I was at the Anja S. Greer Conference on Mathematics and Technology at Phillips Exeter Academy this summer, and I loved it.  Not only did I get to think about math and math teaching all day, but then Happy Hour meant refreshment and more great problems.   Like when Frank Griffin posed this problem:
 
How many regions are formed inside a circle when you draw chords from a given number of points on the circumference? 
 
With one point on the circumference, there are no chords and one region inside the circle. 

With two points on the circumference and there is one chord and two regions. 

Add a third point and draw all possible chords.  How many regions?  There are four regions inside the circle.

How about 4 points?  And so on. At some point there are multiple answers, so we are always looking for the maximum number of regions you can create for a given number of points.  Can you predict how many regions for 10 points? 45? n points?
 
 
That’s the problem.  As I began to work on it, I found myself on the same kind of path I often follow when working on a rich and challenging problem (in fact, it is the reason that we often refer to such problems as “rich”, like a rich vein of ore): not a direct path to the solution, but a branching course, with a lot more trial and error than my students probably imagine math problems are supposed to involve.  I decided to document my path through the landscape of this problem, both for myself and as an example for my students to follow.  So what follows will involve a lot of spoilers.  Stop here if you want to work on the problem on your own.
 
For my students, I want to describe the process of mathematical investigation just a little further.  It is like trying to cross an unfamiliar landscape, where some paths may take you from your camp to a cave on the ridge, but many more will not.  You have to find the right path.  So you look up at the ridge you want to reach, and look at what is just below it.  Then you make a guess:  what looks like a likely path to the top?  Now you have a new task – try to get to the bottom of that path.  In order to get to the bottom of the path you have to cross the river right in front of you.  So now the problem becomes how to get across the river.  You can see some rocks to step on close by, but you can’t see what is beyond that.  So you walk out as far as you can…
 
And that is how it goes.  You keep changing your focus, from far away to the step right in front of you.  You make guesses about what might be useful, and you make discoveries as you journey further along that may or may not change your path.  Your one question – How do I get from here to there? – sprouts numerous smaller tasks and problems, and you try to solve those.  In the meantime, depending on how curious you are and how much time you spend exploring, you get to know the lay of the land, more than just the path you are searching for.  This is how all the various subjects of mathematics (in fact, we often refer to these as “fields of mathematics”) have been discovered and developed, by people following the breadcrumb trail of questions that lead to further questions.  It literally never ends.
 
 [THE SPOILERS START HERE]
So, the first thing that I discovered was that the pattern that appears is not the pattern it appears to be.  1, 2, 4, 8, 16, 31… Wait a second. How does that work?  In looking at the differences between the terms, and looking at those differences, I realized this was not a geometric function, but it was a polynomial of degree 4.  Okay.  That’s weird.  But it also was my first discovery and new idea… or rather my first new question.  This suggests that just because I have 7 terms of 2^n, or even 17, that doesn’t mean I have the function 2^n.  And here come the questions:  is that generally true?  Can I make the change at any term?  Can I change that term to any number?  Can I do it to any power sequence?  This seems to suggest that polynomials can approximate a geometric sequence to arbitrary accuracy… is that useful to anyone?  I don't know, but it is some mathematics that I understand in a different way because I walked here by myself.  I haven't yet walked much farther down that path, but I may come back to that someday.

When we teach math, what are we teaching?  I think of it in layers.
 
On the surface, students learn to solve problems of increasing complexity and sophistication, using mathematical tools of increasing sophistication.  They learn how to multiply, add fractions, solve equations, factor quadratics, etc.  But the goal of math education has to be more than that.  In fact, if that is all we have achieved, we have literally failed.
 
So what is below the surface?  Mathematical thinking.
 
There is a reason that we have fields of study that have endured for thousands of years.  Literature, music, science, philosophy, mathematics all contribute a useful and unique way to view and process the world.  Mathematical thinking is a way to deal with quantities and patterns that can be valuable in many contexts.
 
Broadly, I think there are three areas of mathematical thinking we deal with in primary and secondary education.
 

• First of all we teach numbers.  What is 3, 4/5, -7, or sin(2π)?  Starting with whole numbers, we help students understand what these quantities represent, and what their existence implies.  There are relationships within and between number families that students need to explore and investigate on an ongoing basis.
  

• Secondly we teach operations.  We have to remember that the symbols on the page are code for something else.  Think about what is written here: 4/5.  There is a 4, a line, and a 5.  The symbols don’t tell you anything about what 4/5 means.  When we teach operations like addition or multiplication, we can teach those operations in a purely symbolic way: what to do with the various numbers in different positions on the page.  What does it mean to divide 4 by 5?  What does it look like?  When we multiply two fractions, what does it mean to take 2/3 of 4/5?
  

 
I want to be clear: I am not saying we need to teach students why the various algorithms work (perhaps with the purpose of having them remember them better).  I am saying we want students to understand what the operations we use in mathematics represent.  When we divide 4 by 5, what are we actually doing to the 4, and what role does the 5 play in the operation?  Mathematics describes and uses certain operations on numbers (and algebraic expressions) that our students should explore and understand.

• Finally, we teach generalization.  Sometimes people refer to this as abstraction.  The process of noticing patterns and making up “rules” or routines that apply in general occurs on so many levels – all students have an opportunity to do it on a daily basis in math class.  The process of observing something, finding patterns, and making generalizations, is a process called inductive reasoning.  The concept is powerful but not perfect - the further we get from pure numbers the more carefully it should be applied.  But this is the crown jewel that mathematics and we math teachers have to offer the world.  I would say that there is no point to teaching math if we do not teach this.  And we have important choices to make about our teaching which will determine whether it is present or absent from our classrooms.  If we train students in manipulating symbols on a page, they have no opportunity to experience the process of generalizing, let alone learn to do it themselves.  But if we talk to students about the process and ask them on a daily basis what they see, what they notice, and help them grow in the ability to make conclusions and develop algorithms, we can grow their skills in generalization.
  

 
What is most important?  I think we have to have a zen-like duality in our minds, keeping a close eye on both computational competence and mathematical thinking.  Both measure our progress, but either one alone is not enough.  And they are certainly not mutually exclusive.  Does it hurt students to teach them an algorithm?  Not if you also build a classroom culture in which students understand that doing mathematics means exploring and reasoning and building generalizations for themselves.