

The thing is, math starts out pretty simple.  The original math problems were really, really simple.  For instance, if you had two apples and I gave you two more apples, you would have four apples, which is enough to eat one yourself and give one each to your friends Harry and Sally, with one left over for tomorrow.  That’s math.

We’ve been solving problems like that for a long, long time.  We’ve become really good at solving that kind of problem.  And that’s where it gets complicated.  Because as soon we humans solved a problem like that, another problem would come along which was even more complicated.  Now, instead of there being four apples total, there are hundreds.  And instead of just getting more apples from our friends, we’re trading those apples for milk.  We don’t want to be cheated, so we need to keep track of hundreds of apples and large amounts of milk.  After that, we are trading the milk for land—now we need to know how much land we are buying.  That’s geometry.

Put like this, it may seem kind of overwhelming.  These kinds of problems are those most dreaded and feared of all problems: word problems!  And yet these are exactly the kind of problems which created math in the first place.  Because every time a new problem like this would pop up, people would find a solution.  Hundreds of apples?  We invent counting numbers and ways to write them down.  Buying land?  We invent the idea of area, and discover a lot of useful tricks in geometry.

This has been going on for thousands and thousands of years.  It started simple enough, but we had writing.  We had teachers.  Every generation learned all the old tricks and created a few new ones.  What we try to teach our students in K-12 alone is the result of at least three thousand years of hard work.

Put this way, no wonder math is hard.  That’s a lot to learn.

But this brings us to another question.  Math is hard for everybody, but for some people the effort results in amazing views and fantastic results, while others would rather just forget the whole thing.  Why the difference?  Why do some people get an amazing view from math, while others get out as quickly as they can?

There are many reasons, but I believe that one of the biggest is that a lot of people never see the grand scope of math as I described it here.  Math was originally invented to solve problems.  Lots and lots of really important problems.  Trading apples, buying land, dividing inheritances:  all of these played a big role in creating mathematics.  At some point, people began solving these problems because they were fun to solve, not just because they needed to.  Even then, though, they would set themselves clever problems which needed careful thinking to figure out.  Along the way, they invented lots of tricks to make solving common problems fast and easy, often so that they could get to solving the more important parts of the problem.

This is where math begins to victimize itself.  Because we’ve spent so much time working out how to solve problems quickly, the tricks we’ve created have become very clever.  They are often so clever that there are sometimes many, many steps between the original problems and the tricks we use to solve them.  When people first invented these tricks, the connections to the original problems were obvious to them, but they only invented the tricks after a lot of careful study, thought and practice.  Our ancestors worked hard to invent our modern tricks.

The problems really begin to happen when we realized that the tricks we use to solve problems are in some ways easier to learn than the original problems.  It is very easy to write 2+2=4.  It is a lot harder to write “I had two apples.  I went and picked two more apples.  I now have four apples.”  Also, 2+2 always equals 4, right?  So if we just remember that 2+2=4, we’ll never have to worry about why it works!  It makes life so much easier and faster.  And there are a lot of other tricks we can use to solve problems like this and make actually calculating the answers a lot faster.  There are a lot of good arguments to be made for becoming really good at the tricks which make us faster.

And this is where things go wrong.  The tricks we use to solve math problems are actually kind of boring.  7×8=56?  Yeah, not a real page turner. (x+y)^2=x^2+2xy+y^2? It looks like alphabet soup spilled on your keyboard.  6/1.543=3.88852884? Useless.  By itself, anyway.  If all you do in math is repeat the tricks, you bore the students to tears.  It’s easier to do, but absolutely nobody cares.

Without the context, without the word problems, without the essence of mathematics, the tricks mean nothing.  And because of the thousands of years of history behind the tricks, very few students see through the tricks to understand the original problems the math was meant to solve; the tricks are too clever for their own good.  Math is a victim of its own success; we have made “solving” problems much easier than knowing why we are solving the problem in the first place.

This is a fatal mistake; the math problems are interesting and the math tricks boring.  Without knowing why we need to know the math tricks, mathematics becomes a long slog through useless-seeming facts which go nowhere and do nothing.  With the problems and the context, math is an adventure which can be difficult (that’s a lot of material to learn), but which will result in an amazing view of the world when we finally get to the top.

With the context, with understanding, the difficulty in math becomes a joy and the work becomes a reward.  Children will do the boring work of learning the tricks as a way of getting to the top.  And that is why it is vital to show children the bigger picture.

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